Calorimetry: Fundamentals, Instrumentation and Applications [1 ed.] 3527327614, 9783527327614

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Stefan M. Sarge G€ unther W. H. H€ohne Wolfgang Hemminger Calorimetry

Related Titles Kaletunç, G. (ed.)

Calorimetry in Food Processing 2009 ISBN: 978-0-8138-1483-4, also available in digital formats

Schalley, C. A. (ed.)

Analytical Methods in Supramolecular Chemistry 2012 ISBN 978-3-527-32982-3, also available in digital formats

Stefan M. Sarge, G€ unther W. H. H€ohne and Wolfgang Hemminger

Calorimetry

Fundamentals, Instrumentation and Applications

Authors Dr. Stefan M. Sarge Physikalisch-Technische Bundesanstalt Bundesallee 100 38116 Braunschweig Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for

Dr. G€ unther W. H. H€ohne Mörikeweg 30 88471 Laupheim Germany

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Dr. Wolfgang Hemminger Malerweg 5 38126 Braunschweig Germany

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jV

Contents Preface XIII List of Quantities and Units XV

Introduction

Calorimetry: Definition, Application Fields and Units 1 Definition of Calorimetry 1 Application Fields for Calorimetry 1 First Example from Life Sciences 2 Second Example from Material Science 2 Third Example from Legal Metrology 2 Units 3 Further Reading 4 References 5

Part One

Fundamentals of Calorimetry 7

1 1.1 1.1.1 1.1.2 1.2 1.2.1 1.2.2 1.2.2.1 1.2.2.2 1.3

Methods of Calorimetry 9 Compensation of the Thermal Effect 9 Compensation by a Phase Transition 9 Compensation by Electric Effects 12 Measurement of Temperature Differences 13 Measurement of Time-Dependent Temperature Differences 13 Measurement of Local Temperature Differences 15 First Example: Flow Calorimeter 15 Second Example: Heat Flow Rate Calorimeter 15 Summary of Measuring Principles 16 References 17

2 2.1 2.1.1 2.1.2

Measuring Instruments 19 Measurement of Amount of Substance 19 Weighing 20 Volume Measurement 20

VI

j Contents 2.1.3 2.1.4 2.2 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.1.4 2.3.1.5 2.3.1.6 2.3.2 2.4

Pressure Measurement 21 Flow Measurement 21 Measurement of Electric Quantities 21 Measurement of Temperatures 22 Thermometers 23 Liquid-in-Glass Thermometers 23 Gas Thermometers 24 Vapor Pressure Thermometers 24 Resistance Thermometers 25 Semiconductors 26 Pyrometers 26 Thermocouples 27 Chemical Composition 29 References 29

3 3.1 3.1.1 3.1.2

Fundamentals of Thermodynamics 31 States and Processes 31 Thermodynamic Variables (Functions of State) 31 Forms of Energy, Fundamental Form, and Thermodynamic Potential Function 34 Fundamental Form 35 Thermodynamic Potential Function 35 Equilibrium 38 Reversible and Irreversible Processes 41 The Laws of Thermodynamics 42 The Zeroth Law 42 The First Law 42 The Second Law 42 The Third Law 43 Measurement of Thermodynamic State Functions 43 Phases and Phase Transitions 47 Multiphase Systems 47 Phase Transitions 50 Gibbs Phase Rule 52 Measurement of Variables of State during Phase Transitions 56 References 59

3.1.2.1 3.1.2.2 3.1.3 3.1.4 3.1.5 3.1.5.1 3.1.5.2 3.1.5.3 3.1.5.4 3.1.6 3.2 3.2.1 3.2.2 3.2.3 3.2.4

4 4.1 4.2 4.3 4.4 4.5 4.6

Heat Transport Phenomena 61 Heat Conduction 61 Convection 64 Heat Radiation 65 Heat Transfer 67 Entropy Increase during Heat Exchange 67 Conclusions Concerning Calorimetry 68 References 71

Contents

5 5.1 5.2 5.3 5.4

Surroundings and Operating Conditions 73 The Isothermal Condition 74 The Isoperibol Condition 75 The Adiabatic Condition 75 The Scanning Condition 76 Reference 79

6 6.1 6.1.1 6.1.2 6.1.3 6.2 6.2.1 6.2.2 6.2.3 6.3

Measurements and Evaluation 81 Consequences of Temperature Relaxation within the Sample 81 First Example: Chemical Reaction 81 Second Example: Biological System 82 Third Example: First-Order Phase Transitions 83 Typical Results from Different Calorimeters 86 Adiabatic Calorimeters 86 Isoperibol Calorimeters 89 Differential Scanning Calorimeters 93 Reconstruction of the True Sample Heat Flow Rate from the Measured Function 101 Reconstruction of the Temperature Field for Negative Times 101 The Convolution Integral and Its Validity 102 Solution of the Convolution Integral 105 Obtaining the Apparatus Function 106 Application Limits and Estimation of Uncertainty 107 Special Evaluations 109 Determination of the Specific Heat Capacity 109 Determination of the Kinetic Parameters of a Chemical Reaction 109 Determination of Phase Transition Temperatures 111 Determination of Heats of Transition 112 Determination of the Purity of a Substance 114 Determination of the Measurement Uncertainty 115 References 121

6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 6.5

Part Two

Practice of Calorimetry 123

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.8.1

Calorimeters 125 Functional Components and Accessories 125 Heating Methods 126 Cooling Methods 126 Comments on Control Systems 128 Thermostats 131 On the Classification of Calorimeters 131 On the Characterization of Calorimeters 132 Isothermal Calorimeters 134 Phase Transition Calorimeters 134

jVII

VIII

j Contents 7.8.1.1 7.8.1.2 7.8.2 7.8.2.1 7.8.2.2 7.8.2.3 7.9 7.9.1 7.9.1.1 7.9.1.2 7.9.1.3 7.9.2 7.9.2.1 7.9.2.2 7.9.2.3 7.9.2.4 7.9.3 7.9.3.1 7.9.3.2 7.9.3.3 7.9.3.4 7.9.4 7.9.4.1 7.9.4.2 7.9.4.3 7.9.5 7.9.5.1 7.9.5.2 7.9.5.3 7.10 7.10.1 7.10.1.1 7.10.1.2 7.10.1.3 7.10.1.4 7.10.1.5 7.10.2 7.10.2.1 7.10.2.2

First Example: Ice Calorimeter 135 Second Example: Calorimeter with Liquid–Gas Phase Transition 137 Isothermal Calorimeters with Electrical Compensation 141 First Example: Calorimeter according to Tian 142 Second Example: Isothermal Titration Calorimeter 143 Third Example: Isothermal Flow Calorimeter 144 Calorimeters with Heat Exchange between the Sample and Surroundings 144 Isoperibol Calorimeters with Uncontrolled Heat Exchange 145 First Example: Classic Liquid (or Mixing) Calorimeter 145 Second Example: Combustion Calorimeter 148 Third Example: Drop Calorimeter 151 Isoperibol Calorimeter with Controlled Heat Exchange 154 First Example: Activity Monitor 159 Second Example: Large-Volume Battery Calorimeter 160 Third Example: Calvet Calorimeter 161 Fourth Example: Whole-Body Calorimeter 169 Isoperibol Flow Calorimeter 170 First Example: The Picker Calorimeter 175 Second Example: Flow Calorimeter for High-Pressure and HighTemperature Measurements 176 Third Example: Gas Combustion Calorimeter 177 Fourth Example: Microchip Flow Calorimeter 177 Calorimeters with Linear Temperature Change of the Surroundings 178 First Example: Heat Flow Differential Scanning Calorimeter 179 Second Example: Power-Compensated Differential Scanning Calorimeter 183 Third Example: Privalov Calorimeter 185 Calorimeters with Nonlinear Temperature Change of the Surroundings 186 First Example: Temperature-Modulated DSC 187 Second Example: Stepscan Differential Scanning Calorimetry 189 Third Example: Advanced Multifrequency TMDSC 189 Adiabatic Calorimeters 190 Calorimeters with a Thermally Isolated Sample 190 First Example: Nernst Calorimeter 191 Second Example: Low-Temperature Calorimeter 192 Third Example: AC Calorimeter 194 Fourth Example: 3v Technique 195 Nernst’s Method with a Contactless Energy Supply 196 Calorimeters with Zero Temperature Difference against the Surroundings 197 First Example: Adiabatic Reaction Calorimeter 198 Second Example: Adiabatic Flow Calorimeter 199

Contents

7.10.2.3 7.10.2.4 7.10.3 7.10.3.1 7.11 7.11.1 7.11.1.1 7.11.1.2 7.11.2 7.11.2.1 7.11.2.2 7.11.2.3 7.11.2.4

Third Example: Adiabatic Whole-Body Calorimeter 199 Fourth Example: Adiabatic Scanning Calorimeter 200 Quasi-adiabatic Calorimetry by Sudden Heat Events 201 Example: Pulse Heating Calorimeter 201 Other Calorimeters 202 Reaction Calorimeters 202 First Example: Reaction Calorimeter 203 Second Example: Accelerating Rate Calorimeter (ARC) 204 Special Calorimeters 206 Photocalorimeters 206 Pressure Calorimeters 206 Pressure Perturbation Calorimeter 206 Cement Calorimeter 207 References 207

8 8.1 8.1.1 8.1.2 8.2 8.2.1 8.2.2 8.3 8.3.1 8.3.1.1 8.3.2 8.3.2.1

Recent Developments 213 Microchip Calorimetry 214 First Example: Thin-Film Differential Scanning Calorimeter 216 Second Example: Low-Temperature AC Nanocalorimeter 217 Ultrafast Calorimetry 217 First Example: Ultrafast Nanocalorimeter 218 Second Example: Flash Differential Scanning Calorimeter 220 Extreme Ranges of State 220 High Pressure 221 Example: Power-Compensated High-Pressure DSC 222 High Temperature 222 Example: Levitation Calorimetry on Nickel, Iron, Vanadium, and Niobium 223 Strong Magnetic Fields 224 Example: Influence of Magnetic Fields on Point Defects 224 Plasma Surroundings 224 Example: Calibration Using a Laser Beam 224 Calorimetry as an Analytical and Diagnostic Tool 225 First Example: “Artificial Nose” 225 Second Example: Infection Diagnostics 225 References 226

8.3.3 8.3.3.1 8.3.4 8.3.4.1 8.4 8.4.1 8.4.2

9 9.1 9.1.1 9.1.2 9.1.3 9.1.4 9.1.5 9.1.6

Calorimetric Measurements: Guidelines and Applications 229 General Considerations 229 Sensitivity (DX/Q or DX/DF) 230 Noise (dQ or dF) 230 Linearity (Xout ¼ K
Xin) and Linearity Error (dK/K) 232 Apparatus Function (fapp(t)) 232 Accuracy and Total Error ({Qmeasured – Qtrue}/Qtrue) 233 Repeatability and Random Uncertainty (DQ/Q) 235

jIX

j Contents

X

9.2 9.2.1 9.2.2 9.2.2.1 9.2.2.2 9.2.3 9.2.3.1 9.2.3.2 9.2.4 9.2.4.1 9.2.4.2 9.2.4.3 9.2.5 9.2.6 9.3 9.3.1 9.3.1.1 9.3.1.2 9.3.1.3 9.3.1.4 9.3.1.5 9.3.1.6 9.3.2 9.3.2.1 9.3.2.2 9.3.2.3 9.3.2.4 9.3.2.5 9.3.2.6 9.3.3 9.3.3.1 9.3.3.2 9.3.3.3 9.3.3.4 9.3.3.5 9.3.3.6 9.3.4 9.3.4.1 9.3.4.2 9.3.4.3 9.3.4.4 9.3.4.5 9.3.4.6 9.3.5

Conclusion 235 Guidelines to Calorimetric Experiments 235 Definition of the Problem to be Investigated 236 Selection of the Proper Calorimeter 237 Calorimeter Requirements 237 Selection of the Calorimeter 238 Testing of the Calorimeter 239 Calibration 239 Other Testing 242 Performing the Experiment 243 Preparation of the Sample 243 Calorimetric Measurement 244 Evaluation of the Measurement 245 Interpretation of the Results 245 Uncertainty Estimation 246 Calorimetric Applications 246 Example from Material Science 247 Definition of the Problem to be Investigated 247 Selection of the Calorimeter 247 Calorimetric Experiments 248 Evaluation of the Measurements 248 Interpretation of the Results 251 Uncertainty Estimation 252 Examples from Biology 256 Definition of the Problem to be Investigated 256 Selection of the Proper Calorimeter 256 Calorimetric Experiments 257 Evaluation of the Results 258 Calorimetry on Hornets 258 Uncertainty Estimation 259 Example from Medicine 259 Definition of the Problem to be Investigated 259 Selection of the Proper Calorimeter 259 Calorimetric Experiment 260 Evaluation of the Measurements 260 Interpretation of the Results 260 Uncertainty Estimation 261 Example from Chemistry 261 Definition of the Problem to be Investigated 262 Selection of the Proper Calorimeter 262 Calorimetric Experiment 263 Evaluation of the Measurements 263 Interpretation of the Results 264 Uncertainty Estimation 265 Example from Combustion Calorimetry 265

Contents

9.3.5.1 9.3.5.2 9.3.5.3 9.3.5.4 9.3.5.5 9.3.5.6

Definition of the Problem to be Investigated 265 Selection of the Proper Calorimeter 265 Calorimetric Experiment 267 Evaluation of the Measurements 267 Interpretation of the Results 268 Uncertainty Estimation 268 References 269 Index 271

jXI

jXIII

Preface Fifty years ago, anyone interested in the measurement of heat had to build a calorimeter of his or her own. Thirty years ago, when the book “Calorimetry”1) was originally published, the change from self-made calorimeters to instruments that are produced commercially had just begun. This change has now been completed. Today a large variety of instruments are commercially available. Owing to new production techniques and particularly to the development of electronics and computers, these calorimeters permit accurate and reliable measurements within short intervals of time. It is not surprising, therefore, that calorimetry has become a standard measuring procedure in many branches of natural science as well as in production and quality control. This book is intended to help readers to understand the basics of calorimetry and to find their way in the ever-growing market of commercial instruments. During the three decades that have passed, huge progress in calorimetry has been made concerning the techniques and the instrumentation alike. Because of missing special literature or textbooks, there developed an increasing desire for another basic monograph about calorimetry that would take these developments into account. In the present book, we describe the state of the art of modern calorimetry and today’s instrumentation almost completely. Despite the risk that some of the calorimeters described here will be obsolete in a few years, we decided not to deny readers basic and concise information about all the apparatus that they can buy today. Another objective of the new book is to promote the application of calorimetric procedures in various fields of research, providing practical advice and examples for this purpose. In accordance with these considerations, this book is intended for scientists considering the use of calorimeters, senior students engaged in heat measurements, and technicians working in the field of thermal analysis or calorimetry. To achieve these objectives, we have written all the chapters and sections in such a way as to emphasize principles and problems. The measuring examples and instruments described were selected in accordance with this view. Crucial items, such as the evaluation of measuring curves, are treated in detail and with reference to particular commercial calorimeters. Readers are instructed about criteria for the 1) Hemminger, W., H€ ohne, G. (1984) Calorimetry. Fundamentals and Practice, Verlag Chemie, Weinheim.

XIV

j Preface evaluation of calorimeters to enable them to select the ones that best suit their purposes. Sometimes we have included in our discussion classical calorimeters that are no longer marketed as such but are perhaps available in an improved version. We have done this when necessary in order to explain certain principles or to show special applications. For the same reason, numerous self-made calorimeters are described, and some hints for their construction are given. For further details, readers are referred to the literature. The information provided may be of value when certain experimental requirements are not met by commercial instruments or these instruments are oversized for the work involved. Because of the rapid progress of electronic measuring and control techniques as well as data processing, we have made no attempt to cover these aspects in detail. We have not attempted to provide a comprehensive review of the special literature, but it is our hope that we have not overlooked any important instruments or procedures. Regrettably, non-English literature could only be partly considered.

Braunschweig and Laupheim, 2013

Stefan M. Sarge G€ unther W. H. H€ohne Wolfgang Hemminger

jXV

List of Quantities and Units (Bold symbols describe vector quantities.)

Symbol

Name

Unit

a a A A A b c c ci C C Cc Cht d d e e E f f F g

(Relative) activity Reciprocal heat capacity Area Helmholtz energy Preexponential factor Half-width Constant (general) Specific heat capacity Sensitivity coefficient Electric capacitance Heat capacity Convection coefficient Coefficient of heat transfer Degree of freedom Distance Error Specific energy Energy Frequency Force Fraction melted Acceleration due to gravity (9.81 m s 2) Gibbs energy Thermal conductance Specific enthalpy Enthalpy Volumetric superior calorific value

1 KJ 1 m2 J (m3 mol 1)n 1 s 1 s Depends Jg 1K 1 Output unit/input unit F JK 1 Wm 2K 1 Wm 2K 1

G G h H Hs,V

m Same as corresponding quantity J kg 1 J s 1 N 1 ms 2 J WK 1 J kg 1 J kWh m

3

XVI

j List of Quantities and Units I J Jq k k k0 K K K12 KF KQ l L m M n n N NGr p p P q q Q Q r r R R

Rth s S S t T u U U v v

Electric current Heat flux Heat flux field Coverage factor Rate constant Preexponential factor Calibration factor Missing heat of fusion Empirical radiation exchange coefficient Heat flow calibration factor Heat calibration factor Length Angular momentum Mass Molar mass Amount of substance Order of reaction Number of entities Grashof number Pressure Momentum Power Electric charge Specific heat Electric charge Heat Correlation coefficient Position vector Electric resistance Molar gas constant ((8.31446210.0000075) J K 1 mol 1) Thermal resistance Standard deviation Entropy Seebeck coefficient Time Temperature Uncertainty Voltage Internal energy flow rate Velocity

A Wm Wm

2 2

(m3 mol 1)n 1 s 1 (m3 mol 1)n 1 s 1 Output unit / input unit J m 2 1 or W K 1 1 or J K 1 s m Js kg kg mol 1 mol 1 1 1 Pa kg m s 1 W C J kg 1 C J 1 m V J K 1 mol 1

1

KW 1 Same as corresponding quantity JK 1 VK 1 s K (or  C) Same as corresponding quantity V J m3 s 1 ms 1

List of Quantities and Units

V wc W x x

Volume Degree of cystallinity Work Mole fraction Position coordinate (general)

m3 (or l) 1 J 1 m

Greek Symbols

a a b hdyn l m w f F P r s s sB

q Q t v v

Cubic expansion coefficient Degree of reaction Heating rate Dynamic viscosity Thermal conductivity Chemical potential Phase shift Electric potential Heat flow rate Peltier coefficient Density Standard deviation Surface tension Stefan–Boltzmann constant ((5.6703730.000021)  10 8 W m 2 K 4) Temperature Temperature Time constant Angular velocity Angular frequency

K 1 1 K s 1 (or K min 1) Pa s Wm 1K 1 J mol 1 rad V W V kg m 3 (or g cm 3) Same as corresponding quantity Jm 2 Wm 2K 4



C C s rad s rad s 

1 1

Indices and Subscripts

A act am app C cal clb comb cond cryst eff

Amplitude Activation Amorph Apparatus Container Calorimeter Calibration Combustion Condensate Crystallized Effective

jXVII

XVIII

j List of Quantities and Units el exp F F fin fus g I I in ini liq M m max n out p r ref R Rp s S S sln T th trs V vap w W

Electric Experimental Furnace (surroundings) Final state Final state Fusion Glass transition Inflection Initial State Inlet Initial state Liquid Measuring system Molar Maximum Standard conditions Output At constant pressure Reaction Reference state Reference Response Superior Sample Surface Solution At constant temperature Thermal Transition At constant volume Vapor, vaporization Water Wall

1

Introduction Calorimetry: Definition, Application Fields and Units

Definition of Calorimetry

Calorimetry means the measurement of heat. In the past, the term heat was associated with various concepts. Nowadays one no longer speaks of different energies (e.g., heat energy, electrical energy, and kinetic energy) coexisting in a substance or system independent of one another. According to the modern view, there is only one single energy (the internal energy) stored in a body, which – only during an exchange – appears in a variety of energy forms such as heat energy, electrical energy, or kinetic energy. Accordingly, the form of energy known as heat can only be conceived as coupled with a change of energy. Heat is always associated with a heat flow. In other words, heat is the amount of energy exchanged within a given time interval in the form of a heat flow. Calorimeters are the instruments used for measuring this heat. Application Fields for Calorimetry

Calorimetry has been well known for centuries as a very effective method in natural sciences. The precise measurement of heat capacity, heat of fusion, heat of reaction, heat of combustion, and other caloric quantities has built the basis for progress in thermodynamics and classical physical chemistry. The classical methods of calorimetry have not changed very much during the past century, and the scientific interest in and the knowledge about them have dropped accordingly. Only the progress in microelectronics and computer science during the past few decades has made it possible to develop new types of calorimeters and open new fields of application. As a result, there is now an increasing interest in calorimetry as a very easy and powerful method for different kinds of investigation. Modern calorimetry is successfully used in many fields, such as material science, life sciences (biology, medicine, and biochemistry), pharmacy, and food science, for quality control, safety investigations, and the determination of the energy content of fuels. Some examples illustrating this are presented below.

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

2

Introduction Calorimetry: Definition, Application Fields and Units

First Example from Life Sciences Disregarding, for example, frictional heat, every living organism produces heat because of the chemical reactions involved in the metabolism of the cells. Depending on the temperature and other parameters (atmosphere, nutrients, and respiration), the heat production differs. If the temperature and the surroundings are kept constant, the heat production remains constant, too. Different microbiological organisms such as microbes and bacteria may produce different amounts of heat, but a single cell produces – under the above conditions – a characteristic amount of heat. If this quantity is known for certain organisms – say, bacteria – it is possible with a microcalorimeter to determine the amount of bacteria in a sample from the heat flow rate they produce (approximately 1–3 pW per cell, James, 1987). Furthermore, it was shown that the heat flow–time profiles of bacteria in a suitable nutrient solution are very specific and can be used to identify a bacterial species with a microcalorimeter much faster (within 15–24 h) than with a traditional culture medium (Trampuz et al., 2007). Possible applications of this calorimetric method could be the quick testing of the possible contamination of donated blood products or the faster identification of the bacterium causing blood poisoning and the more successful treatment of very dangerous sepsis. Second Example from Material Science Polymer processing frequently implies molding of nonamorphous polymers that often crystallize partially on cooling. Unfortunately, crystallization changes the material properties of certain polymers dramatically for the worse and should therefore be avoided. This can be achieved by quick cooling into a supercooled state (below the crystallization range). To investigate the crystallization behavior of such polymers, quick heating and quick cooling are required. Unfortunately, the heating and cooling rates of differential scanning calorimeters (DSCs) are limited to less than 100 K min 1 because of the thermal inertia of the equipment. For many polymers, this is not fast enough to avoid crystallization and to come to a really amorphous state. To overcome this problem, so-called chip calorimeters have recently been developed with the help of modern chip processing technology. Such ultrasmall calorimeters have a very small mass and very good thermal conductivity and, therefore, nearly no thermal inertia. The very low time constants of such calorimeter chips together with very small (1 ng) and very thin (10 mm) samples allow heating and cooling rates up to 106 K s 1 (Minakov et al., 2007). At heating and cooling rates higher than 103 K s 1, recrystallization can be avoided for several polymers, and the melting kinetics along with the superheating behavior and their influence on material properties can be investigated with such a chip calorimeter. Third Example from Legal Metrology The commercial value of fuels in general and fuel gases in particular depends on the amount of energy contained in a given amount of fuel. Traditionally, this amount of energy per unit amount, the calorific value, has been determined by gas calorimetry (Hyde and Jones, 1960). In recent times, gas chromatography

Introduction Calorimetry: Definition, Application Fields and Units

has been used to infer the calorific value of natural gas from its composition. But biogases and other nonconventional fuel gases such as landfill gas and shale gas contain components not included in natural gas. As a consequence, gas chromatography fails or becomes prohibitively expensive. As a simple, quick, and cheap method for the determination of the calorific value of such gases, gas calorimetry is again applied, this time computer controlled and fully automatically (Haug and Mrozek, 1998). In this particular calorimeter, fuel gas flow and combustion airflow are controlled by nozzles. The two gas streams are mixed and burned. The resulting heat is transferred to a constant flow of air, whose temperature is measured at the entrance and the exit of the heat exchanger. This temperature increase is a measure of the energy content of the gas. To account for the influence of different fuel gas properties on the flow through the nozzle, the density of the gas is measured and used for converting the primary output of the calorimeter, the Wobbe number, to the desired quantity, the inferior calorific value of the fuel gas. Units

As illustrated by the earlier examples, there is no direct method for the measurement of heat. Consequently, heat has to be determined by means of its effects. The older unit quantity of heat – the calorie – was therefore defined in terms of a measurement instruction: One 15  C calorie (cal15 ) is the amount of heat required to raise the temperature 1) of 1 g of water from 14.5 to 15.5  C under standard atmospheric pressure. Because heat is merely one form of energy, as are the electrical and mechanical energies (Mayer, 1842; Joule, 1843; Colding, 1843, according to Dahl, 1963), a special unit for heat is unnecessary. Today, in the International System of Units (SI), heat is expressed in the unit of energy: 1 J ¼ 1 Nm ¼ 1 Ws

Conversion between the old unit (cal) and new SI unit (J) is made as follows: 1 cal ¼ 4:1868 J;

1 J ¼ 0:2388459 cal

The latter is the International Steam Table calorie (calIT), one of the two “calories” still in use of a number of older calories (e.g., the “15  C water calorie” corresponding to 0.9996801 calIT) (Stille, 1961). The second calorie still in use in some parts of the world is the US National Bureau of Standards calorie (calNBS or calthermochem): 1 calNBS ¼ 4:1840 J ðexactlyÞ;

1 J ¼ 0:239006 calNBS

1) Temperature is treated here as a directly measurable quantity, which, strictly speaking, is not the case.

3

4

Introduction Calorimetry: Definition, Application Fields and Units

Further Reading Numerous monographs on calorimetry have been published:

Calorimetry, Blackwell Scientific Publications, Oxford. McCullough, J.P. and Scott, D.W. (eds) (1968) Experimental Thermodynamics, vol. 1, Calorimetry of Non-Reacting Systems, Butterworths, London. Rossini, F.D. (ed.) (1956) Experimental Thermochemistry. Measurement of Heats of Reaction. Interscience Publishers, New York. Roth, W.A. and Becker, F. (1956) Kalorimetrische Methoden zur Bestimmung chemischer Reaktionswärmen, Friedrich Vieweg & Sohn, Braunschweig. Sengers, J.V., Kayser, R.F., Peters, C.J., and White, H.J., Jr. (eds) (2000) Experimental Thermodynamics, vol. 5, Equations of State for Fluids and Fluid Mixtures, Elsevier, Amsterdam. Skinner, H.A. (ed.) (1962) Experimental Thermochemistry, vol. II, Interscience Publishers, New York. Sorai, M. (ed.) (2004) Comprehensive Handbook of Calorimetry and Thermal Analysis, John Wiley & Sons, New York.  Sunner, S. and Mansson, M. (eds) (1979) Experimental Chemical Thermodynamics, vol. 1, Combustion Calorimetry, Pergamon Press, Oxford. Swietoslawski, W. (1946) Microcalorimetry, Reinhold Publishing Corp., New York. Wakeham, W.A., Nagashima, A., and Sengers, J.V. (eds) (1991) Experimental Thermodynamics, vol. 3, Measurement of the Transport Properties of Fluids, Blackwell Scientific Publications, Oxford. Weber, H. (1973) Isothermal Calorimetry, Peter Lang, Frankfurt. Weir, R.D. and de Loos, Th.W. (eds) (2005) Experimental Thermodynamics, vol. 7, Measurement of the Thermodynamic Properties of Multiple Phases, Elsevier, Amsterdam. White, W.P. (1928) The Modern Calorimeter, American Chemical Society Monograph Series No. 42, The Chemical Catalog Company, New York. Zielenkiewicz, W. and Margas, E. (2002) Theory of Calorimetry, Kluwer, Academic Publ. Dordrecht.

Brown, M.E. (ed.) (1998) Handbook of Thermal Analysis and Calorimetry, vol. 1, Principles and Practice, Elsevier, Amsterdam. Brown, M.E. and Gallagher, P.K. (eds) (2003) Handbook of Thermal Analysis and Calorimetry, vol. 2, Applications to Inorganic and Miscellaneous Materials, Elsevier, Amsterdam. Brown, M.E. and Gallagher, P.K. (eds) (2008) Handbook of Thermal Analysis and Calorimetry, vol. 5, Recent Advances, Techniques and Applications, Elsevier, Amsterdam. Calvet, E. and Prat, H. (1963) Recent Progress in Microcalorimetry, Pergamon Press, Oxford. Cheng, S.Z.D. (ed.) (2002) Handbook of Thermal Analysis and Calorimetry, vol. 3, Applications to Polymers and Plastics, Elsevier, Amsterdam. Eder, F.X. (1983) Arbeitsmethoden der Thermodynamik, Bd. II, Thermische und kalorische Stoffeigenschaften, Springer, Berlin. Eucken, A. (1929) Energie- und W€armeinhalt, in Handbuch der Experimentalphysik, Band 8, 1. Teil (eds W. Wien and F. Harms), Akademische Verlagsgesellschaft, Leipzig. Goodwin, A.R.H., Marsh, K.N., and Wakeham, W.A. (eds) (2003) Experimental Thermodynamics, vol. 6, Measurement of the Thermodynamic Properties of Single Phases, Elsevier, Amsterdam. Haines, P.J. (ed.) (2002) Principles of Thermal Analysis and Calorimetry, The Royal Society of Chemistry, Cambridge. H€ ohne, G.W.H., Hemminger, W.F., and Flammersheim, H.-J. (2003) Differential Scanning Calorimetry, 2nd edn, Springer, Berlin. Hyde, C.G. and Jones, M.W. (1960) Gas Calorimetry, 2nd edn, Ernest Benn, London. Kemp, R.B. (ed.) (1999) Handbook of Thermal Analysis and Calorimetry, vol. 4, From Macromolecules to Man, Elsevier, Amsterdam. LeNeindre, B. and Vodar, B. (eds) (1975) Experimental Thermodynamics, vol. 2, Experimental Thermodynamics of Non-Reacting Fluids, Butterworths, London. Many physics books contain separate chapters Marsh, K.N. and O’Hare, P.A.G. (eds) (1994) Experimental Thermodynamics, vol. 4, Solution dedicated to calorimetry:

References Maglic, K.D., Cezairliyan, A., and Peletsky, V.E. (1984) Compendium of Thermophysical Property Measurement Methods, vol. 1, Survey of Measurement Techniques, Plenum Press, New York, pp. 457–685. Maglic, K.D., Cezairliyan, A., and Peletsky, V.E. (1992) Compendium of Thermophysical Property Measurement Methods, vol. 2, Recommended Measurement Techniques and Practices, Plenum Press, New York, pp. 409–545. Oscarson, J.L. and Izatt, R.M. (1986) Calorimetry, in Physical Methods of Chemistry, vol. VI, Determination of Thermodynamic Properties (eds B.W. Rossiter and R.C. Baetzold), 2nd edn, Wiley-Interscience, New York, pp. 573–620. Spink, H. and Wads€o, I. (1976) Calorimetry as an analytical tool in biochemistry and biology, in Methods of Biochemical Analysis, vol. 23 (ed. D. Glick), John Wiley & Sons, New York, pp. 1–159. Warrington, S.B. and H€ohne, G.W.H. (2008) Thermal analysis and calorimetry, in Ullmann’s Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim. Monographs on calorimetry with reference to special topics: Beezer, A.E. (ed.) (1980) Biological Microcalorimetry, Academic Press, London. KaletunSc , G. (ed.) (2009) Calorimetry in Food Processing: Analysis and Design of Food Systems, Wiley-Blackwell, Ames. Koch, E. (1977) Non-Isothermal Reaction Analysis, Academic Press, London.

Kubaschewski, O. and Alcock, C.B. (1979) Metallurgical Thermochemistry, 5th edn, Pergamon Press, Oxford. Ladbury, J.E. and Doyle, M.L. (eds) (2004) Biocalorimetry 2: Applications of Calorimetry in the Biological Sciences, John Wiley & Sons, New York. Three series of international conferences dedicated to calorimetry take place regularly. Their presentations are partly published in special issues of different journals or as separate proceedings: The European Conference on Thermal Analysis and Calorimetry. The International Conference on Chemical Thermodynamics (until 2006 known as IUPAC Conference on Chemical Thermodynamics). The International Conference on Thermal Analysis and Calorimetry. Several journals publish original contributions on thermal analysis, calorimetry, and experimental thermodynamics: International Journal of Thermophysics (Springer, Berlin). Journal of Thermal Analysis and Calorimetry (Springer, Berlin). Journal of Chemical Thermodynamics (Elsevier, Amsterdam). Netsuo Sokutei (Calorimetry and Thermal Analysis) (Nihon Netsu Sokutei Gakkai, Tokyo). Thermochimica Acta (Elsevier, Amsterdam).

References Dahl, F. (1963) Ludvig A. Colding and the conservation of energy. Centaurus, 8, 174–188. Haug, T. and Mrozek, C. (1998) Temperaturstabilit€at und Anzeigegeschwindigkeit bei Verbrennungskalorimetern, gwf-Gas Erdgas, 139, 7–12. Union Instruments (2009) Data Sheet CWD2005 www.unioninstruments.com/fileadmin/documents/ Analysis%20systems/Datasheets/

CWD2005_en_datasheet_2009_03.pdf (October 10, 2010). Hyde, C.G. and Jones, M.W. (1960) Gas Calorimetry, 2nd edn, Ernest Benn, London. James, A.M. (1987) Calorimetry, past, present, future, in Thermal and Energetic Studies of Cellular Biological Systems (ed. A.M. James), Wright, Bristol, p. 4. Joule, J.P. (1843) On the calorific effects of magneto-electricity, and on the mechanical

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Introduction Calorimetry: Definition, Application Fields and Units value of heat. Philos. Mag., 23, 263–276, polymers. Thermochim. Acta, 461, 347–355, 435–443. 96–106. € ber die Mayer, J.R. (1842) Bemerkungen u Stille, U. (1961) Messen und Rechnen in der Kr€afte der unbelebten Natur. Ann. Chem. Physik, 2nd edn, Springer, Braunschweig, Pharm., 42, 233–240. p. 113. Minakov, A.A., van Herwaarden, A.W., Trampuz, A., Salzmann, S., Antheaume, J., and Wien, W., Wurm, A., and Schick, C. (2007) Daniels, A.U. (2007) Microcalorimetry: Advanced nonadiabatic ultrafast a novel method for detection of microbial nanocalorimetry and superheating contamination in platelet products. phenomenon in linear Transfusion, 47, 1643–1650.

