Chemical kinetics аnd electrochemistry: еducational manual 9786010430464

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AL-FARABI KAZAKH NATIONAL UNIVERSITY

A. K. Ospanova G. A. Seilkhanova

CHEMICAL KINETICS AND ELECTROCHEMISTRY Educational manual

Almaty «Qazaq University» 2017

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UDC 666.1/2 (075.8) LBC 35.41 я 73 O-86 Recommended for publication by the decision of the Academic Council of the Faculty of Chemistry and Chemical Technology, Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol №3 dated 07.12.2017)

Reviewers: Doctor of chemical sciences, professor D.Y. Korulkin Candidate of chemical sciences, ass. professor A.K. Galeeva

O-86

Ospanova A.K. Chemical kinetics аnd electrochemistry: еducational manual / A.K. Ospanova, G.A. Seilkhanova. – Almaty: Qazaq university, 2017. – 136 p. ISBN 978-601-04-3046-4 In the еducational manual presents the theoretical and practical aspects of chemical kinetics and electrochemistry. Much attention is paid to the important section on the problems of catalysis. Modern views on the nature of homogeneous and heterogeneous catalysis are considered. And the features of the influence of the catalyst on the rate of chemical reactions are given. The problems of the theory of solutions of strong and weak electrolytes, the thermodynamics of electrochemical processes are considered. The еducational manual is intended for students studying in chemical and chemical-technical specialties, and can also be used by undergraduates, doctorants, teachers of higher educational institutions of the Republic of Kazakhstan. Published in authorial release.

UDC 666.1/2 (075.8) LBC 35.41 я 73 © Ospanova A.K., Seilkhanova G.A., 2017 © Al-Farabi KazNU, 2017

ISBN 978-601-04-3046-4

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BASIC PRINCIPLES OF CHEMICAL KINETICS THE SUBJECT AND TASKS OF CHEMICAL KINETICS, THE REACTION RATE, THE BASIC POSTULATE OF CHEMICAL KINETICS, THE ORDER AND MOLECULARITY OF CHEMICAL REACTIONS Chemical kinetics is one of the important sections of physical chemistry. Chemical kinetics is a science that studies the patterns of chemical reactions in time and their mechanism. Chemical kinetics considers the dependence of the rate of chemical reaction on the concentration of reagents, temperature, environment properties, and other factors. One of the important tasks of chemical kinetics is the clarification of chemical reaction mechanism, the relationship between the rate of the processes and the structure of the molecules of the reacting substances. Usually, the laws of chemical thermodynamics are used for studying chemical processes. For example, for the reaction: ν1Α1+ν2Α2+…+νiΑi=ν1′Α1′+ν2′Α2′+…+νk′Αk′. a) Spontaneous condition ∆G 0, x moles of substance A reacted, remained (a − x) . Then the equation (3) can be written as 1 a . K = ln t a−x

(4)

The last equation is often used for practical calculations. The dimension of the rate constant of the first order chemical reaction can be expressed from equations (3.4) and it is equal to dim{K1} = [time-1]. Time can be expressed in h, min, s, etc. In the International System of Units (SI) time is represented in seconds (s). Knowing the initial concentration а and the current one (а − х ) , we can calculate the rate constant using equation (4). The current concentration of the reagent can be determined by potentiometric, conductometric, spectrophotometric methods, methods of volumetric titration, etc. For the first order reactions, it is possible to use proportional to the concentration, values: pressure, optical density, electrical conductivity, etc. The rate constant of the first order chemical reaction can also be found by graphical method. For this, we represent the equation (4) as: ln(a − x) = ln a − kt (equation of a line) . Graphical dependence ln(a − x) from t is represented in Figure 1, from which follows K = tgα . The third method of determining the rate constant is through the half-life or half-transformation period, t 1 . 2

t 1 – the time after which half of the initial substance will react, 2

i.е. at t1 / 2 c = с0 . 2 Then 1 c 1 0,693 K= ln 0 = ln 2 = , whence t 1 = 0,693 . t1 c t1 t1 K 2 2 2 2 13

(5)

Figure 1. Dependence of ln(a-x) on time

As can be seen from the last equation, the half-life of the first order chemical reaction does not depend on the concentration. Thus, knowing t1 / 2 , it is possible to find the rate constant, and conversely. The equation (3) or (4) can be written in exponential form: с = c0 e − Kt .