7

Part One Fundamentals of Calorimetry

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Methods of Calorimetry This chapter provides a brief outline of the principles of heat measurement. A classification scheme will be developed on the basis of simple examples. A more detailed treatment of the procedures and calorimeters involved can be found in the second part of the book.

1.1 Compensation of the Thermal Effect

The heat released from a sample during a process flows into the calorimeter and would cause a temperature change of the latter as a measuring effect; this thermal effect is continuously suppressed by compensating the respective heat flow. The methods of compensation include the use of “latent heat” caused by a phase transition, thermoelectric effects, heats of chemical reactions, a change in the pressure of an ideal gas (Ter Minassian and Milliou, 1983), and heat exchange with 1) a liquid (Regenass, 1977). Because the last three methods are confined to special cases, only the compensation by a physical heat of transition and by electric effects are briefly discussed here. 1.1.1 Compensation by a Phase Transition 2)

Around 1760, Black (Robison, 1803) realized that the heat delivered to melting ice serves for a transition from the solid to the liquid state at a constant temperature. Indeed, although the melting of ice requires a steady supply of heat, the temperature of the ice–water mixture only begins to rise after all the ice has melted. Black is said to have been the first to have used this “latent heat of fusion” of ice for the measurement of heat. His “phase transition calorimeter” was very simple. He placed a warm sample in a cavity inside a block of ice and sealed the cavity with an

1)For example, “Bench Scale Calorimeter” developed by Ciba-Geigy Ltd., Switzerland, and commercialized by Mettler-Toledo (Schweiz) GmbH, Switzerland, as Reaction Calorimeter RC1. 2)Black only reported his findings verbally; see Encyclopaedia Britannica (2003) or Ramsay (1918). Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Methods of Calorimetry

Air in

Air out

Working equation: ∆Q = qtrs · ∆m ∆Q Heat produced by the sample ∆m Mass of molten ice qtrs Specific heat of fusion of ice To be determined ∆Q To be measured ∆m Known qtrs

Figure 1.1 Calorimeter of Lavoisier and Laplace (according to Kleiber, 1975).

ice sheet. After the sample had assumed the temperature of ice, he determined the mass of the melted ice by weighing. The principle of this method is that the heat DQ exchanged with the calorimeter is not measured as a heat flow but causes a phase transition in a suitable substance (e.g., ice). If the specific heat of transition qtrs of the respective substance is known, the heat involved can be determined because it is proportional to the mass of the transformed substance Dm: DQ ¼ qtrs
Dm

The mass of the transformed substance Dm is determined either directly by weighing or indirectly (e.g., by measuring the volume change due to the difference between the densities of the two phases). The first usable calorimeter involving a phase transition – the “ice calorimeter” – was developed by Lavoisier and Laplace (1780). Figure 1.1 schematically shows the design of this device. The sample chamber is completely surrounded by a double-walled vessel containing pieces of ice. This inner ice jacket is surrounded by a second double-walled vessel filled with an ice–water mixture (outer ice jacket). The whole system is in thermal equilibrium at 0  C. The basic idea in this calorimeter is that the measuring system proper (i.e., the inner ice jacket) is insulated by the outer jacket, in which any disturbing influence of heat from the environment on the inner ice jacket is compensated by an ice–water phase transition in the outer jacket. Only heat released inside the sample chamber serves for the melting of ice in the inner ice jacket. Because there is no temperature difference between the inner and outer jackets, no heat exchange between them takes place. Lavoisier and Laplace designated the measured heat as the “mass of melted ice.” The specific heat capacities of solids and liquids, as well as heats of combustion and the production of heat by animals, were measured this way. These measurements were carried out in

1.1 Compensation of the Thermal Effect

Figure 1.2 The density of water as a function of temperature.

winter when low environmental temperatures allowed experiments to be carried out over longer periods of time (up to 20 h) in order to measure relatively small heat from animals, for example. The ice calorimeter of Lavoisier and Laplace is very bulky; moreover, the inner ice jacket must be carefully prepared before each test. In addition, it suffers from a systematic error that stems from the influx of relatively warm air at the lid. This air cools down in the calorimeter, thus releasing heat, and escapes with the downward flow of ice water. Another systematic error that affects the accuracy results from the fact that the water layer located in the inner jacket between the pieces of ice may attain a local temperature of several degree C, depending on the magnitude of the heat and the rate of its release. These layers may rise because of density differences (Figure 1.2) and transfer part of their heat to the outer jacket (lid). This latter heat thus escapes being measured. Bunsen (1870) was the first to describe an ice calorimeter that was free from these errors and allowed precise and reliable measurements (see Section 7.8.1.1). A disadvantage of all phase transition calorimeters stems from the fact that the experimental temperature is determined by the transition temperature. Consequently, a variety of experimental temperatures can only be obtained in such calorimeters by using substances other than water. Very high sensitivity can be attained by using the liquid–gas phase transition (e.g., liquid nitrogen–gaseous nitrogen). The advantages of phase transition calorimeters lie in their relatively simple construction, their great sensitivity, and the possibility of enclosing the calorimeter in a vessel in which a phase transition identical to that occurring in the calorimeter takes place. This approach compensates for disturbances from the surroundings and creates “adiabatic” conditions (see Section 5.3). For quantitatively determining heats with phase transition calorimeters, the specific heat of transition of the phase changing substance must be known. To calibrate such a calorimeter, a known amount of (electrical) energy Eel is supplied to the inside of the calorimeter via a heating wire, and the mass of substance undergoing the transition is measured. The energy supplied divided by the mass of

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Working equation: qsln · m = ∫ U(t)·I(t)dt qsln Heat of solution of the salt (endothermic) m Mass of dissolved salt U Voltage I Current t Time Boundary condition: Constant temperature T0 of the calorimeter liquid To be determined qsln To be measured U(t), I(t) Known m

1 Reactant (salt), 2 stirrer, 3 thermometer, 4 calorimeter liquid, 5 electric heater Figure 1.3 Br€ onsted’s calorimeter (shown schematically).

substance transformed gives the specific heat of transition: qtrs ¼

E el Dm

1.1.2 Compensation by Electric Effects

This method was applied for the first time by von Steinwehr (1901) and Br€ onsted (1906). This measuring procedure is best illustrated by the experiment performed by Br€onsted. The device shown schematically in Figure 1.3 served for the measurement of the endothermic heat of solution of a salt in water. An amount of salt is dissolved endothermically in a Dewar vessel containing water, and the contents are steadily mixed by means of a stirrer. An electric heater yields a heat output that is regulated so as to leave the solution temperature unchanged. If the R voltage U(t) and the current I(t) are constantly recorded, then DQ ¼ UðtÞ
IðtÞdt (the electrically generated compensatory heat) equals the heat of solution of the salt. A major advantage of this procedure is that the conditions of heat transfer to the surroundings (heat leakage; see Chapter 4) remain unchanged during the measurement. Consequently, a highly sensitive but not necessarily calibrated thermometer can be used as a zero change indicator. The only correction to be made is to the heat generated by the stirrer; this is determined separately in a blank run. Because a resistor can only produce heat, this method is restricted to the measurement of endothermic effects. The compensation of exothermic effects is possible, in principle, by the use of electric cooling using the Peltier effect. The applications of Peltier cooling are discussed in greater detail in Chapter 7. Calorimeters involving an electrical compensation of the thermal effect are

1.2 Measurement of Temperature Differences

advantageous because the calibration of the calorimeter is necessary only for the determination of the sensitivity of the apparatus, and the electrical quantities voltage U and current I can be measured with great accuracy.

1.2 Measurement of Temperature Differences

Every exchange of heat needs a temperature difference to enable a heat flow. Therefore, the measurement of heats and heat flow rates can be reduced to the measurement of temperature differences (i) as a function of time, that is, DT ¼ Tðt2 Þ Tðt1 Þ, inside a calorimeter, or (ii) as a function of position, DT ¼ Tðx 2 Þ Tðx 1 Þ, along a heat conducting path. 1.2.1 Measurement of Time-Dependent Temperature Differences

The oldest method for the indirect determination of heat consists of measuring the change of temperature of a given mass of water after the introduction of a hot sample. This approach to the measurement of heat emerged with the development of reproducible, graduated thermometers (approximately since 1700); it is based on the principle that a given heat, DQ, always changes the temperature of a given mass of water by the same amount, that is, DT ¼ DQ/Cw. This is true only if the heat capacity of water Cw does not depend on temperature, which is only roughly correct. A way out of this difficulty would be to confine the use of the calorimeter to a narrow temperature range, say, between 14.5 and 15.5  C, to give a historical example. By means of this “mixing calorimetry” method, Wilcke (1781) and Crawford (1788) determined the specific heat capacities of various substances. Calorimeters with a liquid calorimeter substance can be made in a variety of designs (see Chapter 7). Using a version known as a combustion calorimeter, Crawford (1788) also measured the heats of combustion of various substances and compared these with the heat generated by a guinea pig. The calorimeter substance used for this purpose does not have to be water. Other liquid and even solid substances are also suitable. If the calorimeter substance is a solid (usually a metal), the calorimeter is referred to as “aneroid” (nonliquid). In all calorimeters based on this principle (Figure 1.4), the heat exchange between the sample and the calorimeter substance alters the temperature of the latter from T(tini) ¼ Tini to T(tfin) ¼ Tfin. The change of temperature DT ¼ Tfin Tini is measured. The quantity of heat to be determined is DQ ¼ Ccal
DT. The “heat capacity” Ccal of the calorimeter, which represents the sum of the heat capacity of the calorimeter substance and the heat capacities of other instrument components (stirrer, thermometer, vessel) involved to a greater or lesser extent in the temperature change DT, must be known. This “heat capacity” is an instrument-

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1 Methods of Calorimetry

Working equation: CS · (TS — Tfin) = Ccal · (Tfin — Tini) CS Heat capacity of the sample Ccal Heat capacity of the calorimeter TS Initial temperature of the sample Tini Initial temperature of the calorimeter substance Tf Final temperature of the calorimeter substance To be determined CS To be measured Tfin, Tini, TS Known Ccal

1 Sample, 2 stirrer, 3 thermometer, 4 calorimeter liquid Figure 1.4 Calorimeter for the measurement of a time-dependent temperature difference (classical calorimeter with liquid calorimeter substance).

associated factor (determined by a proper calibration procedure) that can assume a variety of values depending on the experimental conditions (e.g., the magnitude of DT, the duration of the investigation, etc.). Moreover, part of the entire heat DQ0 escapes directly into the surroundings by heat transfer via “heat leaks” and does not contribute to the temperature change DT. This heat loss must be determined. Hence, there is the need for a calibration of the calorimeter. For this purpose, a known quantity of heat DQ is introduced, and the resulting temperature change DT is measured to get Ccal ¼ DQ/DT. The effective “heat capacity” of the calorimeter Ccal determined in this manner represents a calibration factor valid only for specific conditions of the respective experiment. The historic term for this quantity is “water value,” namely, the mass of water possessing the same heat capacity as that of the calorimeter components, partly or entirely, involved in the temperature change. Thus, the water value reflects the sensitivity of the calorimeter because it indicates the heat necessary to obtain a given temperature change. A large water value consequently means low sensitivity, and vice versa. The term “energy equivalent” is also used. The term “heat capacity” will be used here in the sense of an instrument-associated or calibration factor. The experimental determination of this factor is best performed using electric heating because electric energy can be conveniently released at the desired site and measured with great accuracy. The advantage of these calorimeters is their simple construction. However, precise determinations require the use of special arrangements (see Chapter 7). One of the major applications of calorimeters with measurement of temperature differences as a function of time is the determination of specific heat capacities by measuring the rise of the sample temperature following the supply of a known amount of electric energy. A number of versions of these devices are described in Chapter 7.

1.2 Measurement of Temperature Differences

1.2.2 Measurement of Local Temperature Differences

This method consists of the simultaneous measurement of temperature at two positions, x1 and x2, usually as their difference. It is illustrated by the following examples. 1.2.2.1 First Example: Flow Calorimeter Two liquids capable of reacting with one another and possessing the same known temperature T1 (at position x1) flow into a reaction tube (Figure 1.5). There they react. At the measuring position for T2 (i.e., at position x2), where the reaction is assumed to be already completed, the liquid flows out of the tube. The calorimeter operates continuously. With the establishment of a thermal steady state between the liquid-containing reaction tube and the surroundings, a constant temperature difference DT ¼ T(x2) T(x1) ¼ DT(x1, x2) is established that is proportional to the heat of reaction. The proportionality factor has to be determined by proper calibration. This can be done in a subsequent experiment in which the collected reaction product flows with the same flow rate and temperature around an electric heater inside the reaction tube. 1.2.2.2 Second Example: Heat Flow Rate Calorimeter In this type of calorimeter, a sample container is connected to a thermostat via a certain heat conducting body (e.g., a bar) (Figure 1.6). Initially, the entire device has the temperature T0. However, the occurrence of a thermal process (reaction) in the sample alters its temperature. This generates an equalizing heat flow rate W ¼ dQ/dt through the bar and under ideal circumstances only through the bar (no heat leaks). In the steady-state case, the heat flow rate between two adjacent cross sections of the bar is associated with a temperature difference DT ¼ W/G (see Section 4.1), where G is the thermal conductance between the sites where the

Working equation: qr · ∆m = K · ∆T = K · (T(x2) — T(x1)) qr Specific heat of reaction ∆m Mass reacted K Calibration factor T Temperature x1, x2 Position coordinates To be determined qr To be measured ∆m, ∆T Known K

1 , 2 Reaction educts, 3 reaction product Figure 1.5 Calorimeter for the measurement of a local temperature difference (flow calorimeter).

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1 Methods of Calorimetry

3 2

T(x1) 1 T(x2)

Working equation: Q = K(T0) · ∫∆T(t)dt = K(T0) · ∫(T(x2,t)—T(x1,t))dt Q Heat produced by the sample K(T) Calibration factor T Temperature t Time Position coordinates x1, x2 To be determined Q To be measured ∆T(t) Known K(T0)

1 Heat conduction path with temperature measurement points, 2 sample, 3 thermostat Figure 1.6 Calorimeter for the measurement of a local temperature difference (heat flow calorimeter).

temperature is measured located at a distance Dx from one another. Thus, G ¼ l
A/Dx (where l is the thermal conductivity of the bar material and A the crosssectional area of the bar). In the steady state, the temperature difference between two adjacent cross sections of the bar is thus proportional to the heat flow rate (for non-steady-state conditions, see Section 4.1). To determine an unknown heat flow rate, this temperature difference is measured as a function of time: DT(t) ¼ T(x2, t) T(x1, t). If the entire system reverts to the constant temperature T0, the entire heat has flowed through the bar. The R total heat exchanged through the bar can be determined from the integral DTðtÞdt together with the thermal conductivity and the geometry of the bar. This, however, applies only to the ideal case in which the entire heat flows only through the bar without any losses caused by radiation, convection, or parasitic heat transfer (heat leakage). Under actual experimental conditions, there is always a certain leakage of heat, and the integral R DTðtÞdt is proportional to the heat exchanged with an unknown proportionality factor K(T), Rwhich must be determined by proper calibration, whereupon Q ¼ KðT 0 Þ
DTðtÞdt. 1.3 Summary of Measuring Principles

A brief overview of measuring principles is given below. i) Measurement of the heat exchanged by compensation, that is, suppression of any temperature change of the calorimeter caused by the thermal effect of the sample. 1) Compensation by a phase transition and measurement of the mass of transformed substance.

References

2) Compensation by electric cooling (Peltier effect) or heating (Joule effect) and measurement of the respective electric energy. Compensation principle By endothermic effect By exothermic effect Phase transition (solid–liquid; Phase transition (liquid–solid; liquid–gaseous) gaseous–liquid) Electric cooling (Peltier effect) Electric heating (Joule effect) ii) Measurement of the heat exchanged by measurement of a temperature difference. 1) Measurement of a time-dependent temperature difference and of the effective heat capacity of the calorimeter. 2) Measurement of a local temperature difference along a well-defined heat conducting path and of a calibration factor. This classification covers all the types of calorimeters that are of relevance in practice. It provides the basis for Chapter 7, which describes instruments operating in accordance with these methods. It is noteworthy that any exact measurement of heat consists essentially of the measurement of electric energy or is traceable to electric energy determinations because the latter form of energy is easy to release, can be measured with great accuracy, and is directly connected to the base unit of the SI (Systeme international d’unites) for the electric current, the ampere. Accordingly, all calorimeters are calibrated either directly by the use of electricity or by means of precisely known heats of reaction or transition, which in turn are measured in electrically calibrated or electrically compensated calorimeters.

References Br€ onsted, J.N. (1906) Studien zur chemischen Affinit€at. II. Z. Phys. Chem., 56, 645–685. Bunsen, R. (1870) Calorimetrische Untersuchungen. Ann. Phys. Chem., 141, 1–31. Crawford, A. (1788) Experiments and Observations on Animal Heat and the Inflammation of Combustible Bodies; Being an Attempt to Resolve These Phenomena into a General Law of Nature, 2nd edn, Johnson, London. Encylopaedia Britannica (2003) The New Encyclopaedia Britannica, vol. 2, Encylopaedia Britannica, Chicago, pp. 251–252.

Kleiber, M. (1975) The Fire of Life. An Introduction to Animal Energetics, Robert E. Krieger Publishing Company, Huntington, NY. Lavoisier, A. L. and De Laplace, P.-S. (1784) Memoire sur la chaleur. Mem. Acad. Sci. Année 1780, 355–408. Ramsay, W. (1918) The Life and Letters of Joseph Black, M.D., Constable, London. Regenass, W. (1977) Thermoanalytische Methoden in der chemischen Verfahrensentwicklung. Thermochim. Acta, 20, 65–79. Robison, J. (ed.) (1803) Lectures on the Elements of Chemistry Delivered in the University of Edinburgh, by the Late

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1 Methods of Calorimetry Joseph Black, M.D., William Creech, Edinburgh. Ter Minassian, L. and Milliou, F. (1983) An isothermal calorimeter with pneumatic compensation – principles and application. J. Phys. E: Sci. Instrum., 16, 450–455.

€ ber die von Steinwehr, H. (1901) Studien u Thermochemie sehr verd€ unnter L€osungen. Z. Phys. Chem., 38, 185–199. Wilcke, J.C. (1781) Om eldens specifica myckenhet uti fasta kroppar, och des afm€atande. Kongl. Swenska Wetensk. Acad. Handl., II, 49–78.

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2 Measuring Instruments Various calorimeters are described in Chapter 1. As pointed out in that chapter, heat cannot be measured directly but only indirectly through its effects. Before a more detailed treatment of calorimeters in the second part of this book, the measuring instruments required in calorimetry will be discussed. Two categories of instruments are introduced here. The first category comprises instruments that provide a quantitative measure of the changes of quantities that are the consequences of exchanged heats, and this discussion is provided in a manner that is as simple and precise as possible. The second category describes instruments necessary for sample preparation, for the calibration of a calorimeter, and for performing a calorimetric experiment correctly. Sufficient precision and calibration capability are essential requirements necessary for the traceability of the measurement results to the International System of Units (SI). Total measurement uncertainties for the final result of the calorimetric experiment in the order of 10 2 are common, 10 3 is state of the art, and lower uncertainties require high expenditure for the calorimeter and ancillary instruments and high effort in performing the measurements.

2.1 Measurement of Amount of Substance

First, the amount of substance investigated in any calorimeter must be precisely known to obtain quantitative caloric data. The sample mass is normally determined with a balance. Section 1.1.1 describes calorimeters in which the exchanged heat serves for the phase transition of a substance (e.g., ice into water). Thus, the measurement of heat is reduced to the measurement of the amount of a substance. This quantity can be measured either directly by weighing or indirectly by determining the changes of volume or pressure associated with the phase transition. For other calorimeters, the volume and/or the pressure of a substance must be known, too.

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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2 Measuring Instruments

2.1.1 Weighing

Analytical balances can have a high degree of accuracy, a range of seven orders of magnitude, and a resolution down to 10 7 g, working in an electronic compensation manner. Mass comparators compensate for most of the weight by means of reference weights and thus allow weighing with very high resolution in a narrow range. Many models permit the continuous registration of weight changes in the course of measurement. The accuracy of these instruments is so high that nonsystematic weighing errors can be neglected in the analysis of errors involved in calorimetric measurements because other error sources, such as heat leaks or an incomplete separation of the transformed substance, are nearly always dominant. Nevertheless, correct mass determinations require some effort. Electrostatic charges, magnetic forces, adsorption or desorption especially of water, other contaminations, air drafts caused by temperature differences between the sample and the balance, and so on must be excluded. In mass determinations, a buoyancy correction is necessary when the density of the sample differs from that of the calibration weights. For further information, see, for example, Debler (2000). 2.1.2 Volume Measurement

Volume measurement (i.e., measurements of volume changes) can be easily carried out by the displacement of an incompressible medium (e.g., mercury). Small volume changes result in a large displacement of the meniscus of a mercury column in a capillary tube. This approach provides high resolution, and by careful execution (constant cross section of the capillary, elimination of parallax, avoidance of any temperature differences), it also ensures accuracy. Because temperature changes affect the volume of all substances and the thermal expansion coefficient depends also on temperature, strictly isothermal conditions must be created to obtain good repeatability as well as satisfactory calibration. The main drawbacks and sources of error of such displacement instruments stem from the necessarily nonisothermal experimental conditions. The conventional liquid-in-glass thermometer is, in effect, an instrument for the measurement of volume changes. It will be discussed in greater detail later in the description of instruments for the measurement of temperature. If the volume change takes place at constant temperature, as is the case with phase transition calorimeters, the volume change can be measured with high accuracy. Volume changes of 10 6 cm3 can easily be read on a thin capillary. (A volume change of this magnitude, using 5 cm3 of mercury, reflects a temperature change of 0.001 K.) The creation of isothermal conditions is crucial for an accurate determination of volume changes. For the most direct link to the SI, it is advantageous to weigh the amount of liquid or gas. Some of the most precise calorimetric measurements have been performed with weighing of the fluid; for example, the determination of the heat capacity of sapphire, a widely used calibration material, has been performed with a

2.2 Measurement of Electric Quantities

Bunsen ice calorimeter by weighing the amount of molten water (Ginnings, Douglas, and Ball, 1950). The amount of methane burned for the measurement of its heat of combustion has been determined by direct weighing of a gas cylinder (Dale et al., 2002). 2.1.3 Pressure Measurement

The considerations for nonisothermal conditions apply also to the measurement of pressure changes, which usually consist of measurement of volume changes. However, pressure changes are far more conveniently measured by means of pressure sensors. Here, too, the pressure measurement is based on a deformation, however small and elastic, of the sensor. The displacement is measured by resistive, capacitive, or inductive transducers whose output signals are received and displayed by means of carrier frequency measuring bridges. At an uncertainty level of 0.1–1%, these instruments provide a continuous recording of the pressure course at a resolution of less than 1 Pa in the atmospheric pressure range (105 Pa). 2.1.4 Flow Measurement

Some flow calorimeters (continuous calorimeters) make use of air as a heat transfer medium; in other cases, gases or liquids react with each other or are products of the reaction. In the latter case, a possible approach to the measurement of amounts of substances consists in allowing the newly formed phase (usually a gas) to leave the system via a flow meter. Here the flow rate provides a measure of the quantity of substance transformed per unit time. Usually a pressure difference is the measurand as in capillary flow meters or is caused by the back pressure of the measuring instrument; however, the possibility of pressure rises (caused by a “buildup”) in the vessel must be taken into account. Other techniques for measuring amounts of gas make use of displacement gas meters, turbine meters, or ultrasonic meters. In these cases, the volume flow is the measured quantity. For measuring the mass flow, Coriolis or thermal mass flow meters can be used. In any case, it is very difficult to reduce the uncertainty of flow measurements below approximately 1%. This can only be achieved in exceptional cases when great effort is made to calibrate the meter with fluids of similar and known thermophysical properties (e.g., heat capacity, thermal conductivity, viscosity, density, etc.).

2.2 Measurement of Electric Quantities

A precisely known heat can be released with relative ease by means of an electric current flowing through a resistance. For determining the heat released, it is

21

22

2 Measuring Instruments

sufficient to measure the potential difference and current at the resistance and the time the current flows. It is possible to display voltages and currents with a resolution of 10 6 and more by means of modern digital voltmeters. A relative accuracy of 10 4 is easy to achieve. However, this applies first and foremost to the measuring instrument. For a highly accurate measurement of low voltages, it is necessary to eliminate the influence of contact resistances and thermal electromotive forces (EMFs) in the circuit. If only one measurement (the voltage) is to be made, it is advisable to use a constant current source where the current can be adjusted and stabilized electronically to 10 5. The heat released at the resistance equals the product of the current, the potential difference (voltage) at the resistance, and the time interval during which the current flows (a correction for the heat released in the current carrying wires must, however, be taken into consideration): Q ¼ U  I  t ð1 J ¼ 1 V A sÞ

The time interval involved is usually measured by means of an electronic clock that is triggered by the current. The resulting uncertainty of the time measurement is usually not limited by the clock but by the jitter of the trigger signal and is rarely better than 10 ms. Thus, electrically generated heat can be released and measured without much expense and with a relative uncertainty of 10
3 to 10
4. This illustrates the efficiency of calibration with electrically created heat and explains its extensive application in calorimetry.

2.3 Measurement of Temperatures

The measurement of a change of temperature in a body as a result of an exchange of heat has remained the fundamental approach of calorimetry from its earliest days to the present time. Similar to heat, temperature changes can only be measured indirectly, that is, by means of their effects. These may consist of a change of volume, resistance, the spectral distribution of emitted light, or the contact potential of metals. All of these measurements refer, in effect, to differences, which raises the question of the zero point of temperature and of the temperature scale. Thermodynamics has shown the existence of an absolute temperature and its independence from any thermometer. For the present purposes, it is sufficient to state that there is a thermodynamic temperature scale, but the experimenter has nothing to do with it. And the experimenter is not concerned with the primary standards derived from this scale. The commercial measuring instruments used in experimental work are based on or calibrated against such standards. Their accuracy with respect to the standards is an essential requirement, as are the possibility of calibration, the constancy of sensitivity, and so on. The relationships involved will not be discussed in greater detail here. Briefly, all temperature measurements are associated with a thermometer whose principles and operation must be understood before entering into a discussion of possible

2.3 Measurement of Temperatures

errors in the course of measurement (for further reading, see Eder, 1981; Quinn, 1990; Nicholas and White, 2001; Bernhard, 2004). 2.3.1 Thermometers 2.3.1.1 Liquid-in-Glass Thermometers Liquid-in-glass thermometers were the earliest devices used for the measurement of temperature (Table 2.1). They operate on the basis of a change of the volume (length, thickness) or pressure of a body as a result of a change of temperature. Variously graduated liquid-in-glass thermometers were already in production as long ago as 1700 (Fahrenheit, 1709; Reaumur, 1730; Celsius, 1742). The classical thermometer displays the volume change of a liquid (usually alcohol or mercury) located in a capillary glass

Table 2.1

Methods of temperature measurement.

Type

Range

Resolution

Liquid-in-glass thermometers

200---500 K

To 10

Gas thermometers

2---700 K

10

2

K

Vapor pressure thermometers

1---100 K

10

2

K

Resistance thermometers Metals 15---1300 K Semiconductors 0---600 K (thermistors)

To 10 To 10

4

4 6

K

K K

Thermocouples

1---3300 K

To 10 4 K (to 10 7 K in piles)

Pyrometers

>900 K

To 10

2

K

Needs

Characteristics

Isobaric, isothermal conditions Isochoric or isobaric conditions, manometer or dilatometer Manometer

Large heat capacity and sluggishness, numerous error sources Relatively large volume, high sluggishness

Constant power source and voltmeter or bridge Voltmeter or bridge, thermostat

Calibrated pyrometer or blackbody radiator

Applicable only in certain temperature ranges, depending on the liquid Standardized temperature sensors; small and rapid; small measuring uncertainty with metals; thermistors tend to age Cannot be manufactured strictly reproducibly; contact sites and inhomogeneities generate additional voltages; minute in size and quick in operation; negligible measuring errors Contactless measurement; emission dependent on the surface

23

24

2 Measuring Instruments

container as a function of temperature. Because thermal expansion coefficients are usually related to temperature in a nonlinear manner, such thermometers have to be calibrated point by point, taking the following factors into account. Before a reading is taken, some time must be allowed until the entire thermometer substance and the glass container have assumed the temperature to be measured and the corresponding mechanical equilibrium. Because glass also has a thermal expansion coefficient, its increase in volume leads to an increase in the volume available to the thermometer liquid. Consequently, various temperatures are read from the same thermometer according to the depth to which it is immersed in the bath whose temperature is to be measured. Most liquid thermometers are calibrated for the case of a fully immersed thermometer body. Moreover, the external pressure affects the internal volume of the thermometer and consequently its reading. The main cause of erroneous measurements is an inhomogeneous distribution of temperature. Homogeneity of temperature sets in rather slowly. The hysteresis of the thermal expansion of the glass also has to be taken into account. An accurate measurement therefore requires long measuring times; in other words, the liquid thermometer is a thermally sluggish measuring system, particularly on decreasing temperatures. Because of these error sources, liquid thermometers are rarely used nowadays for precision measurements (Hall and Leaver, 1959). 2.3.1.2 Gas Thermometers Gas thermometers, which likewise operate on the basis of a volume change, are still extensively used for accurate measurements, in particular for measurements on the thermodynamic temperature scale. Here, the thermometer substance is a gas that is almost ideal, whose expansion coefficient equals 1/273 K 1 and is independent of temperature. From the ideal gas equation pV ¼ nRT (where p is pressure, V is volume, n is amount of substance, R is universal gas constant, and T is temperature), temperature can be inferred, either in terms of a pressure measurement at constant volume or as a volume measurement at constant pressure. Both procedures are precise and reliable (see Section 2.1). Here, too, however, attention must be given to errors resulting from changes of the container volume (e.g., owing to changes of the ambient pressure). 2.3.1.3 Vapor Pressure Thermometers Vapor pressure thermometers constitute a special measuring device used mostly in low-temperature calorimetry. These instruments measure the vapor pressure of a known liquid substance – usually a liquefied gas – which is always associated with a given temperature. Owing to the close and definite relationship between temperature and the vapor pressure of a given substance (Figure 2.1), this method permits accurate measurements of temperature. All thermometers described above occupy a relatively large volume and require a long time interval to reach a steady-state distribution of temperature. In other words, they are hardly suitable for local or rapid measurements of temperature.

2.3 Measurement of Temperatures

Figure 2.1 Vapor pressures p of various gases as a function of temperature T.

2.3.1.4 Resistance Thermometers Temperature affects the electric conductivity of metals and semiconductors. Consequently, it can be determined by measuring the electric resistance and referring to a calibration table. Platinum is particularly suitable for resistance thermometers owing to its high melting point and remarkable chemical inertness, which results in highly reproducible measurements. The use of appropriate manufacturing techniques results in quite small platinum resistance thermometers (as small as 1 mm in diameter) that respond rapidly to temperature changes owing to their small heat capacity. Aged platinum wire can easily be calibrated owing to its chemical inertness. Platinum resistance thermometers (Pt-100 thermometers) are usually made with a resistance of 100 V at 273 K. The resistance values are tabulated as a function of temperature and can be approximated by a second-order polynomial, the Callendar equation (Callendar, 1887). The manufacturers generally indicate the extent to which the Pt-100 sensor corresponds to the standard. Absolute uncertainties of the order of 0.5 K in a range up to 250  C can usually be obtained without a subsequent calibration; more accurate measurements require either a recalibration or the use of a calibrated, and accordingly more expensive, sensor. Because the resistance of platinum changes by only about 0.4% per kelvin (in the room temperature range), the resistance (i.e., the current and the potential difference at the measuring sensor) must be measured with high precision (see Section 2.2). Moreover, attention must be given to the internal resistance of the conduits (e.g., bridge circuits) as well as to parasitic thermal EMFs at solder and terminal points and the internal heating of the platinum resistor by the measuring current. This internal heating up of the resistance thermometer imparts it a temperature higher than that of the region to be measured and thus creates a heat source that may cause large errors in the calorimeter. An advantage of standardized Pt-100 sensors results from the existence of a number of measuring instruments that display temperature

25

26

2 Measuring Instruments

directly with a resolution as small as 10 3 K but with absolute and linearization errors ranging from 0.5 to 1 K. These instruments are at any rate suitable for relative measurements. 2.3.1.5 Semiconductors Semiconductors (negative temperature coefficient (NTC) resistances, thermistors) possess resistances with much greater temperature coefficients than metals; moreover, their resistance decreases exponentially with temperature and can be described by the Steinhart–Hart equation (Steinhart and Hart, 1968). Consequently, semiconductor resistors can also serve as thermometers, especially in view of their minute size (the lower limit being at present about 0.03 mm3, with a mass of approximately 2  10 4 g). The time constant for the equalization of temperature can be brought down to 0.1 s. Thermistors are characterized by their resistance at 25  C, which is usually between 1 and 50 kV. At room temperature, their temperature coefficient amounts to approximately
4% per kelvin. Their upper temperature is generally limited to approximately 250  C. A disadvantage of semiconductors is their poor repeatability with time; these instruments tend to age and have to be calibrated more frequently. Moreover, because of the product heterogeneity, thermistors are interchangeable (as is the case with the Pt-100 thermometers) only in selected cases and in a limited temperature range (0.1 K between 0 and 70  C). Furthermore, considerations analogous to those for metallic resistance thermometers apply, and the measuring methods and error sources are the same. Temperature changes of 10
2 K can be readily measured by means of sufficiently accurate digital voltmeters. For a higher resolution, precision resistance measurements must be made; these can detect temperature changes as small as 10
6 K (W€ urz and Grubi9c, 1980), but the time required for this is rather long, offsetting the advantage provided by the low time constant of the measuring sensor. To sum up, it can be said that resistance thermometers provide accurate and relatively quick readings. They are used mainly for temperatures below 1300 K. Resistance thermometers can be easily integrated in electronic circuits. However, the internal heating by the measuring current must always be considered. 2.3.1.6 Pyrometers Pyrometers measuring the spectral intensity of the radiation emitted by hot surfaces allow the determination of their temperatures, provided the emissivities of the surfaces are known. At low temperatures (approximately
100 to 400  C), where the intensity of the radiation is small, the total emitted radiation is measured. The measurement uncertainty is on the order of 1% of the total temperature. At higher temperatures, the radiation is measured in one or two narrow wavelength bands, and sometimes the signal is directly compared with the radiation of a blackbody radiator or a calibrated tungsten light source of the same temperature. The pyrometric determination of temperature is often the only possible approach at very high temperatures; in calorimetry, however, its significance is limited.