Where, x = a (1 − e − kt ) or (a − x ) = ae − kt , i.е. concentration х – concentration of a reaction product. 2. The unilateral reactions of the second order Consider the reaction of the second order. There are two possible types of reactions: a) 2А→В b) А+В→С. Consider the cases а) and b), when с А = cB . The basic postulate of chemical kinetics for them will be written: (6) ω = Kc AcB = Kc A2 ; − dc = Kc A2 . dt Integrate the equation (6): c

t

dc ∫c c 2 = − K ∫0 dt . 0 14

The solution of this equation gives: express K : K=

1 1 − = Kt , from which we c c0

1 (c0 − c ) . t c0 ⋅ c

(7)

We rewrite the equation (7) in convenient for calculation form: at t = 0, c0 = a, at t > 0 x moles of А has reacted, and remained

(а − х ) . Then

c0 − c = a − (a − x) = x and the equation (7) will be

written: K=

1 x . t a(a − x )

(8)

The dimension of the rate constant of the second order can be determined by the last equation: dim{K 2 } = [conc]-1[time] -1. Graphical analysis of equation 1 = 1 + Kt is used to determine с c0 the rate constant of the second order, the dependence 1/с =f (t) is linear, which is shown in Figure 2, where K = tgα . The half-life for the second-order reactions can be considered only for the case с А = cB .

t1 = 2

2 ( n −1) − 1 . (n − 1) K n ⋅ c0n −1

(9)

1/C

α t Figure 2. Dependence of 1/С on time

15

This is a general formula that links t 1 and K with chemical 2

reactions of any order with equal concentrations of the initial components. Then, for n = 2 , based on the equation (7), t1 = 2

1 . 1 or K= Kc0 t 1 ⋅ c0

(10)

2

In this case, the half-life depends on the initial concentration of the substance. Now consider the case when n = 2 , but c A ≠ c B .

ω = Kc AcB = −

dc . dt

(11)

Suppose that at t = 0 c A = a, cB = b . At t > 0 x – the concentration of the reaction product, then (а − х ) and b − x current (remaining) concentrations of components A and B, respectively. Taking these designations into account, the equation (11) will be dс dx written: − = K (c A − x ) ⋅ (cB − x ) or = K (a − x ) ⋅ (b − x ) . We dt dt x t dx will integrate this equation: ∫ = − K ∫ dt . The solution ( ) ( ) a − x ⋅ b − x a,в 0 of this equation gives: K=

1 b(а − х ) . ln t (a − b ) а(b − х )

(12)

This is the kinetic equation for chemical reactions of the second order at c A ≠ cB . 3. The unilateral reactions of the third order Consider the third-order reaction, when c A = cB = cC : А+В+С→D+L. 16

In this case −

c t dc dc = K ⋅ c 3 and ∫ 3 = − K ∫ dt . The solution of dt c c 0 0

this equation gives:

1 1 1 (c02 − c 2 ) . 2 Kt − = = K , where c 2 c02 2t c02 ⋅ c 2

(13)

This is the kinetic equation for the third order reactions with equal concentrations of the initial components. For the considered reaction K connected with half-life ( t 1 ) by 2

the following relation: t1 = 2

3 , 2 Kc 02

(14)

and for the third-order reaction at c A ≠ cB ≠ cC the half-life does not make sense. Dimension of the rate constant dim{K 3} = [conc]-2[time] –1, general formula for determining the dimension of the rate constant for reactions of different orders: dim{K n } = [conc]-n+1[time] –1 . 4. The unilateral reactions of zero order In practice, zero-order reactions are often used, i.е. n=0. The basic postulate for them is written ω = Kc 0 = K . From this, it follows that the rate of such reactions does not depend on the concentration. This is possible when the excess concentration of initial substance is constant, or when specific types of reactions occur, such as photochemical reactions, when the rate depends on the intensity of the luminous flux. Using the previous approach, we can write: dc − =K, dt c

t

c0

0

then, dividing the variables, we integrate ∫ dc = − K ∫ dt ; and obtain

17

c −c x . − Kt = c − c0 or K = 0 = t t

(15)