2.3 Measurement of Temperatures

2.3.2 Thermocouples

An electric circuit made from two different materials (wires) develops an electric potential difference when the two junctions are at different temperatures. It must be noted that this thermoelectric potential is not caused by the temperature dependence of the contact potentials between the two materials but by the thermally activated diffusion of electrons along the inevitable temperature gradients in both materials. This phenomenon (the Seebeck effect) can be used for the measurement of temperature: two wires made of different metals are soldered or welded together (Figure 2.2), and junction 1 (reference junction) is brought to a known constant temperature. The temperature difference to junction 2 creates a thermoelectric potential that can be measured. The temperatures of all other junctions (e.g., those leading to the measuring instrument) must be equal (although not necessarily constant with time) because otherwise additional thermoelectric potentials would emerge. In expensive thermocouples (e.g., platinum– rhodium), the conductors leading from the thermocouple to the measuring instrument or thermostat are made of the so-called extension wires. These consist of copper–nickel alloys possessing, within a given temperature range, the same thermoelectric potential as the metal they replace. Their junction temperatures must be maintained within the required range, which can always be done without difficulty. If the thermoelectric potential is measured with modern high-impedance instruments (voltmeters), the voltage drop because of current flow is negligibly small (see Figure 2.2). Thermocouples are made of a large variety of metal combinations. They have different thermoelectric potentials that are variously related to temperature (SAB ¼ dUth(T)/dT, Seebeck coefficient for materials A and B; see Figure 2.3). The reproducibility and accuracy of different thermocouples vary, as does the ease of calibration. In the presence of thermal gradients, chemical inhomogeneities or mechanical stresses in the thermo wires lead to interfering additional (b) T 2

(a) T 2

(c) T 2 A

B

V

A

A

A

A

A C

V

B

V

C

B

C B

T1

A

C

T1

Figure 2.2 Principles of measurement of temperature by means of thermocouples. A, B, and C are different metals. (a) Simple arrangement for measurement of temperature difference T2 T1. (b) Measurement of

B

B

B

B

T1

temperature T2 with extension wires and thermostat for constant and known reference temperature T1. (c) Thermopile for measurement of temperature difference T2 T1 with higher sensitivity.

27

28

2 Measuring Instruments

Figure 2.3 Thermovoltage Uth of different thermocouples as a function of temperature H.

voltages that may distort the measurement (parasitic thermovoltage). Maintaining a constant temperature at the reference junction is somewhat inconvenient. The accuracy of the thermostat temperature used for this purpose directly affects that of temperature measurement at the other junction. The thermostats commonly used for this purpose are of the ice-water type or are electronically controlled solid-state devices. Despite the somewhat more complex maintenance of a constant temperature at the reference junction whenever a higher accuracy is required, thermocouples are rather commonly used because they can be manufactured by simple techniques (soldering, welding) in minute dimensions (welding points 0.2 mm in diameter on wires 0.08 mm in diameter). Owing to their negligible heat capacity, thermocouples provide a quick and exact temperature reading that involves the loss of a minute amount of heat. Because a thermocouple represents a voltage source, which can be measured in a currentless manner, internal warming or cooling can be avoided. As can be seen from their principle of operation (Figure 2.2), thermocouples are particularly suitable for the measurement of temperature differences, there being no need for a thermostat for the reference junction. Series connection into thermopiles, comprising up to 1000 thermocouples, permits the rapid measurement of temperature differences of 10 7 K, which is not achieved by other measuring techniques. Thus, thermocouples constitute ideal measuring instruments for the determination and continuous monitoring or control of minor differences of temperature. They are consequently used in many calorimeters. At temperatures above 1300 K, thermocouples surpass resistance thermometers for absolute temperature measurements because the precision of the latter instruments deteriorates rapidly in this range. The measurement uncertainty is on the order of 1% of the absolute temperature. It should be mentioned that the passage of electric current in a thermocouple creates a temperature difference between the junctions. The magnitude of this temperature difference is proportional to the magnitude of the electric current. Thus, the temperature rises at one of the junctions and drops at the other (see

References

Chapter 7). This cooling (the Peltier effect) can be used for the compensation of heat released in a sample, as in some calorimeters (see Chapter 7).

2.4 Chemical Composition

It is obvious that the sample under investigation should be thoroughly characterized with respect to its chemical composition (and its physical state, for example, phase). Many analytical principles are available for this purpose; the appropriate technique must be chosen according to the species sought and the level of concentration desired. But it must be taken into account that any material in contact with the sample or the calorimeter, even the surrounding atmosphere, can affect the result. Only in contactless calorimetry can this be avoided (Chekhovskoi, 1984). This is obvious, for example, in combustion calorimetry, in which air is used as an oxidizing agent. Any combustible component of the air influences the experiment. Applying a gas sensor with a flame ionization detector serves for detecting combustible contaminants. In high-temperature calorimetry, even minute traces of oxygen or water can react with the sample or the calorimeter. In such cases, filters known from gas chromatography can be used to purify the purge gas to the desired level.

References Bernhard, F. (ed.) (2004) Technische Temperaturmessung, Springer, Berlin. Callendar, H.L. (1887) On the practical measurement of temperature. Experiments made at the Cavendish Laboratory, Cambridge. Philos. Trans. R. Soc. Lond. A, 178, 161–230. Chekhovskoi, V.Ya. (1984) Levitation calorimetry, in Compendium of Thermophysical Property Measurement Methods. Vol. 1. Survey of Measurement Techniques (eds K.D. Maglic, A. Cezairliyan, and V.E. Peletsky), Plenum Press, New York, pp. 555–589. Dale, A., Lythall, C., Aucott, J., and Sayer, C. (2002) High precision calorimetry to determine the enthalpy of combustion of methane. Thermochim. Acta, 382, 47–54. Debler, E. (2000) Determination of mass in practice, in Comprehensive Mass Metrology (eds M. Kochsiek and M. Gl€aser), WileyVCH, Weinheim, pp. 400–429.

Eder, F.X. (1981) Arbeitsmethoden der Thermodynamik. Band I. Temperaturmessung, Springer, Berlin. Ginnings, D.C., Douglas, T.B., and Ball, A.F. (1950) Heat capacity of sodium between 0 and 900  C, the triple point and heat of fusion. J. Res. Natl. Bur. Stand., 45, 23–33. Hall, J.A. and Leaver, V.M. (1959) The design of mercury thermometers for calorimetry. J. Sci. Instrum., 36, 183–187. Nicholas, J.V. and White, D.R. (2001) Traceable Temperatures. An Introduction to Temperature Measurement and Calibration, 2nd edn, John Wiley & Sons, Chichester. Quinn, T.J. (1990) Temperature, Academic Press, London. Steinhart, J.S. and Hart, S.R. (1968) Calibration curves for thermistors. Deep Sea Res., 15, 497–503. W€ urz, U. and Grubi9c, M. (1980) An adiabatic calorimeter of the scanning ratio type. J. Phys. E: Sci. Instrum., 13, 525–529.

29

31

3 Fundamentals of Thermodynamics Processes involving heat exchange can be described phenomenologically using the tools of thermodynamics without recourse to atomic interpretations. Work in the field of calorimetry requires basic knowledge of thermodynamics for a better understanding of what actually has to be measured. As an example, two researchers engaged in measuring heat exchanged during a process would obtain two different results if the calorimeter used by one of them measures the heat of reaction at constant volume, whereas that of the other permits measurements at constant pressure only. Knowledge of thermodynamics helps explain this fact and enables the interconversion of the results. However, to a person not versed in thermodynamics, it may appear inconceivable that a measured heat should depend on extraneous factors such as the constancy of pressure or volume. Without sufficient knowledge of thermodynamics, there is a considerable risk in interpreting the results obtained. Thermodynamics is too wide ranging and complex to be dealt with in detail here. We shall therefore confine ourselves to an outline of essential aspects relevant to calorimetry. Some basic mathematical knowledge is, however, needed to understand the thermodynamic background and the equations presented. For a more thorough study and a better understanding necessary for calorimetry, readers are referred to textbooks of thermodynamics or physical chemistry (Falk and Ruppel, 1976; Adkins, 1983; Zemansky and Dittman, 1997; Callen, 1985; Keller, 1977; Lebon, Jou, and Casas-V
azquez, 2008).

3.1 States and Processes 3.1.1 Thermodynamic Variables (Functions of State)

A body or substance placed in a calorimeter constitutes a thermodynamic system. Such a system can be characterized by indicating the boundary conditions relative to the surroundings and the values of all relevant physical quantities, namely, temperature, pressure, and volume (for solids, the stress and deformation tensors), Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

32

3 Fundamentals of Thermodynamics

the number of particles or the amount of substance of the various chemical reagents, electric charge, electric field strength, energy, and so on. These quantities are referred to as thermodynamic variables (parameters) of the system. The thermodynamic boundary conditions specify the quantities that can be exchanged between the system and its surroundings and those that cannot traverse the “wall” of the boundary of the system. The above variables are external variables because they can, in principle, be set externally. However, not all external variables can be set at the same time independent of one another because various quantities are interrelated by equations of state, depending on the type of system. Thus, a change of the volume of a certain amount of gas also affects its pressure. In contrast to these external variables, there are also internal variables, which are not set externally but constitute a function of the external variables. What belongs to this category of internal variables are the orientation or order parameters (e.g., parameters of dipoles in an electric field and of crystallites of macromolecules produced by deformation of a body), the population of quantum states, and lattice parameters arising from allotropic transitions. If all variables of a system have a fixed value, the system is said to be in a given thermodynamic state; if they are allowed to change, we are dealing with a process. Every process involves an exchange of energy in different forms such as heat, magnetic energy, kinetic energy, displacement energy, and chemical energy between different parts of the system (phases or substances) or between the system and its surroundings (e.g., the calorimeter). The quantities that depend only on the initial and final states of a process but not on its path are designated state variables or functions of state. These include external variables that characterize or clearly define a given state, namely, temperature, pressure, volume, and electric field strength. Energy, momentum, and angular momentum are likewise state functions owing to the universal validity of the laws of conservation. The forms of energy exchanged as energy flows during a process (e.g., work, heat) are generally not state functions. On the other hand, the overall energy exchanged with the surroundings or between subsystems in a process – as a difference of state functions – is, of course, unrelated to the path by which the process proceeds if the initial and final states are the same. If the exchanged energy of a process is split into different energy flows, this distribution may vary in accordance with the specific path of the process. For this reason, work and heat are in general not state functions, that is, the amount of heat exchanged in a process depends on the respective path of the process. These relationships can also be presented in a mathematical form: external variables of a system can be treated as mathematical variables. The reduction of the total number of all conceivable parameters by subtracting the number of all laws (equations of state) that relate them yields the number of independent variables (degrees of freedom) of a system. Which of the state variables can then be regarded as independent and which are to be treated as dependent according to the indicated laws is a matter of free choice; the decision must be based on the expediency of the

3.1 States and Processes

case at hand. A successful selection yields concise formulas and relationships. The corresponding independent variables are designated standard variables. The n independent variables can be conceived as an n-dimensional space known as the space of states. Every state characterized by a set of n-values of the variables represents a point in this space. Consequently, a process constitutes a line (path) in this space. State functions depend only on the values of the variables of the respective point. They can be represented as the exact or total differential of the variables forming the space of states. In terms of mathematics, let x1, x2, . . . , xn be the independent variables and Z(x1, x2, . . . , xn) a state function; then the total differential is dZ ¼

@Z @Z @Z dx 1 þ dx2 þ


þ dx n @x 1 @x2 @xn

ð3:1Þ

Any quantity DX exchanged in a process can be represented as an integral along a path L that the process takes (line or path integral, see mathematical textbooks): DX ¼

∲

dX

ðLÞ

Generally, the exchanged quantity DX depends on the respective path. If the quantity DX is a state function difference DZ, the line integral depends only on the initial state I and the final state F, but not on the path L of the process: DZ ¼

∲

dZ ¼ ZF  Z I

ð3:2Þ

ðLÞ

For these reasons, the line integral over any closed path (symbolized with a closed circle) for a state function is always zero, þ DZ ¼ dZ ¼ 0 because the state function gets the same value upon returning to the starting state. Thus, for the temperature T as a state variable, þ DT ¼ dT ¼ T F  T I and dT ¼ 0 ∲ ðLÞ

In other words, the same state is always associated with the same temperature. It can be shown that the algebraic linkage of state functions provides a definition of other state functions. Thus, the three state functions energy E, pressure p, and volume V yield a new state function H, known as enthalpy H ¼ E þ p  V. Many other state functions can be defined in this manner. A few of such “artificial” state functions besides enthalpy have proven their significance in thermodynamics. These functions, also known as thermodynamic potential functions or thermodynamic potentials, are discussed in Section 3.1.2. It is noteworthy that these new definitions

33

34

3 Fundamentals of Thermodynamics

do not affect the number of independent variables, because every new state function is associated with yet another relationship and the difference between the number of variables and the number of relations does not change. If two identical systems that are in the same state (i.e., all their relevant variables are equal) are united into a single system, it can be seen that some variables (e.g., temperature, pressure, field strength) remain unchanged while others (e.g., volume, number of particles, charge, energy) assume a double value. The variables of the first category are known as intensive, and those of the second category as extensive. It is advisable to select independent variables for a given system that are either all intensive or all extensive. It is recommended that extensive variables be selected for systems that are not in equilibrium (see Section 3.1.3). For calorimetric purposes, the variables of choice are usually temperature, pressure, and volume (or the stress and deformation tensors for solids) and in some cases also electric field strength and charge (in low-temperature physics, for example). Energy exchanged in processes is measured in calorimetry mainly in the form of heat. However, other forms of energy must also be measured in certain experiments, such as the quantity of displacement energy (mechanical work) in deformation calorimetry, and in some cases also surface energy and compression energy if they change during an experiment. 3.1.2 Forms of Energy, Fundamental Form, and Thermodynamic Potential Function

For a given state, energy remains constant by definition; in other words, a process must take place if energy is to be exchanged. In an exchange with another system (or with the surroundings), energy transfers in specific forms, namely, as heat, kinetic energy, chemical energy, electric energy, and so on. These pass or flow from one system to another or to the surroundings. The various forms of energy flow only during the process that involves their exchange – never in a state. Accordingly, the forms of energy can be formulated mathematically as differential quotients per unit time, that is, as heat flow rates, or disregarding the denominator of the differential quotient, in the form of a differential (not necessarily a total differential). With regard to all known forms of energy, the energy change can be presented as a product of an intensive variable and the differential of an extensive one. Examples of energy forms Displacement energy (work) (f force, r position vector) Kinetic energy (v velocity, p momentum) Rotational energy (v angular velocity, L angular momentum) Compression energy (p pressure, V volume) Surface energy (s surface tension, AS surface area) Electric energy (w electric potential, Q electric charge) Chemical energy (m chemical potential, N number of particles) Heat (T temperature, S entropy)

f  dr v  dp v  dL p  dV s  dAS w  dQ m  dN T dS

3.1 States and Processes

For heat, the last form of energy listed, the extensive variable S, the entropy, is also a state function. In the case that any energy form is a total differential, the quantity of exchanged energy of this form is the simple definite integral over the respective form. If the differentials of the energy forms are not total ones (the normal case), the curve (or line) integral has to be calculated; in such a case, the result depends also on the path of the process and not only on its initial and final states. Regrettably, the form of energy dealt with in calorimetry, namely, heat, does not constitute a total differential. The heat exchanged in a process may vary according to the path of the process even if the initial and final states are the same, as illustrated by a well-known example (Gay-Lussac): if an approximately ideal gas is allowed to flow into an evacuated vessel, there is no exchange of heat with the surroundings and the temperature remains unchanged. However, if the same gas is allowed to expand slowly by means of a piston to the same final state, the system must exchange a quantity of heat. In particular, the energy introduced must exactly compensate the quantity of expansion energy obtained by means of the piston. 3.1.2.1 Fundamental Form The customary method for the description of processes consists in writing down all possible, mutually independent forms of energy for a given system – there are as many as there are independent variables – and obtaining their sum. This sum represents the change of the energy of the respective system undergoing any conceivable process: dE ¼ i1  de1 þ i2  de2 þ    þ in  den

ð3:3Þ

where i1    in are intensive variables, e1    en are extensive variables, and im  dem is the mth energy form. The expression (3.3) is known as the Gibbs fundamental form of the system (it represents a special Pfaffian form, as it is called in differential calculus). Every process that can be performed by the system must satisfy this differential form. On the left-hand side is the total differential of energy (because energy is a state function!); the energy forms located on the right-hand side of the equation generally do not constitute total differentials, although they can often be summed up into total differentials (see later). 3.1.2.2 Thermodynamic Potential Function All systems capable of exchanging energy in the same forms have the same Gibbs fundamental form. The fundamental form of systems that can exchange heat, compression energy, and chemical energy (i.e., systems commonly encountered in calorimetry) is as follows for m kinds of substances: dE ¼ T  dS  p  dV þ

m X

mi  dN i

ð3:4Þ

i¼1

If the system is in a particular state, all variables are constant. In particular, all differentials of the extensive variables are zero. Hence, it follows from

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3 Fundamentals of Thermodynamics

Eq. (3.4) that dE ¼ 0, namely, that E is constant. Consequently, E must be a function of the extensive variables (i.e., a state function). The expression E ¼ E (e1, e2, e3, . . . , en), known as the thermodynamic potential function, precisely characterizes the system. It follows from the mathematical analysis that the intensive variables conjugated to the independent, extensive variables can be derived from the latter if the thermodynamic potential function is known. Because the energy is a state function, dE is a total differential; from Eq. (3.1), it therefore follows that dE ¼

@E @E @E  de1 þ  de2 þ    þ  den @e1 @e2 @en

A comparison with Eq. (3.3) shows that @E ¼ i1 ðe1 ; e2 ; . . . ; en Þ; @e1

@E ¼ i2 ðe1 ; e2 ; . . . ; en Þ; . . . ; @e2

@E ¼ in ðe1 ; e2 ; . . . ; en Þ @en

In view of the large choice of extensive variables that can be selected as independent variables in thermodynamics, the relevant independent variables should always be indicated as such in order to avoid misunderstandings. A thermodynamic potential function provides a complete description of the system. However, energy can only constitute the thermodynamic potential function of a system if the independent, extensive variables of the system are selected as standard variables. Other thermodynamic potential functions would result if other variables – all intensive or some of them intensive and others extensive – are selected as independent. Example from calorimetry: For a system that – during a process – exchanges heat and compression energy, the fundamental form reads as follows: dE ¼ T  dS  p  dV

ð3:5Þ

Where entropy S and volume V are the extensive variables and temperature T and pressure p are the conjugated intensive variables (the signs indicated follow from the definition of the positive direction of the energy flow, namely, one directed into 1) the system). Thus, the thermodynamic potential of this system is E ¼ E(S, V) ; if this function is known, the system can be regarded as completely described. But the variables S and V cannot be controlled in a simple, direct manner. Thus, it would be preferable to select the pressure p and the temperature T as variables because they can easily be set experimentally. Which quantity is then the proper thermodynamic potential? The above fundamental form is dE ¼ T  dS  p  dV

1) In thermodynamic textbooks, the energy E is usually named internal energy with the symbol U.

3.1 States and Processes

Our purpose is to establish a thermodynamic potential function Z(T, p), which satisfies the following differential form: dZ ¼ S  dT þ V  dp

ð3:6Þ

This is carried out in two steps: Step one: Addition of the identity d(p  V) ¼ V  dp þ p  dV to Eq. (3.5): dH  dðE þ p  VÞ ¼ T  dS þ V  dp

ð3:7Þ

The independent variables S and p are located on the right-hand side, whereas the corresponding thermodynamic potential is on the left-hand side. The function H ¼ E þ p  V is named enthalpy. Obviously, with this choice of variables S and p, energy is not a thermodynamic potential function because E is not solo on the left-hand side of this differential form with the variables S and p. Step two: Subtraction of the identity d(T  S) ¼ S  dT þ T dS from Eq. (3.7): dG  dðH  T  SÞ ¼ S  dT þ V  dp

ð3:8Þ

The function G ¼ H  T  S ¼ E þ p  V  T  S is termed the Gibbs function or Gibbs energy; it represents the thermodynamic potential function sought in Eq. (3.6) for the independent variables T and p. For the sake of completeness, the thermodynamic potential function must also be derived for the fourth possible combination of variables. For this purpose, the above identity must be subtracted from Eq. (3.5), yielding dA  dðE  T  SÞ ¼ S  dT  p  dV

ð3:9Þ

The function A ¼ E  T  S is known as the Helmholtz function or Helmholtz energy; it represents the thermodynamic potential function for the independent variables T and V. If one starts from a fundamental form other than that of Eq. (3.5), that is, from systems capable of also exchanging other forms of energy, the result would be a thermodynamic potential function obtained in a similar manner but for another set of selected independent variables. However, in calorimetry, the above-mentioned thermodynamic potential functions are usually sufficient. Calorimetry often consists in finding the thermodynamic potential function that provides a complete description of the respective system. Once one thermodynamic potential function has been found, all the others can be calculated using the definitions H EþpV GHST AEST

The equations for the determination of the respective dependent variables can be obtained from the thermodynamic potential function by partial differentiation with respect to the independent variables.

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3 Fundamentals of Thermodynamics

3.1.3 Equilibrium

A system is in thermodynamic equilibrium if all measurable parameters (variables) maintain a constant value after being allowed to reach a final state without external influences. Equilibrium is the final state of a system possessing free variables. In the state of equilibrium, no exchange takes place within the system or between the latter and its surroundings. To confirm the existence of equilibrium, it is not sufficient to test the constancy of all variables; it must also be proven that every variable said to be independent and free in the particular system could actually reach its final state without impediment. As an example, a vehicle located on a sloping road is not in equilibrium because it tends to roll down. If the brakes of the vehicle are engaged, their friction prevents the vehicle from rolling down and creates a metastable state, which is not an equilibrium state with regard to the potential energy (the thermodynamic potential function of that system). Any system is characterized thermodynamically by indicating the free variables and the boundary conditions. The boundary conditions also show whether a (normally free) variable is kept artificially constant. If a gas is enclosed in a given volume, the boundary of the system prevents any expansion of the gas into the general atmosphere. Within the closed system, the gas can then be regarded as being in equilibrium if the variables that remain free (e.g., pressure, temperature, energy) are invariant. The importance of an exact definition of the system in question and its boundary conditions in any discussion of equilibrium can be illustrated in the following example. Let us imagine a heat-permeable vessel that is divided in a 2 : 1 ratio by a heat-permeable partition. The smaller chamber contains a given amount of oxygen (characterized by the number of particles), and the larger chamber contains double the amount of hydrogen. After a certain time interval, one finds that pressure and temperature are constant and equal in the two chambers. This state is one of equilibrium for the given system. If the system is changed by removing the partition between the two gases, these will mix with one another and a new state will emerge (oxyhydrogen). This state, however, is not one of equilibrium (unless chemical reactions are ruled out by formulating another boundary condition), but a metastable state. A state of equilibrium would only be reached if the oxyhydrogen reaction 2H2 þ O2 ! 2H2O (liquid) þ 570  103 J is triggered (by a spark), thus allowing the number of O2, H2, and H2O particles – which are likewise variables of the system – to attain their equilibrium values, which involves the release of considerable energy. The above example shows how difficult it is to ascertain that a system is actually in a state of equilibrium. Yet many statements in thermodynamics apply strictly to equilibrium states (e.g., the law of mass action, the constancy of the melting point of a pure substance at a constant pressure). False equilibriums invariably involve a source of error that may be greatly misleading in caloric measurements. Thus, the measured heat of transition can only be regarded as a difference of state variables if the crystal modification created by the solidifica-

3.1 States and Processes

∆

∆

G ∆actG ∆rG

Gibbs function Gibbs energy of activation Gibbs energy of reaction (difference of Gibbs energy between state 1 and state 2)

Figure 3.1 Gibbs function of a system. 1: metastable state, 2: stable equilibrium state.

tion of a substance is really the equilibrium crystal form. Only in this case the measured heat of transition permits a correct calculation of the effect of pressure on the melting temperature (via the Clausius–Clapeyron equation). If a solid exists in a number of modifications (polymorphs), its cooling from the melt may yield any one of these or a mixture of them, depending on the rate of cooling. Transformation into the real equilibrium form may be so slow as to simulate a stable equilibrium, although the system is still far from achieving such a state. In thermodynamics, a decisive criterion exists for the existence of real equilibrium: the thermodynamic potential function is always at a minimum in a state of equilibrium. Thus, if the thermodynamic potential function of the system is known, the extreme value (equilibrium state) can be calculated and compared with the measured values of the system variables. For the example of the oxyhydrogen system described above, the Gibbs energy (the thermodynamic potential function) would be a hyperplane in a multidimensional space of states with two minima: one of them, the relative one (the oxyhydrogen gas mixture), is only metastable; the other absolute minimum (the water) characterizes the equilibrium state of the system. Figure 3.1 shows a one-dimensional simplification with a similar functional path. The system remains at the relative minimum (1) until the Gibbs function rises by DactG owing to the introduction of energy (activation energy) by the ignition spark, whereupon the system overcomes the barrier and reaches the absolute minimum (2), thus releasing the introduced activation energy together with an additional amount of energy (corresponding to DrG). States at which the thermodynamic potential function is at a relative minimum are designated metastable states. They remain stable and simulate an equilibrium state up to the introduction of the activation energy. This can take place by supplying the system locally with the necessary amount of energy (spark). If the activation energy is not too large, the thermal motions of atoms may suffice to overcome the activation threshold and trigger the reaction (thermally activated processes).

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To sum up, it can be stated that a system (characterized by the independent variables and the boundary conditions) is in equilibrium if the corresponding thermodynamic potential function is at a minimum. There are metastable states, characterized by a relative minimum of the thermodynamic potential function, that can exist for any length of time provided the necessary activation energy is not introduced. A state that is stable under specific boundary conditions may, by a further reduction of the thermodynamic potential function, pass into a more stable state if the boundary conditions of the system are changed. Stable and metastable states must satisfy the extreme condition that the differential of the thermodynamic potential function must be zero, namely, dG ¼ 0, dA ¼ 0, dH ¼ 0, dE ¼ 0 (depending on the choice of independent variables). Because the extreme value of the thermodynamic potential function is at a minimum, all second derivatives must be greater than zero. It follows from the extreme value condition that in the state of equilibrium, the intensive variables belonging to the freely exchangeable forms of energy are equal throughout the system and in any subsystems. Thus, when a system composed of n subsystems and capable of exchanging heat, compression energy, and chemical energy is in a state of equilibrium: T ¼ const: and T ¼ T 1 ¼ T 2 ¼ T 3 ¼    ¼ T n p ¼ const: and p ¼ p1 ¼ p2 ¼ p3 ¼    ¼ pn m ¼ const: and m ¼ m1 ¼ m2 ¼ m3 ¼    ¼ mn

where T is the temperature, p is the pressure, and m is the chemical potential. Another condition, although less obvious, is of major importance: the entropy S is always at a maximum in an isolated system in equilibrium. The requirement for an isolated system means that there must be no exchange of energy whatsoever with the surroundings. This requirement can always be met by modifying the boundary conditions so as to combine the system with the surroundings to which an exchange of energy in some form occurs – in other words, by considering the overall entropy of the system and the surroundings. It is noteworthy that a system, even when not in equilibrium, always has definite extensive variables: as for the intensive variables, they are – as a rule – definite in a stable or metastable state only. Hence, the preference is for extensive quantities as independent variables. Another concept to be explained in this connection is the so-called steady state. This is not a state of equilibrium. In the steady state, a steady flow of energy occurs between the system and its surroundings in those forms of energy that can be exchanged by the system, and this occurs such that the variables of the system remain constant with time. In other words, the flow of energy and particles into and out of the system are equal. Here a steady-state process creates the false impression of a state. In such cases, the intensive variables are, of course, not constant in space because otherwise no flow of energy would be possible; moreover, the entropy increases during steady-state processes.

3.1 States and Processes

3.1.4 Reversible and Irreversible Processes

Processes that take place spontaneously in a system (i.e., without interference from the surroundings) on a path in the space of states in only one direction are named irreversible. All processes that can occur in the universe are irreversible. It is characteristic of them that, in general, only their final state is one of equilibrium and that the energy forms of the respective fundamental form are not sufficient for a description of the system and the process. During the process, the system often breaks down into subsystems between which an exchange of additional forms of energy may take place. Within the system there appear, for example, turbulences, currents, fields, and so on, which vanish by the time the process is over. Intensive variables are usually not defined in such cases, and a path in the space of states is therefore not to be indicated in all cases. Under such circumstances, an integration of the Gibbs differential equation is impossible because the processes involved cannot be described by means of classical thermodynamics. A reversible process is only possible as a mental exercise; it requires, at every moment of the process – that is, at every point of the path – the existence of equilibrium with all its implications. In principle, a process cannot take place under such conditions unless the variables undergo infinitesimal variations along the path, in which case the process would last for an indefinitely long period. The advantage of reversible processes is, however, that all forms of energy exchanged along a reversible path can be calculated because all variables are always defined, so that the path is known. There is no entropy production in reversible processes. Consequently, all the entropy change of the system is calculated only from the quantity of heat exchanged with the surroundings. The performance of reversible, cyclic processes as a thought experiment has contributed a great deal toward the understanding of the principles involved in thermodynamics. If a real process is to be carried out as reversibly as possible in the laboratory, it must be conducted very slowly and therefore without major changes (and gradients) of the variables. With reference to calorimetry, this means in particular that temperature differences must be small and the change of variables slow to allow the necessary relaxation processes to take place in the system. Under such circumstances, the real path of the process in the space of states differs only slightly from the ideal reversible path (which represents a sequence of equilibriums). Suitable measures such as stirring or the use of small dimensions of apparatus assist in creating a nearly reversible path for the process. To ascertain if the process actually approximates its reversible limit, it is necessary to conduct a series of experiments in which the process is carried out more and more slowly, so that the parameters can be extrapolated to an infinitely slow path and the errors estimated. Thus, a reversible process is an idealized process that can only be carried out as an approximation, although it allows the calculation of all the exchanged quantities. Together with the second law (see Section 3.1.5.3), these findings also yield conclusions on irreversible, that is, real processes.

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3.1.5 The Laws of Thermodynamics

The laws of thermodynamics are based on experience. As with all axioms, they cannot be proven, although there can be no doubt as to their validity or applicability. 3.1.5.1 The Zeroth Law Every thermodynamic system in a state of equilibrium possesses one intensive variable of state equal within the system: the temperature. In nonequilibrium states and in irreversible processes, temperature – as with all intensive variables – is generally not defined. 3.1.5.2 The First Law Energy is conserved; it can neither be destroyed nor created but only exchanged in a variety of forms. This “energy conservation law,” based on experience, was tacitly assumed in the preceding sections; otherwise, the fundamental form would be inconceivable and the physical quantity known as energy would not comply with the mathematical requirements. From the first law, it follows that energy is a state function. The total energy exchanged does not depend on the path the process performs; rather, it depends only on the initial and the final states. 3.1.5.3 The Second Law The second law can be formulated in a number of ways. All these formulations are equivalent; they merely represent different versions of the same physical principle. This can be proven theoretically. For practical purposes, one can therefore use the version that best fits the specific situation.

i) Formulation by Clausius It is impossible to transfer heat from a colder to a warmer reservoir without causing changes in the surroundings. Example: A refrigerator consumes electric energy from the surroundings in order to transfer heat from its interior to the warmer “reservoir” kitchen. ii) Formulation by Thomson It is impossible to construct a periodically working machine that does nothing except doing work and cooling a single heat reservoir. Example: A ship cannot obtain propulsion power by only cooling the ocean. iii) Formulation by Carath
eodory In the neighborhood of any arbitrary state of a physical system, there are other states that cannot be attained adiabatically (i.e., without an exchange of heat). iv) Formulation According to Falk and Ruppel Entropy cannot be destroyed, but only exchanged or created. In other words, the entropy of the universe increases all the time because all real processes are connected with an entropy production.

3.1 States and Processes

v) Formulation According to Keller a) Every simple thermodynamic system possesses, in any equilibrium state, an extensive state variable S (entropy): S ¼ S0 þ

dQ ∲ T

ðLR ¼ reversible pathÞ

ð3:10Þ

ðLRÞ

b) If a system passes from state 1 to state 2, Sð2Þ  Sð1Þ 

ð2

dQ T

ðintegrated along any path LÞ

ð3:11Þ

1

where dQ is the heat exchanged between the system and a reservoir of temperature T. In the terms used here, formulation (b) means that in irreversible processes a change of entropy comes not only from the exchanged quantity of heat but also from additional sources. In other words, every heat exchanged in a process is connected with an entropy change (see Eq. (3.10)), but not every entropy change is connected with a heat exchange! Only in the extreme case of an ideal, reversible process can the change of entropy be calculated strictly from the exchanged heat. To calculate the entropy change between two states, one has to perform a reversible process from state 1 to state 2. In this case, the equality sign applies in Eq. (3.11). As an example, the entropy of mixing of two noninteracting liquids cannot be determined in a calorimeter. 3.1.5.4 The Third Law

Formulation by Nernst At T ! 0 K, the entropy of a homogeneous system tends toward zero if the system is in states that satisfy the stability requirements and the intensive variables of the system have finite values. As a consequence, entropy cannot take on any negative values. It follows from the third law that for every extensive variable e and every conjugated intensive variable i, the following equations apply at T ¼ 0 K:   @e  @i  ¼ 0 and ¼0 @T T¼0;i;... @T T¼0;e;...