In order to determine a rate constant of zero order graphically, we can write the equation (15): Kt = c0 − c or c = c0 − Kt , which represents linear dependence с= f (t) (Figure 3), K = tgα

.

α t Figure 3. Dependence of concentration on time

Dimension dim{K 0 } = [conc]1[time] –1 and the rate constant linked to half-life by the following relation: K=

c0 . 2t 1

(16)

2

It is impossible to compare the rate constants of chemical reactions of different orders, but we can compare the kinetic curves at the same с0 and K , Figure 4. C

n=3 n=2 n=1 t Figure 4. Kinetic curves for reactions of different orders

18

The greater the curvature of the dependence, the higher the order of the corresponding reaction. 5. The unilateral reactions of the n-th order Many chemical processes proceed through a complex mechanism, and orders of these reactions can be expressed by integer or fractional numbers. Let А1+А2→В+С proceeds by a n-th order. Then the rate can be dc expressed: ω = Kc n = − . dt After conversion, it is convenient to write: dcn = − Kdt . c Integrating this equation in the range from t=0 to t, from c0 to с, obtain: 1  1 1  1  1 1 . −  n −1 − n −1  = Kt or K =   (n − 1)  c c0  t (n − 1)  c n −1 c0n −1 

(17)

From the last equation, the current concentration of the reagent can be calculated at any time if the rate constant (K) and the order of the reaction (n) are known: c=

[1 + c

n −1 0

c0

(n − 1)Kt ]1 / n −1

.

(18)

The derived general equations (17, 18) can be used for reactions of the n-th order for any values of n, except for the first-order reactions.

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METHODS OF DETERMINING THE ORDER AND RATE CONSTANTS OF SIMPLE CHEMICAL REACTIONS IN CLOSED SYSTEMS Methods for determining the reaction order are divided into two large groups: 1) integral 2) differential. 1. Consider the integral methods. Integral methods are based on the use of kinetic equations in integral form, which are divided into three groups. a) substitution method In this method, the basic kinetic equations for the chemical reactions of different orders are used: n=1 k = 1 ln c0 t

c

x 1 c − c at с =с n=2 K = 1 А В = 0 t a(a − x ) t c0 ⋅ c

2 2 n=3 K = 1 (c0 − c ) and for n=3, сА=сВ= сС etc. 2 2

2t c0 ⋅ c

The essence of the method based on the fact, that in these equations we substitute the concentration values at each moment of time, and determine the rate constant K . If the obtained values of K are constant within the experimental error, i.e. do not change over time, consequently, the investigated chemical reaction proceeds according to the regularities of the reaction order in which concentration values were substituted in the kinetic equation. b) graphical method In this case, we use the basic kinetic equations for all values of n, in the following form: n=1, ln c = ln c0 − Kt ; n=2, 1 = 1 + Kt ; с c0 1 1 n=3, = + 2 Kt . c 2 c02 20

The essence of the method is that the reaction order is equal to the order corresponding to the graph in case of obtaining a linear dependence, constructing the dependence of different concentration function on time (in accordance with the above equations). For example, if we consider the first-order chemical reaction, the constant rate (K= tg α) can be determined by tangent of an angle (α) (Fig.1). 1/C