Thus, the thermal expansion coefficient @V=@T, the thermal tension coefficient @p=@T, and the heat capacities @E=@T and @H=@T are all equal to zero at the absolute zero. Consequently, the temperature T ¼ 0 K can in principle not be reached. 3.1.6 Measurement of Thermodynamic State Functions

The only forms of energy (cf. Section 3.1.2) usually dealt with in calorimetry are heat, compression work, and chemical energy. The measurement of state functions will be described for such cases in what follows. For a different combination of

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variables, the fundamental form reads as follows: X dEðS; V; N i Þ ¼ T  dS  p  dV þ mi  dN i and furthermore

dHðS; p; N i Þ ¼ T  dS þ V  dp þ

P

mi  dN i P dAðT; V; N i Þ ¼ S  dT  p  dV þ mi  dN i P dGðT; p; N i Þ ¼ S  dT þ V  dp þ mi  dN i

Case 1: Without Chemical Reactions In the calorimetry case, when heat is to be measured and the composition is kept constant (with the boundary condition that no chemical reactions take place, that is, dNi ¼ 0), it follows that the fundamental form simplifies: dEðS; VÞ ¼ T  dS  p  dV dHðS; pÞ ¼ T  dS þ V  dp dAðT; VÞ ¼ S  dT  p  dV dGðT; pÞ ¼ S  dT þ V  dp

If, for example, the measurement is made at constant volume, that is, with the boundary condition dV ¼ 0, then dEV ¼ T dSV is the differential of exchanged heat; on the other hand, ð@E=@TÞV ¼ CV is the heat capacity at constant volume. By combining these equations, we obtain dSV ¼

CV ðTÞ  dT T

and SV ðTÞ ¼

ð

C V ðTÞ dT þ S0 : T

Furthermore, we get dE V ¼ CV  dT

so that ð E V ðTÞ ¼ C V ðTÞdT þ E 0 :

The energy and the entropy of the system can be determined from the heat capacity at constant volume CV(T). For measurements at constant pressure, that is, with the boundary condition dp ¼ 0, we get in a similar way dH p ¼ T  dSp

3.1 States and Processes

and with ð@H=@TÞp ¼ Cp , the heat capacity at constant pressure, it follows that dSp ¼

Cp ðTÞ  dT T

and Sp ðTÞ ¼

ð

Cp ðTÞ dT þ S0 T

With regard to enthalpy, dH p ¼ Cp ðTÞ  dT

so that ð H p ðTÞ ¼ C p ðTÞdT þ H 0

In the case of constant pressure, entropy and enthalpy can be determined from the heat capacity at constant pressure Cp(T). The measurement of heat capacities as a function of temperature permits the calculation of all thermodynamic functions; thus, the missing state functions can be calculated from SV(T), Sp(T), EV(T), and Hp(T) if the change of pressure or volume dependent on temperature is also measured. The thermodynamic potential function corresponding to the selected free variables is thus obtained. The system is now completely described because the still missing dependent variables can be calculated as partial derivatives of the respective thermodynamic potential function. Case 2: Chemical Reactions The measurement of the state functions of chemical reactions can either start from the boundary condition that the volume is constant (the reaction takes place in an autoclave or Berthelot bomb) or that the pressure is constant (the reaction takes place at atmospheric pressure in the laboratory). In the former case (dV ¼ 0), it follows from the law of conservation of energy that the quantity of isothermally exchanged heat equals the chemical energy of the reaction: dE V;T ¼ 0 ¼ T  dSV;T þ

X

mi  dN i

i

Hence, the heat of reaction reads: Dr Q V;T ¼ T  Dr SV;T ¼ 

ðF X I

mi  dN i

i

where DrSV,T is the entropy exchanged with the heat DrQV,T. I and F characterize the initial and final state of the chemical reaction with the respective concentrations of the components.

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3 Fundamentals of Thermodynamics

Furthermore, if volume and temperature are kept constant: X dAV;T ¼ mi  dN i i

Hence, we get the Helmholtz energy of reaction: Dr AV;T ¼ Dr Q V;T ¼ T  Dr SV;T

For reactions at constant pressure and temperature, it follows in a similar way that X mi  dN i dH p;T ¼ 0 ¼ T  dSp;T þ i

Hence, the heat of reaction reads Dr Q p;T ¼ T  DSp;T ¼ 

ðF X I

mi  dN i

i

and because dGp;T ¼

X

mi  dN i

i

the Gibbs energy of reaction reads Dr Gp;T ¼ Dr Q p;T ¼ T  Dr Sp;T

If the reactions were carried out reversibly, the total entropy of the whole system would not change; hence, the quantity of heat exchanged between the subsystems would correspond to an exchange of entropy without any additional generation of entropy. If the process is conducted irreversibly in an isolated system, the total entropy must, however, increase. Because entropy is a state variable, the final state is, as a matter of principle, different from that of a reversible process. At any rate, such an undefined final state is of no significance for the measurement of DrAV,T, and DrGV,T if the boundary conditions T ¼ constant and V ¼ constant or p ¼ constant for the initial and final states are complied with because A and G are state functions and as such independent of the possibly changing entropy S, if T is kept constant. These special boundary conditions stipulate that the final state is only undefined with regard to the entropy; all the other state functions and variables are equal and independent of the path of the process – irrespective of its being reversible or irreversible. In particular, the quantity of exchanged heat (owing to the conservation of energy) does not depend on whether this exchange is reversible (e.g., strictly isothermal) or irreversible (e.g., with local temporary temperature gradients). This fact emphasizes the importance of the above boundary conditions in practical calorimetry and the importance of the Gibbs and the Helmholtz energy as state functions. The entropy, on the other hand, which is very important in irreversible thermodynamics, can be disregarded in equilibrium calorimetry.

3.2 Phases and Phase Transitions

3.2 Phases and Phase Transitions 3.2.1 Multiphase Systems

The term phase applies to a homogeneous subsystem of a composite (heterogeneous) system. The phases of a heterogeneous system differ from one another in at least one chemical or physical property. A phase may consist of a single kind of particle (molecule, ion, or atom) or represent a mixture (solution) of different kinds of particles. Examples: Homogeneous systems (single-phase systems): All gases and mixtures of gases, water, ice (single crystal), mercury, beer (without foam or bubbles), NaCl (single crystal), solutions. Heterogeneous systems (multiphase systems): Water together with ice at 0  C, oil– water emulsions, some copper–zinc alloys (brass), beer (with foam and bubbles). Pure substances may occur in a variety of phases, depending on the boundary conditions. Thus, increasing the temperature of a solid at constant pressure causes its fusion and finally its vaporization to a gas. In pure substances, phases correspond to the states of aggregation. In a heterogeneous system, on the other hand, a number of phases in the same state of aggregation may coexist (see earlier examples). Here, we shall describe the conditions under which a multiphase system is in equilibrium; in other words, when does a homogeneous system dissociate into different phases? These issues are dealt with in the theory of heterogeneous equilibriums, which occupies an important place in all branches of chemistry and in materials science. For the present purposes, we shall only outline the thermodynamic laws on which this theory is based. A system is in a stable state if its thermodynamic potential function has a minimum at the specified boundary conditions. The most commonly occurring boundary conditions stipulate that the pressure p and the temperature T are specified from the exterior. In such case, the Gibbs function G is a thermodynamic potential function. The following considerations, although based on this particular choice of variables, can be applied in an analogous manner to other thermodynamic potential functions with other free variables. A single-component system, such as metallic copper, will remain in a given phase as long as its Gibbs function at the given p and T is smaller than the Gibbs function of any other conceivable phase. In the case of copper at normal pressure and at T < 1357 K, the crystalline solid state is in this respect more favorable relative to the liquid state. But above 1357 K, the liquid state has a smaller Gibbs function. The solid and liquid phases may coexist at the melting point. Only at the melting point are the specific Gibbs functions of both phases equal. Indeed, from the equilibrium condition stipulating a minimal energy at any state, it follows that if the Gibbs function of one of the phases (e.g., the solid state) were smaller, the system would pass spontaneously to that state, releasing Gibbs energy in that process. This means that two phases could never coexist, contrary to reality. It follows that in a stable,

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3 Fundamentals of Thermodynamics

(a)

G T 1 2

Gibbs function Temperature G(T ) for the solid phase G(T ) for the liquid phase

G H S ∆trsS ∆trsH

= H —T⋅S Enthalpy Entropy Entropy change at Ttrs Enthalpy change at Ttrs

(b)

Figure 3.2 Gibbs function, enthalpy, and entropy as a function of temperature in a onecomponent system undergoing a phase transition.

heterogeneous, one-component system, the thermodynamic potential functions must be equal in all phases. In mathematical terms, this means that stable multiphase systems occur at those combinations of variables where the hypersurfaces of the thermodynamic potential function of the different phases intersect in the space of states. In the case of a single variable (e.g., temperature), the hypersurfaces of the Gibbs function of the different phases are curves that intersect at the phase transition point; the relationships obtained are shown in Figure 3.2. In contrast to the Gibbs function, which passes steadily from one function to the other at the phase transition, enthalpy and entropy show a jump, with DtrsH ¼ T  DtrsS, which follows from the requirement DtrsG ¼ DtrsH Ttrs  DtrsS ¼ 0. We shall now explain in greater detail how knowledge of the thermodynamic potential function of a single-phase, one-component system makes it possible to determine the conditions under which the system breaks up into different phases. We start from the stability condition and show the manner in which the thermodynamic potential function of the two-phase equilibrium can be constructed. As an example, let us select energy as the thermodynamic potential function for such a system that can only exchange heat with its surroundings. The respective boundary

3.2 Phases and Phase Transitions

E S A, B I St

Energy Entropy Stable states Inflection points States of the system

Figure 3.3 Thermodynamic potential function for a two-phase system.

conditions are, therefore, that the volume and the number of particles are kept constant. Such a system can be, for example, a certain quantity of gas in a heatpermeable, closed container. From the equilibrium condition, it follows (see textbooks of thermodynamics) that for a homogeneous system to be stable, @TðS; x 2 ; . . . ; xn Þ >0 @S

ð3:12Þ

Here, the entropy S and the concentrations x2, . . . , xn are the independent variables. In other words, the temperature of a stable system is a monotonically increasing function of entropy. From Eq. (3.12) and the system already described, the following condition can be deduced: @2E >0 @S2

ð3:13Þ

because temperature, being a dependent intensive variable of the energy function, is the partial derivative of energy with respect to entropy T ¼ @E=@S. Equation (3.13) indicates that the function E(S) of a stable system must always be curved toward the energy axis. Figure 3.3 shows schematically how the thermodynamic potential function would look in this case. The behavior of this system will be followed by changing the variable S (which is monotonically connected with temperature T). The first part of the curve to the point of inflection I1 has a positive curvature, so that the system could be stable in this zone. Between I1 and I2, the thermodynamic potential function has a negative curvature, which means that the system is unstable. From I2 onward, it could be stable again. It can be exactly deduced that the system breaks down spontaneously at the state St3 into two subsystems (phases), one of which is in state A and the other in state B, whereupon the sum of the energies of the two subsystems is smaller (namely, on the dashed line for the same entropy S) than the energy of the hypothetical state St3 (on the solid curve). It can be demonstrated that this is also true of the state St2 or St4, that is, in the regions A to I1 or I2 to B, but the system can only break down into the two subsystems (characterized by the states A and B) if a certain activation energy is supplied. Hence, the single-phase state is metastable in these regions.

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Remark: When heating a substance, the metastable region A to I1 characterizes the region of “superheating” of phase A. On cooling, the metastable region B to I2 characterizes the region of “supercooling” of phase B. 3.2.2 Phase Transitions

How would a substance possessing the thermodynamic potential function E(S), as shown in Figure 3.3, behave on heating and cooling? Because the temperature of a simple system constitutes a monotonic function of entropy (Eq. (3.12)), we can imagine entropy substituted by the intensive variable temperature. We start from a state possessing a high entropy (or, respectively, temperature) and remove entropy (in the form of heat) from the system in a reversible manner whereupon the system cools down until it reaches state B. If heat is extracted further, nuclei in the state A should form suddenly. But spontaneous splitting of a system into two phases in this manner does not always occur because the energy of the system has to be first increased in order to provide – for example – the surface energy of the nuclei (droplets or crystallites). If this “activation energy” is not supplied, the system remains in single phase B up to I2. The energy of the homogeneous system (solid curve in Figure 3.3) exceeds – in the region B to I2 – the sum of the energies of the two possible phases (dashed line); such states are named metastable. With the formation of the first nuclei of phase A (droplets or crystallites) at point B, the quantity of phase B (gas or melt) diminishes and that of phase A increases accordingly; however, the temperature of the two phases in equilibrium remains constant until everything is transformed. Because the temperature T ¼ @E/@S is the slope of the function E(S), the two-phase system proceeds along a straight line, namely, the tangent BA (Figure 3.4). At point A, the entire substance is already completely in state A. If more entropy (heat) is removed, the temperature must again decrease because the system is again homogeneous.

E S ∆trsE ∆trsS

Figure 3.4 Change of energy during a phase transition on cooling.

Energy Entropy Energy change during transition Entropy change during transition

3.2 Phases and Phase Transitions

E S I

Energy Entropy Inflection point

Figure 3.5 Change of energy in the spontaneous breaking up of a system into two phases.

The quantities DtrsE ¼ E(B)  E(A) and DtrsS ¼ S(B)  S(A) are the transition energy and the transition entropy, respectively. Both constitute important quantities in calorimetry. The reverse procedure that involves heating of the system (starting from phase A) is, of course, also possible in a similar manner. If the activation energy for the phase separation in state A (on heating) or B (on cooling) is not provided, the process follows a different course (Figure 3.5). Starting at B (on cooling), the system remains at the thermodynamic potential function until I2 whereupon it separates spontaneously (also in the absence of activation energy) into two subsystems, namely, the states A and B. At the same moment, the temperature, which is lower at I2 than at B, rises again to the equilibrium value (at B) for the transition. Further cooling causes the process to assume the course along the tangent toward A as described above. Similar considerations apply to the heating of a system from a state of lower entropy (phase A) toward the state B. The case described above occurs whenever the activation energy for the formation of a new phase, which is still high at point B or point A, is not provided. This occurs, for example, when the thermal energy resulting from the motion or vibration of atoms is not large enough to provide the activation energy for the formation of a new phase. The phenomena resulting in this case are referred to as supercooling and superheating, respectively. Such effects as in Figure 3.5 were omitted in the equilibrium diagram sketched in Figure 3.2a. The existence of any supercooling or superheating would mean that the system remains some distance beyond the transition temperature on the G(T) curve of the previously stable phase before it drops down to the lower G(T) curve and transition takes place. For the liquid–solid transition, supercooling is actually the rule, in particular for very pure phases, but it is seldom extended from B to the limit of stability I2 as in Figure 3.5. The activation energy that is required for the formation of the new phase decreases rapidly with the increasing distance of the solid and the dashed curve. Either thermal energy or mechanical agitation is ultimately sufficient to cause the transition. It is known, for example, that very pure liquid tin can be kept as a supercooled melt at room temperature in ampoules with an extremely clean

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surface. But the phenomenon of supercooling is also known to occur during the gas–liquid transition; an example is the sudden appearance of mist. Superheating and supercooling during phase transitions must always be considered in calorimetry, particularly as sources of error for such quantities as phase transition temperature, transition energy, and transition entropy. The above considerations apply to a system composed of one pure substance (i.e., one single kind of molecule or atom) transforming from one phase to another at a given temperature. Such systems are referred to as one-component systems. If the system consists of two components (e.g., NaCl and H2O), its description is more complex. Thus, if a certain amount of NaCl is dissolved in H2O at 80  C so as to form a saturated solution, the result would be a homogeneous system (solution). Lowering the temperature would result in crystallization of solid NaCl in an increasing amount from what is then a supersaturated solution, thus creating a second solid NaCl phase. Liquid and solid phases coexist at equilibrium. Further cooling of the system would eventually freeze the remaining solution, forming two solid phases in contact with one another (ice and NaCl). The obvious difference from one-component systems lies here in the fact that the phase transition takes place not at a given constant temperature but over a temperature range. Within this range, two phases coexist, whose proportions vary with temperature. In our example, a change of the concentration of NaCl in the solution occurs simultaneously. A new variable is necessary for the description of systems consisting of more than one component. One possibility is to indicate the amount of substance or the number of particles of the components involved. This, however, is often not advisable because the sought properties of the system frequently depend on relative variables only. A more efficient and acceptable approach consists in the introduction of a new intensive variable, the concentration, expressed as particle density N/V. For a system consisting of the two components A and B, the concentration (mole fraction) is defined as the relative number of particles: xB ¼

NB NA þ NB

This definition suffices as the other concentration, xA ¼ NA/(NA þ NB), can be obtained from the equation xA þ xB ¼ 1. Thus, only one additional variable, the concentration variable xB, is required for a two-component system. In the presence of K0 components, K0  1 additional variables are necessary for describing the system. The concentrations so defined are intensive variables. As the number of components increases, the description of the system becomes increasingly more complicated, and its representation is quite difficult. 3.2.3 Gibbs Phase Rule

The thermodynamic laws required for the description of systems stem from the above-mentioned extreme principle (see Section 3.2.1) and the properties of the

3.2 Phases and Phase Transitions

corresponding thermodynamic potential function. The equilibrium requirements reduce the number of independent variables according to the number of components and phases. The Gibbs phase rule provides a link between the number of intensive variables that can be freely chosen for a system and further parameters of the system. It states that F i ¼ K 0  Ph þ M E  Rch

ð3:14Þ

where Fi is the number of intensive variables that can be freely selected (which are continuously variable) in a given range, K0 is the minimum number of components or species (molecules, atoms) necessary to describe the system, Ph is the number of phases, that is, of subsystems that are themselves homogeneous, ME is the number of energy forms, excluding chemical energy, and Rch is the number of possible chemical reactions. In many systems of interest to calorimetry, the only relevant forms of energy other than chemical energy are heat and compression energy, so that ME ¼ 2. Consequently, the phase rule becomes F i ¼ K 0  Ph þ 2  Rch

ð3:15Þ

Examples : i) One-component system (K0 ¼ 1), for example, water. If no chemical reaction can take place (Rch ¼ 0), it follows from Eq. (3.15) that Fi ¼ 3  Ph. If one phase is present – say, the gaseous phase (water vapor) – we obtain Fi ¼ 3  1 ¼ 2

Thus, we have two degrees of freedom; two intensive variables can be freely selected (e.g., pressure and temperature). However, this only applies to a limited range of pressure or temperature. A major increase of pressure or a corresponding lowering of temperature would create the liquid-phase water. The graphic representation of these relationships yields the well-known p–T diagram for water. Here, the different single-phase regions are marked off by curves along which the two neighboring phases are in equilibrium at given values of p and T. Figure 3.6 shows such a general p–T diagram. If the system consists of two phases – water and water vapor, for example (i.e., boiling water) – Fi ¼ 3  2 ¼ 1. Only one intensive variable can be freely selected in the case of equilibrium between two phases. If temperature is thus selected, the pressure (i.e., the vapor pressure at that selected temperature) is fixed. The connection between the two variables is a curve in the p–T diagram (curve 3 in Figure 3.6). The slope of the equilibrium curve between pressure and temperature is defined in the Clausius–Clapeyron equation: dp Dtrs S ¼ dT Dtrs V

Here, the entropy of transition DtrsS ¼ SB  SA and the volume change DtrsV ¼ VB  VA involved in the phase transition A ! B depend on temperature.

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p T 1 2 3

Pressure Temperature Sublimation curve Fusion curve Boiling curve

Figure 3.6 p---T diagram for a one-component system, shown schematically.

The Clausius–Clapeyron equation is the analytical expression (in the form of a differential equation) of the above-mentioned curve that separates the onephase region water from the one-phase region vapor in the p–T diagram (Figure 3.6). In the presence of three phases – ice, water, and water vapor – no intensive variable can be freely selected because Fi ¼ 3  3 ¼ 0. In a one-component system, three phases can coexist only at one pair of pressure and temperature values. This point is the so-called triple point in the p–T diagram (Figure 3.6) where the three phase equilibrium curves cross. ii) Two-component systems (K0 ¼ 2), for example, the solvent water (A) and the solute sugar (B). Because Rch ¼ 0 (no chemical reaction), it follows from Eq. (3.14) that Fi ¼ 4  Ph. The intensive concentration variable xB is additionally required for the definition of this system. Thus, xB ¼ Nsugar/(Nsugar þ Nwater). If a single phase is present, for example, liquid, then Fi ¼ 4  1 ¼ 3. Consequently, the pressure, the temperature, and the concentration xB can be freely selected within a certain range, that is, they can be specified externally without causing the formation of a new phase. Three thermodynamic degrees of freedom exist. If two phases exist simultaneously – liquid and gaseous, for example – then Fi ¼ 4  2 ¼ 2. If the temperature is predetermined (thus “using up” one degree of freedom), the pressure becomes a function of the only remaining free variable, namely, the concentration xB of the dissolved substance. The quantitative relationship can be formulated with regard to dilute solutions; it shows that, in general, the vapor pressure decreases with increasing concentration. If the pressure is constant, for example, at atmospheric pressure, the corresponding equilibrium temperature (i.e., the boiling point) becomes a function of the concentration xB of dissolved substance B; at low concentrations xB, the boiling temperature rises with increasing concentration. Similar conclusions can be formulated with regard to the phase equilibrium between a solid body and a liquid. Thus, the freezing point at constant pressure

3.2 Phases and Phase Transitions

decreases with rising concentration. The laws applicable to these cases on the basis of thermodynamic considerations are valid only at low concentrations (Raoult’s laws). They will not be dealt with here. At any rate, the pronounced effect of concentration on the phase transition temperatures Ttrs of multicomponent systems is a matter of major importance in calorimetry. Because clear-cut analytical laws can only be derived in rare cases, the relationship involved is commonly shown in a graphic form (phase diagram) on the basis of empirical data. To sum up, a two-phase, one-component system possesses a single degree of freedom; pressure and temperature are functionally correlated, and they yield the boiling curve in the p–T diagram. The “two-phase region” of a one-component system has a single dimension (i.e., it is a curve). A two-phase binary system has two degrees of freedom; pressure, temperature, and concentration are functionally interrelated and graphically, that is, in the p–T–x diagram, they form a two-dimensional surface in three-dimensional space. Obviously, a greater number of components do not lend themselves to a graphic presentation of coexisting states of several phases. Thus, whereas three dimensions (p, T, x) are necessary for K0 ¼ 2, K0 ¼ 3 would require four dimensions (p, T, x1, x2), which cannot be represented graphically. This difficulty can only be overcome by reducing the number of variables, that is, by considering the system at p ¼ constant or at p ¼ constant and T ¼constant (sections through a hypersurface). In this case, the system can be presented twodimensionally on flat paper (at K0 ¼ 2 by means of T and x) (Figure 3.7). For ternary systems (K0 ¼ 3), a commonly used and rather instructive “three-dimensional” presentation in the plane of the paper can be used (Figure 3.8). Here xA þ xB þ xC ¼ 1, that is, indicating the two concentrations xA and xB is actually sufficient at K0 ¼ 3, although it is useful to indicate xC also. The above requirement is taken into account in the selection of the equilateral triangle for the representation of the possible concentrations. It is noteworthy that phase diagrams are valid only for stable (or long-lived metastable) equilibriums. As pointed out previously, this necessitates a very slow

T A Ttrs B Ttrs

xB

Temperature Transition temperature of component A Transition temperature of component B Concentration of component B

Figure 3.7 Change of transition temperatures in a two-component system with unlimited mutual solubility of A and B.

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3 Fundamentals of Thermodynamics

T I Ttrs xI I

Temperature Transition temperature of component I Concentration of component I = A, B, C

Figure 3.8 Change of transition temperatures as a function of the concentrations in a threecomponent system with limited mutual solubility of the components A, B, C.

process. If the process is conducted at a finite, fast rate (as is the case with larger heating or cooling rates in a calorimeter), these diagrams are only approximately correct. In practical terms, this means, for example, that the phases observed at a given temperature do not possess the concentrations indicated in the diagram or that there may even be “unexpected” phases that, according to the equilibrium diagram, should not exist at all. 3.2.4 Measurement of Variables of State during Phase Transitions

When a one-component system (K0 ¼ 1) passes from phase 1 to phase 2, the respective thermodynamic potential function – corresponding to the boundary conditions – is, as mentioned earlier, continuous if the transition is performed reversibly. But because phase transitions normally involve a change of the microscopic structure, entropy usually changes abruptly by DtrsS1 ! 2 (cf. Figure 3.2b). This corresponds to a heat of transition DtrsQ ¼ Ttrs  DtrsS1 ! 2. Because the phase transition from a lower to a higher temperature (melting, boiling) is usually associated with a transition from a more ordered to a less ordered state, that is, an increase of entropy (according to the principles of statistical thermodynamics), the latent heat of the transition must be added to the system. Conversely, cooling involves the release of this latent heat. Let us first consider a phase transition at constant pressure p and constant temperature T. If only compression energy is exchanged in addition to heat, the fundamental form (Eq. (3.5)) is dE ¼ T  dS  p  dV

If a process is executed, then ∲ ðLÞ

dE ¼

∲ ðLÞ

TdS 

∲ ðLÞ

pdV

ð3:16Þ

3.2 Phases and Phase Transitions

where L indicates the path from state 1 to state 2. By definition, state 1 consists only of phase 1 and state 2 only of phase 2. If a reversible path for the transition is selected at constant intensive parameters p and T, the expression (3.16) adopts the following form in accordance with Eq. (3.2): ð2 1

ð2

ð2

dE ¼ T  dS  p  dV 1

1

Integration yields the transition energy: Dtrs Eðp; TÞ ¼ E 2  E 1 ¼ T  ðS2  S1 Þ  p  ðV 2  V 1 Þ

where (provided S2  Sl ¼ DtrsS and V2  V1 ¼ DtrsV) T  DtrsS is the exchanged transition heat DtrsQ and p  DtrsV is the exchanged transition work DtrsW. Thus, DtrsQ and DtrsV have to be measured to determine DtrsE. For the transition at constant p and T, the other functions of state are as follows: dH ¼ T  dS þ V  dp

ðfrom Eq: ð3:7ÞÞ;

from which it follows, as dp ¼ 0, that Dtrs Hðp; TÞ ¼ T  Dtrs S ¼ Dtrs Q

Similarly, from Eq. (3.9), dA ¼ S  dT  p  dV

from which it follows, as dT ¼ 0, that Dtrs Aðp; TÞ ¼ p  Dtrs V ¼ Dtrs W

and on the basis of Eq. (3.8), dG ¼ S  dT þ V  dp

so that, as dp ¼ dT ¼ 0, Dtrs Gðp; TÞ ¼ 0

The last statement is trivial (see Figure 3.2). As already shown, the thermodynamic potential function is always continuous in a phase transition. Thus, for phase transitions at constant p ¼ ptrs and T ¼ Ttrs, the following points can be summarized: If ptrs is predetermined, it follows that Ttrs ¼ Ttrs(ptrs); further, Dtrs Eðptrs ; T trs Þ ¼ Dtrs Q þ Dtrs W Dtrs Hðptrs ; T trs Þ ¼ Dtrs Q Dtrs Aðptrs ; T trs Þ ¼ Dtrs W Dtrs Gðptrs ; T trs Þ ¼ 0 Dtrs Sðptrs ; T trs Þ ¼

Dtrs Q T trs

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3 Fundamentals of Thermodynamics

where DtrsQ and DtrsW are, respectively, the heat and the work exchanged in the course of the reversible transition and Ttrs is the transition temperature at the indicated pressure. The measured heat of transition DtrsQ corresponds to the enthalpy change under the boundary condition given (p ¼ constant). Because all the intensive variables are defined, the measurement results can be readily converted to a defined normalized final state (where the extensive quantities are proportional to the size of the system). It should be repeated here that the thermodynamic potentials (left-hand side quantities in the above equations) are state functions that depend on the initial and final states only but not on the path of the process. Heat and work of transition, however, are generally no functions of state and may depend on the path the process follows. Constant (ambient) pressure is the usual boundary condition for calorimetric experiments, and in this case – and only in this case – the equality DtrsH(ptrs, Ttrs) ¼ DtrsQ is valid. Because there are calorimeters that operate at a constant volume, for example, combustion calorimeters (Berthelot bombs), the situation of V ¼ V0 ¼ constant must also be considered. Under this boundary condition, the heat exchange also leads to a phase transition (e.g., the boiling of a liquid following the input of heat). In such cases, the phase transition necessarily involves a change (increase) of pressure. This change of pressure alters the temperature of the system during the phase transition in the two-phase equilibrium because the intensive variables are linked together because of Gibbs phase rule. In our example, the result of the pressure increase would be an increase in boiling temperature. Consequently, we cannot integrate all state functions because the relationship p ¼p(T, V0, N) would have to be exactly known for the path of the process followed by the respective amount of substance in the system. However, measurement of the reversibly supplied transition heat DtrsQ and of the pressure difference Dtrsp ¼p2  p1 yields, according to the first law, Dtrs EðN; V 0 Þ ¼ Dtrs Q Dtrs HðN; V 0 Þ ¼ Dtrs Q þ V 0  Dtrs p

These quantities, however, are different from the values of DtrsE(ptrs, Ttrs) and DtrsH(ptrs, Ttrs) obtained at constant p and T, and the measured heat of transition at constant volume DtrsQ is not equal to DtrsH. Indeed, the final states differ markedly from one another in these two cases. In particular, quantities measured at constant volume depend also on the amount of substance because at constant volume, the pressure is a function of the enclosed mass of substance. Consequently, the results of the measurements of one-component systems at constant volume may be worthless because the measured values generally cannot be converted to a standard final state. This is also true if measurements are performed in hermetically closed crucibles; if the pressure changes during the measurement, the measured heat of transition is not equal to the enthalpy difference. Measurements of combustion heats in the Berthelot bomb can be useful (notably, as comparable standard heats) if one starts from defined (and very high) oxygen pressures, so that the pressure

References

change occurring in the course of the measurement remains negligible. Nevertheless, we are dealing here with the heats of chemical reactions rather than energy changes associated with phase transitions. The conclusions from measurements at constant pressure or constant volume are, however, the same. The above brief review of the principles of thermodynamics was intended to strongly indicate to experimenters that work in calorimetry necessitates a thorough understanding of the following: 1) What happens in the system that exchanges heat with the calorimeter? 2) What boundary conditions govern this exchange? 3) To which thermodynamic state variable does the measured heat correspond? It is, of course, not necessary to understand thermodynamics in detail, but it should be borne in mind that the primary quantity heat, measured in calorimeters, is not a function of state, but depends on the path the calorimetric experiment follows even if the initial and final states of the system are the same. Only the above-defined functions of state are independent of the process path.

References Adkins, C.J. (1983) Equilibrium Keller, J.U. (1977) Thermodynamik der Thermodynamics, 3rd edn, irreversiblen Prozesse. Teil 1. Thermostatik und Cambridge University Press, Grundbegriffe, De Gruyter, Berlin. Cambridge. Lebon, G., Jou, D., and Casas-V
azquez, J. Callen, H.B. (1985) Thermodynamics and an (2008) Understanding Non-Equilibrium Introduction to Thermostatistics, John Wiley & Thermodynamics, Springer, Berlin. Sons, New York. Zemansky, M.W. and Dittman, R.H. (1997) Falk, G. and Ruppel, W. (1976) Energie und Heat and Thermodynamics, 7th edn, McGrawEntropie, Springer, Berlin. Hill, New York.

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4 Heat Transport Phenomena An exchange of heat takes place in all calorimeters. Heat transport phenomena and the associated temperature gradient are therefore of major importance for the understanding of the processes that occur in calorimeters and for assessing the reliability of the obtained data because the sample and the temperature sensor are necessarily separated in space. Accurate calorimetry is only possible if this spatial separation does not generate any measuring errors and if the exchanged heat is fully recovered without incurring any losses caused by “heat leaks.” Here the term heat leak refers to heat exchanged with the surroundings along a path not detected by the measuring sensor and therefore “lost” by the calorimeter. Only the essential aspects of heat transport are mentioned in this chapter. Detailed discussions can be found in the literature (Jakob, 1949; Carslaw and Jaeger, 1959; Holman, 2010; € sik, 1985). OziS

4.1 Heat Conduction

Heat conduction is the exchange of heat by the transport of vibrational states (phonons) without any concomitant transport of mass by flow or diffusion. Consequently, pure heat conduction occurs only in solid bodies. The differential equation that defines this phenomenon has the form of a transport equation; its solution characterizes the temperature field that causes the heat exchange. The differential equation is div grad T ¼

r  cp @T  l @t

ð4:1Þ

where r is the density, cp is the specific heat capacity, and l is the thermal conductivity. The term for a heat source has been omitted from Eq. (4.1). The solution of this differential equation yields the temperature field T(r, t). The heat flux field can thus be calculated according to Eq. (4.2): Jq ¼

W ¼ lðTÞ  grad T A

ð4:2Þ

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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4 Heat Transport Phenomena

To the left is the heat flux vector Jq (heat flow rate W per unit area A) and to the right is the spatial change of the temperature field. The thermal conductivity l combines both of these vectors, which makes it a tensor. In general, l depends on the direction inside a solid (depending on its crystal structure); moreover, it is a function of temperature. Equation (4.1) can be solved in a closed form in only a few special cases because the initial and boundary conditions, that is, the temperature field at the initial time and the temperature of the boundary, as well as the geometry of the arrangement are included in the solution. Somewhat less complex are the relationships in the one-dimensional treatment; under such circumstances, Eqs. (4.1) and (4.2) become @ 2 T r  cp @T ¼  l @x 2 @t W ¼ l  A 

ð4:3Þ

dT dx

ð4:4Þ

The solution T(x, t) of the first differential equation enables the solution W(x, t) of the second equation to be calculated. Figure 4.1 shows graphically the solution of the following simple one-dimensional problem: at time t0, a bar of temperature Tini is brought in contact with a block (furnace) possessing an infinitely high thermal

T TF

Temperature Temperature of the furnace (thermostat) Tini Initial temperature of the bar x Position coordinate t Time t0 Time of contact with the furnace t1, t2, ... Times after the establishment of thermal contact

Figure 4.1 Solution of the heat conduction equation for a bar and the schematic representation of the resulting temperature field inside the bar after the contact with the furnace.

4.1 Heat Conduction

J = Φ/A Heat flux Φ Heat flow rate A Area x Position coordinate t Time

Figure 4.2 Schematic representation of the heat flux through two different cross sections of the bar as a function of time (for geometric arrangement, see Figure 4.1).

conductivity and a constant temperature TF. Heat flows into the bar, and a timedependent temperature field develops in the bar until the latter assumes – after a certain time interval – the temperature of the block. Figure 4.2 shows the heat flux through two cross sections of the bar as a function of time. It can be seen that the “sudden” event “establishment of thermal contact” brings about a rather “smeared” response at the end of the bar. As a general principle, heat conduction causes the “smearing” of originally steplike temperature and heat events within the sample. If the exact (unsmeared) time course of temperature or heat flux changes is needed, the heat conduction equation has to be solved for the specific initial and boundary conditions of the respective arrangement. An exact solution is, however, seldom possible. For this purpose, there are a number of numeric procedures of proven merit. One method is the finite element method (FEM): a body of any shape (as well as heterogeneous) is subdivided into sufficiently small subunits and the original three-dimensional body is substituted by a network of corresponding “mass points” connected by massless, heat-conducting “bars.” For such a network of n subunits, a set of equations can be formulated on the basis of the conservation of energy. This system of equations makes it possible to calculate the temperature field as a function of time on the basis of a given temperature field. For a sufficiently fine subdivision of the body, it is possible to find a suitably exact solution of the heat conduction problem. Several FEM software packages are available on the market. The ANSYS1 and the COMSOL1 systems are often used by calorimeter-developing corporations. The software can be used to define the time-dependent heat flux curves at any site of the calorimeter system and also to indicate the temperature field if this is necessary for the purposes of the discussion and for a better understanding including heat leaks of all kinds. According to a simplified view, which, however, applies to a steady state only (i.e., W independent of time), the heat flow rate through a bar is proportional to the temperature difference at the two boundary surfaces: W ¼ l  A 

DT 1  DT ¼ Dx Rth

ð4:5Þ

63

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4 Heat Transport Phenomena

(where Rth is the thermal resistance of the bar and l is a constant) because the conservation of energy requires that the heat flow rate on the output side must be the same as on the input side. Equation (4.5) is very important for calorimetric practice. Because heat only flows in the presence of a temperature gradient, a heat exchange cannot take place under isothermal conditions. Conversely, any temperature difference in a solid body causes a heat flow. Thus, if a temperature difference appears between a part of the calorimeter and its surroundings – at the sample holder, for example – a certain amount of heat flows unmeasured to the surroundings and a “heat leak” results. The leak heat flow rate is, according to Eq. (4.5), proportional to the temperature difference DT and inversely proportional to the thermal resistance Rth. This has practical consequences for the construction of calorimeters: the thermal resistance between the sample holder and the heat measuring sensor should be as low as possible, whereas the thermal resistance between the sample holder and the surroundings should be large. In general, heat leaks can be reduced in two ways: 1) By providing greater heat resistance to the surroundings, that is, insulation of the calorimeter. 2) By a small temperature difference relative to the surroundings, that is, measurement under adiabatic or quasi-adiabatic conditions. Equation (4.5) serves also as a basis for the measurement of heat flow rates by measuring a (local) temperature difference. This measurement, however, is only correct in the presence of steady-state conditions and when the thermal resistance does not depend on temperature.

4.2 Convection

Heat exchange by convection differs essentially from heat exchange by thermal conduction. Heat exchange by convection means heat transport by flow in liquids or gases. Every element of the volume of the flowing medium absorbs heat at the site of higher temperature and transports it to a colder site. Figuratively speaking, the “heat capacity” of the flowing medium is loaded at a warm site and transported to a cold site, where it is unloaded. Such transport requires a flow. A distinction can be made between two categories of flow as follows: i) Free convection in which the flow is generated by the temperature field itself, that is, by the resulting density differences in the medium. Warm air has a lower density than cold air and rises from the source of heating (e.g., room heating, chimney). ii) Forced convection, namely, a flow caused by a pressure gradient generated in the medium by stirring or pumping (e.g., central heating in a building, water cooling of a motor). Forced convection is the main mechanism for heat transport in flow calorimeters.

4.3 Heat Radiation

Free convection usually represents a disturbing factor in calorimeters because it carries heat in an uncontrolled manner (causing a heat leak). In practice, it does not lend itself to exact calculation because it depends on numerous parameters (e.g., geometry, nature of the medium) and may lead to abrupt changes by means of sudden turbulences. Thus, convection constitutes an irreproducible interference and has to be reduced whenever possible. The so-called “Grashof number” NGr provides an estimate of the effect of free convection in comparison with pure heat conduction: N Gr ¼

a  r2  g 3  l  ðT 1  T 2 Þ g2dyn

where a is a cubic expansion coefficient, r is the density, gdyn is the dynamic viscosity, g is the acceleration due to gravity, l is the width of the flow space, and T is the temperature. As a general rule, free convection with Grashof numbers of up to 1000 can be neglected in practice; at NGr ¼ 2000, heat exchange by convection accounts for about 5% of the heat flow rate by conduction of a fluid medium at a given temperature difference. It follows from the above relationship that free convection can be reduced to a minimum by using (i) a small distance l (in a suitably designed apparatus), (ii) a low density r (gas at reduced pressure), or (iii) a high viscosity gdyn (choice of the fluid medium). Precise caloric measurements should be carried out in vacuum if possible. Good thermal insulation, however, can only be obtained by establishing a high vacuum and ensuring that the free path lengths of the gas particles exceed the dimensions of the device. As previously pointed out, free convection is invariably associated with heat conduction; these two heat exchanging mechanisms cannot be separated from one another with the tools of measuring techniques. As a special case, the heat transfer from a wall (area A; temperature TW) to a fluid (temperature T0) is fairly well described by Newton’s law of cooling: W ¼ Cc  A  ðT W  T 0 Þ

where the heat flow rate W is related to the overall temperature difference TW  T0 by means of a coefficient of convection heat transfer Cc. This empirical coefficient depends on the thermophysical properties of the fluid, the category of convection, and the flow rate. Newton’s law of cooling should not be confused with Eq. (4.5)!

4.3 Heat Radiation

All bodies, including those in thermodynamic equilibrium, constantly emit electromagnetic radiation. This so-called “heat radiation” depends on the absolute temperature of the body and on its surface structure as characterized by its emittance. At the same time, the body always absorbs heat radiation originating

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from other bodies around it. Bodies at constant temperature have a zero energy balance, that is, the absorbed and emitted radiation heats are equal. But in the presence of a temperature difference between two bodies, a heat exchange via radiation takes place whereby the warmer body emits more energy than it receives from the colder one and vice versa. This heat transmission can be described by the following law:
 W=A2 ¼ sB  A1  K 12  T 41  T 42 ð4:6Þ

where W/A2 is the heat flux absorbed by body 2 per unit area, s B is the Stefan–Boltzmann constant (a universal constant of nature) equal to 5.67  108 W m2 K4, A1 is the surface area of body 1, K12 is an empirical coefficient, and T is temperature in kelvin. The difficulty in applying this formula lies in the determination of the value of the coefficient K12. The latter depends on the geometry of the radiating body, the temperatures T1 and T2, and the “directional characteristics” of the radiating surfaces, as well as their spectral emittance, absorptance, and reflectance (i.e., their “color”). Obviously, the heat flux depends to a major extent on the empirical coefficient K12, which at best can be given only approximately even when the geometry is known and the surface properties are defined. Heat radiation increases sharply with temperature because the absolute temperature in the formula appears to the fourth power. If the temperatures T1 and T2 differ only slightly from one another, which is usually the case in calorimetry, the exchanged heat flow rate DW can be calculated with sufficient accuracy according to the following formula, obtained from Eq. (4.6) by linear approximation of the respective Taylor series: DW ¼ 4  sB  A1  A2  K 12  T 3  DT

In calorimeters, heat loss by radiation, therefore, depends largely on temperature even in the presence of a constant temperature difference between the measuring system and its surroundings. This is in effect a temperature-dependent “heat leak.” Losses by radiation can only be ruled out when the temperature difference between the system and its surroundings has been eliminated. This can be achieved by enveloping the measuring system with temperature-controlled radiation shields with a temperature equal to that of the system proper. If two bodies of different temperature are separated by a gas, heat is exchanged in general by radiation, convection, and conduction. If the gas does not absorb in the frequency range of the thermal radiation (which is normally the case with air but not with CO2), the transport mechanisms in the gas and the radiation can be regarded as independent of one another. The total heat exchanged represents the sum of heat transported through the medium (via conduction and convection) and by radiation. In other words, the overall thermal conductance (i.e., the reciprocal thermal resistance) is obtained as the sum of the individual thermal conductances of the respective transport mechanism.

4.5 Entropy Increase during Heat Exchange

4.4 Heat Transfer

Heat exchange at the phase boundary between two solid bodies of different temperature in contact with each other can only take place as long as the two bodies differ in temperature. As in the case of heat conduction, the heat flux is proportional to the temperature difference, provided the latter is not too large. 1) Thus, the heat flux can be described as follows: W=A ¼ Cht  ðT 2  T 1 Þ

On the left-hand side is the heat flux; Cht, the surface coefficient of heat transfer, is an empirical constant that depends very much on the nature of the heat transport as well as on the surface structure, in case of two solid bodies on the contact pressure, and on the presence and nature of a fluid medium (gas or liquid) between the two bodies. This is of importance in calorimeters in which the tested substance is put into special containers (crucibles) that are then placed inside the calorimeter. If no measures are taken to ensure well-defined, reproducible heat transfer, the temperature difference involved in the heat exchange between the measuring system and the sample (or respectively, the crucible) may differ from one measurement to another, so variations will occur in the temperature as shown by the sensor relative to the actual temperature of the sample. As a consequence, the measured heat quantity may also differ, leading to an uncertainty of the result.

4.5 Entropy Increase during Heat Exchange

Heat conduction, as well as heat radiation and convection, is an irreversible process. Heat can only flow from a place of higher temperature to a place of lower temperature, that is, the flow of heat cannot take place in the absence of a temperature difference or a local temperature gradient. This leads to the conclusion that the entropy must increase. As can be shown by the case of steady-state heat flow through a bar, if the heat flow rate into the bar at one side (at temperature T1) W1 ¼

dQ 1 dS1 ¼ T1  dt dt

and the heat flow rate out of the bar on its other side (at a temperature T2 < T1) W2 ¼

dQ 2 dS2 ¼ T2  dt dt

1) This type of equation also describes the heat transfer between a solid body and a gas or a liquid (Newton’s law of cooling) (see Section 4.2).

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it follows from the conservation of energy that dQ 1 dQ 2 ¼ ; dt dt

hence

T1 

dS1 dS2 ¼ T2  dt dt

ð4:7Þ

But because T1 > T2, it follows that dS1 dS2 < dt dt

Thus, entropy connected with the exchanged heat has increased during the process. The entropy associated with any heat flow rate is not constant. Flowing heat generates entropy; the entropy generated during the flow through the bar can be calculated from Eq. (4.7) as follows: dS2 dS1 T 1  T 2 dQ T 1  T 2  W  ¼ ¼ dt dt T 1  T 2 dt T1  T2

Thus, the larger the temperature difference and the heat flow rate, the greater the entropy production. It should be mentioned that all heat is connected with a certain entropy, but the necessary entropy production during irreversible processes may occur without any heat exchange of the system.

4.6 Conclusions Concerning Calorimetry

The heat transport phenomena conduction, radiation, convection, and heat transfer are of central importance in all calorimeters. On the one hand, the occurrence of a temperature difference causes a heat flow and thus creates the possibility of heat losses toward the surroundings (heat leaks) – namely, heat flows not detected by the measuring sensor and therefore not measured by the calorimeter. On the other hand, no heat exchange can take place in the absence of a temperature difference. The experimenters find themselves in a dilemma: to be measured, heat must be made to flow, but every heat exchange is associated with temperature differences that create errors in measurement (e.g., heat leaks). There are two possible ways out of this dilemma: (i) the adiabatic calorimeter and (ii) the twin device. In the adiabatic calorimeter (see Section 5.3), suitable measures are taken to prevent any escape of heat from the measuring system to the environment; for this purpose, the measuring system is surrounded by a shield that has exactly the same temperature. Any heat exchange with the measuring system brings about a change in its temperature. The shield must be so adjusted as to match exactly this temperature change in order to ensure that the unknown heat exchange with the environment is reduced to a tolerable magnitude. Figure 4.3 shows an adiabatic calorimeter schematically. The practical construction of this instrument is

4.6 Conclusions Concerning Calorimetry

TF

2 TF 1

Control unit T F = TM

TM

TM

Temperature of the furnace (surroundings) Temperature of the measuring system Measuring system Furnace (surroundings) Heating element

3

Figure 4.3 Schematic design of an adiabatic calorimeter.

expensive (see Section 7.10.2) and its operation very laborious; however, the quality of the obtained results is not even approximated by any other calorimeter. The second possibility of reducing the effects of the inevitable heat leakage is the so-called “twin” device (Figure 4.4). In this long-known tricky measuring technique, two systems are made as equal as possible to one another and are operated in common surroundings in a symmetrical arrangement. Now, if the heat leaks and thus the inevitable measuring errors inherent in the two systems are made to be of equal magnitude, they can be offset by differential measurements. With regard to calorimeters, this means that two calorimetric measuring systems that are as closely equal to one another as possible are 1

3

2

4

Heat flow rate signals Generation of the difference signal Calorimeter 1 Calorimeter 2 Sample Reference

Differential heat flow rate Figure 4.4 Schematic design of a twin calorimeter.

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4 Heat Transport Phenomena

placed symmetrically in common surroundings. If the heat leaks between the measuring systems and the homogeneous surroundings are equal in the two systems, outer disturbances such as temperature fluctuations in the surroundings affect the two measuring systems to the same extent, at least in a first approximation, so that the effects of these disturbances are mutually offset owing to the differential signal. Of course, this does not apply to disturbances or processes that affect only one of the two measuring systems. If the two systems were placed in exactly the same manner and quite symmetrically with regard to the surroundings, heat flow rates from or to each of the two measuring systems would be equal; this means that the measured differential signal would be zero. Now the sample to be studied is placed in one of the two systems. If the second system is left empty, the symmetry of the twin measuring system would be greatly disturbed (e.g., different heat capacities, different surface relationships). As a result, heating the twin calorimeter would produce a major difference between the two heat flow rates. To avoid these types of errors, such studies are usually made by placing the sample in one of the measuring systems, and – in the other measuring system – a reference equal in all respects to the test sample, but showing no reaction in the relevant temperature range. In such a case, the measured signal zero appears under ideal circumstances before and after the reaction interval. But identity between the test sample and the reference sample can hardly be expected in practice. Indeed, the requirement that the reference sample must remain inactive over the relevant temperature range implies that its chemical and physical structures have to be different from that of the test sample. For this reason, the reference sample also differs outside the reaction interval, so at best it can only be approximately equal to the test sample. As a result, the differential signal outside the reaction is normally different from zero. An important example is the melting of a substance in the calorimeter to measure the heat of fusion. The reference sample must not melt at the temperature of fusion of the tested substance, which means that it must consist of a different substance. Consequently, the heat capacity of the reference sample will be different, not only in magnitude but also in its dependence on temperature. By suitable weighing, the heat capacity of the reference sample can be made equal to that of the test sample but only at one given temperature. Outside this temperature there will always be a difference between these two heat capacities and therefore a nonzero differential signal (baseline). In particular, the unavoidable change of the heat capacity in the course of melting does not take place in the inert reference sample; hence, the common appearance of a baseline step change during measurement of a fusion process in twin calorimeters (see Section 6.2.3). This consideration does not include the effects of different thermal conductivities and the surfaces of the sample and the reference (see Section 6.2.3). Despite these drawbacks, it is always better to include a reference sample than to leave one of the measuring systems in twin calorimeters empty. Indeed, working with the reference measuring system empty may cause the heat flow rate difference (baseline) to be much larger than the additional difference brought about by the

References

sample reaction. As a result, the desired quantity (the reaction heat flow rate) causes a small change in a large magnitude (the basic heat flow rate caused by the asymmetry) and, consequently, will be inaccurately determined. The requirements concerning twin calorimeters can thus be summed up as follows: i) Two measuring systems that are as equal to one another as possible and are placed symmetrically in homogeneous surroundings. ii) The test and the reference sample must be as closely equal as possible with regard to heat capacity, geometry, thermal conductivity, heat transfer to the measuring system (including the placement within the systems), and other factors liable to cause thermal asymmetry. Another important implication arising from the discussion of heat transport phenomena will be discussed in detail in Chapter 6 but should be mentioned here. Indeed, the measured signal recorded by the calorimeter does not reflect exactly the time course of the underlying process inside the sample. Because the measuring probe that detects the heat flow rate is necessarily located at some distance from the sample, the heat exchange between the sample and the probe necessary for the measurement requires a temperature difference, and this requires some time. Thus, both the temperature change and the time course of processes in the sample are, in effect, not exactly reproduced by the calorimeter but are distorted; the measured signal is “smeared.” To sum up, to yield repeatable results and to be capable of calibration, a calorimeter must be constructed in such a way that the entire heat exchange of the sample with the surroundings takes place in a defined manner through the measuring system only. In any uncertainty analysis (see Section 6.5), heat transport phenomena and their consequences must be taken into account.

References Carslaw, H.S. and Jaeger, J.C. (1959) Conduction of Heat in Solids, 2nd edn, Oxford University Press, London. Holman, J.P. (2010) Heat Transfer, 10th edn, McGraw-Hill, New York.

Jakob, M. (1949) Heat Transfer, vols I and II (1957), John Wiley & Sons, New York. € sik, M.N. (1985) Heat Transfer: A Basic OziS Approach, McGraw-Hill, New York.

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5 Surroundings and Operating Conditions A distinction between a “measuring system” on the one hand and its “surroundings” (thermostat, furnace) on the other can be made in every calorimeter (Figure 5.1). This distinction must be made in functional terms in view of the fact that the temperature-measuring sensors, the insulation, the stirrer, and other components of the calorimeter belong partly to the measuring system and partly to the surroundings. Parts that constitute or traverse the boundary surface belong to the measuring system insofar as they are involved in one way or another in the changes caused by the sample reaction. The surroundings of the measuring system determine the conditions of operation but also such disturbances as heat leakage. In turn, the surroundings must be sealed off with regard to the environment of the outer world (i.e., the laboratory); otherwise, the latter may affect the surroundings, introducing disturbances and errors in caloric measurements. Let the homogeneous temperature of the surroundings (e.g., a furnace) be TF and that of the measuring system TM. The temperature designated by TM is the one actually measured – for example, the temperature of the calorimeter substance or of the wall of the sample container. Depending on the behavior of TF and TM (constant, equal, or changing), various types of calorimeters are distinguished (e.g., isothermal, adiabatic). The boundaries between the separate parts, that is, between the outer world, the surroundings, and the measuring system, permit heat exchange (e.g., in isothermal calorimeters) or prevent such exchange (e.g., in adiabatic calorimeters). This discussion does not include the transfer of other forms of energy across the system boundaries and into the measuring system, such as mechanical work through a stirrer or electrical energy along feed lines. The operating conditions of calorimeters are defined first and foremost with regard to an ideal state. For this reason, the designations isothermal and adiabatic as used here are not in strict accordance with the concepts of thermodynamics. In calorimetric practice, it would be more appropriate to use the terms quasi-isothermal and quasi-adiabatic. There is a common tendency to use thermodynamic concepts even when the ideal conditions required by them are not complied with. This fact must be kept in mind, in particular in the uncertainty analysis.

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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5 Surroundings and Operating Conditions

Isothermal condition: Rth Very small TF = TM = constant Adiabatic condition: Rth Very large TF = TM Isoperibol condition: Rth Defined TF = constant TM = TM(t) TM Temperature of the measuring system TF Temperature of the furnace Rth Thermal resistance

Environment,

furnace (surroundings),

measuring system,

thermal resistance

Figure 5.1 Schematic design of a calorimeter.

5.1 The Isothermal Condition

In calorimeters operating isothermally, the surroundings and the measuring system always have the same constant temperature, that is, TF ¼ TM ¼ constant (see Figure 5.1). In phenomenological thermodynamics, the isothermal state is established by having the system exchange heat via an infinitesimally low thermal resistance Rth with surroundings of infinite heat capacity. This is not feasible in calorimetry. Consequently, isothermal operation necessitates a compensation of the heat flow released from the sample. This can be achieved by a phase transition (passive measuring system) or by thermoelectric effects (active measuring system). Under strictly isothermal conditions, TF and TM must remain constant in time and space, but then no heat would flow. As pointed out previously, the term isothermal should not be interpreted too strictly. There are no truly isothermal conditions in the measuring system of a compensation calorimeter, least of all in the sample. Constant temperature in time and space cannot be expected because any heat transport from the sample to the substance undergoing transition would be impossible in the absence of temperature differences. Similar considerations apply to calorimeters involving electric compensation with regard to the heat transport between the sample, the temperature sensor, and the heater or the cooler. The magnitude of the temperature difference depends on the quantity of heat delivered per time unit by the sample surface, the thermal conductivities of the substances that surround the sample (vessel materials), and their geometry. In calorimeters involving electric compensation, the insulation of the temperature sensors and of the heating or cooling elements causes additional local temperature differences.

5.3 The Adiabatic Condition

Despite these limitations, the designation “isothermal” is commonly used with regard to calorimeters where TF ¼ TM ¼ constant.

5.2 The Isoperibol Condition 1)

The isoperibol condition refers to the use of a calorimeter at constant temperature surroundings (thermostat) with a possibly changing temperature of the measuring system. The thermal resistance Rth between the measuring system and the surroundings is of finite magnitude in such calorimeters (Figure 5.1), whereas it is infinitesimally small in isothermal calorimeters and infinitely large in adiabatic ones (see below). Because of the existence of a finite, defined thermal resistance Rth between the measuring system and the surroundings, there must be a heat exchange that depends in a definite manner on TM and TF only (see Sections 4.1 and 4.4). Because TF is kept constant in an isoperibol operation, the heat flow will be a function of TM only. The relation is generally linear in this regard for small temperature differences; it can be determined by calibration. It is noteworthy that the temperature of the measuring system (TM) changes by heat exchange with the surroundings until an equilibrium is established. This process necessitates a certain interval of time. Constant generation of heat in the measuring system (e.g., from a living organism) brings about a constant temperature TM after a certain time of operation. If the generation of heat is stopped, TM finally becomes equal to TF (see Section 6.2.2). The heat flow between the measuring system and the surroundings depends on the temperature difference (TF  TM) and the effective thermal resistance Rth. If the thermostat surrounds the measuring system perfectly, the reaction heat is completely transferred into the surroundings (no heat losses). Even for accurate measurements, it is not absolutely necessary to keep heat losses as low as possible. What is more important is that these heat losses be repeatable, depending on the temperature difference between the measuring system and its surroundings; in this case, they can be determined exactly by (electric) calibration (see Section 6.2.2). Of course, large heat loss reduces the sensitivity of the calorimeter to a considerable extent.

5.3 The Adiabatic Condition

Calorimeters can also be operated in an adiabatic manner. In this case, under ideal circumstances, no heat exchange whatsoever occurs between the measuring system

1) The term “isoperibol” (uniform surroundings) was introduced by Kubaschewski and Hultgren (1962).

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and its surroundings. There are three ways of meeting this requirement as satisfactorily as possible. i) The sample reaction takes place so rapidly that no appreciable quantity of heat can leave or enter during the measuring interval. ii) The measuring system is separated from the surroundings by an “infinitely large” thermal resistance, that is, thermally insulated in the best possible way. iii) The temperature of the surroundings is so controlled as to be in every moment equal to that of the measuring system, that is, TF(t) ¼ TM(t). Only the third possibility can be considered for calorimeters in which heat of different magnitudes and release rates has to be measured. In the general case (e.g., in flow calorimeters), the temperature of the surroundings must match that of the measuring system throughout and with a negligible time lag. This requirement presents no problems if the heat is released at a slow or moderate rate. The advantage of the adiabatic method is most pronounced in the calorimetric investigation of such reactions. However, difficulties arise in the presence of surroundings of a large surface area and abrupt changes in the temperature of the measuring system. Thermal resistances (insulations) between such components as temperature sensors, jacket material (surroundings), and heating or cooling elements on the one hand and the time required for the equilibration of the temperature in the jacket on the other result in a certain thermal inertia that restricts the possibilities of the adiabatic operation of calorimeters in the case of rapid reactions. Electronic control is the only method used today to establish adiabatic conditions; other methods are only of historical interest.

5.4 The Scanning Condition

The term “scanning” applies to those cases in which the temperature of the surroundings or of the measuring system is changed in a predetermined manner with respect to time. If heat is supplied to the surroundings (furnace), the measuring system follows this temperature rise with a time lag because the heat exchange with the surroundings takes place through a nonzero thermal resistance Rth (Figure 5.2). In most cases, the furnace is heated linearly: TF ¼ TF,ini þ bt, where TF is the temperature of the surroundings, TF,ini is the initial temperature of the surroundings, b is the heating rate, and t is the time. This type of scanning operation is commonly used in differential scanning calorimeters (DSCs) (Section 7.9.4). Another possibility consists in providing the measuring system with an internal heating facility so as to ensure that its temperature always remains equal to the temperature of the surroundings: TF(t) ¼ TM(t); in other words, we

5.4 The Scanning Condition

Scanning condition: Rth Small, but defined TF = TF(t) = TF,ini + β · t TF Temperature of the furnace TF,ini Initial temperature of the furnace TM Temperature of the measuring system Rth Thermal resistance β Scanning rate t Time

Control unit Temperature sensor,

thermal resistance,

furnace (surroundings),

measuring system,

furnace heater

Figure 5.2 Schematic design of a calorimeter with scanning of the surroundings.

are dealing with an adiabatic scanning operation (Figure 5.3). This type of operation involves considerable electronic control equipment. In other cases, the temperature of the surroundings remains constant (Figure 5.4), whereas the measuring system actually composed of two separate measuring systems is heated linearly with time. Each of the two separate measuring systems has a controlled heater that brings it to a temperature identical

Adiabatic scanning condition: Rth Very large TF (t) = TM(t) = TF,ini + β · t TF Temperature of the furnace TF,ini Initial temperature of the furnace TM Temperature of the measuring system Rth Thermal resistance β Scanning rate t Time

Control unit Temperature sensor,

thermal resistance,

furnace (surroundings),

measuring system,

heaters

Figure 5.3 Schematic design of a calorimeter with adiabatic scanning.

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Control unit Temperature sensor,

thermal resistance,

furnace (surroundings),

Isoperibol scanning condition: Rth Defined TF = constant TM(t) = TM,ini + β · t TF Temperature of the furnace TM Temperature of the measuring system TM,ini Initial temperature of the measuring system Rth Thermal resistance β Scanning rate t Time

measuring system,

heaters

Figure 5.4 Schematic design of a calorimeter with isoperibol scanning.

to that of the other measuring system. The sought heat exchanged with the sample is obtained from the difference in energy inputs into the two systems. One of the reasons for the extensive use of scanning calorimeters is the possibility of selecting different working temperatures. But the main advantage lies in the fact that many reactions (e.g., phase transitions, order processes, chemical reactions) are thermally activated and that kinetic data of the reactions can also be obtained. The essential operating conditions of calorimeters can be summarized as follows: Isothermal condition: T M ¼ T F ¼ constant Calorimeters involving compensation of the thermal effect by phase transition or thermoelectric effects. Isoperibol condition: T F ¼ constant; T M ¼ T M ðtÞ Calorimeters involving the measurement of a time-dependent temperature TM(t) or temperature difference DT(t). Adiabatic condition: T M ¼ T F Calorimeters involving the measurement of a time-dependent temperature or a compensation of the thermal effect by thermoelectric effects. Scanning condition: T F ¼ T F ðtÞ or T M ¼ T M ðtÞ with T F ¼ constant Calorimeters involving the measurement of a temperature difference (heat flow calorimeters) or with a compensation of the thermal effect by thermoelectric effects (power compensation calorimeters).

The operating conditions described can be used as major characteristics for the classification of calorimeters.

Reference

Reference Kubaschewski, O. and Hultgren, R. (1962) Metallurgical and alloy thermochemistry, in Experimental Thermochemistry, vol. II

(ed. H.A. Skinner), Interscience Publishers, New York, pp. 343–384.

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6 Measurements and Evaluation Calorimetric measurement and data processing (evaluation) as well as calorimeter control are nowadays carried out electronically and with the help of a computer. In most cases, the computer ultimately presents the result of the measurement graphically for the sake of clarity and in order to make any change of test values readily visible. Various types of calorimeters, depending on their particular design, ultimately yield characteristic data that – for all their specific traits – share a number of common properties to be discussed in this chapter. The specific features inherent in the various designs are dealt with in the description of individual calorimeters (see Chapters 7 and 8). We first consider the consequences of heat transport phenomena, which take place inside the measuring systems or substances to be studied with the calorimeter. Afterward we shall discuss additional effects associated with the specific type of calorimeter used. This chapter also provides an outline of methods for the elimination of heat conduction effects by mathematical means and ends with the presentation of some special methods of evaluation, including the assessment of the measurement uncertainty.

6.1 Consequences of Temperature Relaxation within the Sample 6.1.1 First Example: Chemical Reaction

Let the system intended to be studied in the calorimeter be a solution inside a vessel. After the introduction of a second substance, a reaction takes place that has to be followed quantitatively with regard to the production of heat. Obviously, the reaction begins at the site of contact of the two substances and then spreads by diffusion. The heat produced (or consumed) brings about a change of temperature, which in turn causes a heat flow and other effects. A sensor (thermometer) located within or outside the reaction vessel detects a temperature change that occurs with some time lag relative to the reaction proper and can be only loosely correlated with Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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α t t0

Degree of reaction Time Time of start of reaction

T Tini Tfin ∆T

Temperature of probe Initial temperature Final temperature Overall temperature change

Figure 6.1 Time-dependent chemical reaction in an isolated container and the resulting temperature change.

the course of the chemical reaction because of the uncontrollable character of such phenomena as diffusion, convection, and heat conduction in the liquid. However, if sufficient time is allowed for all equalization processes to go to completion, it becomes evident that the overall temperature change DT ¼ Tfin  Tini (Figure 6.1) is closely related to the overall heat of reaction Qr: DT ¼ Q r =C

where C is the apparent heat capacity of the system (the so-called water value). Insofar as diffusion is concerned, stirring can accelerate the relaxation processes. But even in the case of an ideally rapid mixing of the reagents, heat transport is necessarily hindered and delayed by the vessel walls or by the insulation of the thermometer. Moreover, the very act of stirring generates additional heat (and entropy), thus perhaps distorting the measurement. 6.1.2 Second Example: Biological System

The heat generated by the heart muscle of an animal cannot be measured by placing the entire test animal in a calorimeter. Indeed, all organs produce heat, and the calorimeter cannot distinguish between heat sources. Although seemingly trivial, this example is instructive because it applies to any system that comprises heat flows of various locations – in other words, to all cases in which the heat

6.1 Consequences of Temperature Relaxation within the Sample

generation is even remotely dependent on a specific site. Such local heat flows can never be measured by means of an externally located sensor, which, of course, can only determine the total heat flow of the system. If the heat generated by the heart muscle is to be measured, the sensor must be placed directly on the heart muscle or inside it – a rather far-reaching intervention in the system “animal.” Similar considerations apply to other inhomogeneous systems. 6.1.3 Third Example: First-Order Phase Transitions

As shown in Section 3.2.2, a phase transition takes place by heat exchange with constant intensive variables such as the temperature. If a homogeneous sample is heated continuously, the result would have to be an abrupt transformation at the transition temperature, which at this moment would involve an infinitely large heat flow rate through the surface of the sample (Figure 6.2). In reality,

H ∆trsH T Ttrs

Enthalpy Transition enthalpy Temperature Transition temperature

Cp ∆trsCp

Heat capacity Transition heat capacity change

Φ

Heat flow rate as recorded by the calorimeter

Figure 6.2 Enthalpy, heat capacity, and actually measured heat flow rate during a first-order phase transition.

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Φ Heat flow rate T Temperature Tfus Fusion temperature λ Thermal conductivity Heating rate of the left margin: 1 K min-1

Figure 6.3 Model calculation of heat flow rates into unilaterally heated layers of 1 mm thick indium and paraffin during the melting process.

however, heat transport to and through the phase transformed requires both the time and a temperature gradient, so that the heat flow through the surface remains finite and follows the course shown in Figure 6.2. Such heat flow rate functions can be calculated with suitable computer software as described in Section 4.1. To illustrate what is mentioned above, some characteristic results of finite element calculations are presented in the following figures. Figure 6.3 shows the heat flow rate through the boundary surface for a case in which a metal plate (indium) and a paraffin plate, each 1 mm thick, are heated on one side at a linear rate of 1 K min1 starting a little below the melting point. Figure 6.4 presents the corresponding temperature profiles inside the plate during the steady-state heating of the stable phase and in the course of melting. The dotted line at xfus marks the “heat-consuming” phase boundary between the two phases, which acts as an adiabatic shield impervious to heat. As a result, the layers of substance situated to the right of this site have a nearly constant temperature. The differences in the plotted curves stem from the thermal conductivities, which differ from one another by more than two orders of magnitude. The metal that has a higher thermal conductivity allows a higher heat flow rate at smaller temperature differences and accordingly melts more rapidly. Figure 6.5 shows calculated melting curves of paraffin plotted for two layers of different thickness under identical conditions. As could be expected, the thinner layer melts more quickly; the curves are nearly identical in their initial parts. Figure 6.6 reveals the effect of the geometry of the sample on the form of the heat flow rate–time function.

6.1 Consequences of Temperature Relaxation within the Sample

∆T

= T(x = 0) — T(x) Temperature difference with regard to the left side of the layer (x = 0) x Position coordinate xfus Position of the phase boundary Heating rate of the left margin: 1 K min-1

Figure 6.4 Temperature fields inside unilaterally heated plates of indium and paraffin. Top: In the case of a steady-state heat flow. Bottom: During the course of melting (for geometric arrangement, see Figure 6.3).

These examples show that the actually measured heat flow rate–time course depends to a considerable degree on the geometry, layer thickness, and thermal conductivity of the substance involved. Because of the time-consuming heat transfer, the measured heat flow rate function is always different from the expected “sharp” transition (see Figure 6.2).

Φ T Tfus

Heat flow rate Temperature Fusion temperature of paraffin d Thickness of layer Heating rate of the left margin: 1 K min-1

Figure 6.5 Calculated heat flow into two unilaterally heated paraffin plates (0.1 and 1 mm thick) during melting.

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Φ T Tfus

Heat flow rate Temperature Fusion temperature of paraffin Heating rate of the left margin: 1 K min-1

Figure 6.6 Calculated heat flow into melting paraffin samples of cylinder–symmetric and flat geometry.