α1

≈ 1/C

2

α2

≈ lnC

α3 t

t

Figure 1. Dependence of various functions of the concentration of initial substances on time for reactions of different orders

c) method for determining the reaction order to the halftransformation time t 1 2 As is well known, t 1 is related to the rate constant K for 2 chemical reactions of different orders in the following way: n=1, t 1 = ln 2 ( t 1 is not depend on c0), 2

n=2, t 1 = 2

K1

2

1 ( t is depend on 1 ), 1 2 c0 K 2 c0 21

n=3, t 1 = 2

1 3 ( t 1 is depend on 2 ). 2 c0 2 K 3 c02

Several experiments are carried out at different initial concentrations of the reagent to determine the reaction order by this method and determine t 1 , and it is easy to install n by the character of the 2

dependence of t 1 on c0 . In this case, the definition is carried out by 2

one of the initial substances, the rest should be in excess, i.е. thus, find a particular order of the investigated substance. Then, knowing the private orders, you can determine the overall reaction order (see above). This method was proposed by A.V. Rakovski. A continuation of this method is the method of Ostwald-Noosa. The following equation was used to determine the reaction order by the method of Ostwald-Noosa t 1 = f (c0 ) : 2

t1 = 2

2 ( n −1) − 1 . (n − 1) K n ⋅ c0n −1

Taking the logarithm ( n −1) −1 2 − (n − 1) lg c0 . lg t 1 = lg 2 K (n − 1)

of

this

equation,

Figure 2. Dependence of lg t 1 = f( lg c0 ) 2

22

obtain

If we plot the graph of the dependence of lg t 1 on lg c0 , we 2

obtain a linear dependence, from which it is possible to find the tangent of an angle α (Figure 2), tgα = n − 1 or n = 1 + tgα This method is often used in determining the reaction order, when they proceed through a complex mechanism. The reaction order in this method can also be determined by calculation. Dividing t ′1 by t 1// , these half-lives, corresponding to two 2

2

experiments at different initial concentrations of reagent c0/ and c0//

t 1/ we obtain a convenient formula for calculating n:

t

2 // 1 2

 c // =  0/  c0

  

n −1

;

taking logarithm and transforming the resulting equation, obtain the equation of Noosa: t 1// lg / 2 t1 2 . (1) n = 1+ / c0 lg // c0 2. Differential methods In these methods, the kinetic equations are used in differential form. Differential methods are divided into two groups: a) Ostwald excess method For the reaction n1A+n2B→n3C make two experiments. Let the basic postulate for this reaction will be written:

ω = Kc An ⋅ cBn . 1

2

(2)

In the first experiment let cB>>cA and then the equation (2) will take the form: ω = K / c An1 , where K / = K ⋅ c Bn2 . Define n1 (a particular order for substance A) by any above-described method. In the second experiment let cA>>cB and then ω = K // ⋅ cBn 2 , K // = Kc An2 . Similarly define n2 (a particular order for substance В). 23

Then the general order of the reaction is n = n1 + n2 . b) Van't Hoff method The method is based on graphical analysis of the equation ω = Kc n (general form of the equation for the rate of chemical reaction). Take the logarithm of this equation: lg ω = lg K + n lg c (equation of the line), and the reaction order ( tgα = n ) can be found by tangents of an angle of the dependence lg ω = f (lg c ) . lgW A

tgα = n

α

lgK lgC Figure 3. Determination of the order of reaction by the method of Van't Hoff

There are two graphical types of this method. Option 1. The initial reaction rates (reaction rates determined at the initial part of the kinetic curves) are determined at different initial reagent concentrations (experiments are carried out experimentally at different initial concentrations of reagent). Tangents are made to each point c1, c2, c3 to determine the reaction rate (Figure 4), and find the angular coefficients of these lines. Then dc1 = ω1 dt dc tgα 2 = − 2 = ω 2 dt tgα1 = −

tgα 3 = −

dc3 = ω3 . dt

24

C C3

C2 C1 t Figure 4. Kinetic curves for different initial concentrations of reagents

Next, the logarithms of the obtained rates are plotted on the graph of the dependence lg ω = f (lg c) and determine the order of the reaction (Figure 3). The obtained order is denoted by nc and called concentration or true. The resulting intermediate products do not affect the process rate with this definition of the reaction order. Option 2. The data of one experiment is used to determine the reaction order, i.е. one kinetic curve is obtained, and the reaction rate is determined at different times from the start of the reaction. For this, tangents are made to different points on the curve and the reaction rate at different times is determined by tangent of the angle:

tgα1 = −

dc dc , tgα 2 = − 1 etc. (Figure 5). dt 1 dt2

C

α1 α2 α3 t Figure 5. Kinetic curve for one initial concentration

25

Then the graph of the dependence lg ω = f (lg c) is plotted, and the time order of the reaction tgα = nτ is determined by tangent of the angle of this dependence (Figure 3). Formation of intermediate compounds, consequently the rate of chemical reaction can influence on the value of the time order of a reaction. nc and nτ – can be different, if intermediate substances distort the course of the reaction. If nτ > nc , i.е. the reaction rate decreases faster with time, then the intermediate substances inhibit the process. If nτ < nc , in this case, the reaction rate decreases less than expected. Consequently, the investigated process undergoes activation, i.e. autocatalysis. Half-life method is the least accurate in determining the reaction order, as it gives the time order of reaction, an average between nc and nτ . Integral methods make it possible to determine the general order of the reaction, but it is not true; the determination uses the results of only one experiment, in addition, this method is a "trial and error" method. Inaccuracy in determining the order of reactions by differential methods is due to possible errors in making tangents to certain sections on the kinetic curves. Thereby, a complex approach is necessary to establish the order of chemical reactions, i.е. combination of the results of several methods using computer processing of the obtained data.

26

INFLUENCE OF TEMPERATURE ON THE RATE OF CHEMICAL REACTION One of the important factors affecting the rate of chemical reaction is temperature. For the reaction n1A+n2B→n3C+n4D the basic postulate of chemical kinetics will be ω = K ⋅ c An1 ⋅ cBn 2 . According to the presented formula, if the temperature affects the rate, it must affect either the concentration, or the order of a reaction, or the rate constant. As the temperature increases to 10 °С, the number of collisions increases only to 1.2-1.4%, consequently, the temperature, practically, does not affect the concentration of substances, and the order of a reaction does not change. Hence, the temperature should affect the rate constant of a chemical reaction ( K ). Van't-Hoff established an empirical rule in 1879: the temperature increase for every 10 °С leads to the increase in the reaction rate in 2-4 times (for simple reactions). At the same time, the effect of temperature on the rate constant has been emphasized, and was quantitatively represented in the following way: Κ Τ 2 = KT1 ⋅ γ

T2 − T1 10

= KT1 γ

∆T 10

,

(1)

where γ – the temperature coefficient, which shows how many times the reaction rate increases with increasing temperature by 100С.

γ =

∆T 10

K2 K1

(2)

or

γ =

K T +10 ≈ 2 ÷ 4. KT

(3)

γ – approximately allows to evaluate the effect of temperature on the rate of the process. Later, Van't Hoff quantitatively characterized the relationship between the rate constant and temperature in 1887. He assumed that 27

the change in the rate constant of chemical reaction on temperature is the same as the change in the equilibrium constant of this reaction on temperature. For this purpose, Van't Hoff analyzes the isochore equation: d ln K c ∆U = dT RT 2

(4)

is applied to the effect of temperature on the rate of chemical reaction, where ∆U – the thermal effect of the reaction at a constant volume (V=const). Consider the reversible reaction А⇔Β, then K c = K1 , where K2

K c – the rate constant, K1 , K 2 – the rate constants of direct and reverse reaction. Taking into account this ratio, we will rewrite the equation (4): d ln K1 d ln K 2 ∆U E E − = = 12 − 22 2 dT dT RT RT RT E d K ln d ln K E 2 1 or = 2 2 + const . = 1 2 + const and RT dT dT RT

In general, it can be expressed for any reaction in the following way: d ln K E (5) = + const . dT RT 2 Arrhenius developed the idea of Van't Hoff about the influence of temperature on the rate of chemical reaction and proved that practically const=0 and the equation (5) can be written: d ln K E = a2 . dT RT

– Arrhenius differential equation

28

(6)

In addition to the differential equation, Arrhenius proposed an equation in integral form: E (7) ln K = − a + ln A0 , RT where Еa and А0 – were called by Arrhenius activation energy and a pre-exponential factor, which have a certain physical meaning. The basic Arrhenius equation is equation (7) written in the following form: K = A0 ⋅ e

− Ea

RT

.

(8)

In the derivation of equation (6), Arrhenius has not made any assumptions, therefore it is the most general equation, but in deriving equation (7), he assumed that the activation energy is independent of temperature (in a narrow temperature range), and can be used for calculations. In the derivation of all these equations, Arrhenius made several assumptions. The first assumption: not all molecules can react, but only those that are in active modification, therefore, the act of formation of the reaction product must be preceded by the act of activation. The formation of active modification is due to a collision in which molecules absorb energy in the thermal form, because these processes are endothermic, and according to Arrhenius this can be written as follows 2А→В (usually): А+А+ЕА →А*+А*→В activation is due to thermal energy. The fraction of active molecules under normal conditions is low and their energy distribution obeys the Boltzmann equation, but their number increases significantly with increasing temperature. The Boltzmann multiplier (the number of active modifications) is determined by the formula:

n = n0e

− Ea

RT

,

where n – active particles, n0 – total number of particles. 29

Active molecules are those, which have excess energy of kinetic, translational motion, increased kinetic energy of the rotational, vibrational motion of atoms and groups of atoms in molecules. Therefore, the value of Ea – Arrhenius calls the activation energy. The second assumption: the formation of active modifications is a reversible process and the law of mass action is applied to it, i.e. the concentration of active modifications corresponds to thermodynamic equilibrium, and it can be expressed through the equilibrium constant Кс: А⇔А* and K c =

[A ] . [A] − [A ] *

*

The third assumption: the concentration of active modifications is small and it does not affect the concentration of initial molecules, * i.е. K c = A or A* = K c ⋅ [ A] . The Van't Hoff isochore equation [A] for Кс can be written: d ln K c ∆U a , = dT RT 2

[ ]

[ ]

where ∆Uа – the heat of formation of active molecules. The forth assumption: active modification is transformed into the reaction product at the rate that does not depend on temperature, while the temperature affects only the rate of formation of active modifications. Using these assumptions, Arrhenius derives his equation in a differential form: For the reaction А⇔В, the rate expressed through the product of the reaction will be written

dB dt

= const[ A * ] = const ⋅ K c [ A]

(according to the 3 assumption). 30

(9)

On the other hand

dB = K [ A] dt or

(10)

K [A] = const ⋅ K c [A] ,

and

K = const ⋅ K c .

(11)

Taking the logarithm of the equation (11):

ln K = ln const + ln K c . Differentiate with respect to T and obtain: d ln K dT

=

d ln K c , dT

(12)

but, because ∆U d ln K c E d ln K E = a2 . = = a 2 , then 2 dT RT RT dT RT

(13)

Activation energy ( Ea ) – the heat of formation of 1 mole of active modifications, has the energy dimension (J). Consider the following reaction to determine the physical meaning of the activation energy: А+В⇔С+D. Imagine the energy course of this reaction in the following way, Figure 1: E 2 − E1 = U 1 − U 2 = ∆U , where Е1 and Е2 the activation energies of direct and reverse reactions, respectively. From the presented figure it is seen that the activation energy is a potential barrier that particles must overcome in order to interact, this is its physical meaning. The activation energy is some excess amount of energy (in comparison with the average) that a molecule must have to react, or the minimum energy necessary for the molecules to collide, and then to interact. 31

E

E1

E2

1 U

U2

Reaction energy profile Figure 1. The energy change of a system during a chemical reaction

The activation energy of the direct and reverse reactions is related to the thermal effect of the reaction ∆Η by the following ratio: E 2 − E1 = ∆H (Figures 2а и 2b), ∆Н>0 – endothermic reaction, ∆Н