6.2 Typical Results from Different Calorimeters

The thermal relaxation effects already outlined must be kept in mind in any analysis of the characteristics of the data output of calorimeters regardless of the particular type of instrument involved. 6.2.1 Adiabatic Calorimeters

Under ideal circumstances, the system (sample) to be tested in an adiabatic calorimeter (see Section 5.3) is totally insulated against any heat exchange with the surroundings. Any heat generated or consumed by the sample in the measuring system during a process brings about a change of its temperature. Figure 6.7 illustrates this relationship in the case of an exothermic process. The temperature– time function drawn reproduces rather accurately the constant generation of heat assumed between t1 and t2. Time lags and rounded-out shapes reflect heat transport phenomena between the system and the temperature sensor outside the sample. The entire heat Q produced by the sample can be calculated from the measured temperature difference DT: Q ¼ C  DT

The heat capacity C of the measuring system can be easily determined by calibration using electric energy. It consists of the heat capacities of the sample, the

6.2 Typical Results from Different Calorimeters

E t t1, t2 Q

T

∆T C

Energy of the calorimeter system (without sample) Time Start and end of the exothermic process Total heat of the process

Temperature of the total system (sample and calorimeter) = —Q/C Overall temperature change of the system Heat capacity of the system

Figure 6.7 Exothermic process involving constant heat generation with the corresponding measured curve obtained with an ideal adiabatic calorimeter.

container, and the thermometer taken together. Under ideal conditions, the slope of the temperature–time curve (Figure 6.7) is proportional to the heat flow rate of the process. If the measured function (lower curve) does not deviate too much from the real temperature change (which is proportional to the upper curve), we get after differentiation

WðtÞ ¼

dQ ðtÞ  dt

C


dTðtÞ dt

Thus, the heat flow rate W(t) can, under favorable conditions (i.e., low thermal lag), be obtained directly from the measured curve T(t) if the apparent heat capacity is known. In reality, ideal adiabatic conditions cannot be expected because total thermal isolation of the measuring system from the surroundings is not possible. As a consequence, both the initial and the very final temperatures of the measuring system are equal to the temperature of the surroundings (thermostat) TF. In other words, the temperature change DT resulting from the process inside the sample will not be constant in time. Even the smallest heat leakage causes an equalization of the measured temperature with the surroundings temperature. The leakage heat flow rate is, as a rule, proportional to the temperature difference T TF between the measuring system and its surroundings. Thus, the temperature change of the

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system for TF ¼ constant (thermostat) is dTðtÞ / TðtÞ dt

This is a simple differential equation that can be integrated:  t TðtÞ ¼ DT  exp  þ T F t where DT ¼ Tmax  TF is the initial temperature difference of the measuring system. The temperature follows an exponential function whose “time constant” t depends on the magnitude of the heat leakage and the apparent heat capacity of the measuring system, but it can be easily determined experimentally. If at t ¼ 0, a short heat pulse is given (e.g., by means of electric current in a resistor), we obtain in ideal conditions the temperature function shown in Figure 6.8. The time constant can be determined from the width of the function at the ordinate TF þ DT/e (e Euler’s number). In real cases, the time interval of the reaction is not this short, and the thermometer reacts with a certain time lag to the temperature change, leading to characteristic deviations from the ideal case as shown in Figure 6.9. The real initial temperature change DT can be determined from the measured function T(t) in the following way: The ideally thought intersection point Tmax is constructed at that moment of time t0, where the integral average value of the initial temperature change lies (i.e., where the areas A1 and A2 are equal). This is the point of time to be selected if the entire heat of the process were released at once (procedure of Regnault and Pfaundler) (Pfaundler, 1866; White, 1928). In the vast majority of cases, this moment can with sufficient accuracy be assumed to lie at the inflection point TI of the temperature function; otherwise, to get higher precision, the areas A1 and A2 must be made equal (Eucken, 1929; Eder, 1956).

T TF Tmax ∆T t τ

Temperature Temperature of surroundings Maximum temperature after heat pulse Temperature change on heat pulse Time Time constant of the exponential temperature relaxation

Figure 6.8 Temperature–time function of a “poorly adiabatic” (isoperibol) calorimeter after a short exothermic heat pulse.

6.2 Typical Results from Different Calorimeters

T TF Tmax ∆T t t1, t2 t0

Temperature Temperature of surroundings Constructed maximum temperature Assumed sudden temperature change Time Start and end of the heat production process, resp. Assumed moment of a similar but pulse-like heat event (areas A1 and A2 equal)

Figure 6.9 Reconstruction of the “true” temperature increase in an adiabatic calorimeter with heat leakage.

Summary Measured quantity:  Temperature of the measuring system as a function of time Evaluation result:  Heat calculated from the correctly determined temperature change DT (with known apparent heat capacity of the measuring system)  Heat flow rate from the differentiated temperature function (with known apparent heat capacity of the measuring system)  (Apparent) Heat capacity from the correctly determined temperature change DT (with known electrically generated heat) 6.2.2 Isoperibol Calorimeters

In isoperibol calorimeters (see Section 5.2), the measuring system is coupled via defined heat conduction paths with surroundings that have constant temperature as well as good thermal conductivity (thermostat). In such calorimeters, all processes can – with some approximation – be reduced to a heat exchange between the system to be investigated and an isothermal heat reservoir of infinite capacity. Figure 6.10 shows the construction principle of such a calorimeter and a simplified, one-dimensional representation of the heat conduction path between the isothermal surroundings and the sample being examined. If an abrupt (pulse-like) heat-generating event occurs in the sample at the moment t0, the temperature T of the measuring system (calorimeter vessel) undergoes the change shown in 2 and the sample ○ 3 have a much higher thermal Figure 6.11 if both the vessel ○

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Thermostat (surroundings) Calorimeter vessel Sample Heat conductive connections

Figure 6.10 Top: Schematic representation of an isoperibol calorimeter. Bottom: Onedimensional simplification of the thermal path. 4 . The temperature jump DT is conductivity than the heat-conductive paths ○ proportional to the quantity of heat produced in the sample: DT ¼a  Q. The constant a (the negative reciprocal heat capacity of the device) is best determined by calibration using electric heating of the measuring system. The function T(t),

Φ t t0

T TF ∆T A

Heat flow rate inside the calorimeter Time Timing of the heat pulse

Temperature of the calorimeter vessel Temperature of the thermostat (surroundings) Temperature jump after heat pulse Area between temperature function and extrapolated baseline

Figure 6.11 Measured temperature–time function of an isoperibol calorimeter after a sudden heat production (heat pulse) inside the calorimeter.

6.2 Typical Results from Different Calorimeters

according to which the temperature of the system approaches that of the surroundings, depends on the nature and magnitude of heat transport between the measuring system and the isothermal surroundings; in other words, it is a function of the apparatus used. For pure heat conduction, it is an exponential function (see Section 7.9.2). The relationship between the measured temperature and the heat flow rate to the thermostat is, however, defined; consequently, the area A between the curve T(t) and the dotted baseline is proportional to the heat produced at t0. If at t ¼ t0 a constant heat flow rate is thought to be “switched on,” the course of the temperature will be as shown in Figure 6.12. The asymptotically reached final temperature of the calorimeter vessel characterizes the steady state where the heat produced equals the heat flowing out of the vessel. The latter is a well-defined function of the temperature difference between the system and the thermostat. Consequently, Tmax also represents a well-defined function of the heat flow rate W ¼ dQ/dt generated in the system from t ¼ t0 on. If a constant heat flow is switched on at time t0 and discontinued at time t1, the relationship obtained will be as shown in Figure 6.13. The temperature function consists of the two components described above. The maximal change Tmax  TF of the function T(t) depends here not only on the heat production rate W but also on the time interval t1  t0. In the absence of an exact knowledge of the relationship involved (see Section 7.9.2), one can only estimate the area A between the temperature–time function and the baseline that is proportional to the total produced heat Q ¼ W0  (t1  t0). Φ Φ0 t t0

T TF Tmax ∆T

Heat flow rate inside the calorimeter Heat production rate in calorimeter vessel Time Time of beginning of the heat production

Temperature of the calorimeter vessel Temperature of the thermostat (surroundings) Final temperature of the calorimeter at constant heat production rate Equilibrium temperature change

Figure 6.12 Measured temperature–time function of an isoperibol calorimeter for a constant heat production starting at time t0.

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Φ Φ0 t t0, t1 Q

T TF Tmax A

Heat flow rate inside the calorimeter Heat production rate in calorimeter vessel Time Time of beginning and end of the heat production, resp. = Φ0 · (t1 — t0) Total heat produced Temperature of the calorimeter vessel Temperature of the thermostat (surroundings) Maximum temperature in the calorimeter vessel Area between temperature function and extrapolated baseline

Figure 6.13 Measured curve of an isoperibol calorimeter for a constant heat flow of finite duration.

The form of the measured function T(t) differs so much from the actual course of the real thermal event W(t) in the system (upper Figure 6.13) that the “desmearing” procedure described in Section 6.3 provides the only way of reconstructing this true heat production W(t) from the measured one. 4 in Figure 6.10) We have assumed so far that the heat conduction path (○ represents the dominant factor for the measuring process and that the heat equilibration processes inside the system and in the vessel are negligible. This is normally not the case, and even with an abrupt production of heat, the temperature has a “smeared” course as shown in Figure 6.14. Here, the course of the temperature change T1(t) is determined by the heat transport phenomena in the system, whereas T2(t), as noted above, depends on the heat conduction path between the measuring system and the thermostat. The area A between the temperature–time function and the baseline is proportional to the heat produced at t0, which, can thus be readily calculated from the temperature curve traced. In the case of any production of heat with flow rate W(t) in the sample, the plotted curve changes in accordance with the relationships shown in Figures 6.12 and 6.13. It is noteworthy that the course of the initial temperature change T ¼ T1(t) (Figure 6.14) depends not only on the apparatus used but also on the tested substance inside the calorimeter and its heat conduction properties. As a rule, a reconstruction of the real heat production W(t) of the sample is only possible if T1(t) does not vary much from one measurement to the next. This can be brought about by providing a heat 4 in Figure 6.10) with a thermal resistance large enough to conduction path (○

6.2 Typical Results from Different Calorimeters

Φ t t0

T TF A

Heat flow rate inside the calorimeter Time Time of the heat pulse

Temperature of the calorimeter vessel Temperature of the thermostat (surroundings) Area between temperature function and extrapolated baseline

Figure 6.14 Measured temperature change of an isoperibol calorimeter for a “heat pulse.”

ensure a – relative to T1(t) – slow decline of T2(t) with time. Then any change of the slope of T1(t) can often be neglected, and we obtain – as an approximation – nearly the relationships shown in Figure 6.11, but at the price of a large time constant (i.e., inertia) of the calorimeter and the necessity of a mathematical desmearing of the measured curves for obtaining the true heat flow rate W(t) produced in the sample. Summary Measured quantity:  Temperature of the measuring system as a function of time Evaluation result:  Heat calculated from the area A (with known – reciprocal – apparent heat capacity of the measuring system)  Heat flow rate function from the measured temperature function (after proper correction of the influences of thermal inertia)

6.2.3 Differential Scanning Calorimeters

Scanning calorimeters are often calorimeters in which the surroundings of the measuring system (furnace) are heated or cooled at a constant rate during the

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T TF β t TM TS x

Temperature = β · t Temperature of the heating element (furnace) Heating rate Time Temperature of the temperature sensor Average temperature of the sample Position in a linear simplification of the measuring system

1 Heating element, 2 heat conduction path, 3 sample container, 4 sample, 5 temperature sensor Figure 6.15 Schematic representation of the measuring system of a scanning calorimeter in a linear simplification with the corresponding temperature field in the presence of a steadystate heat flow.

measurement. These instruments always have a twin design (see Section 7.9.4). Figure 6.15 shows in a one-dimensional simplified form the path of the heat flow and the temperature field formed at steady state on the sample side of the twin measuring systems. The temperature measured at the sample container in the heating mode is always lower than the furnace temperature (unless no strongly exothermic reactions take place in the substance), and yet higher than the temperature of the substance itself, owing to heat transport phenomena. The 1) temperature difference between the furnace and the measuring site TF  TM in the steady state determines the heat flow rate along the heat conduction path (see Eq. (4.5)): W¼

A  lðTÞ  ðT F  T M Þ Dx

ð6:1Þ

where A is the cross-sectional area, l(T) is the temperature-dependent thermal conductivity, and Dx is the length of the heat conduction path. Figure 6.16 shows how the temperatures of the heater and the measuring site change with time. A sudden exothermic transition occurring at the temperature Ttrs causes an abrupt temperature rise in the substance, which, in turn, leads with some delay to a rise of the temperature TM of the measuring site. As a result, the heat flow arising from the heater diminishes with regard to the steady-state condition, and the temperature rises only slowly until the reestablishment of the previous steady-state conditions after a certain time interval Dt1. In the event of an 1) This is a rather stringent requirement, which is surely not fulfilled in the case of sudden thermal events in the system (sample).

6.2 Typical Results from Different Calorimeters

T T0 TM TF t

Temperature Start temperature Temperature of measuring site Temperature of heating element (furnace) Time

Ttrs Transition temperature ∆t1, ∆t3 Temperature relaxation times to reestablish steady state conditions ∆t2 Time needed to complete phase transformation

Figure 6.16 Temperature–time course in the scanning calorimeter system with test substance without phase transition (top), test substance with a sudden exothermic transition (middle), and test substance with an endothermic phase transition (bottom).

endothermic first-order phase transition at Ttrs, the temperature of the substance will remain unchanged until the transition is completed. TM remains practically constant (if the thermal resistance between the temperature sensor and the substance is negligible). The heat flow rate, however, increases owing to the increase of TF  TM. If the transition is completed after the time interval Dt2, the temperature difference exceeds that obtained under steady-state conditions, thus causing a larger heat flow until the previous steady-state conditions are restored by a more rapid temperature rise within the time interval Dt3. Figure 6.17 shows the result of such measurements originating from those of Figure 6.16 by subtracting the furnace temperature TF from TM of the sample system. The curves representing exothermic and endothermic processes clearly differ to a marked extent from one another. The delay Dt0 in recording the transition results from the heat transfer process between the substance and the temperature sensor. The other distortion in time (the time intervals Dt1, Dt2, and Dt3) depends on the nature and quality of the heat conduction path between the heater and the temperature sensor; in other words, it is associated with the apparatus used.

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∆T TM TF

= TM — TF Temperature of measuring site Temperature of heating element (furnace) t Time ∆t0 Time interval between the beginning of the transition in the sample and the sensor reaction ∆t1, ∆t3 Temperature relaxation times to reestablish steady state conditions after transition ∆t2 Time needed to complete phase transformation A Area between measured function and baseline

Figure 6.17 Idealized differential temperature–time functions measured with a differential scanning calorimeter. Sample with sudden exothermic transition (top) and sample with endothermic phase transition (bottom).

Because we are dealing with a twin device, the actually measured quantity is not the temperature difference plotted in Figure 6.17 but the temperature difference between the temperature sensors of the two calorimetric systems. Most disturbances from the surroundings and heat leaks are compensated for in a fully symmetrical twin system. Here the curve obtained depends only on processes based on the different nature of the two substances. From Eq. (6.1), it follows that A1  l1 ðTÞ  ðT F1  T M1 Þ Dx 1 A2 W2 ¼  l2 ðTÞ  ðT F2  T M2 Þ Dx 2

W1 ¼

The subscripts designate the respective calorimetric system, that is, 1 for the sample and 2 for the reference side. Twin calorimeters are usually built so that the same heater serves both individual measuring systems, these being made as closely equal to one another as possible (see Section 4.6). Consequently, A1 ¼ A2 l1 ¼ l2

Dx 1 ¼ Dx2 ¼ Dx T F1 ¼ T F2 ¼ T F

so that W1  W2 ¼ 

A  lðTÞ  ðT M1  T M2 Þ Dx

or in abbreviated form DW ¼ K W ðTÞ  DT

ð6:2Þ

6.2 Typical Results from Different Calorimeters

The difference between the heat flow rates to the two measuring systems, the differential heat flow rate, is proportional to the (negative) measured temperature difference between the sample and reference sides. At any rate, the “calibration factor” KW(T) is usually not constant over large temperature ranges, and the respective calibration function has to be determined empirically in most cases. As noted in Section 4.6, one of the measuring systems, the sample side, is filled with the substance to be tested, but the other measuring system, the reference side, remains empty or contains a reference material. The thermophysical properties of the reference material must be known. To fulfill the symmetry condition, its heat capacity should at least be able to compensate that of the sample in the vicinity of the transition temperature (reaction temperature) of the sample. In the temperature range of the transition (reaction), the reference material should, of course, show no transition whatsoever – that is, it must be inert (inactive). With such conditions, the measured functions of a differential scanning calorimeter (DSC) transform from those of Figure 6.17 to those plotted in Figure 6.18. Here it should be mentioned that the heat flow rate into a system on an endothermic event is defined as positive in thermodynamics, but the temperature difference is negative in such a case. According to Eq. (6.2), the areas A in Figure 6.17 between the DT curves and the line that would be recorded by the device in the absence of any transition (baseline) are proportional to the heat Q of the process taking place in the sample. Consequently, ð ð Q ¼ DWdt ¼ K Q DT  ðtÞdt where DT  is DT with subtracted baseline.

∆Φ t Q

Differential heat flow rate Time Total heat of the thermal effect

Figure 6.18 Idealized differential heat flow rate measurement of a differential scanning calorimeter for a sample with sudden exothermic transition (top) and a sample with endothermic phase transition (bottom).

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∆Φ t Q

Differential heat flow rate Time Total heat of the thermal effect

Figure 6.19 Measured curve of a differential scanning calorimeter for an endothermic process (dashed: baseline; dotted: extrapolated from steady-state regions).

This relationship, however, applies only if the calibration factor KQ can be regarded as constant in the respective temperature range and the steady-state conditions of heat transfer are not distorted during the thermal event in the substance. The relationships involved are shown in an ideal form in Figure 6.18. Perfect symmetry can hardly ever be attained in a twin calorimeter. Thus, the DW curve emerges as shown in Figure 6.19, where the following can be observed: 1) The baseline, even outside the reaction time, does not coincide with the abscissa DW ¼ 0 nor is it straight, because l1(T) and l2(T) are not strictly equal; moreover, the inert substance (reference material) in system 2 and the test substance (sample) in system 1 do not possess exactly the same heat capacities and thermal conductivities. 2) Extrapolation of the baselines from the “left” and “right” sides into the reaction range yields a step at the phase transition because of the difference in the specific heats of the two phases and also as a result of the fact that heat transfer between the substance and the container (sample holder) may change considerably at the phase transition. Because this brings about a change of the temperature profile in the steady-state situation (see Figure 6.15), there appears a difference in the TM1 function and consequently in DT and DW. There are a series of procedures for finding a “correct” baseline. Because detailed knowledge of the actual causes of the baseline step change is necessary to present an exact theory of the construction of the baseline, a discussion is largely superfluous (Hemminger and Sarge, 1991). It must therefore be assumed that a step change in the baseline always implies a systematic error in the heat of transition thus determined. Obviously, good repeatability and linearity of the baseline represent essential features of a DSC. If the true heat flow rate function produced in the sample W(t) (or W(T)) is to be reconstructed from the measured DW function recorded, then a “desmearing” process, as described in Section 6.3, is necessary. Another type of DSC, the power-compensated differential scanning calorimeters (see Section 7.9.4.2), follows in principle the model shown in Figure 6.15, but instead of one, there are two furnaces for the sample and the reference system, 1 is so controlled that the respectively. The current in the heaters (furnaces) ○ average temperature of the two systems always matches the preset temperature,

6.2 Typical Results from Different Calorimeters

Φ ΦS(T) Φ0(T) T β t ΔΦ g(T) cp m Te

Ttrs A

Heat flow rate Heat flow rate function with sample Heat flow rate function without sample = β ⋅ t + T0 Program temperature Heating rate Time = ΦS — Φ0 Heat flow rate difference = cp(T) ⋅ m ⋅ β Baseline function Specific heat capacity of the sample Mass of the sample Constructed transition temperature (extrapolated onset temperature) Transition temperature Area between measured function and baseline

Figure 6.20 Endothermic phase transition measured with a differential scanning calorimeter with power compensation.

which normally changes proportionally with time. In addition to the basic heating of the two measuring systems, a second current is added to the heater, which largely compensates the measured temperature difference. The output is the respective compensation power DW, which is proportional to the temperature difference between the sample and the reference measuring system. As a result, the measured functions for phase transitions are still of the form illustrated in Figure 6.17, but the sign is inverted and the ordinate is the differential power DW between the sample and the reference furnaces. Figure 6.20 shows the measured heat flow rate function of a power-compensated DSC containing a substance in the sample crucible and another one with an empty crucible. The difference between the two measurements directly yields the heat flow rate into the sample because possible influences of asymmetries of the two systems are eliminated. In reality, the heat flow rate is measured as a function of time t. Because the heating electronic circuit always keeps the measured temperature T equal to the desired program temperature, the output signal can also be the heat flow rate as a function of temperature as there is a linear relationship T ¼ b  t þ T0

ð6:3Þ

where b is the heating rate and T0 is the initial temperature. The total heat Q of the process occurring in the sample on heating is determined from the peak area of the compensation heat flow rate as a function of time (with the baseline g(t) subtracted): Q¼

tð fin

tini

ðDWðtÞ  gðtÞÞdt

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It is, however, proportional to the peak area A in Figure 6.20, where the heat flow rate difference is plotted as a function of temperature: Q¼

1  b

Tðfin

ðDWðTÞ  gðTÞÞdT ¼

1 A b

T ini

Both types of differential scanning calorimeters are mostly used to investigate transitions and reactions occurring at a certain temperature. In addition to the heat of reaction or transition, even the temperature of the event is of interest and has to be determined from the measurement. Of course, in a heating run, the measured temperature is somewhat higher than that of the sample because the temperature sensor lies outside the latter and heat transport necessitates a temperature gradient (see temperature profile, Figure 6.15). The difference between the measured and the actual temperatures of the sample Tmeas  Tsample ¼ DT(b) depends on the thickness and material of the sample container and the heat transfer conditions as well as on the heating rate (see temperature profile in Figure 6.15). Heat transport requires only a few seconds in samples of a few milligrams placed in metal containers with very thin walls. It is evident from Eq. (6.3) that the difference DT is less than 1 K in this case even at a heating rate of 20 K min1. Constructing the transition temperature by extrapolation of the left edge of the measured curve down to the baseline g(T) as in Figure 6.20 would obviously be a very good approximation. Another important quantity that can easily be evaluated from the measurement of a DSC is the specific heat capacity of any substance positioned in the measuring system on the sample side. From the definition of the heat capacity, CðTÞ ¼

dQ dT

together with Eqs. (6.3) and (6.2) as well as division by the mass m, we get csample ðTÞ ¼

dQ sample dt 1 DW K W ðTÞ  DT   ¼ ¼ dt dT m b  m bm

ð6:4Þ

The specific heat capacity of the sample follows from the differential heat flow rate after division by the heating rate (in K s1) and mass if the equipment is symmetrically constructed and the sample is the only difference between both systems. Remark: Here it is assumed that the empty DSC measurement yields a zero heat flow rate function. If this is not true, the respective measurement must be subtracted from the sample run as in Figure 6.20. For details, see Section 9.3.1 and H€ohne, Hemminger, and Flammersheim (2003, pp. 147–153). The specific heat capacity, the temperature, and the transition heats can be obtained directly from the measured curve with relatively high accuracy. In principle, however, the DW(T) curve recorded with a DSC does not correspond exactly to the heat flow rate generated in the sample, but reflects the processes

6.3 Reconstruction of the True Sample Heat Flow Rate from the Measured Function

occurring in the substance smeared in time. To obtain more exact information on the time dependence of the heat flow rate produced in the sample, the measured curve has to be desmeared. In all scanning calorimeters, any change of heat transfer between substance and container (or between container and temperature sensor) creates changes of the steady-state temperature fields, which, owing to the heat capacities always involved in the process, bring about a change of the measured heat flow rate. Such changes often produce the same effect on the measured functions as do the heats of transition. The only way of distinguishing between the two is by performing repeated measurements, whereupon the changes of heat transfer clearly occur randomly rather than at specific temperatures. Summary Measured quantity:  Differential temperature or electrical compensation power as a function of time Evaluation result:  Differential heat flow rate between sample and reference system (with known calibration function) as a function of time or temperature  Specific heat capacity of the sample as a function of temperature  Heat of reaction or transition calculated from the peak area (from the differential heat flow rate function after subtraction of the baseline)  Temperature of reaction or transition 6.3 Reconstruction of the True Sample Heat Flow Rate from the Measured Function

We have shown that calorimeters, regardless of their design, necessarily present a distorted picture of the heat exchange involved in the process under investigation. The ubiquitous heat conduction path from the site of the event to the measuring probe causes a smearing of the measured function in time. The cause and manner of smearing vary in accordance with the type of calorimeter involved, as shown in the preceding sections. Moreover, the smearing lends itself to a simple mathematical description only in a few exceptional cases, such as the isoperibol Calvet calorimeter (see Section 7.9.2.3). Here we shall consider the basic relationship between the event and its recording, regardless of the specific type of apparatus used, as well as mathematical procedures for reducing the smearing of the measured function. 6.3.1 Reconstruction of the Temperature Field for Negative Times

Consider a fly placed in a calorimeter vessel (or, for that matter, a very rapid chemical reaction). The question arises as to whether the heat generation of the

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fly’s muscles during a wing beat can be resolved in time. The answer is: certainly not, because the time constant of the measuring system would at least have to be of the same order of magnitude as the duration of a wing beat. Suitable calculations show that for time constants on the order of milliseconds, the heat transmission path between source and probe must be only a fraction of a millimeter. In principle, the respective differential equation, the heat flow equation, can also be solved for negative times if the boundary and initial conditions are known – that is, the measured values obtained at the end of any heat conduction path make it possible to reconstruct initial processes backward in time, but only with certain limitations owing to the statistical nature of heat relaxation. If the signal is smaller than the statistic background noise (e.g., Brownian motion of air molecules) or smaller than the measuring error, it can neither be separated from them nor traced back in time by mathematical means. Indeed, the heat flow oscillations produced by the beats of the fly’s wings fuse into a uniform average heat flow within a short distance and a brief time interval owing to the motion of air molecules. Thus, the information from the fly’s wings is totally lost and cannot be reconstructed. Formally speaking, the statistical fluctuations of the measured quantity (noise) would have to be included in the heat flow equation. But with regard to negative times, this would soon yield temperature fluctuations at the site of the event far in excess of those generated by the effect to be measured. At any rate, a reconstruction of the true event back in time by means of the heat flow equation is an extremely laborious task even if the probe receives signals markedly above the level of statistical fluctuations. 6.3.2 The Convolution Integral and Its Validity

For the solution of measuring problems in optics and other fields of physics, there is a simple mathematical procedure (i.e., the theory of linear response) that makes use of the overall behavior of the apparatus in defined processes in order to calculate unknown complex processes from the measured function. Here we shall derive this relationship in order to define the conditions under which this desmearing procedure can be applied. We shall formulate the laws in a general manner using the variable x as in mathematics. Consider an abrupt (pulse-like) event taking place in the apparatus at x1. Using the Dirac delta function d(x),

dðx  x 1 Þ ¼

with

1 ð

1



1 0

for x ¼ x1 for x 6¼ x1

dðx  x1 Þdx ¼ 1

6.3 Reconstruction of the True Sample Heat Flow Rate from the Measured Function

an abrupt (pulse-like) event at the moment x1 with the “weight” Q1 can be described as follows: gðxÞ ¼ Q 1  dðx  x1 Þ

with

1 ð

1

gðxÞdx ¼ Q 1

For this “event function” g(x), the apparatus yields a measured “response function” h(x) for which the same relationship applies (the calibration factor is thought to be one), namely, 1 ð

1

hðxÞdx ¼ Q 1

From h(x), one gets the so-called “apparatus function” f(x0 ) by normalization and shifting the abscissa. This is known in response theory as the transfer function or Green’s function. Thus, hðx  x1 Þ f ðx Þ ¼ Q1 0

with

1 ð

1

f ðx 0 Þdx 0 ¼ 1

Figure 6.21 shows this relationship graphically.

Real event: g(x) = Q1 · δ(x — x1)

Measured function: h(x) = Q1 · f(x — x1) ∫ h(x)dx = Q1

Apparatus function: f(x') = h(x — x1)/Q1 ∫ f(x')dx' = 1 x' = x — x1

Figure 6.21 Definition of the apparatus function f(x 0 ).

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Real event: g(x) = Q1 · δ(x — x1) + Q2 · δ(x — x2)

Measured function: h(x) = h1(x) + h2(x) = Q1 · f(x — x1) + Q2 · f(x — x2)

Figure 6.22 Construction of the measured curve for two pulse-like events according to the superposition principle.

In the case of two independent pulse-like events, h(x) represents the sum of two response functions, each of them referring to a single event; this is conditional on the superposition principle applying to the examined physical phenomena (Figure 6.22). For n pulse-like events, the apparatus thus traces the following function: hðxÞ ¼

n X i¼1

Q i  f ðx  xi Þ

ð6:5Þ

A continuous function g(x) can be approximated by a large number m of pulse-like events if the function is divided into a series of stripes Dxi wide and g(xi) high, whereupon Qi ¼ Dxi  g(xi) is the area of the ith strip and the response function for an infinite number of the smallest strips can be presented as follows: hðxÞ ¼

lim

Dx!0;m!1

m X i¼1

Dxi  gðxi Þ  f ðx  xi Þ

This represents an integral, namely, hðxÞ ¼

1 ð

1

gðx 0 Þ  f ðx  x0 Þdx0

ð6:6Þ

also known as the convolution integral (or convolution product) and is often written in abbreviated form as follows: _ hðxÞ ¼ gðxÞf ðxÞ or hðxÞ ¼ gðxÞ  f ðxÞ

6.3 Reconstruction of the True Sample Heat Flow Rate from the Measured Function

The integral equation (6.6) reveals the mathematical relationship between the measured function h(x), the real event g(x), and the apparatus function f(x) based on the following conditions: i) The response of the apparatus to a pulse-like event must be repeatable and normalizable with regard to the initial value, the area, and all experimental variables. ii) The apparatus must superimpose the responses to any number of pulse-like events (linear behavior). For common calorimeters, these conditions are usually fulfilled. 6.3.3 Solution of the Convolution Integral

If the apparatus function is known (its derivation will be described later), the unknown true event g(x) can be calculated from the measured function h(x) by solving the above integral equation (6.6). There are essentially two methods for this purpose: the Fourier transform and the recursion method. Both require numerical calculations with a computer. The Fourier transform represents an integral operation: rffiffiffiffiffiffi 1 ð 1 f ðyÞ  eixy dy Fðf ðxÞÞ   2p 1

Applied to the convolution integral (6.6), it yields (see textbooks of mathematics) FðhðxÞÞ ¼ FðgðxÞÞ  Fðf ðxÞÞ ðconvolution theoremÞ

Thus, the convolution product turns into an ordinary product in Fourier space that can be solved for FðgðxÞÞ ¼ FðhðxÞÞ=Fðf ðxÞÞ

The sought function is obtained by an inverse Fourier transform: gðxÞ ¼ F1 fFðgðxÞÞg ¼ F1 fFðhðxÞÞ=Fðf ðxÞÞg

This method can be applied in all cases. The Fourier transform procedure is included commonly in mathematical software products and is easily available for everyone. The drawbacks of this procedure lie in its laborious course and abstract nature because the calculations are performed in Fourier space. Those who lack experience in numerical Fourier transforms are advised to study some “pitfalls” such as the “break-off effect” and the “sampling theorem,” both obtained by numerical treatment. Under specific conditions, this simulates periodicities and fluctuations that do not reflect any actual processes in the sample. Furthermore, the signal-to-noise ratio turns worse; for further details, readers are referred to the literature (Bracewell, 2000; Davies, 2002).

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The recursion method for solving the convolution integral (6.6) is of minor importance. It starts from the following recursion formula _ g n ðxÞ ¼ g n1 ðxÞ þ fhðxÞ  f ðxÞg n1 ðxÞg ¼ g n1 ðxÞ þ fhðxÞ  f ðxÞ  g n1 ðxÞg ð6:7Þ starting with the zeroth approximation: g0(x) ¼ h(x). The difference between the “reconvoluted,” still inaccurate synthetic function hn(x) ¼ f(x)  gn1(x) and the measured function h(x) is used in an additive manner for a simple correction of the approximation. The recursion formula (6.7) does not converge for all event functions. Abrupt changes and steps (on–off effects and similar phenomena) generate oscillations of the approximation function that diverge rapidly. In the case of the smooth curves commonly encountered in calorimetry, the procedure converges quickly and without problems. 6.3.4 Obtaining the Apparatus Function

For desmearing the measured curve as described above, one must first see whether the apparatus in question meets the requirements of linearity and the superposition principle detailed in Section 6.3.2; the next stage is to obtain the apparatus function. For this purpose, an event that is as pulse-like as possible is generated in the measuring system, and the corresponding output curve is obtained. The following phenomena can be used for obtaining the event curve: Exothermic processes:  Pulse-like Joule heating (current pulses in small resistors)  Current pulse in thermocouples (Peltier effect)  Quick introduction of a hot particle Endothermic processes:  Crystallization of supercooled pure substances  Quick introduction of a cold particle  Current pulse in thermocouples (Peltier effect) The measuring arrangement must be identical to that used for the measurement to be desmeared. First of all, the linearity of the calorimeter is checked by varying the magnitude of the heat pulse while keeping all the other conditions constant. The measured functions obtained are then “normalized” (by dividing the function values by the respective quantities of released heat). If the calorimeter behaves linearly, the normalized curves must coincide. A good match means that the normalized measured curve obtained should be used as the apparatus function. In the event of differences that are neither systematic nor too large, the average curve may be used for desmearing. At any rate, the deviations must be included in the error analysis and the result of the desmearing procedure interpreted accordingly with caution. Furthermore, two or more pulse-like events should be generated and

6.3 Reconstruction of the True Sample Heat Flow Rate from the Measured Function

Φ T T’ Tfus f(T ) g(T) h(T)

Heat flow rate Temperature = T — Tfus Melting temperature of pure substance Apparatus function Calculated desmeared function Measured function

Figure 6.23 Example of the desmearing procedure applied on the melting behavior of an octadecane sample measured by means of a differential scanning calorimeter.

the response of the apparatus compared with the curve obtained according to Eq. (6.5) in order to test the validity of the superposition principle. If the apparatus operates in a linear and repeatable manner, desmearing according to the convolution integral is an exact method capable of producing the true heat flow rate of the respective sample. Figure 6.23 illustrates this in the case of the melting curve of octadecane on the basis of a measured curve obtained by means of a DSC. The increase of the resolution in temperature is obvious and shows the pretransition more clearly, whereas the fluctuations (noise Dg) of the 2) desmeared heat flow rate function increase accordingly . The quality of the desmearing can be evaluated by comparing the reconvoluted _ function hðx Þ ¼ gðxÞf ðxÞ with the measured function. 6.3.5 Application Limits and Estimation of Uncertainty

The desmearing approach is necessary whenever the heat flow changes of the sample in time are within the same order of magnitude as the time resolution of the calorimeter, the latter being roughly equal to the halfwidth of the apparatus function. If the expected changes of the heat flow are much slower, the measured function provides a good approximation of the actual relationships and a desmearing would be to no avail. Finally, an estimation of uncertainty should be presented. The uncertainty (noise) of the desmeared function depends, of course, on the uncertainty of the measured function h(x) and the apparatus function f(x), respectively. But the relationship involved is not a simple one; here it will be outlined only briefly (see textbooks on Laplace transform and transfer theory). In calorimetry, accuracy in the x-direction 2) From transfer theory follows that the product of Df and DT never decreases on desmearing.

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(mostly the time) is usually much greater than in the y-direction (temperature or heat flow rate). Consequently, f ðxÞmeasured ¼ f ðxÞtrue  Df ðxÞ gðxÞmeasured ¼ gðxÞtrue  DgðxÞ hðxÞmeasured ¼ hðxÞtrue  DhðxÞ

The fluctuations (noise) around the true functions are generally not dependent on the x value and can be approximated by the average, which is taken as constant: Df ðxÞ  hDf i DgðxÞ  hDg i DhðxÞ  hDhi

The introduction of these in Eq. (6.6) yields the following uncertainty estimate after some computation: 1 ð gðxÞdx jhDg ij  jhDhij þ Q
jhDf ij; where Q ¼ 1

In other words, the uncertainty (noise) of the desmeared function g(x) is much larger than the uncertainty of the measured function h(x). Because of these reasons, the desmearing procedure should only be performed if really needed. The fluctuations of the apparatus function Df result, of course, in a fluctuation Db of the halfwidth (Figure 6.24). It can be shown that the halfwidth of the steeper arm of the apparatus function measures the resolution of the instrument along the abscissa (usually the time; see “Theory of Calvet’s Calorimeter” in Section 7.9.2.3). Accordingly, the fluctuation of the halfwidth provides a measure of the resolution of the desmeared measured curve along the abscissa. f(x) ∆f

Apparatus function Uncertainty of the apparatus function b1 Half-width of the steep arm b2 Half-width of the less-sloped arm ∆b1, ∆b2 Uncertainties of the half-widths obtained for the same ∆f.

Figure 6.24 Example of an apparatus function f(x) with the range of the uncertainty of its values and the resulting uncertainties of the halfwidths.

6.4 Special Evaluations

Summary  The apparatus function is the normalized response of the apparatus on a pulselike event inside the sample (the halfwidth of it is a measure for the time constant of the apparatus).  The measured function is the convolution product of the real heat flow rate of the sample and the apparatus function.  The true heat flow rate function of the sample is available from the measured heat flow rate function by deconvolution, but at the price of an increasing uncertainty (noise). 6.4 Special Evaluations 6.4.1 Determination of the Specific Heat Capacity

The measurement of specific heat capacities is a first step toward thermodynamic quantities. The highest accuracy is achieved if a known mass of the substance to be investigated is placed into a strictly adiabatic calorimeter and the temperature change resulting from the introduction of a known heat is measured. This is done in practice by the production of electric heat, which is determined by accurate measurements of current, voltage, and time. The specific heat capacity can be calculated from the simple relationship cp ¼ Qp/(DT m) if measured at constant pressure, or cV ¼ QV/(DT m) if measured at constant volume (where Q is the introduced heat, DT is the temperature change, and m is the mass). If the selected value of Q is so small that the specific heat capacity can be regarded as constant in the range of the respective temperature change DT, this approach can achieve an uncertainty in the order of a few thousandths or better. With a modern DSC, it is also possible to measure specific heat capacities (see Sections 6.2.3, 7.9.4, and 9.10.1). The uncertainty in this case can vary between 1 and 5% depending on the calorimeter in question. Measurements of specific heat capacities are of major importance because the thermodynamic state functions can be calculated from them (see Section 3.1.6). 6.4.2 Determination of the Kinetic Parameters of a Chemical Reaction

Every chemical reaction involves an exchange of heat. The respective heat flow rate is proportional to the reaction rate. Thus, the course of the reaction can be followed from the measured heat flow rate function. In the case of quick reactions, where the reaction heat is released within time intervals of the order of magnitude of the time constant of the calorimeter, the measured curve desmeared according to Section 6.3 must be taken as a basis for further evaluations. In reactions that proceed much more slowly than the time constant of the calorimeter, the

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evaluations outlined below can be taken directly from the measured curve. These efforts are aimed at determining the kinetic parameters of the chemical reaction. Any chemical reaction such as A þ B ! C þ D þ    follows a certain kinetic law: da ¼ kðTÞ  f ðaÞ dt

ð6:8Þ

where a is the degree of reaction, k(T) is a rate constant, and f(a) is a function describing the particular reaction model. Some examples are given in the following: Chemical reaction of nth order : f ðaÞ ¼ ð1  aÞn Autocatalytic reaction :

f ðaÞ ¼ að1  aÞ

Phase boundary reaction :

f ðaÞ ¼ nð1  aÞðn1Þ=n

where n is the reaction order. For many thermally activated reactions, the rate constant follows the Arrhenius law:   E act kðTÞ ¼ A  exp  ð6:9Þ RT where A is the preexponential factor and Eact is the activation energy as the kinetic parameters of the reaction, and R is the molar gas constant. The degree of reaction at the moment t is proportional to the heat of reaction Q(t) produced up to this moment with Qr the total heat of reaction: aðtÞ ¼

QðtÞ Qr

The reaction rate is proportional to the heat flow rate: da dQ 1 WðtÞ ¼ ¼  dt dt Q r Qr

Combining these results with Eq. (6.8) and using the proper model, for example, the Arrhenius approach, yields an equation that connects the measured quantities W(t), Qr, and T(t) with the kinetic parameters n, A, and Eact. Such an equation can only be solved numerically. Nowadays powerful software is available that does the job of determining the kinetic parameters from the measured heat flow rate function. But to get reliable results, the proper kinetic model must be selected first. The theory behind it is not simple, and a lot of experience is necessary to handle the rather complex kinetic software. In principle, caloric measurements thus shed light on the kinetic data of the examined reactions. However, every “true” reaction equation must provide a complete description of the measured curve. If the kinetic parameters A, Eact, and n for a given reaction equation are obtained by one of the above procedures or by any other method of nonisothermal reaction analysis (Kissinger, 1957; Freeman and Carroll, 1958; Ozawa, 1970; Carroll and Manche, 1972; Koch, 1977), the heat flow rate function thus calculated must match the measured curve. At any rate, this

6.4 Special Evaluations

requirement is not sufficient because within the framework of measuring and computational uncertainties, the measured curve can be described just as well or just as poorly by means of reaction equations of various function types. In the absence of exact knowledge on the individual steps of the reaction and the validity of the laws involved and the Arrhenius equation (6.9), the kinetic parameters obtained can only be of a formal descriptive nature without any physical–chemical significance. In view of these problems, there is considerable controversy on the procedures for the calculation of reaction kinetics data from calorimetric (or general thermoanalytical) measurements, their applicability, and the significance of the data obtained. On the other hand, the kinetic evaluation of caloric measurements is very helpful in simpler tasks of chemical kinetics, such as studies of the chemical stability of strongly endothermic compounds often used in the chemical industry. For kinetic analysis, commercial software packages exist (e.g., from Netzsch, Germany, or AKTS, Switzerland) that enable users to try different kinetic models as well as “model-free” kinetic evaluations. Such software is, however, rather complex, and a lot of experience is needed to achieve reliable results and then draw the right conclusions from the measurements. 6.4.3 Determination of Phase Transition Temperatures

Scanning calorimeters, either adiabatic or isoperibol, single or twin design, allow the determination of the heat capacity of the sample as a function of temperature. Consequently, these instruments can be used to determine the temperature of a phase transition. The same holds for calorimeters, with which the temperature of the sample is not increased continuously but stepwise. For the first-order phase transitions, such as melting, the enthalpy versus temperature function shows a step (see Section 3.2.1). If the enthalpy–temperature function is determined experimentally, this step is broadened into a sigmoid function and the phase transition temperature is defined as that temperature where the areas A1 and A2 between the measured curve and linearly extrapolated straight lines are equal (Figure 6.25).

H ∆trsH T Ttrs

Enthalpy Transition enthalpy Temperature Transition temperature constructed so that area A 1 = A2

Figure 6.25 Construction of the phase transition temperature Ttrs for a broadened (smeared) phase transition.

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Figure 6.26 Determination of the (fictive) glass transition temperature. Solid lines are from measured data of a glass transition with enthalpy relaxation; dotted lines are linearly extrapolated. The glass transition temperature is thermodynamically defined

cp T

Specific heat capacity Temperature

H Tg

Enthalpy Glass transition temperature constructed so that area A2 = A1 + A3

by the intersection point of the extrapolated enthalpy functions from the low-temperature (glassy) and high-temperature (rubbery) states, that is, where areas A1 plus A3 equal area A2.

The heat capacity, as the first derivative of the enthalpy, goes to infinity at the phase transition temperature. In reality, this pulse-like event is smeared into a more or less sharp peak (Figure 6.2). Traditionally, the phase transition temperature is assigned to the extrapolated onset temperature at zero heating rate. The determination of the phase transition via the enthalpy–temperature function is particularly worthwhile for phase transitions of higher order or glass transitions. Because in these cases the enthalpy–temperature function shows only a change in slope, the phase transition temperature is defined as the intersection of the two extrapolated straight lines (Figure 6.26). This is true even if the heat capacity function is superimposed by a so-called annealing peak as is often the case with enthalpy relaxation processes such as the glass transition of polymers. Because the enthalpy is the integral of the heat capacity, the “true” transition temperature (the “fictive glass transition temperature”) can easily be inferred from the heat capacity versus temperature function by the proper addition of areas (Richardson, 1989, Richardson, 1997). Figure 6.26 depicts the algorithm. 6.4.4 Determination of Heats of Transition

Determination of heats and enthalpies of transition is one of the major fields of application of calorimetry. Because in the strict thermodynamic sense only the

6.4 Special Evaluations

first-order phase transitions are accompanied by an enthalpy of transition (see Section 3.2.2), only these cases are treated here. The enthalpy of transition is defined as the enthalpy difference between the high- and the low-temperature phases (Figure 6.25). However, from the enthalpy–temperature function, enthalpy differences between the high-temperature phase b and the low-temperature phase a at any temperature T can be calculated, for example, also for a substance that undergoes a phase transition of higher order: Dtrs HðTÞ ¼ H b ðTÞ  H a ðTÞ

The enthalpy of transition becomes temperature dependent when Hb(T) and H (T) have different slopes (Figure 6.25). The enthalpy as a function of time is readily available from, for example, drop calorimetry experiments or from adiabatic calorimeters with incremental temperature increases. Scanning calorimeters, however, furnish the heat capacity of the sample. In these cases, the phase transition shows as a peak and the enthalpy of transition is calculated by integration of the peak area. Traditionally, this is done after constructing a proper baseline under the peak between the start and the end of the peak. The definition of the start and the end of the peak and the shape of the baseline under the peak are somehow arbitrary, particularly when the phase transition is accompanied by a heat capacity change. The enthalpy change at the transition temperature Ttrs can be calculated from the heat capacity curve by   Dtrs HðT trs Þ ¼ H b ðT 2 Þ  H a ðT 1 Þ  Hb ðT 2 Þ  H b ðT trs Þ a

 ½Ha ðT trs Þ  H a ðT 1 Þ

This is demonstrated in Figure 6.27. It also becomes clear from the figure that it is not necessary to determine the absolute heat capacity; the area differences can be calculated from a single experiment, and subtraction of the zero line is not needed for this purpose. Remark: In reality, a scanning calorimeter measures heat flow rates rather than heat capacities (which are assumed to be proportional to the heat flow rate). The

cp T Ttrs A A1 A2, A3 ∆trsH

∆trsH = A

Specific heat capacity Temperature Transition temperature Area Total area (grey) Partial areas (hatched) Transition enthalpy

∆trsH = A1 – A2 – A3

Figure 6.27 Determination of the phase transition enthalpy by integration of the heat capacity curve with an arbitrarily constructed baseline (left) and a thermodynamically constructed baseline (right).

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peak integration yields a heat of transition that equals the enthalpy of transition only in the case of constant pressure (see Section 3.1.6). 6.4.5 Determination of the Purity of a Substance

The basis of all methods for the determination of purity by calorimetry is the observation that two or more components can interact with each other and influence their melting behavior. A particularly simple description can be derived for systems in which the components are completely miscible in the liquid state and completely immiscible in the solid state, that is, eutectic systems. The coexistence between the solution and the solid impurity is described by ln a1 ¼

ðT

T fus

Dfus H 1 dT RT 2

ð6:10Þ

where a1 is the activity of component 1 in the solution, DfusH1 is the enthalpy of fusion of component 1, R is the molar gas constant, T is the temperature, and Tfus is the melting temperature of pure component 1. The resulting lowering and broadening of the melting point is independent of the nature of the impurity and depends only on its activity. This phenomenon has long been used to assess the purity of a substance by cryoscopy or calorimetry (Johnston and Giauque, 1929; Glasgow et al., 1957; Baldan et al., 2009). The advantage of this method is that it is fully thermodynamically founded (Prigogine and Defay, 1954) and thus is regarded as one of the few absolute methods for purity determination (Milton and Quinn, 2001). The disadvantage is that it is limited to impurities of such substances that form eutectic systems with the main component and that there is no information about the identity of the impurity. The melting curve of a eutectic system at low impurity concentration can be described by Van’t Hoff’s law of melting point depression, which is derived from Eq. (6.10) after the introduction of some simplifications and the assumption of an ideal mixture (activity coefficient equal to 1): T ¼ T fus 

RT 2fus x2 Dfus H

where x2 is the concentration of the impurity 2. The concentration of the impurity 2 is proportional to the inverse fraction melted 1/F and the impurity concentration in the original mixture, x 2 , that is, T ¼ T fus 

RT 2fus  1 x Dfus H 2 F

The fraction melted is given by the ratio of the partial heat of fusion at temperature T, DtrsH(T), and the total heat of fusion DtrsH. The quantities T, DfusH(T) and DfusH are directly available from calorimetry and can be determined by step-by-step melting or by scanning the temperature.

6.5 Determination of the Measurement Uncertainty

When plotting T as a function of 1/F, the slope of the resulting straight line renders x 2 and the intersection with the ordinate Tfus. A problem to be taken into account in scanning calorimetry is the thermal inertia of the apparatus, which results in smeared melting peaks. In addition, the eutectic peak often remains undetected. As a consequence, a certain amount of heat is missing when determining the partial and the total heats of fusion. These problems lead to a curved T ¼ f(1/F) function. Better linearity can be achieved after proper desmearing (see Section 6.3) and by adding an adjustable parameter K to the measured heats of fusion: F¼

x 2 Dfus HðTÞ þ K ¼ x2 Dfus H þ K

Another less known method calculates the impurity directly from the shape of the measured melting curve of the impure substance, taking account of the smearing effect by proper corrections (Bader, Schawe, and H€ ohne, 1993). The calorimetric purity determination is a rather fast and comfortable method, but the above-mentioned limiting assumptions restrict the precision and uncertainty of the results compared with, say, chromatographic methods that are available nowadays.

6.5 Determination of the Measurement Uncertainty

It is impossible to determine the true value of a quantity by measurement. Even if the measurements are performed with all conceivable error sources eliminated and repeated infinitely often, the measurements would show a dispersion around a mean value. This dispersion indicates a range of values within which the true value lies with a certain probability. It is the purpose of the uncertainty determination process to quantify this distribution in terms of widths and shape. A prerequisite for any uncertainty determination is the traceability of the result. Traceability is a property of a measurement result by which it can be related to an accepted standard through an unbroken chain of comparison, each having a known uncertainty. This standard is usually a base unit or a derived unit of the SI (Systeme international d’unites). In cases where this is not possible, an accurate realization of the unit must suffice. Traceability guarantees comparability. The comparability of measurements between different places on earth is achieved by comparisons performed by the National Metrology Institutes (NMIs). The comparability over time was given in the past because the NMIs have realized and maintained the units for long times. Currently, there is an effort to define the base units as far as possible on the basis of quantum effects and fundamental physical constants, which are, to our knowledge, stable in time. Uncertainty determinations should be performed according to accepted standards or guidelines, for example, according to the Guide to the Expression of

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Figure 6.28 Definition of systematic and random errors in a measurement series (DIN 1319-1, 1995). The dashed curve displays the scatter of a number of

measurements of the same quantity. The expected value m is the arithmetic mean of all individually measured values x.

Uncertainty in Measurement (GUM) (JCGM 100, 2008), which has been issued jointly by a number of international organizations (BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML). This approach ensures comparable uncertainty statements. Every measurement process renders values that are not centered at the true value but show some offset from it. These differences are often called errors. There are two types of error, systematic and random. A systematic error is a constant offset, whereas a random error is different between subsequent measurements, and this difference cannot be predicted. One approach of expressing the information gained from an experiment is to provide a best estimate of the measurand and information about systematic and random error values (in the form of an error analysis; see, for example, Bevington and Robinson, 2003; Taylor, 1997). Another approach, the 3) GUM approach, is to express the result of a measurement as a best estimate of the measurand together with an associated measurement uncertainty, which combines systematic and random errors on a common probabilistic basis. Figure 6.28 explains graphically the definitions for random and systematic errors and the corrections applied to the latter. Figure 6.29 shows the treatment of systematic and random errors in the course of an uncertainty analysis. The GUM defines the measurement uncertainty as a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that

3) See JCGM 100 (2008).

6.5 Determination of the Measurement Uncertainty

Figure 6.29 Systematic and random errors and their treatment in the determination of the measurement uncertainty. (According to Hernla, 1996.)

could reasonably be attributed to the measurand. The following are the claims of the GUM:    

Its methodology is universally applicable. The uncertainty quantity is useful in practice. The uncertainty quantity is internally consistent. The uncertainty quantity is transferable to further uncertainty determinations.

It is a requirement of the GUM that all known systematic errors are corrected by an estimate of the correction term. This estimate will have an uncertainty even if the estimate is zero. Depending on the method for determining uncertainty components, the GUM distinguishes between type A and type B evaluations. Type A evaluations are based on a statistical analysis of a measurement series; type B evaluations cover all other methods based on the available knowledge. Both components are combined and treated in the same way. The process of uncertainty determination consists of several steps (Eurolab, 2006; JCGM 104, 2009): 1) Defining the output quantity y of the experiment (the measurand). 2) Identifying the n input quantities xi on which the output quantity depends. 3) Developing a measurement model y ¼ f(xi) that relates the output quantity to the input quantities. 4) Quantifying the main uncertainty contributions. Each of the input quantities receives a standard uncertainty u(xi) either in the form of a standard deviation of a measurement series (type A evaluation) or a standard deviation of a reasonable distribution of values (type B evaluation). In case of a normal

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(Gaussian) distribution of the measurement values in a type A evaluation, the experimental standard deviation s can be used as an estimate for the dispersion of the values and the arithmetic mean value as an estimate for the value of the input quantity:

s¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP um u ðx i  xÞ2 ti¼1 m1

ð6:11Þ

with m P

xi i¼1  x¼ m

ð6:12Þ

The standard uncertainty u(x) of this input value is then given by s uðxÞ ¼ pffiffiffiffi m

ð6:13Þ

For type A evaluations, the degrees of freedom shall be calculated. In the simple case of m independent observations, the degrees of freedom equal m  1. However, when only a small number of observations have been made, the use of the Student’s t-distribution is more appropriate than a Gaussian distribution (see textbooks of statistics). 5) Determining the sensitivity coefficients ci for the dependence of the output quantity F on the input quantities xi as the mathematically derived differential quotients c i ¼ @F=@x i or the experimentally determined difference quotients ci ¼ DF=Dx i . 6) Determining and quantifying of correlations between the uncertainties of input quantities. Correlations occur when two input quantities xi and xj are not independent of each other or depend on a third quantity xk. These correlations are quantitatively taken into account by their covariances u(xi, xj). The covariances can also be expressed by the product of the individual uncertainties u(xi) and u(xj) and the correlation coefficient r between xi and xj, u(xi, xj) ¼ u(xi)  u(xj)  r(xi, xj) (see Taylor, 1997, pp. 209–221). 7) Propagating the distributions of the input quantities through the measurement model to the probability distribution of the output quantity to obtain the combined standard uncertainty (Gauss’s error propagation law): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n  2 n n1 X X uX @F  @F @F

ð6:14Þ   u xi ; xj  uðx i Þ2 þ 2 uðyÞ ¼ t @x @x @x i i j i¼1 j¼iþ1 i¼1 Another approach for propagating the uncertainties of the input quantities uses the Monte Carlo method. Here the input quantities are simultaneously numerically varied, and their influence on the output quantity is analyzed (JCGM 101, 2008).

6.5 Determination of the Measurement Uncertainty Example of an uncertainty budget for uncorrelated input quantities (Kacker, Sommer, and Kessel, 2007). Table 6.1

Quantity

Value xi

Standard uncertainty u(xi)

Degrees of freedom d

Sensitivity coefficient ci

Uncertainty contribution ci u(xi)

Input quantity 1 Input quantity n Output quantity

x1 xn y

u(x1) u(xn)

d(x1) d(xn) d(y)

c1 cn

c1u(x1) cnu(xn) u(y)

8) Calculating the estimate of the measurand according to the measurement model. 9) Multiplying the standard uncertainty u by a coverage factor k to obtain an expanded uncertainty U in which the true value of the measurand lies with a specified probability. 10) Reporting of the results of the analysis, the uncertainty budget, giving all necessary information to recapitulate it, for example, in the form of a table (Table 6.1). Example: Determination of the Heat of Fusion by Means of DSC Let us assume that the task is to determine the heat of fusion of an unknown sample by means of differential scanning calorimetry. The calorimeter is first calibrated by using a certified sample of indium of mass mclb ¼ 10.12 mg as the calibration material. The certificate states an enthalpy of fusion of Dfushclb ¼ (28.64  0.06) J g1. Four measurements are performed and give the following results (peak areas Aclb): 284.15, 281.39, 263.49, and 276.04 mJ. The mean value is Aclb ¼ 276:27 mJ. The calibration factor KQ of the instrument is calculated according to the following equation: KQ ¼

mclb  Dfus hclb ¼ 1:049 Aclb

Then four measurements on the sample of mass mS ¼ 7.98 mg are performed that give the following peak areas AS: 345.45, 377.18, 368.33, and 357.75 mJ. The mean value is AS ¼ 362:18 mJ. The heat of fusion qfus is calculated according to qfus ¼ K Q

AS mclb  Dfus hclb AS  ¼ ¼ 47:61 mJ mS mS Aclb

The uncertainties of the input quantities are as follows: Masses (mclb, mS): The standard uncertainties of the weighings have been determined previously according to an accepted guideline (EURAMET, 2009) and amount to u(mS) ¼ u(mclb) ¼ 0.05 mg. This is a type B uncertainty with infinite degrees of freedom.

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Enthalpy of fusion of the calibration material Dfushclb: The certificate gives the following information: Dfushclb ¼ (28.64  0.06) J g1, where the uncertainty u is given with a coverage factor k ¼ 2 for 95% probability. Thus, the standard uncertainty u(D fushclb) equals 0.03 J g1. Again, this is a type B uncertainty with infinite degrees of freedom. Calibration measurements (peak area Aclb): The standard uncertainty u(Aclb) is calculated by a type A evaluation from the standard deviation of the measurements (Eq. (6.13)): pffiffiffi uðAclb Þ ¼ sðAclb Þ= 4 ¼ 4:58 mJ

Sample measurements (peak area AS): The standard uncertainty u(AS) is likewise calculated: u(AS) ¼ 6.85 mJ.

The standard uncertainty of the result is calculated according to Eq. (6.14) using the measurement model equation for qfus, first neglecting any correlations: uðqfus Þ ¼

(

  2   @qfus 2 @qfus @qfus 2  uðmclb Þ2 þ  uðDfus hclb Þ2 þ  uðAclb Þ2 @mclb @Dfus hclb @Aclb )1=2     @qfus 2 @qfus 2 2 2  uðAS Þ þ  uðmS Þ þ @AS @mS

which results in a standard uncertainty of u(qfus) ¼ 1.26 J g1. The result of this uncertainty calculation can be expressed in the form of Table 6.2. Because of the few degrees of freedom, the coverage factor for a 95% probability is taken from the Student’s t-distribution, k(95%) ¼ 2.43. Therefore, the final result of the measurement would be expressed as qfus ¼ (47.6  3.1) J g1. However, because the weighings of the sample and the calibration material are performed with the same balance, the weighing data are correlated. It is assumed that the correlation is 100%, that is, r ¼ 1 and u(mclb, mS) ¼ u(mclb)  Table 6.2 Result of uncertainty calculation.

Quantity

Value xi

Standard uncertainty u(xi)

Degrees of freedom d

Sensitivity coefficient ci

Uncertainty contribution ci u(xi)

mclb Dfushclb Aclb mS AS qfus

10.12 mg 28.64 J g1 276.27 mJ 7.98 mg 362.18 mJ 47.61 J g1

0.05 mg 0.03 J g1 4.58 mJ 0.05 mg 6.85 mJ 1.26 J g1

1 1 3 1 3 7

4.7 mJ mg2 1.7 0.17 mg1 6.0 mJ mg2 0.13 mg1

0.24 J g1 0.05 J g1 0.79 J g1 0.30 J g1 0.90 J g1

References

u(mS)  1. This correlation reduces the uncertainty of the result; applying Eq. (6.14) yields u(qfus) ¼ 1.20 J g1. On the other hand, the correlation reduces the degrees of freedom further to 5, increasing the coverage factor k(95%) to 2.65. The final result is thus qfus ¼ (47.6  3.2) J g1. This uncertainty determination is by far not complete. The model could be refined by the following:    

Choosing different integration limits for the peak area determination. Using different baseline constructions. Treating the calibration factor as temperature and mass dependent. Taking other possible factors into consideration. Further examples of uncertainty determinations are given in Chapter 9.

References Bader, R.G., Schawe, J.E.K., and H€ohne, G.W. H. (1993) A new method of purity determination from the shape of fusion peaks of eutectic systems. Thermochim. Acta, 229, 85–96. Baldan, A., Bosma, R., Peruzzi, A., van der Veen, A.M.H., and Shimizu, Y. (2009) Adiabatic calorimetry as support to the certification of high-purity liquid reference materials. Int. J. Thermophys., 30, 325–333. Bevington, P.R. and Robinson, D.K. (2003) Data Reduction and Error Analysis for the Physical Sciences, 3rd edn, McGraw-Hill, Boston. Bracewell, R.N. (2000) The Fourier Transform and Its Applications, 3rd edn, McGraw-Hill, Boston. Carroll, B. and Manche, E.P. (1972) Kinetic analysis of chemical reactions for non-isothermal procedures. Thermochim. Acta, 3, 449–459. Davies, B. (2002) Integral Transforms and Their Applications, 3rd edn, Springer, New York. DIN 1319-1 (1995) Fundamentals of Metrology – Part 1: Basic Terminology, Beuth, Berlin. Eder, F.X. (1956) Moderne Memethoden der Physik, Vol. 2, Thermodynamik, VEB Deutscher Verlag der Wissenschaften, Berlin.

Eucken, A. (1929) Energie- und W€armeinhalt, in Handbuch der Experimentalphysik, Vol. 8, Teil I (eds W. Wien and F. Harms), Akademische Verlagsgesellschaft, Leipzig. EURAMET (2009) Guidelines on the calibration of non-automatic weighing instruments. Calibration Guide EURAMET/ cg-18/v.02. Eurolab (2006) Guide to the evaluation of measurement uncertainty for quantitative test results. Eurolab Technical Report 1/ 2006, Paris. Freeman, E.S. and Carroll, B. (1958) The application of thermoanalytical techniques to reaction kinetics. The thermogravimetric evaluation of the kinetics of the decomposition of calcium oxalate monohydrate. J. Phys. Chem., 62, 394–397. Glasgow, A.R., Jr., Ross, G.S., Horton, A.T., Enagonio, D., Dixon, H.D., Saylor, C.P., Furukawa, G.T., Reilly, M.L., and Henning, J.M. (1957) Comparison of cryoscopic determinations of purity of benzene by thermometric and calorimetric procedures. Anal. Chim. Acta, 17, 54–79. Hemminger, W.F. and Sarge, S.M. (1991) The baseline construction and its influence on the measurement of heat with differential

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6 Measurements and Evaluation scanning calorimeters. J. Therm. Anal., 37, 1455–1477. Hernla, M. (1996) Meunsicherheit und F€ahigkeit. Qualit€at und Zuverl€assigkeit, 41, 1156–1162. H€ ohne, G.W.H., Hemminger, W.F., and Flammersheim, H.-J. (2003) Differential Scanning Calorimetry, 2nd edn, Springer, Berlin. JCGM 100 (2008) Evaluation of measurement data: Guide to the expression of uncertainty in measurement. JCGM 100:2008 (GUM 1995 with minor corrections; 1st edn, 2008, corrected version 2010), Bureau international des poids et mesures, Sevres. JCGM 101 (2008) Evaluation of measurement data: Supplement 1 to the “Guide to the expression of uncertainty in measurement”: propagation of distributions using a Monte Carlo method. JCGM 101:2008, Bureau international des poids et mesures, Sevres. JCGM 104 (2009) Evaluation of measurement data: An introduction to the “Guide to the expression of uncertainty in measurement” and related documents. JCGM 104:2009, Bureau international des poids et mesures, Sevres. Johnston, H.L. and Giauque, W.F. (1929) The heat capacity of nitric oxide from 14  K to the boiling point and the heat of vaporization. Vapor pressures of solid and liquid phases. The entropy from spectroscopic data. J. Am. Chem. Soc., 51, 3194–3214. Kacker, R., Sommer, K.-D., and Kessel, R. (2007) Evolution of modern approaches to express uncertainty in measurement. Metrologia, 44, 513–529. Kissinger, H.E. (1957) Reaction kinetics in differential thermal analysis. Anal. Chem., 29, 1702–1706.

Koch, E. (1977) Non-Isothermal Reaction Analysis, Academic Press, London. Milton, M.J.T. and Quinn, T.J. (2001) Primary methods for the measurement of amount of substance. Metrologia, 38, 289–296. Ozawa, T. (1970) Kinetic analysis of derivative curves in thermal analysis. J. Therm. Anal., 2, 301–324. Pfaundler, L. (1866) Ueber die W€armecapacit€at verschiedener Bodenarten und deren Einfluss auf die Pflanze, nebst kritischen € ber Methoden der Bemerkungen u Bestimmung derselben. Pogg. Ann. Phys. Chem., 129, 102–135. Prigogine, I. and Defay, R. (1954) Solution– crystal equilibrium: eutectics, in Chemical Thermodynamics, Longmans, Green, London, pp. 357–367. Richardson, M.J. (1989) Thermal analysis, in Comprehensive Polymer Science, Vol. I, Polymer Characterization (eds C. Booth and C. Price), Pergamon, Oxford, pp. 867–901. Richardson, M.J. (1997) Quantitative aspects of differential scanning calorimetry. Thermochim. Acta, 300, 15–28. Taylor, J.R. (1997) An Introduction to Error Analysis, 2nd edn, University Science Books, Sausalito. White, W.P. (1928) The Modern Calorimeter, American Chemical Society Monograph Series No. 42, The Chemical Catalog Company, New York, pp. 37–43. Manufacturers AKTS Kinetics Software; www.akts.com (December 7, 2012). Netzsch Thermokinetics; http://www.thermsoft.com/english/kinetics.htm (December 7, 2012).

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Part Two Practice of Calorimetry In Part One, the fundamentals of calorimetry were presented in a more general manner. In what follows, we turn toward more practical aspects of calorimetry. To start with, different types of calorimeters are described together with typical application areas. In another chapter, new developments in this field are presented. Finally, guidelines for successful measurements are given followed by some typical applications of modern calorimetry. The aim is to give an overview of what is on the market and to enable the user to decide which type of calorimeter is most suitable for the problem to be solved. Of course, it is not possible to cover the whole area of calorimetric practice and to present all the numerous applications, but we will try to highlight some characteristic examples to enable readers to judge the power of this method using the available instrumentation. Because both newcomers and experienced scientists should profit from these chapters, one or the other compromise concerning the different interests was unavoidable. The literature given may help to deepen readers’ understanding and to extend their knowledge on demand.

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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7 Calorimeters After some comments on the classification of calorimeters, we shall embark in this chapter upon a discussion of the instrumentation of calorimetry in greater detail and depth. The examples presented were selected so as to make the respective operation principle as clear as possible. No attempt was made to describe the newest instrument of a given type; on the contrary, in many cases, a number of simple, “classical” calorimeters were chosen instead. However, efforts were made to cover the currently available types of commercial instruments as fully as possible. Every section devoted to a particular category of calorimeters begins with a few general comments concerning the criteria of the respective grouping. As mentioned earlier (see Chapter 5), we distinguish among the central part of the calorimeter with the sample to be measured, the surroundings of the calorimeter system, and the outer environment (laboratory). For precise measurements, the heat exchange between the sample and the surroundings must be controlled or definitely suppressed. The different measurement principles are explained in Chapter 5.

7.1 Functional Components and Accessories

The calorimeter system comprising the sample, possibly contained in a vessel or crucible, must be heated or cooled or kept at constant temperature in any experiment. The heating method usually used is the electrical one, but other heating methods are also possible. Infrared heating and inductive heating should be mentioned because these methods are used in calorimetry, too. Controlled cooling of the sample is not that easy. One method is to cool the calorimeter system via proper contact to the cold surroundings or with a Peltier cooling element and compensate the uncontrolled strong cooling with controlled additional electric heating to get the desired cooling rate or temperature. Of course, the thermostat plays an essential role as part of the calorimeter; precise calorimetry is in need of surroundings with an accurately controlled temperature. Such essential parts are therefore briefly presented to enable the user to judge the calorimeter.

Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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7.2 Heating Methods

The most common method to heat a sample in a calorimeter uses the Joule effect. A current is sent through a resistance, where the electrical energy is completely transformed into heat if the production of electromagnetic radiation fields can be excluded. If, by proper design, the produced heat stays inside the calorimeter (no heat leakage), the heat flow rate W equals the electrical power P produced in a resistance R: W ¼ P ¼ U  I ¼ R  I2 ¼

U2 R

The electrical quantities (voltage U and current I) can be determined with great precision (see Section 2.2). Sources of error are, however, the wires between the heater and the environment. On the one hand, they form an unavoidable heat leakage (which is larger, the thicker the wire), and on the other hand, every wire is a resistance itself and produces additional Joule heat (which is larger, the thinner the wire). Seldom used in calorimetry are two other heating methods: heating with infrared radiation and inductive heating. These two methods have the advantage that the radiation and electromagnetic energy can be wirelessly transferred to the sample even if the sample is in a vacuum chamber. The disadvantage is that the transferred energy is not easy to determine quantitatively.

7.3 Cooling Methods

One method of cooling the calorimeter system is to put it inside a low-temperature thermostat (cryostat; see later). This method is often used in low-temperature calorimetry and works down to temperatures well below 1 K. Another method of moderate cooling, working at ambient temperature and above, uses the Peltier effect. The principle of operation is shown graphically in Figure 7.1. Two different conductors, A and B, are connected in electric contact with one another (thermocouple; see Section 2.3.2). If the circuit is connected with a current source, one junction becomes warm while the other cools down. The effect is reversible; that is, when the current is reversed, the previously warm junction becomes cold. Moreover, the power absorbed at the cold junction equals that released at the warm one, namely P ¼ P  I

where P is the “Peltier power” and P is the Peltier coefficient of a given pair of metals at a given temperature; P equals the product of the absolute temperature of the respective junction and the thermoelectric power (see Section 2.3.2) of the pair

7.3 Cooling Methods

A, B 1 2

Different metals Amperemeter Current source

P=±Π⋅I P Heating or cooling power at the junction (Peltier power) I Current Π Peltier coefficient I0 Current at which the cooling and heating power at the junction are of equal magnitude Figure 7.1 Peltier circuit and power balance of the Peltier effect.

of conductors. The total quantity of heat absorbed or released at the junction is therefore tð fin DQ ¼ P  IðtÞdt tini

where tini and tfin are the moments at which the current is switched on and off, respectively. The Joule heat released is disregarded here. Because an electric current generates heat in every conductor (including the junction), the total power converted to heat in the “half circuit” corresponding to the respective junction is P ¼ R  I2  P  I

where R is the resistance of the “half circuit” with one junction. The Peltier effect and the Joule heat are superimposed at the junction site. The Joule heat is always positive, that is, exothermic. What matters for practical purposes is the power converted at the junctions. The Joule heat released along the circuit flows, depending on the experimental conditions (the link between the wires and the surroundings), more or less through the junctions as well. For practical purposes, these relationships are characterized for every specific arrangement by a number that represents the “effective” resistance Reff of the cold junction. This resistance is determined as follows. The magnitude of I is set so that the Joule heat (R  I2) generated at the cold junction is compensated by the Peltier cooling effect (P  I) on the site. If

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the current necessary for this purpose is I0, it follows that the cooling power of the cold junction is P cool ¼ P  I  ð1

I=I 0 Þ

Figure 7.1 shows the total power P (cooling and heating) of the cold junction of a Peltier circuit as a function of the current I. The Peltier effect has only a limited application in calorimetry. This can be attributed to the following reasons: i) Cooling can take place only if I < I0 because the Joule heating effect preponderates at I > I0. ii) There exists a maximal cooling power at I ¼ I0/2, namely P max cool ¼ P
I 0 =4

(This means in practice that only small heat flow rates can be compensated because the cooling power is limited by Pmax cool .) iii) Whereas the Peltier power is released at the junctions only, the generation of Joule heat takes place all along the conductor, that is, also between the junctions. This may cause systematic errors (thermal leakage). iv) For a maximal cooling power, I0 must be as large as possible, which means that the effective resistance of the junction has to be low; in other words, a large cross section is necessary. But under such circumstances, the heat conduction between the junctions increases, causing a thermal relaxation that has an adverse effect on the efficiency of the Peltier cooling phenomenon. Such Peltier elements are, however, used in certain calorimeters to enable cooling experiments. In rare cases, the Peltier element is also used for heating purposes.

7.4 Comments on Control Systems

In every calorimeter, the temperature has to be controlled in one or another way. This can only be achieved by means of a control system including sensors, algorithms, and actuators. Automatic controlling necessitates a suitable instrumentation, especially if both slow and quick temperature changes have to be done with the same system. The following basic considerations apply to the control of temperature deviations. Any controller (Figure 7.2) can only begin to operate in the presence of a deviation from a reference or “set value.” In the present case, the temperature (i.e., the “actual value”) of the calorimeter substance or the surroundings (thermostat) changes slightly with regard to the set value. This temperature change, which represents a deviation from the set value, is converted (by means of a suitable thermometric sensor and an amplifier) into an electric voltage, which causes the controller to adjust the heating output (controller output) so as to restore the original temperature and reestablish the equality between the preset and the actual value. The controlling system mostly used is the proportional–integral–derivative (PID) controller. From the input signal – the difference of the set value and the measured

7.4 Comments on Control Systems

Figure 7.2 Scheme of a closed-loop control system.

value – three signals are calculated and combined to give the output signal – the heating power. First, there is the P-part, which is proportional to the input signal; the larger the deviation from the set value, the larger the output signal. Second, there is the I-part, which is proportional to the time integral of the input signal; as long as an input signal is detected, the integral and the output signal change. This Ipart stays constant when the input signal becomes zero (steady-state case). Third, there is the D-part, which is proportional to the time derivative of the input signal; it becomes larger, the larger the change (slope) of the temperature difference in time. The P- and D-parts of the controller contribute to the output signal only as long as there is a difference between the set value and the actual value changing in time. These three components have to be adjusted carefully to get a fast and precise controller. The adjustment needs some experience because the three components influence one another strongly. The required deviation from the set value and the resulting fluctuation of the actual value cannot be reduced below a certain minimum because the sensitivity of the controller has limits of its own because of the characteristics of the controlled system (e.g., thermal inertia, damping, limitation of the set value) and of the amplifier (amplification factor, time constant, noise). The ubiquitous random fluctuations of the controlled quantity (temperature), or, respectively, of the analog voltage, as well as the intrinsic noise of the sensors, are of major importance for an effective maximal sensitivity of the controller. If the sensitivity is made too high, the intrinsic noise of the sensor or the subsequent amplification may trigger the controller output, which unnecessarily complicates the evaluation of the obtained values. In this case, a strong random fluctuation is superimposed on the effective controller output (heating power). Because every control circuit is a system able to oscillate, an excessively high sensitivity causes, in addition, an undamped oscillation of the controller output, often to a very large amplitude. A control circuit can operate most efficiently if the sensitivity and damping of the system are set so

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Φ t t0 ∆t ∆T

P 1 2 3

Heat flow rate of an event Time Thermal event start System delay time Temperature difference between set value and actual value Heating power Creep case Oscillating case Aperiodic limit

Figure 7.3 Control behavior of a feedback system for the compensation of a constant endothermic effect --- starting at t0 --- with differently set damping ratio of the PID controller.

as to establish the so-called aperiodic limit, where the control system has just stopped oscillating. In the event of a higher sensitivity or a lower damping, the system begins to oscillate. On the other hand, if the sensitivity is made too low and the damping too large, it takes a long time for the system (controller and calorimeter) to attain the new values after an exponential function (creep case). This may result in considerable errors of measurement if the reading is made before the system has reached its steady state. Example: Let a constant heating power be consumed in a calorimeter (e.g., by an endothermic process). In such a case, the electric power necessary for compensating this effect will be constant in the steady state, and the output provided by the controller will be equal to the heat consumption of the sample in the absence of heat leakage. Figure 7.3 shows the effect of variously adjusted control circuits in this case. Even with an optimally adjusted controller, such electronically controlled (“active”) systems suffer from the inherent drawback of a high noise level. With modern electronics, however, this disadvantage is kept to a negligible level except in some extreme cases involving the measurement of minute amounts of heat. An electronic control circuit can serve not only for controlling the temperature of a thermostat with a heater but also for cooling by means of the Peltier effect. It is also used for power compensation purposes (see Section 1.1.2).

7.6 On the Classification of Calorimeters

7.5 Thermostats

A thermostat, or cryostat in the case of low temperatures, is a device that keeps the temperature of a substance or container constant. This can be done in different ways. One possibility is to take advantage of the thermodynamic demand that a pure substance in two-phase equilibrium remains at constant temperature as long as the two phases coexist (see Section 3.2.3). This may be a solid and a liquid phase. One example is the ice–water mixture defining the temperature of 0  C, or a liquid and a gas phase such as liquid helium in equilibrium with its vapor at 4.2 K (helium cryostat) or liquid nitrogen at 77.4 K. Another method makes use of active control systems (see earlier) to keep the temperature of a substance or container constant. The advantage of active systems is that every temperature in a wide range can be selected as the set value. Controlling hardware and software makes it possible to construct thermostats that keep any temperature with very high accuracy. There are, however, two problems that restrict the accuracy of thermostats. On the one hand, there is the temperature difference compared with the environment (laboratory); the larger this temperature difference and the worse the insulation of the thermostat, the larger the fluctuation of the temperature inside the thermostat. This problem can be solved by proper insulation; in the case of very large temperature differences compared with the environment (e.g., with cryostats), the performance can be increased by constructing an insulation of several layers (shells) with stepwise changing temperature to divide the large temperature difference into smaller steps. For precision thermostats, every shell in such a multishell thermostat is separately temperature controlled. The other problem concerns the homogeneity of the temperature field inside the thermostat. A very accurate temperature controller makes no sense if the temperature is not that constant all over the volume to be thermostatized. To overcome this problem, the heater should cover the whole surface of the container, which should be made of a material with high thermal conductivity (silver or copper). Another possibility is to fill the thermostat with a liquid and to force temperature homogeneity by proper stirring. In this case, any container inside the thermostat is completely surrounded by a medium at constant temperature. Nowadays, it is possible to build thermostats with long-term temperature stability of 1 mK and below.

7.6 On the Classification of Calorimeters

Calorimeters can be classified on the basis of a variety of criteria (Rouquerol et al., 2008). The choice of a particular classification is a matter of convenience. Every classification is intended to group together and describe calorimeters possessing specific characteristics. Therefore, every classification must be consistent and provide a clear definition of the measuring principle and mode of operation of a given group of calorimeters. This requires as a rule a number of designations, or “code names.” Naming a category of calorimeters after the inventor (e.g., Bunsen

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calorimeter) has its drawbacks because it cannot be applied to all types of calorimeters and would be understood by specialists only. A more general approach to classification can be made in terms of compensating versus noncompensating measuring systems, or “active” versus “passive” ones. But such classification is not sufficiently differentiated for practical purposes. In this book, we have filed calorimeters into three groups, which is more advantageous in our opinion: i) Isothermal calorimeters. ii) Calorimeters with controlled heat exchange between the sample and surroundings. iii) Adiabatic calorimeters. A further classification can be made in every group according to other practical criteria. The instrument can also be described further in terms of technical characteristics that are self-explanatory to the specialist. Examples are the titration calorimeter, bomb calorimeter, gas calorimeter, flow calorimeter, drop calorimeter, heat flow calorimeter, and ice calorimeter. The designation “microcalorimeter” should be avoided because it does not show whether the term “micro” refers to the size of the device, the sample container, or the quantity of heat measured. These designations have gained importance in the course of the history of calorimetry. In our opinion, however, they only serve as auxiliary indications. In every classification, there are certain calorimeters that do not clearly belong to a particular category. In some cases, this is because of the possibility of using these instruments in a variety of ways. Numerous calorimeters exist, and it is not possible to present all of them; in particular, we are not able to list all calorimeters that are commercially available but only typical examples to explain the respective characteristic properties. Of course, it is possible that one and the same calorimeter can be operated in different modes; in such cases, the respective calorimeter is only presented once in that group that fits best to the main applications. Furthermore, it should be emphasized that the published performance data of commercial calorimeters are neither judged nor controlled; the respective data are in the exclusive responsibility of the manufacturers or authors of the respective publication.

7.7 On the Characterization of Calorimeters

Unfortunately, the manufacturers do not characterize the performance of their instruments in a standardized, comparable, and unambiguous way. Instead, each manufacturer uses its own criteria, and often even the definition behind a characterizing term differs. The following list of terms and definitions tries to unify the terms currently in use. Most of the definitions follow the International Vocabulary of Metrology (JCGM 200, 2008) closely.

7.7 On the Characterization of Calorimeters

Accuracy

Measurement error Random error Systematic error Limit of detection Linearity Noise

Precision

Repeatability

Reproducibility

Resolution Sensitivity Uncertainty

Standard uncertainty Combined standard uncertainty Expanded uncertainty Coverage factor

Closeness of agreement between a measured value and the true (or reference) value of a measurand. “Accuracy” is a qualitative, not a quantitative concept. Measured value minus the true (or reference) value. Component of measurement error that varies in an unpredictable manner when repeating the measurement. Component of measurement error that remains constant when repeating the measurement. Measured value with a specified probability of falsely claiming the absence and the presence of an effect. Proportionality between a value of a quantity and the corresponding indication. Random fluctuations of an output signal, expressed as peak, peak-to-peak, average, or root-mean-square (RMS) noise. Closeness of agreement between measured values obtained when repeating the measurement under specified conditions. Measurement precision under repeatability conditions. Repeatability conditions include the same measurement procedure, same operators, same measuring system, same operating conditions, same location, and replicate measurements on the same or similar objects over a short period of time. Measurement precision under reproducibility conditions. Reproducibility conditions include different locations, operators, measuring systems, and replicate measurements on the same or similar objects. The smallest change in a value of a quantity that causes a perceptible change in the corresponding indication. Quotient of the change in an indication and the corresponding change in a value of a quantity. Non-negative parameter associated with a measurement result that characterizes the dispersion of the value that could reasonably be attributed to the measurand based on the available information. Uncertainty of the result of a measurement expressed as a standard deviation. Standard uncertainty of the measurement result when that result is obtained from the values of a number of other quantities. Quantity defining an interval in which the true value of the measurand may fall with a specific level of confidence. Numerical factor for multiplying the combined standard uncertainty in order to obtain the expanded uncertainty.

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7.8 Isothermal Calorimeters

The term isothermal refers to equilibrium thermodynamics, where it means “constant temperature.” Strictly speaking, the temperature of isothermal calorimeters must be kept constant in every part and every moment. But in such a case, no heat transport would occur because heat can only flow when a temperature difference exists (see Chapters 4 and 5). In other words, at least the sample temperature must be more or less different from the calorimeter temperature, and therefore it is more correct to call this type of calorimeter quasi-isothermal rather than isothermal. Furthermore, the heat produced during a reaction in such a calorimeter must be compensated for immediately in one way or the other; otherwise, it would change the temperature of the calorimeter. There are two possibilities to compensate for the heat produced by the sample: by a phase transition or by electrical compensation. In the first case, the amount of heat produced by the sample is proportional to the amount of substance transformed; in the second case, the heat flow rate from the sample is proportional to the electric compensation power. It should be mentioned that calorimeters with the surroundings kept at constant temperature (thermostat) are often named isothermal calorimeters in the literature. This is, however, not correct because the sample and the sample container temperature during the reaction are not constant and furthermore may be very different from the thermostat temperature until heat has been exchanged. Such calorimeters operate under isoperibol conditions (see Section 5.2); we present them in Section 7.9. 7.8.1 Phase Transition Calorimeters

The heat introduced into the calorimeter converts a certain amount of the calorimeter substance from one phase to another. Such is the case with the transformation of ice into water. If the specific heat of transition and the mass of the converted substance are known, the total heat introduced into the calorimeter can be determined. The advantages of phase transition calorimeters lie in their relatively simple design and high sensitivity (especially in the liquid–gas transition), notably in slow reactions, which involve small exchanges of heat per unit time. The accuracy of measurement is high as the “lost” heat is kept to a minimum, that is, the heat that escapes to the surroundings without causing a phase transition. Such heat leakage can be minimized by having the sample (or the reaction vessel) surrounded as fully as possible by the phase to be converted (e.g., ice, liquid nitrogen, vapor), so that the heat originating from the sample serves entirely for the phase transition. Heat losses via mounts or feed lines between the surroundings and the calorimeter are suppressed by providing an insulating jacket in which the same phase transition as in the calorimeter proper takes place. The thus predetermined working temperature constitutes an underlying limitation to

7.8 Isothermal Calorimeters

the applicability of such calorimeters. Because the temperature during the transition in two-phase equilibrium remains unchanged at constant pressure for pure substances, these instruments belong to the category of “isothermally” operating calorimeters. Of course, the compensation of an exothermic or endothermic thermal effect can be achieved by phase transition of a large choice of pure calorimeter substances. Among the substances that have been used for this purpose are water (melting point, 273.15 K ¼ 0  C), diphenyl ether (melting point, 300.05 K ¼ 26.90  C), and nitrogen (boiling point, 77.35 K ¼ 195.80  C at 1013 mbar, for exothermic effects only). Calorimeters suitable for various measuring temperatures can thus be built, within certain limitations, using a variety of working substances. The liquid– gaseous phase transition can also be used for this purpose. The accuracy of a phase transition calorimeter depends on the uncertainty at which the corresponding heat of transition is known. The calorimeters should be calibrated electrically, that is, by the electric generation of heat in the sample, if possible, or at least at the location of the sample. The electric power generated and the total heat involved must simulate the measured effect as closely as possible with respect to time dependence and signal level. 7.8.1.1 First Example: Ice Calorimeter Figure 7.4 shows a phase transition calorimeter containing ice as the calorimeter substance (according to Bunsen, 1870). The glass tube that accommodates the sample – or sometimes a small calorimetric bomb – is fused inside a glass vessel completely filled with pure, air-free water and mercury. The entire device is surrounded by an (isothermal) ice–water mixture. Before the experiment, part of the water surrounding the sample tube is frozen by means of a cooling agent. An ice jacket now envelops the sample tube and is in temperature equilibrium with the surrounding water. Any release or consumption of heat in the sample tube causes

Working equation: Q = qfus ∧ ∆mice Q Heat released by the sample qfus Specific heat of fusion of ice ∆mice Transformed mass of ice To be determined Q or qfus To be measured ∆mice Known qfus or Q

1 Ice—water mixture (thermostat), 2 calorimeter vessel, 3 water, 4 mercury, 5 capillary with mercury column, 6 ice, 7 sample container Figure 7.4 Ice calorimeter (according to Bunsen, 1870).

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the melting or freezing of a corresponding mass of ice or water, respectively. Because ice at 0  C has a density of r ¼ 0.916 72 g cm 3 (Feistel and Wagner, 2006) but the density of water at 0  C is r ¼ 0.999 84 g cm 3 (Wagner and Pruss, 2002), the phase transition alters the volume of the ice–water body contained in the vessel. The smallest volume changes are indicated by the displacement in the level of the meniscus of a thread of mercury enclosed in a capillary that is connected to the mercury in the vessel. Because the heat of fusion of ice is qfus ¼ 333.4 J g 1 (Giauque and Stout, 1936), a heat quantity of 1 J changes the volume by 1/(333.4 (rwater rice)) ¼ 0.036 cm3. The advantage of the Bunsen ice calorimeter over the corresponding device of Lavoisier and Laplace (1780) (see Section 1.1.1) lies in the fact that the only exchange of heat takes place with the ice jacket that envelops the sample container (almost) completely. Thus, there is no risk of any escape of heat by convection from the warmer ice water to the surroundings. This calorimeter is particularly suitable for the measurement of small quantities of slowly released heat. However, accurate measurements of small magnitudes of heat with a margin of uncertainty of about 0.5% or better can only be expected if a number of precautions are taken. Thus, the water and the ice in the vessel must be air free and very pure. The heat released by the sample must cause the formation of only a thin layer of water around the sample tube. Thick or overheated layers of water may cause the ice jacket to melt through or break apart. Despite these precautions, the ice calorimeter in a state of rest shows a “drift,” that is, a slow change of the volume of the ice–water mixture. This drift must be determined before and after every measurement for correction of the result obtained. Its causes vary. They comprise heat leakage, a lowering of the freezing point at the ice–water phase boundary owing to dissolved impurities, and the existence of vertical temperature differences at the phase boundary because of vertical differences of the hydrostatic pressure, which are magnified by the mercury column on top of the device. These effects prevent a complete thermodynamic equilibrium at the ice–water phase boundary, as can be seen from a slow change of the volume and shape of the ice jacket. The volume change of the ice–water mixture is often determined by means of a weighing vessel connected with the mercury capillary rather than by the displacement of the meniscus of the mercury column. A volume change causes a corresponding amount of mercury to be sucked into the capillary or expelled from it. The change of weight of the involved quantity of mercury is proportional to the change of volume. The following relationships apply to ice calorimeters. The heat Q to be measured is proportional to the mass of transformed ice Dmice, the specific heat of fusion qfus being the proportionality factor: Q ¼ qfus
Dmice

The mass of molten ice during the phase transition Dmice must be equal to the increase of the mass of molten water Dmwater: Dmice ¼

Dmwater

7.8 Isothermal Calorimeters

On the other hand, Dm ¼ DV  r

where DV is the volume change and r is the density. These two equations can be summed up as follows: DV ice  rice ¼

DV water
rwater

The measured volume change DV is DV ¼ DV ice þ DV water

Combining these equations yields Q ¼ qfus
DV
rice =ðl

rice =rwater Þ

ð7:1Þ

Thus, the sought heat can be determined from the measured volume change DV. The sensitivity of a phase transition calorimeter increases with the difference between the densities of the two phases and is inversely related to the heat of transition. Applications of the Ice Calorimeter The very first applications of an ice calorimeter were reported by Lavoisier and Laplace (1780) (see Section 1.1.1). They measured the specific heat capacities of solids and liquids, as well as combustion heats and the production of heat by living animals. In the past century, Ginnings, Douglas, and Ball (1950) used a precision ice calorimeter for determining the enthalpy change of sodium between 0  C and temperatures of up to 900  C. For this purpose, sodium enclosed in a steel capsule was dropped from a furnace into the calorimeter. The same procedure was used for determining the heat content of the empty capsule. For the difference, that is, the enthalpy of sodium, the authors indicate a probable uncertainty of 0.02% between 100 and 600  C. Furukawa et al. (1956) determined similarly the change of enthalpy of aluminum oxide, which was dropped into an ice calorimeter at initial temperatures of up to 900  C. As a probable uncertainty of the measured enthalpy, they indicate 0.1% for drop temperatures of about 50  C and 0.01% for drop temperatures from 300  C upward. Lowering the requirements with regard to the precision and accuracy of measurement permits the construction of an instrument of far simpler design. Thus, Vallee (1962) described an ice calorimeter composed entirely of glass and permitting enthalpy measurements with an uncertainty of about 0.5% at a drift of approximately 4.2 J h1. Despite the cited advantages of the ice calorimeter, today there is only a historical interest in this type of calorimeter. 7.8.1.2 Second Example: Calorimeter with Liquid---Gas Phase Transition The solid and the liquid phases (e.g., ice and water) differ only slightly in density, so the ratio of their densities is close to unity. On the other hand, the

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7 Calorimeters Characteristic values for liquid---gas transitions (pressure 1 atm).

Table 7.1

Temperature K

H2O NH3 N2

373 240 77



Density

C

Liquid (g cm 3)

Gas (g cm 3)

100 33 196

0.958 0.681 0.808

0.598  10 0.897  10 4.560  10

3 3 3

Density ratio (liquid/gas)

Latent heat of evaporation (J g 1)

1602 759 177

2256 1368 199

According to D’Ans-Lax (1967).

density ratio between a liquid and a gas is higher by two to three orders of magnitude (see Table 7.1). Calorimeters involving such phase transitions are by the same factor less sensitive to random or systematic volume changes of the measuring system that envelopes the liquid and gaseous phases. This density ratio, however, is not the only factor that determines the sensitivity of calorimeters involving a liquid–gas transition; of further importance is the energy required for the evaporation of a given quantity of liquid. By relating the indicated density ratios to the corresponding heats of evaporation, one can obtain a measure of the sensitivity (see Table 7.1). In analogy to the ice calorimeter, we find from Eq. (7.1) Q ¼ qvap  rliq  rgas  DV=ðrliq

rgas Þ

where qvap is the latent heat of vaporization from the liquid to the gaseous state. Because rliq  rgas, it follows (regardless of the sign) Q  qvap  rgas  DV

If the volume of evaporating gas is not measured at the transition temperature, the gas density changes accordingly under isobaric conditions (gas laws). Thus, if during the transformation of liquid into gaseous nitrogen the volume of gas is measured at 0  C, the ratio between the densities of liquid and gaseous nitrogen yields the more advantageous value of approximately 650 instead of the 177 obtained at 77 K. The requirements for constancy of the temperature at which the volume of gas is measured are not very stringent. Fluctuations by 1 K create density changes of the order of a few thousandths. For example, nitrogen has an expansion coefficient of approximately 3.7  10 3 K 1 in the temperature range 0–100  C. The working pressure has to be exactly maintained in calorimeters involving a liquid– gaseous transition. The ratio between the volume change per unit quantity of introduced heat (a measure of sensitivity) and the pressure sensitivity of the phase transition temperature depends, according to the Clausius–Clapeyron equation (see Section 3.2.3), on the reciprocal temperature of the phase transition (Williams, 1963). On these grounds, the temperature of phase equilibrium between liquid and gaseous nitrogen under normal pressure is, by a factor of approximately 4, less

7.8 Isothermal Calorimeters

sensitive to pressure fluctuation than any other liquid–gaseous transition at room temperature. Applications Involving Liquid---Gaseous Transition Although this calorimeter type is only of historical interest nowadays, one interesting application should, however, be mentioned: Williams (1963) described a deformation calorimeter suitable for use with different liquids. The deformation calorimeter serves to measure the heat released in the sample during a deformation. Calorimeters involving a liquid–gaseous transformation as well as other calorimetric procedures can be used for this purpose. The difference between the performed deformation work and the measured heat represents the energy “stored” in the deformed material. An obvious and common drawback of all measurements with deformation calorimeters stems from the fact that the sought quantity constitutes a minute difference between two measured values that are subject to uncertainties (i.e., deformation work and released heat) and are measured independent of one another and in different ways. If the stored energy accounts for about 10% of the deformation work, the heat and the deformation work must be measured with an uncertainty of 0.5% to determine the stored energy with an uncertainty of 10%. Wolfenden and Appleton (1967) described another deformation calorimeter in which metals are plastically deformed by the application of tensile forces in liquid nitrogen (Figure 7.5). The sample, mounted in a tension test device, is completely surrounded by liquid nitrogen. To keep heat input via the massive metal parts of the tension test device

Working equation: Q = qvap ⋅ ∆mN2 Q Heat released by the tensile test sample qvap Specific heat of evaporation of nitrogen ∆mN2 Mass of evaporated nitrogen To be determined Q To be measured ∆mN2 Known qvap

1 Tensile test device, 2 liquid nitrogen (thermostat), 3 calorimeter vessel, 4 liquid nitrogen (calorimeter liquid), 5 calibration heater, 6 outflow tube for gaseous nitrogen, 7 tensile sample Figure 7.5 Deformation calorimeter involving a liquid---gas transition (N2)(according to Wolfenden and Appleton, 1967).

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dV/dt t V1, V3 V2

Volume flow rate Time Gas volume (evaporated by calibration process) Gas volume (evaporated by deformation process)

Figure 7.6 Flow rate of evaporated nitrogen in a deformation calorimeter involving a liquid---gas phase transition (see Figure 7.5)(according to Wolfenden and Appleton, 1967).

as low as possible, the calorimeter vessel is immersed in liquid nitrogen. The evaporation rate is measured from the flow rate of the evaporated gas. With the establishment of steady-state conditions (i.e., when the evaporation rate or the associated measuring value of the calorimeter attains a stable level or at least a definite, time-independent trend), a certain amount of electric energy is supplied to the calibration heater. This results in a greater evaporation rate and consequently in a higher flow rate during the heating interval (see Figure 7.6). The measurement, which comprises a plastic deformation of the sample, is performed when the flow rate reaches the “baseline” again. The deformation heat that is then released in the sample raises the flow rate of the nitrogen. A second calibration is made after the measuring effect has faded away. Another obvious application of a liquid–gaseous phase transition calorimeter is the determination of the heat of evaporation: a known amount of electric energy is supplied to a liquid that is in equilibrium with its vapor at the boiling point. This causes the evaporation of a certain mass of liquid. The ratio between the introduced energy and the evaporated mass yields the specific heat of evaporation of the liquid used. Jamin (1870) described a simple device for the measurement of such heats of evaporation (Figure 7.7). A vessel containing the test liquid and a heating resistance is located in a second vessel that contains the same liquid in a boiling state. Within a certain time interval, a steady state develops whereby there is a constant rate of evaporation from the inner vessel, which is heated to the boiling point. The condensate formed within a given time interval is weighed. Now if the electric heater is switched on, an additional mass of liquid will evaporate during the same period; this mass is likewise determined by weighing. However, calorimeters of this type suffer from a systematic error: during the measurement, the liquid level in the boiling vessel drops, and the gas volume increases by DV. Part of the obtained vapor thus remains in the calorimeter vessel. The vapor mass contained in the volume DV is Dmvap ¼ rvap  DV

where rvap is the vapor density.

7.8 Isothermal Calorimeters

Working equation: tfin

qvap ⋅ ∆ m = ∫ U(t) ⋅ I(t)dt tini

qvap

Specific heat of evaporation ∆m Mass of liquid evaporated by electric energy U Voltage t Time tini, tfin Times of start and end of the electric heating To be determined qvap To be measured ∆m Known U(t), I(t)

1 Vessel with boiling liquid (thermostat), 2 calorimeter vessel, 3 liquid (sample) 4 condensate, 5 electric heater Figure 7.7 Calorimeter for the determination of heats of evaporation (according to Jamin, 1870).

This mass of vapor was obtained by the application of an energy equal to qvap  Dmvap (qvap is the specific heat of evaporation). Part of the introduced electric energy was spent for this purpose. Accordingly, tð fin   UðtÞ  IðtÞdt ¼ qvap  Dmcond þ qvap  Dmvap ¼ qvap Dmcond þ rvap  DV tini

where U(t) is the voltage, I(t) is the electric current, t is the time, Dmcond is the mass of condensate, and tini and tfin are times of the start and end of the electric heating, respectively. In the case of water, the second term of the above formula can be easily calculated: because 1 g of water occupies a volume of 1 cm3, the density of the steam directly provides the magnitude of the correction term. Steam has a density of about 0.6  10 3 g cm 3 at 100  C (see Table 7.1). Thus, the resulting error lies in the range of several parts per thousand. 7.8.2 Isothermal Calorimeters with Electrical Compensation

If, say, a salt dissolves endothermically in a solvent, the temperature of the calorimeter vessel can be kept constant by a “controlled” supply of electric energy (heating). The total amount of energy thus introduced equals the heat of solution of

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the salt. An exothermic effect can, in principle, be compensated for by electric cooling using the Peltier effect (see earlier and Figure 7.1). Both methods of electrical compensation are used in practice to maintain isothermal or quasiisothermal conditions. A major advantage of the electrical compensation method is that a calibration of the calorimeter becomes unnecessary because the sought heat can be determined directly as electric energy with ease and accuracy. Of course, the calorimeter vessel must be well isolated against the surroundings (thermostat) to avoid sources of error caused by unnoticed heat exchange. One peculiarity of electrical compensation should be mentioned, too: every electronic temperature controller needs a minimal but nonzero temperature difference to react. Furthermore, the heat produced by the sample and that produced by the electrical heater are not at the same place, and there must be some temperature gradient to allow a heat exchange between them. Therefore, such calorimeters operate not in a strictly isothermal but in a quasi-isothermal mode. The temperature is generally kept constant but may differ locally. The deviations depend on the heat flow rate as well as on the quality of the compensation controller. Normally, these differences can be neglected in practice. 7.8.2.1 First Example: Calorimeter according to Tian Tian (1923) introduced the compensation of exothermic processes by means of the Peltier effect (see Section 7.3) to calorimetry. Tian’s calorimeter is shown schematically in Figure 7.8. The calorimeter vessel (sample container), in which the reaction being studied takes place, is connected by means of two separate thermopiles with a thermostat kept constant with a precision of 0.01 K. Heat could flow through the thermopiles between the vessel and the thermostatic surroundings. Because the thermopiles have a relatively high thermal resistance (thin wires), the relaxation of temperature would take rather a long time (large time constant). The surroundings serve as a reference for the measurement of temperature of the vessel, and the temperature

1 2 3

Figure 7.8 Calorimeter (according to Tian, 1923).

Thermostat Thermopiles Calorimeter vessel

7.8 Isothermal Calorimeters

difference between the vessel and the surroundings serves merely as a set point deviation for electric compensation. One thermopile serves to measure the temperature difference, and the other thermopile is used for a total or – experimentally more easily achieved – partial compensation of exothermic processes by means of the Peltier effect. Naturally, endothermic effects can be compensated with the same thermopile by reversing the current. Today this type of calorimeter is only of historical interest. A modern version of such a calorimeter type is presented in the following example.

7.8.2.2 Second Example: Isothermal Titration Calorimeter Becker and Kiefer (1969) described a titration calorimeter for the measurement of heats of mixing of two liquids in a continuous manner in the mole fraction region between x ¼ 0 and x ¼ 0.5. In this device, one of the components is pumped continuously into a calorimeter vessel that is filled completely with the second component. A stirrer ensures thorough mixing, and the overflow is pumped off. The heat generated or consumed in the reaction is compensated for as completely as possible by means of a suitably operated Peltier pile (see Section 7.3). A tracing of the Peltier power is made, and the obtained integral represents the heat of mixing involved. The entire device is placed inside a water thermostat with a temperature constancy of 0.1 mK. According to the authors, heat flow rates of 0.5 W are compensated for so quickly that the temperature change in the calorimeter vessel does not exceed 0.05 K. A side effect generally encountered in calorimetry involving liquids – namely, that the evaporation of part of the liquid may cause major errors in the determination of the heat of mixing owing to the large heat of evaporation – is eliminated here by mixing without any gaseous phase in the overflow vessel. At any rate, the changes of quantities and the resulting changes of concentrations in the overflow vessel must be accounted for. Becker and Kiefer (1969), for example, have managed to determine the heats of mixing of the acetone–chloroform system throughout the concentration range with an uncertainty of less than 1% using only two measurements. High-precision versions of isothermal titration calorimeters (ITCs) are commercially available:

MicroCal “iTC200”

TA Instruments “Nano DSC”

Temperature range: 2–80  C; sample volume: 200 ml. “VP-ITC”: Temperature range: 2–80  C; minimum response time: 20 s; sample volume: 1.4 ml; short-term noise: 2 nW; peak repeatability: