Isotope Effects In Chemistry and Biology 0824724496, 9780824724498

Citation preview

Isotope Effects in Chemistry and Biology

Isotope Effects in Chemistry and Biology Edited by

Amnon Kohen Hans-Heinrich Limbach

Boca Raton London New York

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2449-6 (Hardcover) International Standard Book Number-13: 978-0-8247-2449-8 (Hardcover) Library of Congress Card Number 2005041897 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Isotope effects in chemistry and biology / edited by Amnon Kohen, Hans-Heinrich Limbach. p. cm. Includes bibliographical references and index. ISBN 0-8247-2449-6 1. Chemical reaction, Conditions and laws of. 2. Isotopes. 3. Chemical kinetics. 4. Isobaric spin. 5. Chemical equilibrium. I. Kohen, Amnon. II. Limbach, Hans-Heinrich. III. Title. QD501.I8127 2005 541'.394--dc22

2005041897

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc.

and the CRC Press Web site at http://www.crcpress.com

Preface The unifying theme of this book is the application of isotopic methods to make significant advances in chemistry and biology. Isotopes are atoms with identical nucleic electrical charges and identical electronic properties. Isotopes contain the same number of protons but a different number of neutrons, hence they exhibit different masses and different nuclear spins. Isotope effects can be classified into three categories, i.e., kinetic isotope effects (KIEs), equilibrium isotope effects (EIEs), and anharmonic isotope effects (AIEs). KIEs are the ratio of reaction rates involving reactants that only differ by their isotopic composition. These are one of the only measures that directly probe the nature of the reaction’s transition state, and thus are very useful tools in studies of reactions’ mechanisms. EIEs are the ratio of two isotopes that are distributed between stable populations in thermodynamic equilibrium. These are a unique measure of the difference in the chemical potential of these two populations. AIEs lead to geometric changes of molecules and molecular systems via the anharmonicity of zero-point vibrations. Isotope effects are of substantial importance and utility in many fields of science and technology. The use of isotope effects is prevalent in a wide variety of disciplines. Scheme 1 below summarizes many of the areas that utilize isotope effects. The book’s nine parts and 42 chapters provide a comprehensive review of developments in isotope effects studies to date. The chapters were written by internationally recognized leading researchers in their fields. Authors from 13 countries contributed to the book: Canada, Denmark, France, Germany, Israel, Japan, Poland, Russia, Spain, Sweden, Switzerland, UK, and USA (by alphabetical order).

Geoscience

Marine science

Molecular biology

Organic chemistry

Elucidation of reaction mechanisms

Isotope effects structural dynamic kinetic equilibrium

Physical chemistry Solid state physics

Solid state physics

Cosmology Environmental and ecological science Agricultural science

Engineering applications

Elucidation of molecular structures, interactions dynamics

Inorganic chemistry

Chemical physics

Anthropology

Elucidation of naturally occurring processes

Structural biology Biochemistry Biophysics

Atmospheric science

Medicinal sciences Pharmacological sciences Polymer sciences Material sciences

Solid state physics

Stable isotope separation

Scheme 1. The many applications and implications of isotope effects and their relationship to many fields of science and technology. This Scheme was drawn by Takanobu Ishida and modified by Hans Limbach.

iv

Preface

Subjects range from the physical and theoretical origin of isotope effects to modern uses of these effects in chemical, biological, geological, and other applications. The following Table of Contents clearly emphasizes the multidisciplinary nature of this book. The book starts with the problem of isotope effects on molecular geometries arising from anharmonic vibrations and the consequences for isotope-dependent non-covalent interactions. Chemical bond breaking and formation dynamics are then addressed using the examples of simple molecules in the gas phase, also including the motif of hydrogen transfer. Novel mass independent isotope effects are discussed. The problem of hydrogen transfer, tunneling, and exchange is picked up for condensed matter, ranging from polyatomic molecules to enzymes. When the barrier for hydrogen or proton transfer becomes small, the area of low-barrier hydrogen bonds is reached and explored experimentally and theoretically. A unique application is provided in a chapter devoted to water isotope effects under pressure. Isotope effect studies in organic and organometallic reactions are needed for the understanding of the sessions that follow on isotope effects in more complex enzyme reactions. The book brings together a wide scope of different points of view and practical applications based on our current knowledge at the beginning of the new millennium. Some chapters summarize the perspective of a well-established subject while others review recent findings and ongoing research. It may appear that some of these later items are not consistent with each other. This reflects contemporary conclusions and controversies in the field. We chose to present such studies only in cases where clear scientific arguments and discussion are presented by all relevant authors. This approach demonstrates the way research progresses, and we hope it will enhance the reader’s curiosity and interest.

Editors Amnon Kohen was born in a kibbutz in northern Israel. He received his B.Sc. degree in chemistry in 1989 from the Hebrew University in Jerusalem and his D.Sc. in 1994 from the Technion-Israel Institute of Technology. After that he was a postdoctoral scholar with Judith Klinman at the University of California at Berkeley. In 1999, he moved to the University of Iowa and is currently (2005) an associate professor in the Department of Chemistry. His main interest is bioorganic chemistry, and he enjoys studying the mechanisms by which enzymes activate C– H bonds and N2 triple bonds. His research focuses on the relationship between enzyme structure, dynamics, and catalytic activity. Isotope effects were one of the main tools used by his group in recent years. Hans-Heinrich Limbach was born in Bruehl near Cologne, Germany. He studied chemistry at the Universities of Bonn and Freiburg. He did his doctoral research (Dr. rer. nat.) under the direction of Herbert W. Zimmermann at the University of Freiburg. After his Habilitation he was a visiting scientist with C.S. Yannoni at the IBM Research Laboratory, San Jose and with C.B. Moore at U.C. Berkeley. He is currently a professor of physical chemistry at the Freie Universita¨t Berlin. His research interests include the chemistry of hydrogen and its isotopes in liquids, organic solids, and mesoporous systems up to enzymes, which he is studying by liquid and solid state nuclear magnetic resonance.

Contributors Vernon E. Anderson Department of Biochemistry Case Western Reserve University Cleveland, Ohio

Gleb S. Denisov V.A. Fock Institute of Physics St. Petersburg State University St. Petersburg, Russian Federation

Katsutoshi Aoki Synchrotron Radiation Research Center Kansai Research Establishment Japan Atomic Energy Research Institute Kansai, Japan

Ileana Elder Department of Pharmacology University of Florida Gainesville, Florida

Jaswir Basran Department of Biochemistry University of Leicester Leicester, United Kingdom

Antonio Ferna´ndez-Ramos Department of Physical Chemistry Faculty of Chemistry University of Santiago de Compostela Santiago de Compostela, Spain

Jacob Bigeleisen Department of Chemistry State University of New York Stony Brook, New York

Paul F. Fitzpatrick Department of Biochemistry & Biophysics Texas A&M University College Station, Texas

Adam G. Cassano Center for RNA Molecular Biology Case Western Reserve University Cleveland, Ohio

Perry A. Frey Department of Biochemistry University of Wisconsin-Madison Madison, Wisconsin

W. Wallace Cleland Institute for Enzyme Research and Department of Biochemistry University of Wisconsin Madison, Wisconsin

Yasuhiko Fujii Tokyo Institute of Technology Research Institute for Nuclear Reactors O-okayama, Meguro-ku, Tokyo, Japan

Paul F. Cook Department of Chemistry and Biochemistry University of Oklahoma Norman, Oklahoma Janet E. Del Bene Department of Chemistry Youngstown State University Youngstown, Ohio

Nikolai S. Golubev V.A. Fock Institute of Physics St. Petersburg State University St. Petersburg, Russian Federation Sharon Hammes-Schiffer Department of Chemistry Davey Laboratory Pennsylvania State University University Park, Pennsylvania

viii

Contributors

Poul Erik Hansen Department of Life Sciences and Chemistry Roskilde University Roskilde, Denmark

Amnon Kohen Department of Chemistry University of Iowa Iowa City, Iowa

Michael E. Harris Center for RNA Molecular Biology Case Western Reserve University Cleveland, Ohio

Alexander M. Kuznetsov Department of Chemistry Technical University of Denmark Lyngby, Denmark

Alvan C. Hengge Department of Chemistry and Biochemistry Utah State University Logan, Utah Michael Hippler Department of Chemistry University of Sheffield Sheffield, United Kingdom James T. Hynes Department of Chemistry and Biochemistry University of Colorado Boulder, Colorado Takanobu Ishida Department of Chemistry State University of New York Stony Brook, New York William E. Karsten Department of Chemistry and Biochemistry University of Oklahoma Norman, Oklahoma Philip M. Kiefer Department of Chemistry and Biochemistry University of Colorado Boulder, Colorado

Jonathan S. Lau Department of Chemistry University of California San Diego, California Rene´ Le´tolle l’Universite Pierre et Marie Curie Paris, France Brett E. Lewis The Albert Einstein College of Medicine Bronx, New York Hans-Heinrich Limbach Institut fu¨r Chemie Freie Universita¨t Berlin, Germany John D. Lipscomb Department of Biochemistry, Molecular Biology and Biophysics University of Minnesota Minneapolis, Minnesota Laura Masgrau Department of Biochemistry University of Leicester Leicester, United Kingdom

Judith P. Klinman Department of Chemistry and Department of Molecular and Cell Biology University of California Berkeley, California

Olle Matsson Department of Chemistry Uppsala University Uppsala, Sweden

Heinz F. Koch Department of Chemistry Ithaca College Ithaca, New York

Zofia Mielke Faculty of Chemistry University of Wrocław Wrocław, Poland

Contributors

Dexter B. Northrop Division of Pharmaceutical Sciences School of Pharmacy University of Wisconsin-Madison Madison, Wisconsin Mats H. M. Olsson Department of Chemistry University of Southern California Los Angeles, California Piotr Paneth Institute of Applied Radiation Chemistry Technical University of Lodz Lodz, Poland Charles L. Perrin Department of Chemistry University of California San Diego, California Ehud Pines Department of Chemistry Ben-Gurion University of the Negev Be’er Sheva, Israel Bryce V. Plapp Department of Biochemistry The University of Iowa Iowa City, Iowa Martin Quack Physical Chemistry ETH Zu¨rich Zu¨rich, Switzerland Daniel M. Quinn The University of Iowa Department of Chemistry Iowa City, Iowa Franc¸ois Robert Laboratoire de Mine´ralogie Centre National di Recherche Scientifique Paris, France Emil Roduner Institut fu¨r Physikalische Chemie Universita¨t Stuttgart Stuttgart, Germany

ix

Etienne Roth National des Arts et Me´tiers Paris, France Justine P. Roth Department of Chemistry Johns Hopkins University Baltimore, Maryland Richard L. Schowen Simons Laboratories Higuchi Biosciences Center University of Kansas Lawrence, Kansas Vern L. Schramm The Albert Einstein College of Medicine Bronx, New York Steven D. Schwartz Departments of Biophysics and Biochemistry The Albert Einstein College of Medicine Bronx, New York Nigel S. Scrutton Department of Biochemistry University of Leicester Leicester, United Kingdom Willem Siebrand Steacie Institute for Molecular Sciences National Research Council of Canada Ottawa, Canada David N. Silverman Department of Pharmacology University of Florida Gainesville, Florida Zorka Smedarchina Steacie Institute for Molecular Sciences National Research Council of Canada Ottawa, Canada Lucjan Sobczyk Faculty of Chemistry University of Wrocław Wrocław, Poland C. M. Stevens Naperville, Illinois

x

Contributors

Michael J. Sutcliffe Department of Biochemistry University of Leicester Leicester, United Kingdom

Jordi Villa`-Freixa Grup de Recerca en Informatica Biomedica, IMIM/UPF Barcelona, Spain

Donald G. Truhlar Department of Chemistry and Supercomputing Institute Minneapolis, Minnesota

Arieh Warshel Department of Chemistry University of Southern California Los Angeles, California

Jens Ulstrup Department of Chemistry Technical University of Denmark Lyngby, Denmark

Ralph E. Weston Jr. Chemistry Department Brookhaven National Laboratory Upton, New York

W. Alexander Van Hook Chemistry Department University of Tennessee Knoxville, Tennesse

Max Wolfsberg Chemistry Department University of California Irvine, California

Table of Contents Chapter 1 Theoretical Basis of Isotope Effects from an Autobiographical Perspective ................................................................................................................ 1 Jacob Bigeleisen

Chapter 2 Enrichment of Isotopes .......................................................................................... 41 Takanobu Ishida and Yasuhiko Fujii

Chapter 3 Comments on Selected Topics in Isotope Theoretical Chemistry ........................ 89 Max Wolfsberg

Chapter 4 Condensed Matter Isotope Effects ....................................................................... 119 W. Alexander Van Hook

Chapter 5 Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes ............................................................................. 153 Janet E. Del Bene

Chapter 6 Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure ............................................................................. 175 Katsutoshi Aoki

Chapter 7 Hydrogen Bond Isotope Effects Studied by NMR .............................................. 193 Hans-Heinrich Limbach, Gleb S. Denisov, and Nikolai S. Golubev

Chapter 8 Isotope Effects and Symmetry of Hydrogen Bonds in Solution: Single- and Double-Well Potential ...................................................................... 231 Jonathan S. Lau and Charles L. Perrin

Chapter 9 NMR Studies of Isotope Effects of Compounds with Intramolecular Hydrogen Bonds ................................................................................................... 253 Poul Erik Hansen

Chapter 10 Vibrational Isotope Effects in Hydrogen Bonds ................................................ 281 Zofia Mielke and Lucjan Sobczyk

xii

Table of Contents

Chapter 11 Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics .......... 305 Michael Hippler and Martin Quack

Chapter 12 Nonmass-Dependent Isotope Effects ................................................................. 361 Ralph E. Weston, Jr.

Chapter 13 Isotope Effects in the Atmosphere ..................................................................... 387 Etienne Roth, Rene´ Le´tolle, C. M. Stevens, and Franc¸ois Robert

Chapter 14 Isotope Effects for Exotic Nuclei ....................................................................... 417 Olle Matsson

Chapter 15 Muonium — An Ultra-Light Isotope of Hydrogen ........................................... 433 Emil Roduner

Chapter 16 The Kinetic Isotope Effect in the Photo-Dissociation Reaction of Excited-State Acids in Aqueous Solutions ................................................... 451 Ehud Pines

Chapter 17 The Role of an Internal-Return Mechanism on Measured Isotope Effects .................................................................................................... 465 Heinz F. Koch

Chapter 18 Vibrationally Enhanced Tunneling and Kinetic Isotope Effects in Enzymatic Reactions ......................................................................... 475 Steven D. Schwartz

Chapter 19 Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions ............................................................................................................ 499 Sharon Hammes-Schiffer

Chapter 20 Kinetic Isotope Effects in Multiple Proton Transfer ......................................... 521 Zorka Smedarchina, Willem Siebrand, and Antonio Ferna´ndez-Ramos

Chapter 21 Interpretation of Primary Kinetic Isotope Effects for Adiabatic and Nonadiabatic Proton-Transfer Reactions in a Polar Environment ............. 549 Philip M. Kiefer and James T. Hynes

Chapter 22 Variational Transition-State Theory and Multidimensional Tunneling for Simple and Complex Reactions in the Gas Phase, Solids, Liquids, and Enzymes ...................................................................................................... 579 Donald G. Truhlar

Table of Contents

xiii

Chapter 23 Computer Simulations of Isotope Effects in Enzyme Catalysis ....................... 621 Arieh Warshel, Mats H. M. Olsson, and Jordi Villa`-Freixa

Chapter 24 Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen ......................................................................................... 645 Justine P. Roth and Judith P. Klinman

Chapter 25 Solution and Computational Studies of Kinetic Isotope Effects in Flavoprotein and Quinoprotein Catalyzed Substrate Oxidations as Probes of Enzymic Hydrogen Tunneling and Mechanism ................................ 671 Jaswir Basran, Laura Masgrau, Michael J. Sutcliffe, and Nigel S. Scrutton

Chapter 26 Proton Transfer and Proton Conductivity in Condensed Matter Environment ....................................................................................................... 691 Alexander M. Kuznetsov and Jens Ulstrup

Chapter 27 Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes ............................ 725 Willem Siebrand and Zorka Smedarchina

Chapter 28 Kinetic Isotope Effects as Probes for Hydrogen Tunneling in Enzyme Catalysis ............................................................................................... 743 Amnon Kohen

Chapter 29 Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis ........................................................................................ 765 Richard L. Schowen

Chapter 30 Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions ........................................................................ 793 William E. Karsten and Paul F. Cook

Chapter 31 Catalysis by Alcohol Dehydrogenases ............................................................... 811 Bryce V. Plapp

Chapter 32 Effects of High Hydrostatic Pressure on Isotope Effects .................................. 837 Dexter B. Northrop

Chapter 33 Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase: Proton Transfer through Intervening Water Molecules ................. 847 David N. Silverman and Ileana Elder

xiv

Table of Contents

Chapter 34 Isotope Effects from Partitioning of Intermediates in Enzyme-Catalyzed Hydroxylation Reactions ................................................ 861 Paul F. Fitzpatrick

Chapter 35 Chlorine Kinetic Isotope Effects on Biological Systems .................................. 875 Piotr Paneth

Chapter 36 Nucleophile Isotope Effects ............................................................................... 893 Vernon E. Anderson, Adam G. Cassano, and Michael E. Harris

Chapter 37 Enzyme Mechanisms from Isotope Effects ....................................................... 915 W. Wallace Cleland

Chapter 38 Catalysis and Regulation in the Soluble Methane Monooxygenase System: Applications of Isotopes and Isotope Effects ...................................... 931 John D. Lipscomb

Chapter 39 Secondary Isotope Effects .................................................................................. 955 Alvan C. Hengge

Chapter 40 Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds ........ 975 Perry A. Frey

Chapter 41 Theory and Practice of Solvent Isotope Effects ................................................ 995 Daniel M. Quinn

Chapter 42 Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase ............................................................................ 1019 Brett E. Lewis and Vern L. Schramm

Index .................................................................................................................................. 1055

1

Theoretical Basis of Isotope Effects from an Autobiographical Perspective Jacob Bigeleisen

CONTENTS I. II. III.

From Soddy – Fajans through Urey –Greiff ......................................................................... 1 Equilibrium Systems — General ......................................................................................... 3 Equilibrium in Ideal Gases .................................................................................................. 4 A. Classical and Quantum Mechanical Systems .............................................................. 4 B. The Reduced Partition Function Ratio of an Ideal Gas .............................................. 5 1. Numerical Calculation of the Reduced Partition Function Ratio ........................ 7 C. Corrections to the Bigeleisen –Mayer Equation.......................................................... 8 IV. Isotope Chemistry and Molecular Structure...................................................................... 12 A. The First Order Rules of Isotope Chemistry ............................................................. 12 B. Statistical Mechanical Perturbation Theory .............................................................. 13 C. Polynomial Expansions of the Reduced Partition Function Ratio............................ 14 V. Kinetic Isotope Effects....................................................................................................... 18 VI. Condensed Matter Isotope Effects ..................................................................................... 25 Acknowledgments .......................................................................................................................... 32 References....................................................................................................................................... 33

I. FROM SODDY– FAJANS THROUGH UREY– GREIFF Isotopes were discovered in radiochemical investigations of the decay of the heavy elements. Products were found with different nuclear properties, which could not be separated chemically, but stood in the same place in the Periodic Table; e.g. Radium, 226Ra, an a emitter with a half life of 1600 years, Mesothorium 1, 228Ra, a b emitter with a half life of 5.7 years and Actinium X, 223Ra, an a emitter with a half life 11.7 days. They were named isotopes by Soddy1 from the Greek words isos topos, the same place. Isotopes had the same nuclear charge, but different atomic masses. This was firmly established by the determination of the atomic weights of the lead isotopes which were the end products of the three radioactive series. Lead from the thorium series was found to have an atomic weight of 207.77; lead from the uranium –radium series had an atomic weight of 206.08. Fajans,2 a major figure in radiochemistry, concluded that isotopes had similar, but not identical, chemical properties. Since isotopes have different atomic masses, molecules substituted with sister isotopes (isotopomers) would have different vibrational frequencies. Consequently they would have different heat capacities, entropies, and free energies. After WWI, Lindemann3,4 subsequently known as Lord Cherwell, derived the equations for the differences in vapor pressures

1

2

Isotope Effects in Chemistry and Biology

of isotopes. For a monatomic solid with a Debye frequency distribution, Lindemann found 0

0

‘nðP =PÞ ¼ 9=8ðQ 2 QÞD =T 0

0

2

‘nðP =PÞ ¼ 3=40ðQ 2 2 Q ÞD =T

2

½QD ¼ hnD =k . 2p

½QD ¼ hnD =k , 2p

ð1:1Þ

ð1:2Þ

where Q 0 and Q are the Debye temperatures for the light and heavy isotopomers, respectively. Equation 1.1 and Equation 1.2 were derived for the case that there was a zero point energy associated with a vibration. He calculated the difference in vapor pressures of 206Pb and 208Pb and predicted the ratio of the vapor pressures of 206Pb/208Pb to be 1.0002 at 600 K. A much larger effect of the opposite sign was predicted for the case of no zero point energy. Since no such difference was found, Lindemann correctly concluded that there was a zero point energy. In actual fact, Lindemann’s calculation for the zero point energy case is a factor of ten too large; the correct calculation from Equation 1.2 leads to a result of 0.002%; the light isotope, 206Pb, has the higher vapor pressure. Equation 1.2, which had been derived independently by Otto Stern, provided the incentive for Keesom and Van Dijk5 to achieve a partial separation of the neon isotopes by low temperature fractional distillation. In planning a search for an isotope of mass 2, Urey6 decided to carry out an enrichment of hydrogen of natural abundance by a Raleigh distillation. He used Equation 1.1 to estimate the difference in vapor pressures of H2 and HD. Urey then turned his attention to the question of isotope effects in chemical reactions. He had his student, David Rittenberg, calculate, using quantum statistical mechanics, the equilibrium constant for the exchange reaction H2 þ 2DI ¼ D2 þ 2HI

ð1:3Þ

as a function of temperature.7 Their calculations were confirmed by experiment.8 The method of Urey and Rittenberg was extended to the case of polyatomic molecules by Urey and Greiff.9 For the isotopic exchange reaction AX þ A0 Y ¼ A0 X þ AY

ð1:4Þ

they expressed the equilibrium constant in terms of partition function ratios. K ¼ ðAY=A0 YÞ=ðAX=A0 XÞ ¼ ðQ=Q 0 ÞAY =ðQ=Q 0 ÞAX X expð21i =kTÞ Q¼ i

ð1:5Þ ð1:6Þ

For the partition function ratio of molecules of like symmetry, ðQ=Q 0 Þ; and for which the translation and rotation obeyed classical statistical mechanics, they obtained ðQ1 =Q2 Þ ¼ ðM1 =M2 Þ3=2 ðABCÞ1 =ðABCÞ2

1=2

Y i

ðeðu2i 2u1i Þ=2 Þð1 2 e2u2i Þ=ð1 2 e2u1i Þ

ð1:7Þ

In Equation 1.7 subscripts 1 and 2 refer to the heavy and light isotopes, respectively; M is the molecular weight, A; B; and C are the principal moments of inertia and ui ¼ hui ¼ hni =kT: The terms eðu2i 2u1i Þ=2 and ð1 2 e2u2i Þ=ð1 2 e2u1i Þ are, respectively, the contributions from the zero point energy differences between the light molecule, u2 ; and the heavy molecule, u1 ; and the Boltzmann excitation factors. Urey and Greiff 9 tabulated values of ðQ1 =Q2 Þ for compounds of the light elements as a function of temperature between 273 and 600 K. The values of ðQ1 =Q2 Þ varied from an order of magnitude for the isotopes of hydrogen to a few percent for isotopes of the elements in the first two rows of the periodic table. The ratios decreased with temperature. The equilibrium constant for an isotopic exchange reaction, which is the quotient of two partition function ratios, is of the order of a few percent excepting those reactions which involve the isotopes of hydrogen. Urey was able to utilize small differences in the chemical properties of the light

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

3

elements to separate the isotopes of nitrogen10 through the exchange reaction 15

14 15 þ NH3 ðgÞ þ 14 NHþ 4 ðsolÞ ¼ NH3 ðgÞ þ NH4 ðsolÞ

ð1:8Þ

He utilized similar acid base exchange reactions to concentrate the isotopes of carbon, oxygen, and sulfur. Lewis and MacDonald11 used the redox reaction, 7

Li ðamalgamÞ þ 6 Li ðEtOHÞþ ¼ 6 Li ðamalgamÞ þ 7 Li ðEtOHÞþ

ð1:9Þ

in a tour de force, to effect a partial separation of the lithium isotopes.

II. EQUILIBRIUM SYSTEMS — GENERAL With the discovery of nuclear fission in 1938, Urey joined with a small group of American scientists to develop the nuclear bomb. He was the logical choice to head up the effort to separate the isotopes required for this purpose. He organized the SAM Laboratory at Columbia University. For this purpose he recruited a staff of scientists, engineers, technicians, and support personnel. The staff grew to approximately 1000 by 1944. Research and development at SAM Laboratory led to the construction of two different types of heavy water plants, a boron isotope separation plant and the large 235U separation plant, K-25, constructed in Oak Ridge Tennessee. A number of other research projects were carried out, including the countercurrent gas centrifuge in collaboration with J. W. Beams, and the feasibility of separation of the uranium isotopes by a photochemical process. Earlier, mercury and chlorine isotopes had been separated by photochemical processes. I was recruited in June 1943 to work on the feasibility of a photochemical process for uranium isotope separation. I brought to this assignment experience in spectroscopy and photochemistry of systems at low temperature. It was in this capacity that my collaboration with Maria Goeppert-Mayer began. The most promising uranium compounds which might exhibit a difference in the absorption spectra were the uranyl salts, which had sharp line spectra. Maria Mayer worked on the theory of the spectra. Along with S. S. Hanna, M. L. Schultz, and D. T. Vier, I determined the spectra of most of the compounds reported in the compilation by Dieke and Duncan.12 In November 1943 I was asked by Urey’s assistant, Isidor Kirschenbaum, to calculate the differences in chemical properties of the uranium isotopes for all compounds of uranium through Equation 1.7. The only compound for which there was sufficient data for this calculation was the uranyl ion, UO2þ 2 . The structure of UF6 was unknown at the time. I started my assignment with the objective to calculate each of the terms Q in Equation 1.7 directly as a difference between two small quantities. I simplified the terms i ðeðu2i 2u1i Þ=2 Þð1 2 e2u2i Þ=ð1 2 e2u1i Þ to give 1 þ ðeui 2 1Þ21 dui 2

ð1:10Þ

where dui ¼ u2i 2 u1i : When I discussed my work with Maria Mayer, she volunteered to join me in this approach. I was pleased to accept her help. Her first suggestion was to remove the classical contributions to Equation 1.7 by adding the term ð21=ui Þ to Equation 1.10. The final result, the reduced partition function ratio to be defined in Section III, ðs=s0 Þf ¼ 1 þ

X i

Gðui Þdðui Þ ¼ 1 þ

X 1 þ ðeui 2 1Þ21 2 1=ui dðui Þ 2 i

ð1:11Þ

was directly applicable for the calculation of the isotope fractionation factor, a; to be discussed below. Based on Equation 1.11, Maria Mayer and I were able to predict that chemical isotope separation factors as large as 1.001 were possible for the isotopes of uranium. We showed that

4

Isotope Effects in Chemistry and Biology

simple acid –base reactions, e.g. reaction (Equation 1.8) and the isotopic exchanges between 22 SiF4 and SiF22 6 and SnCl4 and SnCl6 led to small isotope separation factors. For the purpose of these calculations I prepared a table of GðuÞ values as a function of u: A summary of this development was prepared by Maria Mayer for review by Edward Teller13 and is further discussed by Clyde Hutchinson.14

III. EQUILIBRIUM IN IDEAL GASES In the fall of 1946 a complete exposition of the work by Maria Mayer and myself was prepared for publication in the open literature at the suggestion of Willard F. Libby. What follows is a summary of that landmark paper.15

A. CLASSICAL AND Q UANTUM M ECHANICAL S YSTEMS In classical statistical mechanics, the partition function, Q; is ð h f Q ¼ ð1=sÞ e2Hðp;q=kTÞ dp…dq

ð1:12Þ

where h is Planck’s constant, s is the symmetry number, H is the Hamiltonian, k is Boltmann’s constant and T is the temperature. The integral runs over all the coordinates, q; and the momenta, p: We choose Cartesian coordinates. The integration of the momenta leads to QCl ¼ ð1=sÞ

Y i

ð2pmi kT=h2 Þ3=2

ð

e2Vðq=kTÞ dq1 …dq3n

ð1:13Þ

where V is the potential energy of the system of 3n particles and mi are the atomic masses. Within the Born – Oppenheimer approximation the potential energy of a system is isotope invariant. Within this approximation the classical partition function ratio of two isotopomers, Q=Q 0 ; is Y ð1:14Þ ðmi =m0i Þ3=2 ðQ=Q 0 ÞCl ¼ ðs 0 =sÞ i

This result holds for all phases. Since the m’s in Equation 1.14 are atomic masses, they will cancel identically in any chemical reaction or in any phase change. The equilibrium constant for any equilibrium process calculated by classical statistical mechanics is just the ratios of the symmetry numbers. But this is not the isotope separation factor. We will illustrate this by consideration of the hypothetical exchange reaction between dioxygen, 16O2, with an ideal gas of oxygen atoms, 18O. 16

O2 þ 18 O ¼ 16 O18 O þ 16 O

ð1:15Þ

We now define ðs=s 0 Þf ; the reduced partition function ratio of a pair of isotopomers, RFPR, ðs=s 0 Þf ¼ ðQ=Q 0 Þqm =ðQ=Q 0 ÞCl

ð1:16Þ

ðQ=Q 0 ÞCl ¼ ðs 0 =sÞðm=m 0 Þ3=2

ð1:17Þ

For both the O2 and O species

KCl ¼ sð16 O2 Þ=sð16 O18 OÞ=sð16 OÞ=sð18 OÞ ¼ 2 This equilibrium constant corresponds to random distribution of 16O and 18O atoms between the dioxygen molecules and the oxygen atoms. If the atom fraction of 18O in both the oxygen atoms

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

5

and molecules is x; then the relative numbers of the dioxygen isotopomers are: 16

O2 ¼ ð1 2 xÞ2 ;

16

O18 O ¼ 2ð1 2 xÞx;

18

O2 ¼ x2

The ratio of 18O to 16O atoms in both the atomic and molecular oxygen is x=ð1 2 xÞ ¼ g: The classical equilibrium constant assuming random distribution is 2, just the inverse ratio of symmetry numbers. Now consider the quantum mechanical equilibrium constant. For atomic oxygen ðQ=Q 0 Þqm is equal to ðQ=Q 0 ÞCl . The atom fraction of 18O in the atom is x and the atom ratio is x=ð1 xÞ. Let y be the atom fraction of 18O in the O2 isotopomers. The relative numbers of the dioxygen molecules for the quantum mechanical case is O2 ¼ ð1 2 yÞ2 ,

16

O18 O ¼ 2ð1 2 yÞy,

18

O 2 ¼ y2

The atom ratio 18O/16O in the dioxygen molecules, b; is y=ð1 2 yÞ: The quantum mechanical equilibrium constant is now Kqm ¼ ½2y=ð1 2 yÞ =½x=ð1 2 xÞ ¼ 2b=g ¼ a KCl

ð1:18Þ

where a ¼ ½2y=ð1 yÞ =½x=ð1 xÞ is just the isotope fractionation factor, b=g; between the two chemical species, dioxygen and oxygen atoms. The equilibrium constant is a product of a symmetry number ratio and a fractionation factor. The fractionation factor is independent of the symmetry number. RFPR, ðs=s 0 Þf ; is the isotope fractionation factor between a pair of isotopomers and the free gaseous atoms of the isotopes. Since ðs=s 0 Þf for the gaseous atomic species is identically unity, the free gaseous atom is the logical standard state for isotope fractionation studies. Later we will show that ðs=s 0 Þf is always a positive quantity when the ratio ðQ=Q 0 Þ is defined as the heavy isotope divided by the light isotope, always with the superscript 0 or subscript 2. The derivation of the result in Equation 1.18, Kqm ¼ aKCl ; can be generalized for isotopomers of any symmetry. In general, 0

0

0

0

‘n Kqm ¼ ‘nðs=s Þfproducts 2 ‘nðs=s Þfreactants þ ‘nðs =sÞproducts 2 ‘nðs =sÞreactants

B. THE R EDUCED PARTITION F UNCTION R ATIO OF AN I DEAL G AS To derive a relationship for ðs=s 0 Þf of an ideal gas of a pair of isotopomers, we note that in quantum mechanics the partition function is a product of translation, rotation, and vibration. The nuclear spin can be neglected since it follows classical statistics, except for molecules at very low temperature. Under the Born – Oppenheimer approximation, the electronic energy states are isotope invariant. The translation and rotation of the molecules can be treated classically, except for molecules with very light atoms. In that case a correction needs to be introduced for nonclassical rotation. If the zero of the energy scale is chosen as the minimum in the vibrational potential energy, Qvib;qm ¼

Y i

ðexpð2ui =2ÞÞ=ð1 2 expð2ui ÞÞ

Qvib;Cl ¼

Y i

ð1=ui Þ

ð1:19Þ ð1:20Þ

where ui ¼ hni =kT: The product runs over all 3n 2 6 vibrational frequencies. A degeneracy of order gi is counted gi times. From Equation 1.19 and Equation 1.20 one gets for ðs=s0 Þf and its logarithm Y ðu1i =u2i Þeðu2i 2u1i Þ=2 ð1 2 e2u2i Þ=ð1 2 e2u1i Þ ð1:21Þ ðs=s 0 Þf ¼ i

6

Isotope Effects in Chemistry and Biology

0

‘nðs=s Þf ¼

X

0

‘nðui =ui Þ þ dui =2 þ i

X

i

0

‘n 1 2 expð2ui Þ = 1 2 expð2ui Þ

0

0

ð1:23Þ

Gðui Þdðui Þ

ð1:24Þ

‘n cðui Þ ¼ ‘nðui =ui Þ þ dui =2 þ ‘n 1 2 expð2ui Þ = 1 2 expð2ui Þ 0

‘nðs=s Þf ¼

Gðui Þ ¼

X

i

‘n cðui Þ ¼

X

i

ð1:22Þ

1 þ ðeui 2 1Þ21 2 1=ui 2

ð1:25Þ

Equation 1.21 is frequently referred to as the Bigeleisen –Mayer equation. An equation similar to 1.24 was derived independently by Waldmann, who utilized it to give a qualitative analysis of the fractionation of the nitrogen isotopes in the NH3 – NHþ 4 exchanges and the carbon isotopes in the CN2 – HCN exchanges.16 The nomenclature for dui is dui ¼ u 0i 2 ui : Since prime refers to the light isotope, di is always positive. A plot of Gðui Þ as a function of ui is given in Figure 1.1. It is positive for all values of u and runs from Gðui Þ ¼ 0 at u ¼ 0 to Gðui Þ ¼ 12 at u ¼ 1: Thus ‘nðs=s 0 Þf is a positive number. Equation 1.24 is derived from Equation 1.22 for small dui : This approximation has been found to hold remarkably well for isotopomers of hydrogen, which generally have large values of dui at and above room temperature.17 Stretching vibrations have higher frequencies and larger frequency shifts on isotopic substitution than bending frequencies. Thus it follows from Equation 1.24 that stretching motions will be the major contributors to ‘nðs=s 0 Þf : They need not be the most important factor in determining the fractionation factor between two species. Bigeleisen and Mayer then expanded eui in Equation 1.25 in powers of ui up through u3i to give ðs=s 0 Þf ¼ 1 þ ð1=24Þ

X

du2i

ð1:26Þ

i

du2i ¼ u21i 2 u2i : They carried out the summation over the ð3n 2 6Þ vibrational frequencies by the use of Cartesian coordinates to construct the determinant of the H ¼ FG matrix for the molecular 0.5

u /12

0.4

G (u)

0.3

0.2

0.1

0.0

0

10

u

20

FIGURE 1.1 Plot of GðuÞ; Equation 1.25, vs. u: The limiting slope is u=12:

30

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

7

vibrations. X

jH 2 lIj ¼ 0;

½du2 X i ½dli

3n26 3n26

X ¼ ð"=kTÞ2 dli ; X fij dgij ; ¼ ij

X X 0 ½dli 3n26 ¼ ðmi 2 mi Þfii ; i X 0 X ðmi 2 mi Þaii ; d li ¼ i X X ð1=m0i 2 1=mi Þaii d li ¼ i X ð1=m0i 2 1=mi Þaii ðs=s0 Þf ¼ 1 þ ð1=24Þð"=kTÞ2

ð1:27Þ

i

aii is the three dimensional Cartesian force constant for the displacement of the isotopic atom along the Cartesian axes. They applied Equation 1.27 to calculate the reduced partition function ratios for the isotopomers X0 Yn and XYn simply from the symmetrical stretching vibrational frequency. This was a useful approximation at the time when there was a paucity of detailed knowledge about molecular vibration frequencies; the symmetrical stretching frequency in molecules of the type X0 Yn was frequently known from Raman spectroscopy. It follows from Equation 1.22, Equation 1.24, and Equation 1.27 that plots of ‘n cðui Þ and ‘nðs=s 0 Þf vs. 1=T are linear with positive slow at low temperature or large values of u: They approach u ¼ 0 or T ¼ 1 with zero slope. In the region of high temperature both ‘n cðui Þ and ‘nðs=s 0 Þf are linear in 1=T 2 : While ‘nðs=s 0 Þf is a monotonically increasing function of 1=T; ‘n Kqm and ‘na; the equilibrium constant and fractionation factor between two chemical species, respectively, need not be. This results from the different temperature dependencies of the ‘nðs=s0 Þf values of the two different chemical species. Thus cases are known where the heavy isotopomer concentrates in species A at high temperature but in species B at low temperature. The conditions for such a cross over are spelled out clearly by Urey.18 Additional cases have been found in which the tritium/protium fractionation factor is larger than the deuterium/protium fractionation factor at high temperature but is smaller at low temperature.19 Systems which show the cross over will have either maxima or minima in the fractionation factor as a function of temperature. Finally, in the case of exchange reactions involving the oxides of nitrogen, cases have been found in which there are points of inflection in the temperature dependence.20 The formulation of the concept of the reduced partition function ratio in 1943 and its publication in 194715 led the way toward major developments in isotope effects in the second half of the twentieth century. Among these are the research which led to new processes for separation of isotopes,14,21 – 28 the theory of the kinetic isotope effect,29,30 the use of the kinetic isotope effect in the study of biochemical and organic reaction mechanisms,31,32 the theory of condensed phase isotope effects,33 numerous discoveries relating to quantum effects in condensed matter,34,35 and stable isotope geochemistry.36,37 All of these topics are discussed further in this and subsequent chapters of this book. Through the reduced partition function ratio it has been possible to relate isotope effects directly to molecular structure. A brief description of this development is given in Section IV of this chapter. The Bigeleisen – Mayer paper is the most frequently cited reference in the literature of isotope chemistry. 1. Numerical Calculation of the Reduced Partition Function Ratio Numerical calculations of RFPR are made by calculation of the vibrational frequencies of the isotopomers and their substitution into Equation 1.21 and Equation 1.24. Frequently referenced are the tabulation by Urey18 of values of f ; rather than ðs=s 0 Þf ; for isotopomers of hydrogen, lithium, boron, carbon, nitrogen, oxygen, chlorine, and bromine, all between 273 –600 K. This table was

8

Isotope Effects in Chemistry and Biology

prepared in the summer of 1946 in preparation for the Liversidge Lecture before the Chemical Society of London on December 18, 1946, which preceded the proper treatment of the anharmonic correction to the zero point energy of a polyatomic molecule.38 Thus RFPR tabulations for the isotopomers of hydrogen in Urey’s paper in addition to all publications prior to 1969 fail to account properly for anharmonicity. Urey’s tabulation for the self-exchange reaction between H2O and D2O as a function of temperature conflicts with the theorem in Section IV relating the sign of the deviation from the rule of the geometric mean. He tabulates values for K larger than 4.00; whereas it is less than 4.00 at all temperatures. The Urey tabulation opened up the entire field of isotope geochemistry. Urey’s assistant, Lawrence S. Myers, used Equation 1.24 to calculate the RPFR values of all the isotopomers of the light elements other than hydrogen. To assist him in this effort I supplied them with my unpublished GðuÞ tables. With these tables Myers was able to calculate f values with a precision of ^ 0.0001. and Urey noticed that the fractionation factor for the exchange of 18O and 16O between CO22 3 H2O decreased by 0.00025/8C. Now, if one could measure isotope ratios in paleocarbonate samples with a precision of ^ 0.0002, and if the samples retained the record, one could measure paleotemperatures to ^ 0.88C. Isotope ratio measurements of this precision were an order of magnitude beyond the state of the art at the time. Challenges of this type did not discourage Urey. He had met equal challenges before. He knew how to put together the necessary resources, both material and personnel; he had the drive and persistence to reach the objective. Publications 39 and 40 report the successful attainment of the carbonate– water isotopic temperature scale and the measurement of the temperature variation in the life cycle of a belemnite. Urey was deservedly recognized for this achievement with the Arthur L. Day Medal of the Geological Society of America and the V. M. Goldschmidt Medal of the Geochemical Society. During the 1940s and 1950s RFPR values were generally calculated for symmetrical molecules with the aid of vibrational formulae. Mechanical calculating machines were used. The advent of high-speed digital computers changed all of this. Starting in the early 1960s Max Wolfsberg wrote computer programs to calculate ðs=s 0 Þf from isotopic frequencies derived by solution of the Wilson FG matrices for the isotopomers. The programs were also appropriately modified to calculate kinetic and condensed phase isotope effects. He made these programs widely available and instructed scientists in their use. As a result there are now tabulations of RFPRs which are as good as the F matrices utilized in the calculations. F matrices were constructed to fit experimental spectroscopic data. Some examples are calculations by Bron, Chang, and Wolfsberg41 for the molecular species, H2, H2O, H2S, H2Se, and NH3. These calculations are fully corrected for anharmonicity. In the harmonic approximation there are, among other tabulations, those for carbon and hydrogen isotope RFPRs for a series of simple organic molecules by Hartshorn and Shiner42 and nitrogen and oxygen RFPRs in a series of nitro compounds by Monse, Spindel, and Stern.20,43 Subsequently Wolfsberg and coworkers showed that the necessary vibrational frequencies calculated ab initio from solution of the Schroedinger equation could be used to calculate RFPRs.44,45

C. CORRECTIONS TO THE B IGELEISEN – M AYER E QUATION Two corrections are necessary to the Bigeleisen – Mayer equation. The first has already been alluded to. This is the effect of anharmonicity. The corrections for anharmonicity for diatomic46 and polyatomic molecules38 have been derived by Wolfsberg. The principal correction comes from the fact that the zero point energy of a molecule is mass dependent. This dependence plays no role in spectroscopy, since spectroscopic studies measure vibrational energy differences. This correction is of importance only for the isotopomers of hydrogen. It can lead to corrections by at most 1% in the fractionation factors for isotopic exchange between different molecular species containing hydrogen. The second correction comes from the use of the Born –Oppenheimer approximation. There are two types of corrections. The first correction is due to the fact that the nuclear and electronic wave functions are not completely independent. For nuclei of finite mass there is a

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

9

correction to the electronic energy states which arises from the coupling of the electronic and nuclear motions. These corrections have been worked out by Kleinman and Wolfsberg.47 – 49 The corrections are largest for reactions involving dihydrogen and can amount to a few percent for reactions at 300 K. Bardo and Wolfsberg50 have applied both of these corrections to the exchange reaction HDðgÞ þ H2 OðgÞ ¼ H2 ðgÞ þ HDOðgÞ as a function of temperature. The Bardo – Wolfsberg calculations include the complete corrections for nonclassical rotation. A comparison of their calculations with a variety of experimental data is given in Figure 1.2. The experimental data51 – 53 from three independent laboratories fall on the theoretical curve within the limits of experimental error. A second correction to the Born – Oppenheimer approximation was discovered in the analysis of the isotope fractionation factors involving the isotopes 234U, 235U, and 236U from 238U in the exchange reactions between U(III) and U(IV) and between U(IV) and U(VI).54 My colleagues, Takanobu Ishida and Max Wolfsberg, organized a symposium on 6 May, 1989 at the State University of New York, Stony Brook in celebration of my 70th birthday. Alfred Klemm of the Otto Hahn Institut fu¨r Chemie, who came to the symposium with his wife Hannelore, visited me in my office on the preceding Friday afternoon. He was most interested in my opinion of a manuscript which he had received in his capacity as editor of the Zeitschrift fu¨r Naturforschung. The paper reported an anomalous effect in the fractionation of the isotope 235U from 238U in the chemical exchange reaction between U(IV) in aqueous solution and U(VI) adsorbed on an ion exchange resin. The 235U – 238U fractionation factor was reported to be significantly larger than 1.5 times the 236U – 238U fractionation factor. The nonlinearity of the fractionation factor as a function of the mass number of the isotopomer is shown in Figure 1.3.55 The manuscript had received a favorable review from one referee; a second referee recommended against publication. I read the manuscript and recognized that this was an important experimental finding. I made a back-of-the-envelope calculation and rejected the authors’ explanation that the enhancement of the 235U – 238U fractionation factor was due to the nuclear spin of 235U. I recommended publication on the basis of the established record of the authors in measuring uranium isotope fractionation factors. I suggested that the authors’ explanation of the effect be condensed. In due course someone would figure out the origin of the anomaly.

H2O(g) + HD(g) = HDO(g) + H2(g)

In K

1.25

1.00

Suess (1949) CMRRS&V (1954) RHB (1975) BW (1975)

0.75

0.50 1.5

2.0

2.5 3 10 /T

3.0

3.5

FIGURE 1.2 Plot of ‘n K for the exchange reaction HDðgÞ þ H2 OðgÞ ¼ HDOðgÞ þ H2 ðgÞ as a function of temperature. Suess,51 CMRRS&V,52 RHB,53 BW.50

Isotope Effects in Chemistry and Biology

5.6 13

10

5 7.4

Separation Coefficient e ×104

1.8

15

(×10−4)

10

0

234

235

236

237

238

Mass Number

FIGURE 1.3 Plot of the logarithms of the 234U, 235U, and 236U isotope separation factors vs. 238U. 1 ¼ ‘n 238 U=i U IV; aq: = 238 U=i U VI; resin as a function of the isotope mass, mi ; as reduced to 308 K. (Reproduced from Bigeleisen, J., J. Am. Chem. Soc., 118, 3676– 3680, 1996. With permission.)

The following year I made an accurate calculation of the nuclear spin effect in the hypothetical exchange reaction between U(III) and U(VI). On the basis of the information I received from Clyde Hutchison on the hyperfine splitting in Uþ3, I showed that the nuclear spin effect was almost two orders of magnitude smaller than the observed anomaly for that reaction. It was then that I realized that the effect reported by Fujii et al.55 was due to the nuclear field shift. Due to a number of health problems I did not complete my work on this problem until the summer of 1995.54 The nuclear field shift is a shift in the electron energy states in an atom or molecule due to the perturbation resulting from the interaction of those electrons with a high density at the nucleus with the nuclear charge. Such shifts, which are observed in the differences in the electronic spectra of isotopomers, have been known from the atomic spectra of the heavy elements since the 1920s. For ˚ line in the spectra of 235U and 238U is 0.4098 cm21. A shift instance, the shift between the 5028 A of this magnitude corresponds to a correction of 2 £ 1023 to ‘na; the logarithm of the separation factor at 300 K. This is larger than the observed enrichment factor in the 235 –238 U(IV) – U(VI) exchange,1.3 £ 1023 at 300 K. It is important to call attention to the fact that the heavy isotopomer, 238 U, enriches in the U(IV). The vibrational spectrum and reduced partition function ratio of the 12 – 14 From the known separation factor and the reduced partition function UOþ2 2 are well known. ratio one calculates that the U(IV) species would have to have the improbable stretching frequency of 2000 cm21 compared with 860 cm21 for the uranyl ion, UOþ2 2 . It is difficult to rationalize such strong bonding in the solvated U(IV) ion when compared with UOþ2 2 , which has two covalent double bonds in addition to the solvation. Since an electron is bound more tightly to a small nucleus, the light isotopomer, the ground state energy of the heavy isotopomer, will be higher than that of the light isotopomer. This is opposite of the ordering of the zero point energies associated with the molecular vibrations. In the exchange reaction between 235U and 238U 235

UðIVÞaq þ 238 UðVIÞresin ¼ 238 UðIVÞaq þ 235 UðVIÞresin

ð1:28Þ

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

11

the logarithm of the fractionation factor, a ¼ ð238 U=235 UÞðIVÞ=ð238 U=235 UÞðVIÞ; is, after neglect of the anharmonic correction to the molecular vibrations and KBOELE ; the usual mass correction to the Born – Oppenheimer approximation,47 – 49 ‘n a ¼ ‘n a0 þ ‘n Kfs

ð1:29Þ

where 0

0

‘n a0 ¼ ‘nðs=s Þf ðUIVÞ 2 ‘nðs=s Þf ðUVIÞ ‘n Kfs ; the correction for the nuclear field shift is ‘n Kfs ¼ ðkTÞ

21

ðE 0

238

U 2 E0

235

UÞðVIÞ 2 ðE 0

238

U 2 E0

235

UÞðIVÞ

ð1:30Þ

From the definition of a ¼ ð238 U=235 UÞðIVÞ=ð238 U=235 UÞðVIÞ it follows that the nuclear field shift will lead to a preference of the heavy isotope for the chemical species with the smallest number of s electrons or s bonding orbitals. The nuclear field shift in any electronic energy state of an atom or molecule can be written as the product of the electron density at the nucleus and the size and shape of the nucleus. The latter is independent of the chemical species in which the isotope is combined. It is a specific nuclear property for each isotope. Thus the shifts in different compounds of a given element can be related to the shifts in the atomic spectra, which are frequently known, by n o dEi0 UðVIÞ 2 dEi0 UðIVÞ ¼ dTðUðIÞÞi lPðOÞl2 UðVIÞ 2 lPðOÞl2 UðIVÞ =lPðOÞl2 UðIÞ ð1:31Þ where dEi0 is the nuclear field shift energy for the ith pair of isotopomers, dTðUðIÞÞi is the shift in the atomic spectrum for that pair of isotopes, and lP(O)l2 is the electron density at the nucleus. The logarithm of the separation factor for ith isotope from 238U is 2

‘n ai ¼ aðhc=kTÞfsi þ b ð1=24Þð"=kTÞ dmi =238mi

ð1:32Þ

˚ line in the atomic spectrum of UI for the ith isotopomer with where fsi is the field shift of the 5028 A 238 respect to U, a is the field shift scaling factor, and b is aii (U(IV)) 2 aii (U(VI)) of Equation 1.27. The parameters a and b were evaluated from the absolute value of ‘n a (235 –238) at 300 K and the ratio of ‘nað234 – 238Þ=‘nað235 – 238Þ at 433 K.55,56 The result of the analysis of the contributions of the field shift and the vibrational reduced partition function ratio to the overall separation factors for the isotopes 233U, 234U, 235U and 236U from 238U is given in Table 1.1. Comparisons are made with the experimental values.55,56 The absolute values of the experimental values are derived from

TABLE 1.1 Contributions of Vibrational Effects (‘n a0 ) and Field Shifts (‘n Kfs ) to the U(IV) – U(VI) Isotope Separation Factors at 433 K Isotope Pair 236–238 235–238 234–238 233–238 232–238 1 ¼ 104 ‘n a:

104‘n a0

104‘n Kfs

104‘n a; Calculated54

104‘n a; Experimental55,56

1i =1234

238 2 mi/4

22.54 23.82 25.12 26.48 27.65

8.76 14.62 17.46 22.83

6.22 10.80 12.34 16.40

6.2 10.8 (12.3) 16.4 18.3

0.50 0.88 1 1.33 1.49

0.50 0.75 1 1.25 1.5

12

Isotope Effects in Chemistry and Biology

the measured experimental values and the absolute value of the 235 –238 separation factor at 433 K calculated from the measured value at 300 K. The calculated separation factors are in quantitative agreement with the experimental ones. The anomalous behavior of the 233U and 235U isotopomers is due to the fact that these nuclei have quadrupole moments. The nuclear field shift is the largest contribution to the separation factor for all the isotopes. The nuclear field shift resolves the 50-year-old puzzle as to why the heavy isotope enriches in the U(IV) species. In accordance with molecular structure predictions the vibrational effect does lead to a preference of the heavy isotope, 238U, for the U(VI) species. Since the nuclear field shift, which leads to a preference of this isotope for the U(IV) species, is much larger than the vibrational effect, the net result is a preference of the heavy isotope for the U(IV) species. It is interesting that the role of the nuclear field shift in the fractionation of the uranium isotopes was overlooked in 1943 when Maria Mayer and I worked at the SAM Laboratory under the direction of Harold C. Urey. We were principally engaged in the search for isotopic shifts in the spectra of 235U and 238U. In addition to the fact that there was no information about the field shift in any compound of uranium at the time, we never made the connection between that work and our effort to estimate uranium isotope separation factors by chemical exchange reactions. If we had, we probably would not have proceeded with the development of the statistical mechanical theory of isotope chemistry. I often reflect on what direction my career would have taken. I take satisfaction from my recognition of the importance of this Born –Oppenheimer correction to the isotope chemistry of the heavy elements. In 1995 I was able to formulate a correct answer to the problem Harold Urey presented me with in 1943, even if for only one chemical reaction.

IV. ISOTOPE CHEMISTRY AND MOLECULAR STRUCTURE Section III provides the information for the calculation of isotope fractionation factors from the vibrational frequencies of the isotopomers involved in an equilibrium chemical reaction. It is instructive to see what forces within the molecules are responsible for the fractionation. From Equation 1.27 we will derive some general rules, the first order rules of isotope chemistry. Developments in the theory and analysis of the mathematical structure of the reduced partition function ratio have led to powerful methods to dissect the reduced partition function ratio and thus the fractionation factor into the respective contributions from each of the forces in each of the molecules participating in an exchange reaction.17,57 – 61 Bigeleisen and Ishida61 showed that, when isotopic substitution is at an end atom in the molecule, only the uncoupled stretching and bending coordinates contribute significantly to the reduced partition function ratio. For such molecules the bend – bend interactions have small F matrix elements and therefore make small contributions to RFPR. Somewhat larger corrections come from the interaction of the stretching and bending coordinate due to their larger F matrix elements. With an approximate correction to Equation 1.27, the first order quantum correction, they were able to obtain reasonable agreement with exact values of ‘nðs=s 0 Þf for hydrogen isotope substitution in a series of hydrocarbons. After the derivation of the first order rules of isotope chemistry we present two independent general methods for the dissection of RFPR.

A. THE F IRST O RDER R ULES OF I SOTOPE C HEMISTRY We rewrite Equation 1.27 in its logarithmic form 0

2

‘nðs=s Þf ¼ ð1=24Þð"=kTÞ

X i

ð1=m0i 2 1=mi Þaii

ð1:33Þ

Although Equation 1.27 was derived for small u and small du; in Section IV.C we will derive Equation 1.33 for large values of u and du: The first order rules follow from Equation 1.33.57,60

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

13

1. Isotope effects, ‘nðs=s 0 Þf ; depend only on the masses of the isotopic atoms and the force constants bonding the atom at the site of isotopic substitution with other atoms in the molecule, aii : 2. Isotope effects between different compounds occur only when there are force constant changes at the site of isotopic substitution. 3. Isotope effects are additive. a. Isotopic additivity 0

18

0

16

16

0

16

18

16

‘nðs=s Þf ðD2 O=H2 OÞ ¼ ‘nðs=s Þf ðD2 O=H2 OÞ þ ‘nðs=s Þf ðH2 O=H2 OÞ

b. Substituent additivity 0

‘nðs=s Þf ðCHDFCl=CH2 FClÞ

¼ ‘nðs=s0 Þf ðCH2 DCl=CH3 ClÞþ ‘nðs=s0 Þf ðCH2 DF=CH3 FÞ2 ‘nðs=s0 Þf ðCH3 D=CH4 Þ

c. Isotope effects are cumulative (first rule of the geometric mean) 0

0

‘nðs=s Þf ðD2 O=H2 OÞ ¼ 2‘nðs=s Þf ðHDO=H2 OÞ

d. Equivalent isomers have the same isotope chemistry 0

0

0

‘nðs=s Þf ðC6 H4 D2ðoÞ Þ ¼ ‘nðs=s Þf ðC6 H4 D2ðmÞ Þ ¼ ‘nðs=s Þf ðC6 H4 D2ðpÞ Þ

4. The heavy isotopomer concentrates in the chemical species with the strongest bonding (largest vibrational force constants). Rule (2) was also formulated through numerical computation for kinetic isotope effects.62,63

B. STATISTICAL M ECHANICAL P ERTURBATION T HEORY Singh and Wolfsberg59 developed a perturbation method for the dissection of the isotopic reduced partition function ratio from the diagonal elements of the F and G matrix elements plus the sum of corrections from the off diagonal elements. The Hamiltonian of the molecule is written as a sum of diagonal and off diagonal elements. H ¼ H0 þ H1 X 2 ðgii p2þ 2H0 ¼ i fii qi Þ i

H1 ¼

X

i; i,j

ðgij pi pj þ qi qj Þ

ð1:34Þ ð1:35Þ ð1:36Þ

where the qi s are the internal coordinates and the pi s are the conjugate momenta. With this division of the total Hamiltonian and Schwinger perturbation theory Singh and Wolfsberg obtain 0

0

‘nðs=s Þf ¼ ‘nðs=s Þf0 þ CORR

where 0

‘nðs=s Þf0 ¼

X i

‘n cðui0 Þ

ð1:37Þ ð1:38Þ

14

Isotope Effects in Chemistry and Biology 0

0

‘n cðui0 Þ ¼ ‘nðui0 =ui0 Þ þ dui0 =2 þ ‘n 1 2 expð2ui0 Þ =½1 2 expð2ui0 Þ

ð1:39Þ

and ui0 ¼ ð"=kTÞðgii fii Þ1=2

ð1:40Þ

‘n cðui0 Þ is just the reduced partition function ratio of an uncoupled internal coordinate. CORR is a function of both diagonal and off-diagonal F and G matrix elements.59 A comparison will be given of the results for RFPR from calculations through Equation 1.37 along with calculations using finite polynomials with exact calculations.64

C. POLYNOMIAL E XPANSIONS OF THE R EDUCED PARTITION F UNCTION R ATIO In this section I present an alternate to the Singh –Wolfsberg correction to the diagonal element approximation through the use of finite polynomials. We expand ‘n cðui Þ; the exact reduced partition function ratio of an oscillator, in an infinite series of u 2j and obtain the convergent, infinite series58 XX 0 Aj dðu2j ð1:41Þ ‘nðs=s Þf ¼ ½u0i , 2p i Þ j

i

2j 02j dðu2j i Þ ¼ u i 2 ui

Aj ¼ ð21Þ2jþ1 ½B2j21 =2jð2j!Þ ð"=kTÞ2j

ð1:42Þ

B2j21 are the Bernoulli numbers ðB1 ¼ 1=6; B3 ¼ 1=30; B5 ¼ 1=42 etc.). The numerical coefficients of the first three terms are 1/24, 2 1/2880, and 1/181440. Through Equation 1.41 we have extended the validity of Equation 1.33 to values of u0i , 2p: This corresponds to a maximum at 300 K. We can now remove this restriction through a method suggested frequency of 1300 cm21 . by T. Ishida.17 Instead of expanding ‘nðs=s0 Þf in an infinite series, we choose a finite polynomial for the expansion. This expansion replaces Equation 1.41 with XX 0 Wj Aj dðu2j ð1:43Þ ‘nðs=s Þf ¼ ½u0i , 1 i Þ j

i

The modulating coefficients, Wj ; are isotope independent and have values less than one. Equation 1.43 is similar in structure to Equation 1.41. Again it leads to the first order rules of isotope chemistry but without the restriction that u , 2p: Through Equation 1.43 we have now extended the first order rules of isotope chemistry to all molecules at all temperatures. The best values of the modulating coefficients are the WINIMAX coefficients.65 These are MINIMAX coefficients66 weighted appropriately for use in isotope chemistry. The WINIMAX coefficients supercede the prior FOP coefficients.17,67 Equation 1.43 can be used to derive the correction terms which appear in the Wolfsberg method and form the basis of the WIMPER method.64,68 o Xn 0 0 Wj Aj ð"=kTÞ2j d TrðH j Þ 2 TrðH j Þ0 ‘nðs=s Þf ¼ ‘nðs=s Þf0 þ ð1:44Þ j¼1

The terms d½TrðH j Þ 2 TrðH j Þ0 are the sums of the differences of powers of the eigenvalues of the normal coordinates and the internal coordinates, respectively. In Table 1.2 and Table 1.3 we give examples of the approximation of the reduced partition function ratios of a number of molecules by both the Singh –Wolfsberg and the WIMPER (2) method. The Singh – Wolfsberg method converges more rapidly than does WIMPER.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

15

TABLE 1.2 Approximation of Reduced Partition Function Ratios by Perturbation Methods (D/H) at 300 K Isotopomer Pair HDO/H2O CH3D/CH4 HDCO/H2CO C2H3D/C2H4 C2H5D/C2H6 C6H5D/C6H6 out of plane

Method S-W W(2) S-W W(2) W(2) S-W W(2) S-W W(2) W(2)

ln(s/s 0 )f0

ln(s/s 0 )f, 2nd Order

ln(s/s 0 )f, Exact

2.5990 2.5812 2.4957 2.4789 2.2426 2.3066 2.3099 2.4406 2.3935 0.2349

2.6012 2.5821 2.4899 2.4825 2.2261 2.3399 2.2882 2.4123 2.3830 0.2514

2.6012 2.5829 2.4723 2.4926 2.1997 2.3399 2.3204 2.4177 2.3645 0.2656

S-W;59 W(2).64,68 The differences between the S-W and W(2) values for ‘nðs=s0 Þf0 and ‘nðs=s0 Þf ; respectively, are due to the different F matrices used by these authors.

TABLE 1.3 Approximation of Reduced Partition Function Ratios by Perturbation Methods (13C/12C and 18O/16O) at 300 K Isotopomer Pair 16 H18 2 O/H2 O 13

CH4/12CH4

12 H13 2 CO/H2 CO 18 H2C O/H2C16O 13 CCH4/12C2H4 13

CCH6/12C2H6

13

CC5H6/12C6H6 out of plane

Method S-W W(2) S-W W(2) W(2) W(2) S-W W(2) S-W W(2) W(2)

ln(s/s 0 )f0

ln(s/s 0 )f, 2nd Order

ln(s/s 0 )f, Exact

0.0677 0.0673 0.1313 0.1304 0.1580 0.0887 0.1379 0.1440 0.1447 0.1422 0.0339

0.0649 0.0643 0.1134 0.1134 0.1398 0.0893 0.1266 0.1251 0.1270 0.1314 0.0177

0.0648 0.0643 0.1126 0.1113 0.1369 0.0904 0.1299 0.1253 0.1280 0.1253 0.0183

See notes to Table 1.1.

The differences in the values of ‘nðs=s0 Þf0 between the two methods is the result of the usage of different F matrices by these authors. Corresponding differences also exist for the exact values of ‘nðs=s0 Þf : The zero order approximation, ‘nðs=s0 Þf0 ; to the exact values is within 1– 2% for deuterium substitution except for ring structures. On the other hand, heavy atom ‘nðs=s0 Þf0 values differ from the exact values by about 12%, again except for ring structures. The excellence of the diagonal element approximation in the case of deuterium –hydrogen substitution is a consequence of the fact that hydrogen is generally an end atom. The largest corrections to the diagonal element approximation come from the stretch –bend interaction. For this interaction the dgij matrix element is zero. Therefore there is no contribution to the correction at the j ¼ 1 level in the WIMPER method. The difference in this interaction for end and central atom substitution can be seen by

16

Isotope Effects in Chemistry and Biology

comparison of the 13C and 18O effects in formaldehyde, CH2CO, in Table 1.2. The 13C isotopomer has an 11% correction to ‘nðs=s0 Þf0 ; whereas the 18O correction is only 2%. The difference is primarily due to the fact that the 13C isotopomer involves a central atom substitution, whereas the 18 O isotopomer involves an end atom. An additional reason for the better approximation to ‘nðs=s0 Þf at the zero order for hydrogen– deuterium comes from the fact that all atoms have essentially infinite weight compared with that of the hydrogen atom, and to a lesser extent with the deuterium atom. This leads to small coupling of the motion of the hydrogen atom with the motions of any of the other atoms in the molecule.69 In Table 1.4 and Table 1.5 we tabulate the corrections at the j ¼ 1 and j ¼ 2 levels for each type of interaction for deuterium and heavy atom substitution in water and the out of plane vibrations of benzene. In Table 1.6 we show that the ratio ‘nðs=s0 Þf0 =‘nðs=s0 Þfexact deviates from unity by more than 2% only when there is a contribution to ‘nðs=s0 Þfexact at the level j ¼ 1: The j ¼ 1 values are identically zero for end atom substitution. The j ¼ 1 corrections for ring atom substitution are significant, cf. 13C substitution in benzene, Table 1.5. In that case the zero order approximation, ‘nðs=s0 Þf0 ; is 1.85 times the exact value. Yet the final value after correction at the j ¼ 2 level is within 3% of the exact value! This shows the power of the WIMPER (2) method to analyze all the interactions of the internal coordinates in the calculation of the reduced partition function ratio. Skaron and Wolfsberg72 have shown, without solutions to the vibrational problem, from Equation 1.37 how ‘nðs=s0 Þf changes with a change in force constant. A new value of ‘nðs=s0 Þf is calculated directly from the changed force constant and compared with the original value of ‘nðs=s0 Þf : In principle, their method can also be used to derive the absolute contribution of each force constant, F matrix element, to the reduced partition function ratio. Additional information concerning the role of structure to isotope chemistry can be derived from consideration of the j ¼ 2 corrections to ‘nðs=s0 Þf0 : In Appendix 1A we list some of the properties of the eigenvalues which are relevant to the j ¼ 2 corrections. The properties of isotopomers at the j ¼ 1 have been discussed under the first order rules of isotope chemistry. Deviations from the rule of the mean occur at the level of j ¼ 2 and they are primarily due to

TABLE 1.4 Dissection of the Reduced Partition Function Ratios of Water at 300 K Deuterium Substitution (HDO/H2O) Coordinate

Force Constant

ln c(ui0)

OH stretch HOH bend OH £ HOH OH £ OH Total ‘nðs=s0 Þf exact

8.454 0.762 0.237 20.100

2.2200 0.3611 0 0 2.5811

j51 0 0 0 0 0

j52

ln c(ui)

0.0025 0.0005 20.0016 20.0006 0.0009

2.2225 0.3616 20.0016 20.0006 2.5819 2.5829

16 Oxygen Substitution (H18 2 O/H2 O)

OH stretch HOH bend OH £ HOH OH £ OH Total ‘nðs=s0 Þf exact

8.454 0.762 0.237 20.100

ln c(ui0)

j51

j52

ln c(ui)

0.0550 0.0123 0 0 0.0673

0 0 20.0055 0.0003 20.0052

0.0008 0 0.0015 20.0002 0.0021

0.0558 0.0123 20.0040 0.0001 0.0642 0.0643

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

17

TABLE 1.5 Dissection of the Reduced Partition Function Ratios of the Out of Plane Vibrations of Benzene at 300 K Deuterium Substitution (C6H5D/C6H6) Coordinate

Force Constant

ln c(ui0)

C2C –H op wag CC–CC torsion WAG £ TORS 3C’s WAG £ TORS 2C’s WAG £ WAG 2C’s TORS £ TORS 3C’s TORS £ TORS 2C’s TORS £ TORS 2 &C’s Total ‘nðs=s0 Þf exact

0.378 0.086 ^0.088 ^0.043 20.057 20.060 0.008 0.019

0.2346 0 0 0 0 0 0 0 0.2349

j51 0 0 0 0 0 0 0 0 0

j52

ln c(ui)

0.0132 0 0.0054 0.0010 20.0031 0 0 0 0.0165

0.2478 0 0.0054 0.0010 20.0031 0 0 0 0.2514 0.2656

13

C Substitution (13C C5H6/C6H6)

C2C –H op wag CC–CC torsion WAG £ TORS 3C’s WAG £ TORS 2C’s WAG £ WAG 2C’s TORS £ TORS 3C’s TORS £ TORS 2C’s TORS £ TORS 2 &C’s Total ‘nðs=s0 Þf exact

0.378 0.086 ^0.088 ^0.043 20.057 20.060 0.008 0.019

ln c(ui0)

j51

j52

ln c(ui)

0.0226 0.0113 0 0 0 0 0 0 0.0339

0 0 20.0299 20.0081 0.0045 0.0127 0.0008 20.0005 20.0203

0.0020 20.0008 0.0045 0.0011 20.0007 20.0015 20.0001 0.0001 0.0041

0.0246 0.0105 20.0256 20.0070 0.0038 0.0112 0.0007 20.0004 0.0177 0.0183

See Ref. 68.

TABLE 1.6 Total Correction to the Zero Order Approximation of the Reduced Partition function Ratio Isotopomers

ln(s/s 0 )f0/ln(s/s 0 )f Exact

HDO/H2O 16 H18 2 O/H2 O HDCO/H2CO 12 H13 2 CO/H2 CO H2C18O/H2C16O C6H5D/C6H6 planar 13 CC5H6/C6H6 planar C6H5D/C6H6 out of plane 13 CC5H6/C6H6 out of plane

0.999 1.047 1.020 1.112 0.981 0.980 1.295 0.884 1.852

See Ref. 68.

Correction at j 5 1 None Yes None Yes None None Yes None Yes

18

Isotope Effects in Chemistry and Biology

70 the G0 F 0 terms relating to the bending vibrations. be seen from P 2 PThis0 2can readily P the2 trace 2 2 2 0 In Cartesian coordinates d l m m m m Þ ¼ 2 Þa þ 2ð 2 ð of the H matrix. i j mj aji : The i i i i ii i P first term, P i ðm0i 2 2 m2i Þa2ii ; cancels identically for isotopic disproportionation reactions. P P P P The sums, i ðm0i 2 2 m2i Þa2ii and i ðm0i 2 mi Þ j mj a2ji . The term i ðm0i 2 mi Þ j mj a2ji is larger for the isotopomer pair D2O/HDO than for HDO/H2O. Since the sign of A2, in Equation 1.41 and Equation 1.43, is negative ‘n Kqm # ‘n KCl in the harmonic oscillator approximation. Wolfsberg and coworkers38,47 – 50,71 have shown that anharmonic corrections and corrections to the Born –Oppenheimer approximation do not affect this conclusion. In Appendix 1A, I also give a summary of the structure of the j ¼ 2 corrections. For complete details of the individual matrix elements and their application to a variety of isotopomeric molecules consult the original literature.68 Section III and Section IV of this chapter summarize some of the significant advances in the theory of equilibrium isotope effects since the Bigeleisen –Mayer publication. These include the corrections to the harmonic oscillator and Born – Oppenheimer approximations by Max Wolfsberg and coworkers.47 – 49 A second correction to the Born – Oppenheimer approximation is the shift in the electronic energy levels due to the nuclear size and shape.54 Major advances have resulted from the ability to calculate frequencies of large isotopomers through electronic computers. In addition, we now have the capability of calculating these force fields by quantum mechanics. For both of these developments we are indebted to Max Wolfsberg. Our detailed understanding of the role of molecular structure has been advanced by the first and second order rules of isotope chemistry.60,70 The role of individual forces within a molecule to its isotope chemistry has been elucidated through the perturbation theories of Max Wolfsberg59,72 and those developed by Takanobu Ishida and Myung W. Lee64,68 with whom I experienced a joyful and fruitful collaboration.

V. KINETIC ISOTOPE EFFECTS The availability of gram quantities of heavy water in the early 1930s made possible the study of the effects of isotopic substitution on the rates of chemical reactions. Initial studies were directed at testing the recent developments in the theory of reaction rates. In 1931, Eyring and Polanyi73 introduced an empirical method for the quantum mechanical calculation of the energy surface of a three atom system. Shortly thereafter modifications to the classical transition state theory74 to include quantum statistics were derived.75 – 77 In the transition state theory one assumes a rapid reversible equilibrium between the reactants and an energy rich complex of the reacting species — the transition state. The rate of the reaction is simply the rate at which the transition state complex decomposes into products. The rate is Rate ¼ c‡n‡ ¼ kcA cB

ð1:45Þ

k ¼ c‡n‡=cA cB ¼ ðkT=hÞK‡

ð1:46Þ

The rate constant is

where c‡ is the concentration of activated complexes, cA cB is the product of the concentrations of reactants in the rate determining step, n‡ is the frequency with which activated complexes decompose into products, k is the rate constant and K‡ is the equilibrium constant between reactant and transition state molecules. Equation 1.46 is conventionally corrected for a transmission coefficient, k; and for the fact that some reaction occurs by tunneling through, rather than motion over, the energy barrier to the reaction.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

19

The equilibrium constant, K‡; can be written as the quotient of the partition function of the transition state, Q‡; and the product of the partition functions of the reactants. K‡ ¼ Q‡=QA QB …

ð1:47Þ

K‡ can be calculated, in principle, from Equation 1.47. The partition function of the transition state, Q‡; differs in one important way from the conventional partition function of a stable molecule. Of the 3n coordinates in the transition state, 3 are allocated to the translation of the center of mass, 3 (2 in the case of linear transition states) are allocated to the rotation of the transition state, one belongs to the reaction coordinate, n‡; which is an imaginary frequency. This leaves 3n 2 7 ð3n 2 6 in the case of linear transition states), to the internal vibrations. Early applications of Equation 1.46 were made to the absolute and isotopic reactions of the type H þ H2. For these reactions Farkas and Wigner78 gave the explicit formula for the rate constant, k k ¼ ð1=2ÞkðM‡=Ma Mm Þ3=2 ðI‡=Im Þ"2 ð2p=kTÞ1=2 expð2Q=RTÞðNL =1000Þ sinh bnm =½ðsinh bnd Þ2 sinh bns ð1 þ ðbn‡Þ2 =6Þ

ð1:48Þ

In Equation 1.48 k is the transmission coefficient, M‡ is the molecular weight of the H3 complex, Ma is the atomic weight of the hydrogen atom, Mm is the molecular weight of the hydrogen molecule, I‡=Im is the ratio of the moments of inertia of the H3 complex and the hydrogen molecule, Q is the activation energy measured from the bottom of the potential energy surface (PES) of the H2 molecule to the minimum in the PES of the activated complex, NL is Loschmidt’s number, and the hyperbolic functions ðsinhbxÞ21 ¼ 2ðexpð2u=2ÞÞ=ð1 2 expð2uÞÞ; bx ¼ u=2kT: nm ; nd ; and ns are the stretching frequency of H2, the doubly degenerate bending frequency of H3, and its stretching frequency, respectively. They recognized that there were terms both in Equation 1.47 and the ratio of rate constants, k1 =k2 ; that cancelled. Neither Farkas and Wigner nor subsequently Hirschfelder, Eyring, and Topley79 were able to obtain agreement between calculations based on Equation 1.48 and the Eyring– Polanyi PES. Whether this was a problem with the PES or the quantum statistical transition state theory or both was a problem which awaited future solution. Heavy atom kinetic isotope effect studies became possible after WWII with the ready availability of radioactive isotopes and isotope ratio mass spectrometers for the measurement of stable isotope ratios. An early study was the 14C effect in the decarboxylation of malonic acid by Yankwich and Calvin.80 14C singly labeled in the carboxyl group of malonic acid, CH3CH14 2 COOH, can decompose in two ways: to give either 12CO2 or 14CO2. For the ratio of the rate constants to give 12CO2 compared with 14CO2, k4 =k3 ; they reported 1.12 ^ 0.03 at 1508C. For the bromine derivative, with substitution at the methylene position, they reported a ratio of 1.41 ^ 0.08 at 1158C These isotope effects appeared large to me in comparison with what was known about equilibrium isotope effects for carbon, typically 1.03 for 12C vs. 13C. This would translate to 1.06 for 12C vs. 14C. I decided to study the transition state theory of reaction rates. It became immediately obvious that the method used by Pelzer and Wigner and by Eyring could be greatly simplified by the introduction of the concept of the reduced partition function ratio. Starting with Equation 1.46 and Equation 1.47 one finds30 for the ratio of rate constants of light to heavy isotopomers, kL =kH ; kL =kH ¼ ðnL =nH Þ‡ð f =f ‡Þ

ð1:49Þ

apart from corrections due to the transmission coefficient and tunneling. The partition function ratio, f ; has been defined in Equation 1.16. The partition function ratio of the transition state, f ‡; is s 0 =s times the conventionally defined reduced partition function ratio, ðQ=Q0 Þ‡qm =ðQ=Q0 Þ‡Cl : Since the translation, rotation and reaction coordinates of the transition state are treated classically, f ‡ is the

20

Isotope Effects in Chemistry and Biology

product of ð3n 2 7Þ reduced vibrational partition function ratios, Equation 1.21. Each vibrational reduced partition function ratio consists of three factors: the classical correction, ðui =u0i Þ‡=2, the zero point energy term, dui ‡=2; and the Boltzmann excitation term, {½1 2 expð2u0i Þ =½1 2 expð2ui Þ ‡: The product of the ratio of the frequencies of the crossing the barrier, ðnH =nL Þ‡; when combined with the product ðui =u0i Þ‡ of the ð3n 2 7Þ real vibrations is abbreviated MMI. An important feature of kinetic isotope effect and its applications is the fact that the absolute height of the potential barrier of the transition state from the reactants does not enter into the kinetic isotope effect. The absolute value of the height of the barrier is what causes the major problem in the accurate theoretical calculation of absolute rate constants. Equation 1.49 brings to the study of kinetic isotope effects all that we have learned about equilibrium isotope effects. There are the first order rules of isotope chemistry spelled out in Section IV; there is the ability to correlate kinetic isotope effects with molecular structure. It greatly simplifies our understanding of the origin of kinetic isotope effects and the design and analysis of experiments. A complete exposition of the theory and the design of kinetic isotope effect experiments is given in the chapter in Advances in Chemical Physics by Bigeleisen and Wolfsberg30. Equation 1.49 is properly referred to as the Bigeleisen – Wolfsberg equation. More recent expositions of the theory of kinetic isotope effects include the monographs by Melander and Saunders32 and Willi.81 The latter texts include summaries of the major applications of the kinetic isotope effect to the study of reaction mechanisms. The chapter by Max Wolfsberg in this volume gives details about the calculation of kinetic isotope effects from Equation 1.49 by high speed computers. An early application of Equation 1.49 was to calculate the maximum kinetic isotope effects for all elements in the Periodic Table.82 Apart from the statistical factors ðsL =sH Þ and ðsL =sH Þ‡ which appear in the ratio ðf =f ‡Þ; each of these quantities in f and f ‡ is positive and equal to or greater than unity. The maximum kinetic isotope effect in any reaction occurs when f ‡ is equal to unity. This corresponds to a transition state of an assembly of free atoms, with no bonding whatsoever. Inasmuch as there are extensive tables of the reduced partition function ratio, tabulated as f (cf. Ref. 18), and these values are not significantly different for different compounds of a given element, it was possible to set upper limits for kinetic isotope effects for all elements. This amounted to a factor of 1.5 for 12C vs. 14C at 300 K. This cast doubt on the validity of the Yankwich – Calvin experiments on the decarboxylation of malonic acid. In the relative modes for the decarboxylation of labeled malonic acid, f is unity, since both paths involve the same substrate. I carried out a simplistic calculation, using the Slater coordinate to calculate ðnL =nH Þ‡ and neglecting the factor f ‡ for the isomeric transition states, to obtain a theoretical ratio for k4 =k3 of 1.038 for 12C vs. 14C.83 The experiments on the 13C and 14C kinetic isotope effects in the decarboxylation of malonic acid were repeated in several laboratories.84 – 89 A summary of some of the pertinent results along with the initial findings of Yankwich and Calvin is given in Table 1.7. Both Lindsay, Bourns, and Thode85 and Yankwich and Promislow87 found commercial samples of malonic acid were not homogeneous in their isotopic distribution of 13C between the methylene and carboxyl groups. They made appropriate correction for this inhomogeneity. The inhomogeneity explains the discrepancy between the Bigeleisen – Friedman experiment84 and the later results.85,87 Table 1.7 also includes a comparison of the experimental results with theoretical calculations,30,83,90 which include revisions to the use of the Slater coordinate, which was subsequently shown to be physically unrealistic.30,91 Although my back-of-the-envelope calculation of k4 =k3 83 is incorrect as a result of using the Slater coordinate and neglecting of the contribution from the partition function ratio of the vibrational terms in the activated complex, fortuitously it turned out to be of the correct order of magnitude. The results in Table 1.7 and subsequent experiments on kinetic isotope effects involving the isotopomers of carbon and nitrogen showed the power of the reduced partition function ratio to predict even small kinetic isotope effects. The later experiments on malonic acid also confirm the theoretical prediction that the 14C isotope effect is 1.9 times that of the 13C effect. I now return to the question of the failure of the transition state theory to calculate the rate constant for a simple reaction of an atom with a diatomic molecule, H þ H2 and its isotopomers.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

21

TABLE 1.7 Intramolecular Isotope Effect in the Decarboxylation of Malonic Acids Label

T (8C)

14

C 14 C Br 13 C 14 C 14 C 13 C 13 C 13 C 13 C 14 C 14 C 14 C Br

150 115 138 148 138 138 138 140 140 140 140 118

13

138 140 133

C 13 C 13 C 14 13 C/ C (experimental)

100 1

H

Authors

Experiments 14฀ ^ 3 No 41฀ ^ 8 No 2.0฀ ^ 0.1 No 6฀ ^ 2 No 8.7 2 10.5 No 2.0 No 2.6฀ ^ 0.4 Yes 2.99฀ ^ 0.05 Yes 2.73฀ ^ 0.05 Yes 5.47฀ ^ 0.36 No 5.77฀ ^ 0.20 No 6.46฀ ^ 0.24 No

Yankwich & Calvin (1949) Yankwich & Calvin (1949) Bigeleisen & Friedman (1949) Roe & Hellman (1951) Yankwich, Stivers & Nystrom (1951) Lindsay, Bourns & Thode (1952) Lindsay, Bourns & Thode (1952) Yankwich & Promislow (1954) Yankwich & Nystrom (1954) Yankwich & Nystrom (1954) Grigg (1956) Grigg (1956)

Theory 2.0 2.5 1.84 2.0฀ ^ 0.1

Bigeleisen (1949) Bigeleisen & Wolfsberg (1958) Stern & Wolfsberg (1963) Yankwich, Promislow & Nystrom (1954)

14

C/13C ¼ 1.9 C/13C ¼ 1.9 14 13 C/ C ¼ 1.9 14

1 ¼ ðk4 =k3 Þ 2 1; H ¼ correction for homogeneity.

In 1951, W. M. Jones added to the kinetic data on the reaction of chlorine atoms with the isotopomers H2 and D2 data for the reaction with HT as a function of temperature.92 Plots of the relative rate constants for each of these isotopic reactions as a function of 1=T had positive slopes. Contrary to the experimental findings, calculations for this reaction,92,93 similar to the ones for the H þ H2 reaction, gave a negative temperature coefficient for both the isotopomers HD and HT. Jones entertained a suggestion by Magee94 that the transition state Cl – H –H was bent rather than linear. Wolfsberg proposed to examine whether this could account for the discrepancy between theory and experiment. He approached me about the possibility of collaboration on this study. I welcomed the opportunity to collaborate with Max, an expert in quantum chemistry. Bigeleisen and Wolfsberg95 showed that a simple change in the geometry per se could not remove the discrepancy between the theoretical and experimental signs of the activation energy difference in the kinetic isotope effect in the H2/HT þ Cl reaction. A bent transition state differs from a linear one in the moments of inertia and the number of bending vibrations. The rotation of the transition state complex is classical and, therefore, does not contribute to the kinetic isotope effect. Bending vibrations have low frequencies and as such make a minor contribution to the reduced partition function ratio. Given the known properties of the isotopomers of H2, only the magnitude of the real stretching frequency of the H2Cl transition state can affect the activation energy of the kinetic isotope effect of this reaction. On this basis Bigeleisen and Wolfsberg were able to construct linear and bent transition states, which were in reasonable agreement with all the known kinetic isotope effects. The properties of their linear transition state is given in Table 1.8. Additional experimental data were necessary to validate the result. The crucial additional experimental data were the kinetic isotope effect of H2/HD as a function of temperature. These experiments were carried out by Fritz S. Klein, a guest at Brookhaven National Laboratory (BNL) from the Weizmann Institute for Science, in collaboration with Jacob Bigeleisen. Their experiments, as well as those of Jones on H2/HT, did not measure the branching

22

Isotope Effects in Chemistry and Biology

TABLE 1.8 ˚ 21 ) and Vibrational Frequencies (in cm21) of the H2Cl Force Constants (in mdyne A Transition State

Year f (H–Cl) f (H–H) f12 ns nb nL

WTE

BW

1936 LEP 3.69 20.30 0 2489 439 720i

1955 EMP 1.86 20.48 0 1460 (200) 720i

BKWW 1959 LEPS 1.56 0.57 1.50 1348 727 1497i

SPK

ALTG

BW2

1973 LEPS 1.77 0.45 1.46 1357 707 1483i

1996 EMP

2000 ab initio

1358 581 1520i

1360 540 1294i

WTE, Wheeler, A., Topley, B., and Eyring, H., J. Chem. Phys., 4, 178, 1936; BW, Bigeleisen, J. and Wolfsberg, M., J. Chem. Phys., 23, 1535, 1955; BKWW, Bigeleisen, J., Klein, F. S., Weston, R. E., Jr., and Wolfsberg, M., J. Chem. Phys., 30, 1340, 1959; SPK, Stern, M. J., Persky, A., and Klein, F. S., J. Chem. Phys., 58, 5697, 1973; ALTG, Allison, T. C., Lynch, G. C., Truhlar, D. G., and Gordon, M. S., J. Phys. Chem., 100, 13575, 1996; BW2, Bian, W. and Werner, H-J., J. Chem. Phys., 112, 220, 2000.

ratio between HX and XH reacting withPCl to give H and X atoms, respectively. They measured the total kinetic isotope effect, kðH2 Þ= kðHXÞ þ kðXHÞ: At this time Ralph Weston at BNL was exploring the Sato96 modification of the Eyring –Polanyi method of calculating the PES of a three atom systems. He mapped out the PES for the H2Cl transition state as a function of the Sato parameter. The PES obtained by the four man collaboration,97 which represented the best fit to all the experimental data, is give in Table 1.8. Additional experiments were carried out by Bar-Yaacov, Persky, and Klein, Persky and Klein, and by Persky. Stern, Persky, and Klein98 developed additional PESs, both empirical and of the Sato type. A comparison of their results for both an empirical model and a Sato type potential is given in Figure 1.4. The potential parameters for their Sato like potential (LEPS) is given in Table 1.8. Their Sato like PS does not differ significantly from the BKWW surface. Their calculated kinetic isotope effects include tunneling within both the Wigner and Eckart approximations. The average root mean square deviation between either the BKWW or SPK calculated kinetic isotope effects and experiment is less than 15% over the temperature range 245– 445 K It is characteristic of all the PESs subsequent to the WTE surface that they have much smaller H – Cl binding force constants and a small attractive rather than repulsive H – H force constant compared with the WTE surface. Further, the later force fields involve significant potential energy coupling between the H – H and H –Cl stretching coordinates. The results of BKWW and SPK along with the experimental data constituted the best quantitative validation of the transition state theory at the time. Subsequent to Weston’s 1979 review99 of the H2 þ Cl reaction, important new theoretical calculations of the PES of this reaction100,101 have been published along with theoretical values of the absolute rate and the kinetic isotope effect102,103 and experimental data.104 The experimental data for some of the isotopic reactions cover the range from 250 to 3000 K. The theoretical calculations102,103 use quantum dynamics rather than transition state theory. Earlier calculations using quantum dynamics102 with the G3 surface100 differ insignificantly from variational transition state theory with multidimensional tunneling. The theoretical BW surface, designated as BW2,101 given in Table 1.8, is a modified ab initio quantum calculation. It includes an empirical correction to the electron correlation energy to bring the computation of BW1, the ab initio calculation, into quantitative agreement with the dissociation energies of H2 and HCl. A comparison

Theoretical Basis of Isotope Effects from an Autobiographical Perspective 50

23

HHCL TTCL

40 30

HHCL OTCL+TDCL

k1 / k2

20

HHCL ODCL

10

HHCL HTOL+THCL

5

HHCL HDDL+DHCL

2

3.1

3.3

3.5 3.7 3 10 /(T°K)

3.9

1.8

4.1

DHCL HDCL

1.5 1.2

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3

10 /(T°K)

FIGURE 1.4 Comparison of calculated and experimental isotope effects in the H2 þ Cl reaction. Solid lines, empirical PES with Eckart tunneling; dashed lines, empirical Sato PES with Wigner tunneling; points, experimental. (Reproduced from Stern, M. J., Persky, A., and Klein, F. S., J. Chem. Phys., 58, 5697– 5706, 1973. With permission.)

of the theoretical calculations of the absolute rate with the experimental values of the H2 þ Cl reaction is given in Figure 1.5. Also included in Figure 1.5 are calculations based on the G3 and BW1 surfaces. The agreement of calculations from the BW2 surface with the experimental data is excellent. Comparison of the BW2 with the BKWW and SPK surfaces illustrates the utility of deriving PESs from kinetic isotope effects. Recently Michael et al.105 have reported new experimental data for the reactions of D þ H2 and H þ D2 in the temperature range 1150 –21008 K. The absolute rate constants for these reactions now span 8 orders of magnitude! Arrhenius plots of the logarithms of the rate constants vs. 1=T are not linear due to the Boltzmann excitation terms, tunneling and the fact that quantum dynamics are necessary to describe the low temperature rates. A new theoretical calculation of the PES for this important reaction106 is claimed to be accurate to within ^ 10 cal mol21. Theoretical calculations based on this surface which used quantum dynamics rather than transition state theory107,108 are compared with experiment in Figure 1.6. The solid lines are, respectively, the experimental D þ H2 and H þ D2 reactions. The dotted lines, which are hardly distinguishable from the solid lines, are the theoretical calculations. The excellent agreement between theory and experiment brings to a close 75 years of experimental and theoretical work on this reaction, an important milestone in the history and development of gas phase kinetics. The authors of the recent works point out that the new PES for H3 “ranks with earlier solved problems in molecular quantum mechanics, the short

24

Isotope Effects in Chemistry and Biology 1θ−11

k [cm°/s]

1θ−12

1θ−13

surface I surface II

1θ−14

G3 (Ref.1) Miller and Gordon Kumaran et al. Kita and Stedman Lee et al.

1θ−15

Westenberg and de Haas

1

2

3 1000/T [1/K ]

4

FIGURE 1.5 Plot of the absolute rate constant, k; as a function of temperature for the H2 þ Cl reaction. Solid line, calculated from the BW2 potential energy surface. (Reproduced from Manthe, U., Bian, W., and Werne, H-J, Chem. Phys. Lett., 313, 647– 654, 1999. With permission.)

1E-10 1E-11

k /cm3 molecule−1 s−1)

1E-12 1E-13 1E-14 1E-15 1E-16 1E-17 1E-18 1E-19

0

10

20

30 40 10000 K /T

50

60

FIGURE 1.6 Plot of the absolute rate constants for D þ H2 (top) and H þ D2 (bottom) reactions. Solid lines, experimental; dashed lines, theoretical. (Reproduced from Michael, J. V., Su, M-C., and Sutherland, J. W., J. Phys. Chem., 108, 432– 437, 2004. With permission.)

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

25

list which includes the electronic spectra of H2 and He and the vibrational – rotational spectra of H2, þ 108 Hþ 2 and H3 .”

VI. CONDENSED MATTER ISOTOPE EFFECTS The first theoretical treatment of isotope effects, published in 1919,3,4 dealt with the difference in vapor pressures of monatomic isotopomers in a solid with a Debye frequency distribution. The partial separation of the neon isotopes by fractional distillation in 19315 was the first demonstration of an isotope effect on the physical or chemical properties of matter. In the design of their experiment to search for a heavy isotope of hydrogen, Urey, Brickwedde, and Murphy6 made an estimate of the enrichment of an isotope of mass 2 above natural abundance that could be achieved by a Rayleigh distillation of liquid hydrogen. For the calculation of the elementary separation factor, the vapor pressure ratio, they assumed a Debye solid at the triple point. They neglected effects from rotation and internal vibration consistent with the known thermodynamic and spectroscopic properties of liquid and solid hydrogen. It is interesting to call attention to the fact that K. F. Herzfeld pointed out to them (see footnote 16 of Ref. 6) that the asymmetry in a molecule like HD would perturb the rotational energy states of the condensed phase. I first became aware of this insight by Herzfeld in writing this chapter. Progress on the development of a general theory of condensed matter isotope effects (CMIE) started in the late 1950s. During the 1930s and 1940s experimental results on vapor pressure isotope effects (VPIE) and liquid vapor fractionation factors (LVFF) were reported. These results are summarized elsewhere.34,109 CMIE are an order of magnitude smaller than chemical exchange fractionation factors. For instance the logarithm of the vapor pressure ratio of H2O to HDO is 0.07 at 258C. The logarithm of the chemical exchange factor for the reaction HD þ H2O ¼ HDO þ H2 is 1.59, also at 258C. The intermolecular forces, which are largely responsible for the VPIE, are much weaker than the intramolecular forces, which determine the chemical fractionation. Generally, the light isotope has a higher vapor pressure than the heavier one. This is due to its larger zero point energy and smaller heat of vaporization. However, for deuterocarbons the general rule is that the deutero isotopomer has a higher vapor pressure than the protio isotopomer. This was explained by Topley and Eyring110 as due to the red shift of the C – H vibrations of the molecule on condensation. Later it was found that at low temperatures the VPIE is normal, a topic to which we will return. The chapter by W. Alexander Van Hook on CMIE111 in this volume presents a concise summary of the theory of VPIE, nonideal behavior of isotopic mixtures and other interesting thermodynamic and spectroscopic properties of isotopomers. Earlier reviews of the experimental data and theory of CMIE are given in Refs. 34,35,109– 114. The thermodynamic properties related to the CMIE of isotopomers are all related to the free energy differences between the isotopomers in the condensed and vapor phases. These free energy differences are directly related to VPIE. In this chapter I add an historical perspective to the earlier reviews and the chapter by Van Hook. I became interested in the subject of CMIE in the early 1950s in connection with the possibility of large scale production of heavy water by low temperature distillation of liquid hydrogen. The vapor pressures of solid and liquid H2, HD, D2 and T2 were known.115,116 de Boer117 suggested that the VPIE of simple liquids could be explained by a quantum parameter Lp ¼ h=s ðm1Þ1=2 ; where s is the range of the intermolecular force, m is the molecular weight and 1 is the depth of the intermolecular potential. The VPIE was proportional to Lp2 ; in accordance with the work of Herzfeld and Teller.118 The latter follows from the formulation of the VPIE in terms of the Wigner first quantum correction to the classical partition function. Although the de Boer method was successful in predicting the properties of 3He from 4He, it failed for all the isotopomers of H2.119 In particular the de Boer theory, like all theories in which the VPIE depends on the molecular weights and moments of inertia, predicts that the vapor pressure of HD is related to the vapor pressures of H2 and D2 by the relation PðHDÞ3 ¼ PðH2 Þ £ PðD2 Þ2 in contrast to the experimental

26

Isotope Effects in Chemistry and Biology

finding PðHDÞ2 ø ½PðH2 Þ £ PðD2 Þ : The de Boer method also failed to correlate the vapor pressures of the homonuclear diatomic isotopomers H2, D2 and T2. In support of the de Boer method was the claim by Libby and Barter120 that HT and D2 had the same vapor pressures. The lack of a general theory to explain the vapor pressures of isotopomers was summarized by Urey in his compendium “Thermodynamic Properties of Isotopic Substances”18 with the following: “The differences in the vapor pressures of other compounds of protium and deuterium” (other than H2) “have been observed but no satisfactory theory with regard to these differences exists. With the exception of the hydrogens and the neons, none of these vapor pressure differences can be satisfactorily related to the translational or sound vibrations of the solids and liquids. The isotopic compounds of the hydrogens differ in heats of fusion, vaporization, molar volumes, heats of solution and many other ways. The differences in the hydrogen compounds have prompted the study of similar differences between the isotopic compounds of other elements, and though the differences are very much smaller, they show similarities to the differences in the hydrogen compounds.” Concurrent with my work on the heavy water problem, I turned my attention to a reconsideration of isotope effects with small quantum corrections. I realized there was more in Equation 1.33 than was recognized in the original Bigeleisen – Mayer paper.15 The first result was the derivation of the rule of the geometric mean.57 In that paper I showed that within the first quantum correction, the rule of the geometric mean also applied to VPIE. It was a small step to question the validity of the Libby – Barter results on HT. Consider the hypothetical liquids H2, HT and D2 at high enough temperatures such that the first quantum correction is applicable. The mass P effect for all motions according to Equation 1.33 depends on ð1=m0i 2 1=mi Þ where the sum runs over all atoms in the molecule. Within the first quantum correction the mass effect is not a function of the molecular weight, the moment of inertia and the reduced mass. These terms are appropriate for the use of independent translation, rotation and internal coordinates. From the sums P ð1=m0i 2 1=mi Þ; H2 2 HT ¼ ð1 2 13 Þ ¼ 2=3 and H2 2 D2 ¼ 2ð1 2 12 Þ ¼ 1 it is obvious that HT behaves as a light isotopomer compared with D2. It should have a higher vapor pressure than D2. An effect that exists in the first quantum approximation must be manifest at lower temperatures. I, therefore, decided to test this prediction by repeating the Libby –Barter experiment. At this time I was on the staff of the Chemistry Department of BNL. We did have facilities for the production of liquid hydrogen, but it would have taken a year or more to assemble and fabricate the equipment necessary to carry out the experiment. I decided it would be advantageous to carry out the experiment at Los Alamos where they had an outstanding cryogenic laboratory and facilities for handling tritium at any level within the same laboratory. I approached R. D. Fowler, Director of the CMF Division, and E. F. Hammel, Jr., Group Leader of the Cryogenic Section of the CMF Division, in the spring of 1955 about the possibility of doing the experiment at Los Alamos. They supported my proposal; Eugene C. Kerr of the cryogenic group volunteered to join me in the experiments. Approval of the experiment did not require any additional review either by Los Alamos management or AEC Division of Research. I did require approval from BNL management for me to continue as a BNL employee while working at Los Alamos. My entire family have fond memories of our stays in Los Alamos in the summers of 1955 and 1956. The experiments on the LVFF of HT in H2 were carried out in the summer of 1955. Care was taken in the experiments to achieve isotopic equilibrium between the liquid and vapor phases. To achieve this, the vapor phase was bubbled through the liquid. A countercurrent heat exchanger minimized temperature changes due to the gas recirculation. Equilibrium was approached from both enriched and depleted gas phases compared with the liquid. We found the distribution

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

27

coefficient (LVFF) was 2.00 ^ 0.01 at 20 K121 in good agreement with the theoretical calculation R ¼ 2:03:119 The known vapor pressure ratio PðH2 Þ=PðD2 Þ is 2.89 at 20 K. A similar experiment on the LVFF of DT in D2 was carried out in the summer of 1956, where the value of R ¼ ðXDT ÞL =XðDTÞV where X is the mol fraction of DT, at 20 K is 1.303.122 In both our HT and DT experiments the mol fraction of tritium was approximately 1028. The values reported by Libby and Barter from their Rayleigh distillation experiments are 3.0 ^ 0.06 and 2.1 ^ 0.05, respectively. The fractionation factors measured by Bigeleisen and Kerr121,122 have been confirmed by experiments related to the design of fuel cycles for fusion reactors and the decontamination of the heavy water moderator in power reactors. A plot of the experimental data of ‘n R vs. 1=T 2 for the solution of HT in H2 is given in Figure 1.7.122 For comparison I show a value for ‘n R for a dilute solution of D2 in H2 calculated from the vapor pressures of the pure substances after correction for nonideal solution behavior. The major correction comes from the gas imperfection. The deviation of the behavior of the heteronuclear diatomic molecule, HT, from that of a homonuclear diatomic molecule was suggested by Herzfeld.6 The internal vibration in liquid hydrogen differs imperceptibly from that in the gas. The rotation in both the liquid and the gas are free and do not contribute to the LVFF. The difference between the heteronuclear and the homonuclear comes from the coupling of the rotation with the translation in the liquid. Consider two neighboring HT molecules in the liquid. The translational motion of each molecule can be described in terms of its center of gravity. The interaction energy between the two molecules is described by the mutual distance of the center of force, which is measured from the geometric 1.40

1.20

1.00

0.80

0.60

0.40

0.20

0

10 Te (H2)

20

30 4 2 10 / T

40

50

FIGURE 1.7 Fractionation factor of HT between solution in liquid H2 vs. vapor as a function of temperature. R ¼ XHTðliquidÞ =XHTðvaporÞ : Upper dotted line is calculated for a nonideal solution of D2 in H2(liquid). (Reproduced from Bigeleisen, J. and Kerr, E. C., J. Chem. Phys., 39, 763– 768, 1963. With permission.)

28

Isotope Effects in Chemistry and Biology

centers of the two molecules. Thus, in the heteronuclear molecule, there is a torque exerted on the molecule from the fact that these two centers are not coincident. A quantum mechanical calculation of this perturbation was first carried out by A. Babloyantz in Prigogine’s laboratory.123 She calculated the zero point energies for a cell model of harmonic oscillators and a smooth intermolecular potential. Theory shows that the effective mass, Meff ; of an heteronuclear diatomic molecule is related to the mass, M; of the homonuclear molecule by the relation ð1=Meff Þ1=2 ¼ ð1=MÞ1=2 {1 þ a½ðm1 2 m2 Þ2 =m1 m2 }

ð1:50Þ

m1 and m2 are the atomic masses of the heteronuclear molecule. The values of a calculated by Babloyantz were of the correct sign but large compared with values derived from the experimental data. The use of an anharmonic cell model gave good agreement with the value of a derived from the experimental data.124 In considering the theory of CMIE, I concluded that the theoretical work prior to 1955 could be simplified and significant insight brought to understanding the basic origins of these isotope effects by the introduction of the reduced partition function ratio for the condensed phase.33 My first effort was to review the data of Keesom and Haantjes125 on the vapor pressures of the neon isotopes. In the preface to their paper they cite the derivation by Otto Stern of the VPIE for a monatomic substance within the first quantum approximation. The result is ‘nðP 0 =PÞ is proportional to 1=T 2 : To my surprise they never made such a correlation. They plotted ‘nðP 0 =PÞ vs. 1=T as one would expect from the van’t Hoff equation. When I made the 1=T 2 plot I found that the data conformed to 0

2

‘nðP =PÞ ¼ A=T þ C

ð1:51Þ

where C was zero for the liquid, but finite for the solid. An extrapolation of the data for the solid should give C equal to zero at 1=T 2 ¼ 0: Here was an experimental result by an established and well respected scientist from the famous Kammerling-Onnes Laboratory at the University of Leiden that was in disagreement with the most fundamental theoretical results of Lindemann, Stern, and Herzfeld and Teller. I decided that a reinvestigation of the VPIE of the neon isotopes, 20 Ne/22Ne, was called for. I designed a cryostat based on the basic designs of Giauque and Egan126 and Johnston et al.127 The measurements would consist of differential vapor pressure measurements of enriched samples along with the absolute vapor pressure of a sample of natural abundance. My design was based on the fact that there would be small quantities of enriched isotopes available to me; temperature equilibration between samples was of utmost importance as well as temperature stability during the time of measurements. At the Amsterdam Symposium on Isotope Separation in April 1957 I was approached by Jules Gueron, head of the physical chemistry section of CEA at Saclay, France, about the possibility of one of his scientists working with me at Brookhaven. Gueron had as his objective that Etienne Roth work on a problem that would be suitable for a Ph.D. thesis. Etienne was a graduate of the Ecole Polytechnique; his graduate studies were interrupted by WWII. He spent the war years in Montreal as a member of the French team working on the Canadian atomic energy program. Etienne joined me in 1958; we assembled the cryostat and we made differential measurements between a sample of 99.8% 20Ne and one of 72.2% 22Ne. The results were extrapolated to pure isotopic composition with the assumption of Raoult’s Law. Keesom and Haantjes had shown it to be applicable to mixtures of 20Ne and 22Ne. There was curvature in a plot of ‘n½ðP20 NeÞ=ðP22 NeÞ vs. 1=T 2 ; the data extrapolated to ‘nðP20 NeÞ=ðP22 NeÞsolid equal to zero at 1=T 2 equal to zero.128,129 Although our measurements were made with a Model “T” cryostat and differential pressures were measured on millimeter ruled graph paper with oil as the manometric fluid, they had much greater precision than the measurements of Keesom and Haantjes. The data on liquid neon129 were in quantitative agreement with the LVFF determined by Boato et al.,130 after the latter were corrected for gas imperfections.131 Direct confirmation of our work on the VPIE of the liquid was made

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

29

by Furukawa,132 who had highly enriched samples and more precise manometric equipment. The triple point pressure of 20Ne measured by Furukawa agreed within 0.02 mm Hg with that measured by Bigeleisen and Roth. The one measured by Clusius et al.133 differed by 0.20 mm Hg. The zero point energy of solid neon determined from VPIE was in excellent agreement with the theoretical calculation of Bernardes.134 The curvature in the 1=T 2 plot was due to the fact that the lattice vibrations were anharmonic and there were appreciable corrections to the 1=T 2 term from the 1=T 4 term. Etienne returned to Paris at the end of 1959, wrote up his work and presented his thesis to the Faculty of Sciences of the University of Paris on 22 March, 1960. I did not see the thesis prior to the examination nor was I invited to the final examination. Was this because I did not have academic rank? Later I learned that it is not just academic rank that matters in France. It was also a question of where. In the 1960s Etienne sent a junior member of his staff, Marc Dupuis, to the United States for graduate work in theoretical chemistry. He successfully completed his degree with Prof. Lars Onsager at Yale. When Dupuis returned to France and applied for a faculty position, he was asked about his educational credentials. He had a degree from the Ecole Polytechnique, which was an accepted credential. His degree with Onsager was not recognized! Was there anyone in all of France then active in the area of the physical sciences of stature comparable to Onsager? This bureaucratic chauvinism has not diminished the friendship that exists between Etienne and me. The molecular theory of CMIE33 was supported by the work on the isotopomers of hydrogen.121,122,124 The theory also predicted differences in vapor pressures of isotopic isomers. The first system we studied were the isomers 15N14N16O and 14N15N16O. These were shown to differ in their vapor pressures135 due to the hindered rotation in liquid N2O. The next set of isotopic isomers studied were cis-, trans-, and gem dideuteroethylene. The oil differential manometer in the “Model T” apparatus of Bigeleisen and Roth was replaced by a system of interconnected mercury manometers so that the difference between the vapor pressure of C2H4 and trans-dideuteroethyelene and either cis-or gem-dideuteroethylene could be measured simultaneously.136 The order of the vapor pressures were trans . cis ø gem . C2H4. At 120 K the cis isomer has a higher vapor pressure than the gem isomer. The order of these two is reversed at 170 K. The coefficients A and B in the A – B equation, ‘nðP 0 =PÞ ¼ A=T 2 þ B=T; (see Equation 35 of Ref. 111), are different for all three dideuteroethylenes. The differences in the A terms is due to the different moment of inertia of the three isotopomers. That all three liquid dideuteroethylenes have higher vapor pressures than C2H4 in the above temperature range is due to the B terms which have negative values for all three dideutero-isomers. These are due to the red shift of the C – H vibrations in the liquid compared with the gas. In order to dissect the A terms into their translational and rotational components Marvin 137 Stern and Alexander Van Hook measured the LVFF of 13CH12 The 2 CH2 and the VPIE of C2H3D. A terms of these two isotopomers should differ only in their rotational components. Marvin and Aleck convinced me that their data was inconsistent with a value of Bðd 2 2Þ=Bðd 2 1Þ equal to 2. I should have realized this from the fact that the B values for the three d 2 2 isotopomers were different. Once I accepted their analysis, I realized that there was vibrational –rotational coupling in the liquid, which is absent in the ideal gas, and this was responsible for the deviation of the Bðd 2 2Þ=Bðd 2 1Þ ratio from 2. I was making preparation to leave for a year’s sabbatical in Switzerland and I suggested that Marvin and Aleck pursue this problem with Max Wolfsberg, who was independently working on the general problem of the effects of interactions on VPIE.138 Stern, Van Hook and Wolfsberg139 developed the general theory of the perturbation of all modes, including the internal vibrations, in a condensed phase due to the fact that the translation and rotation are mass dependent coordinates. The use of different coordinates for different isotopomers leads to differences in their potential energy values, contrary to the Born– Oppenheimer theorem. They calculated the differences of the internal vibrations between the liquid and vapor, the B values, for each of the three dideuteroethylenes and the deviation of the ratios of these B values to that of the mono-deutero-in quantitative agreement with experiment. One would think that this closed the chapter on the CMIE of the isotopomers of ethylene. When I returned from sabbatical leave in 1963, I realized that further progress in the experimental

30

Isotope Effects in Chemistry and Biology

determination of VPIE would require major improvements in the operation of the “Model T ” and the accuracy that could be attained with that apparatus. The differences between the isomers of dideuteroethylene were 1022%, which was the limit of the measurements. It was a time when major advances in high vacuum techniques, cryogenic equipment and pressure transducers came through the space program. In collaboration with Frank Brooks of the Cryogenic Division of the Accelerator Department of BNL, I designed the “Rolls Royce” cryostat for VPIE to cover the temperature range 4 –300 K. Together with the new differential pressure measuring equipment, a precision of 1023% in VPIE was achieved.140 This was an order of magnitude better than the state of the art at the time. I decided to test this new equipment against our former results with the dideuteroethylenes. We had the original samples through which Ribnikar and Van Hook had determined the differences between the three dideutero-isomers. To our surprise, Bodo Ribnikar, who had returned from Belgrade for his second appointment at BNL, found that the trans isomer had the smallest vapor pressure of three isomers! We had to resolve this discrepancy before we could proceed with any other work. Attempts to purify the samples by bulb-to-bulb condensation did not change the result. We decide to start from scratch and synthesize new samples. By this time we had adopted the method of low temperature gas chromatography introduced by Van Hook141 for the chemical purification of small samples with high recovery. The new samples reproduced the original data of Ribnikar and Van Hook quantitatively. It was apparent that the stored sample of the transdideuteroethylene had picked up an impurity with a vapor pressure smaller than the ethylene. This was then confirmed by gas chromatographic analysis. The journey with the VPIE of the isotopomers was concluded with the complete redetermination of all the data with the “Rolls Royce” equipment.142 The new data showed that whereas the A – B equation accurately fitted the Ribnikar –Van Hook data; the new data showed that there was curvature in a plot of T ‘nð fc =fg Þ vs 1=T; Figure 1.8. Not surprisingly, the translation and rotation in the liquid were not harmonic. This was treated in the final analysis by Van Hook’s method141 of the use of temperature dependent F matrices for all coordinates, internal as well as external. The results of this final analysis,

3.0

Series 1−9 (1967) Series 23−44 (1967) + B, R and VH (1961)

2.0

trans− C2H2D2 / C2H4 T In (fc /fg)

1.0

0 ++

−1.0 −2.0

−3.0

+

+

+

++

500

6

+

+ ++ ++

100

7

+

+

++ + ++

10

8 3 10 /T

1

9

0.1

10

FIGURE 1.8 Plot of T ‘nð fc =fg Þ of trans-ethylene-d2 as a function of temperature. Series 1 – 9 (1967), Ref. 142; BR & VH (1961), Ref. 136. (Reproduced from Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., J. Chem. Phys. 66, 1689– 1700, 1977. With permission.)

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

31

reproduced in Table 4.7 of Van Hook’s chapter, were in quantitative agreement with experiment. I succeeded in reaching the end of this journey only through “the inspiration from teachings of the late Claude W. Dukenfield.”142 I decided that my next experimental study of VPIE should be directed at the isotopomers of argon. There were earlier measurements on the liquid that covered the narrow range 84 –88 K143 and by fractionation between the vapor and the condensed phase.130,144 I had already shown that isotopic equilibrium was not reached between the solid and the vapor in LVFF experiments.129 The “Rolls Royce” was ideally suited for further investigation of the CMIE in argon. This was important in order to relate the CMIE in liquid and solid argon with the extensive theoretical and experimental literature on the structure and thermodynamic properties of liquid and solid argon. Our first results were disappointing. We could not reproduce the absolute triple point pressure of normal argon. Our problem was solved as a result of a discussion with Bob Sherman of the Los Alamos Laboratory. For the absolute pressure in this and subsequent experiments we used a Texas Instrument Bourdon gauge, which we calibrated against a mercury manometer. We had an early production gauge. Bob pointed out that the early gauges had a periodic error in the screw, which turned the quartz capillary. There were large errors from this periodic error, which were superimposed on a general exponential correction. These could only be detected by very close incremental pressure calibrations. We then recalibrated the quartz Bourdon gauge against a calibrated dead weight gauge and found the absolute triple point pressure of normal argon in agreement within ^ 0.02 K of the accepted literature value. During the course of the measurements with the “Rolls Royce” cryostat the temperature difference between the isotopic samples drifted by less than 1024K. Our argon measurements spanned the range from 62.8 K in the solid to 101 K in the liquid, where the absolute vapor pressure of liquid argon is 3.6 atmospheres.145 This was the limit to which we were able to make absolute vapor pressure measurements with the necessary precision. Our VPIE measurements on the liquid were in excellent agreement with those of Clusius et al.,143 which covered a very limited temperature range. The value of the mean second derivative of the intermolecular potential, k72 U p l; in liquid argon agreed well with calculations based on molecular dynamics. Values of k72 U p l calculated from experimental values of the radial distribution function were in poor agreement. It was desirable to measure CMIE of liquid argon up to the critical point. For this purpose a cryostat was built at the University of Rochester, to where I had relocated starting in the fall of the 1968 –1969 academic year. The new cryostat was similar in design to the “Rolls Royce” cryostat. The sample container in the VPIE cryostat was replaced by a container for liquid and a vapor recirculation system similar in design to the one used at Los Alamos for the HT and DT work. Measurements were made of ‘n a for the isotopes 36Ar/40Ar from the triple point to within 0.1 K of the critical temperature, 150 K.146 In the temperature range 84– 100 K, which overlapped the VPIE measurements,145 the LVFF measurements of T ‘n a were systematically 5% lower than those derived from VPIE data. This difference was more than 10 times the precision and accuracy claimed for the VPIE data and 5 times that of the LVFF data. We were prescient when we pointed out in the original paper “We are at a loss to explain this small discrepancy, amounting to 3 £ 1024 in ‘n a; other than to suggest that there may be isotope effects on mixing not considered to date. We do not rule out a systematic experimental artifact which has escaped us.” An artifact related to the effect of recirculation of the vapor was suggested by me in a note I prepared for publication. The referee kindly advised me informally to withdraw my note from publication. He included a preprint147 of his revision of the Prigogine theory of the isotope effect on mixing.148 The revised theory quantitatively explained the difference between the VPIE and LVFF factors. This nonideal behavior on mixing liquid argon isotopes was subsequently directly confirmed by Rebelo and coworkers.149 The scaling law exponent for k72 U p ll 2 k72 U p lv was shown to be equal to that of the density law exponent of the density difference between liquid and vapor.150 The correlation of the CMIE of the condensed rare gases with the structure of the liquids is given in Section II.C.4 of the chapter by

32

Isotope Effects in Chemistry and Biology

W. Alexander Van Hook and references therein. Using integral equation theory Lopes et al.151 have shown the values of k72 U p ll for all the rare gases can be calculated in good agreement with the values derived from VPIE and LVFF data over the entire liquid range. They show that k72 U p ll ; as are thermodynamic properties, is a scalable quantity for the liquid rare gases. Although the Lennard-Jones 12-6 potential suffices to calculate the energy, pressure and mean value of the second derivative of the intermolecular potential, it fails to reproduce the VPIE of solid neon and argon. For these simple solids the VPIE calculated from a 13-6 potential152 reproduces the experimental data.145 Although ab initio calculations of the potential energies of polyatomic molecules suffice to calculate both equilibrium and kinetic isotope effects, such calculations have not even been attempted for condensed phase isotope effects. The condensed phase isotope effects are an order of magnitude smaller than the gas phase ones. The intermolecular potential near the minimum of the intermolecular potential is much more anharmonic than that of the intramolecular potential. There is the interplay of effects, usually of opposite sign, from the internal and external modes. Finally, there are the contributions from coupling between external modes amongst themselves and between these modes with the internal modes. To date almost all calculations of condensed phase isotope effects are carried out within the harmonic oscillator approximation of a cell model, with one molecule per unit cell. Anharmonic corrections have been applied through the use of temperature dependent F matrices. The F matrices are constructed in part from spectroscopic data combined with experimental CMIE data. Progress toward a more general approach than the single molecule per unit cell was made by Myung Lee’s successful calculation of the lattice modes of the isotopomers of CO2 consistent with the structure of the solid, 4 molecules per unit cell.153 The calculations of the VPIE of solid 13C16O2 and 12C18O2 compared with 12C16O2 based on 4 molecules per unit cell differ insignificantly from those with one molecule per unit cell, which agree with experiment.154 Extension of the method was made to the isotopomers of N2O.155A more general treatment of the anharmonic effects on VPIE and LVFF was made by relating the temperature dependence of both the A and B terms in Equation 1.51 to the linear difference between the densities of the liquid and vapor over most of the entire temperature of the liquid.146,150,156

ACKNOWLEDGMENTS My research on the theory of isotope effects began in November 1943 as a classified project at the SAM Laboratory of Columbia University under Contract W 7405-eng-50 with the Manhattan Project. The Argonne National Laboratory kindly served as a conduit for the declassification and open publication of this work in 1947. This chapter summarizes what I have learned over a period of 60 years about the theoretical basis of isotope chemistry. The learning process was greatly enhanced by collaboration with the following colleagues in chronological order: Maria Goeppert Mayera Max Wolfsberg Etienne Roth Marvin J. Sterna James T. Phillips Frederic Mandel Anthony M. Popowicz a

Lewis Friedman Eugene C. Kerr Slobodan V. Ribnikar Takanobu Ishida Carl U. Linderstrom-Langa Yumio Yato Luis P.N. Rebelo

Ralph E. Weston, Jr, Fritz S. Kleina W. Alexander Van Hook William Spindela Myung W. Leea Masahiro Kotaka

Deceased.

From July 1948 through January 1987 my research was supported for the most part by the USAEC and its successors ERDA and DOE. Additional support came from an unrestricted grant from ACS – PRF, and fellowships from the National Science Foundation and the John Simon Guggenheim Foundation. Institutional financial support came from the University of Chicago,

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

33

the University of Rochester and the State of New York, Stony Brook. I had the responsibility and pleasure of mentoring a total of three graduate students during my career. For the conduct of the experiments cited in this chapter of which I am one of the authors, I had the able assistance of the late John Densienski from 1966 to 1968 at Brookhaven National Laboratory and William Watson at the University of Rochester from 1968 to1978. In the preparation of this manuscript I have received assistance from Prof. Amnon Kohen and Todd Fleischmann in the compilation of the bibliography.

APPENDIX 1A. SOME PROPERTIES OF THE G AND H MATRICES AND OF THE EIGENVALUES OF ISOTOPOMERS X X li ¼ mi aii ;

li ¼ 4p2 n2i ; X ij

dl i ¼

X

i

d li ¼

X i

ðm0i 2 mi Þaii ;

X ij

li ¼

X

fij gij ;

ij

fij dgij

ij

dli0 ¼ ðm0i 2 mi Þfii ;

li0 ¼ 4p2 n2i0 ¼ hii0 ¼ gii fii ;

lH 2 lIl ¼ 0;

H ¼ FG; X

Classification of the G0 F 0 G0 F 2 G1 F 1 G2 F 0 G1 F 2 G2 F 2

X

P

dl2i ¼

X i

dli0 ¼ ð1=m0i 2 1=mi Þaii

X j li ¼ TrlHj l

ðm02i 2 m2i Þa2ii þ 2ðm0i 2 mi Þ

X

mj a2ji

j

ð1:52Þ

dl2i in terms expressed in internal coordinates.68,70

¼ g2ii fii2 statistical mechanical correction and kinetic energy coupling. ¼ F matrix coupling of internal coordinates. ¼ correction to the diagonal element approximation of the H matrix. ¼ kinetic energy coupling of internal coordinates. ¼ correction to the diagonal element approximation of the H matrix. ¼ correction to the diagonal element approximation of the H matrix due to the interactions of three, four coordinates.

The superscript is the order of the off diagonal term in the j ¼ 2 correction to the diagonal element approximation. For a complete enumeration of the G and F matrix elements in each of these terms see Appendix A and Equation 4 of Ref. 68.

REFERENCES 1 Soddy, F., The chemistry of mesothorium, J. Chem. Soc., 99, 72 – 83, 1911. 2 Fajans, K., The periodic system of elements, the radioactive transformations, and the structure of the atom, Physik. Z., 16, 456– 486, 1915. 3 Lindemann, F. A., Note on the vapor pressure and affinity of isotopes, Philos. Mag., 38, 173– 181, 1919. 4 Lindemann, F. A. and Aston, F. W., The possibility of separating isotopes, Philos. Mag., 37, 523– 535, 1919. 5 Keesom, W. H. and VanDijk, H., Possibility of separating neon into its isotopic components by distillation, Proc. R. Acad. Sci. Amsterdam, 34, 42 –50, 1931.

34

Isotope Effects in Chemistry and Biology 6 Urey, H. C., Brickwedde, F. G., and Murphy, G. M., Phys. Rev., 39, 164– 165, 1932. Urey, H. C., Brickwedde, F. G., and Murphy, G. M., The hydrogen isotope of mass 2 and its concentration, Phys. Rev., 40, 1 – 15, 1932. 7 Urey, H. C. and Rittenberg, D., Some thermodynamic properties of the H1H2, H2H2 molecules and compounds containing the H2 atom, J. Chem. Phys., 1, 137– 143, 1933. 8 Rittenberg, D. and Urey, H. C., The thermal decomposition of deuterium iodide, J. Am. Chem. Soc., 56, 1885– 1889, 1934. 9 Urey, H. C. and Greiff, L. J., Isotopic exchange equilibria, J. Am. Chem. Soc., 57, 321–327, 1935. 10 Urey, H. C., Thode, H. G., and Gorham, J. E., The concentration of N15 and S34, J. Chem. Phys., 6, 296, 1938. 11 Lewis, G. N. and MacDonald, R. T., Separation of lithium isotopes, J. Am. Chem. Soc., 58, 2519– 2524, 1936. 12 Dieke, G. H. and Duncan, A. B. F., Spectroscopic Properties of Uranium Compounds, McGraw Hill Book Co. Inc., New York, 1949. 13 Mayer, M. G., Equilibrium Constants, SAM Report 4-M-159, Technical Information Division, Oak Ridge, TN April 8, 1944. 14 Hutchinson, C. A. Jr., Chemical Separation of the Uranium Isotopes, NNES, Division III, Technical Information Division, Vol. 3, United States Atomic Energy Commission, Oak Ridge, TN, 1952. 15 Bigeleisen, J. and Mayer, M. G., Calculation of equilibrium constants for isotopic exchange reactions, J. Chem. Phys., 15, 261– 267, 1947. 16 Waldmann, L., The theory of isotope separation by exchange reactions, Naturwissenschaften, 31, 205– 206, 1943. 17 Bigeleisen, J. and Ishida, T., Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys., 48, 1311– 1330, 1968. 18 Urey, H. C., Thermodynamic properties of isotopic substances, J. Chem. Soc., 562– 581, 1947. 19 Kotaka, M., Okamoto, M., and Bigeleisen, J., Anomalous mass effects in isotopic exchange equilibria, J. Am. Chem. Soc., 114, 6436– 6445, 1992. 20 Stern, M. J., Spindel, W., and Monse, E. U., Temperature dependences of isotope effects, J. Chem. Phys., 48, 2908–2919, 1968. 21 Perlman, M., Bigeleisen, J., and Elliott, N., Equilibrium in the exchange of deuterium between ammonia and hydrogen, J. Chem. Phys., 21, 70 – 72, 1953. 22 Spindel, W. and Taylor, T. I., Preparation of 99.8% nitrogen-15 by chemical exchange, J. Chem. Phys., 24, 626– 627, 1956. 23 Clusius, K., Schleich, K., and Vecchi, M., Results of low-temperature research. XXIX. Anomalous large vapor-pressure differences between N14016 and N15O18, Helv. Chim. Acta, 42, 2654– 2666, 1959. 24 Roth, E., Bedhome, A., Lefrancois, B., Le Chatelier, J., and Tillol, A., Heavy water plant of Mazingarbe, Energie Nucleaire, 10, 214– 219, 1968. 25 Woodward, R.W. et al., Y-185 Report of the Y-12 Plant, US Atomic Energy Commission, Oak Ridge, TN, 1949. 26 Palko, A. A., Drury, J. S., and Begun, G. M., Lithium isotope separation factors of some two-phase equilibrium systems, J. Chem. Phys., 64, 1828– 1837, 1976. 27 Seko, M., Miyake, T., Inada, K., and Takeda, K., Uranium isotope enrichment by chemical method, Nucl. Technol., 50, 178– 186, 1980. 28 Dujardin, T. and Longchampt, G., Review of the French Chemex Process, In Proceedings of the International Symposium on Isotope Separation and Chemical Exchange Enrichment, Fujii, Y., Ishida, T., and Takeuchi, K., Eds., pp. 161– 171, 1990. 29 Bigeleisen, J., The relative reaction velocities of isotopic molecules, J. Chem. Phys., 17, 675– 679, 1949. 30 Bigeleisen, J. and Wolfsberg, M., Theoretical and experimental aspects of isotope effects in chemical kinetics, In Advances in Chemical Physics I, Prigogine, I., Ed., Interscience Publishers, New York, pp. 15 – 76, 1958.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

35

31 Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Isotope effects on enzyme catalyzed reactions, University Park Press, Baltimore, MD, 1977. 32 Melander, L. and Saunders, W. H. J., Reaction rates of isotopic molecules, Wiley, New York, 1980. 33 Bigeleisen, J., Statistical mechanics of isotope effects on the thermodynamic properties of condensed systems, J. Chem. Phys., 34, 1485– 1493, 1961. 34 Jansco, G. and Van Hook, W., Condensed phase isotope effects, Chem. Rev., 74, 689– 750, 1974. 35 Jansco, G., Rebelo, L. P. N., and Van Hook, W., Isotope effects in solution thermodynamics: excess properties in solutions of isotopomers, Chem. Rev., 93, 2645– 2666, 1993. 36 Hoefs, J., Stable Isotope Geochemistry, Springer Verlag, Heidelberg-Berlin, 1973. 37 Taylor, H. P. Jr., O’Neill, J. R., and Kaplan, I. R., Stable Isotope Geochemistry: A Tribute to Samuel Epstein, Vol. Special Publication No. 3, The Geochemical Society, Trinity University, San Antonio, TX, 1991. 38 Wolfsberg, M., Correction to the effect of anharmonicity on isotopic exchange equilibria. Application to Polyatomic Molecules, J. Chem. Phys., 50, 1484– 1486, 1969. 39 Urey, H. C., Lowenstam, H., Epstein, S., and McKinney, C. R., Measurement of paleotemperatures and temperatures of the upper cretaceous of England Denmark, and the Southeastern United States, Bull. Geol. Soc. Am., 62, 399– 416, 1951. 40 Epstein, S., Buchsbaum, R., Lowenstam, H., and Urey, H. C., Carbonate – water isotopic temperature scale, Geo. Soc. Am. Bull., 62, 417– 426, 1951. 41 Bron, J., Chang, C. F., and Wolfsberg, M., Isotopic partition function ratios involving hydrogen, water, hydrogen sulfide, hydrogen selenide, and ammonia, Z. Naturforsch, 28, 129– 136, 1973. 42 Hartshorn, S. R. and Shiner, V. J. Jr., Calculation of H/D carbon-12/carbon-13, and carbon-12/ carbon-14 fractionation factors from valence force fields derived for a series of simple organic molecules, J. Am. Chem. Soc., 94, 9002– 9012, 1972. 43 Monse, E. U., Spindel, W., and Stern, M. J., Analysis of isotope-effect calculations illustrated with exchange equilibriums among oxynitrogen compounds, Adv. Chem. Ser., 89, 148– 184, 1969. 44 DeFrees, D. J., Hassner, D. Z., Hehre, W. J., Peter, E. A., and Wolfsberg, M., A priori calculations in isotopic exchange equilibriums, J. Am. Chem. Soc., 100, 641– 643, 1978. 45 Hout, R. F., Wolfsberg, M., and Hehre, W. J., Direct calculation of equilibrium constants for isotopic exchange reactions by ab initio molecular orbital theory, J. Am. Chem. Soc., 102, 3296– 3298, 1980. 46 Wolfsberg, M., Correction to the effect of anharmonicity on isotopic exchange equilibria, In Isotopes and Chemical Processes, Advances in Chemistry Series, 89, Gould, R. F., Ed., American Chemical Society, Washington, DC, pp. 185– 191, 1969. 47 Kleinman, L. I. and Wolfsberg, M., Corrections to the Born – Oppenheimer approximation and electronic effects on isotopic exchange equilibria, J. Chem. Phys., 59, 2043– 2053, 1973. 48 Kleinman, L. I. and Wolfsberg, M., Shifts in vibrational constants from corrections to the Born – Oppenheimer approximation: effects on isotopic exchange equilibria, J. Chem. Phys., 60, 4749– 4754, 1974. 49 Wolfsberg, M. and Kleinman, L. I., Corrections to the Born – Oppenheimer approximation in the calculation of isotope effects on equilibrium constants, In Isotopes and Chemical Principles, Rock, P., Ed., American Chemical Society, Washington, DC, pp. 64 – 76, 1975. 50 Bardo, R. D. and Wolfsberg, M., A theoretical calculation of the equilibrium constant for the isotopic exchange reaction between water and hydrogen deuteride, J. Phys. Chem., 80, 1068– 1071, 1976. 51 Suess, H. E., The equilibrium H2 þ HDO ¼ HD þ H2O and other equilibiria in the system H2-D2H2O, Z. Naturforsch, 4a, 328– 332, 1949. 52 Cerrai, E., Marchetti, C., Renzoni, R., Rosso, L., Silvestri, M., and Villari, S., Chem. Eng. Prog. Symp. Ser., 50, 271, 1954. 53 Rolston, J. H., Den Hartog, J., and Butler, J. P., The deuterium isotope separation factor between hydrogen and liquid water, J. Phys. Chem., 80, 1064 –1067, 1976. 54 Bigeleisen, J., Nuclear size and shape effects in chemical reactions. Isotope chemistry of the heavy elements, J. Am. Chem. Soc., 118, 3676– 3680, 1996. 55 Fujii, Y., Nomura, M., Okamoto, M., Onitsuka, H., Kawakami, F., and Takeda, K., An anomalous isotope effect of uranium-235 in uranium(IV)-uranium(VI) chemical exchange, Z. Naturforsch, 44, 395– 398, 1989.

36

Isotope Effects in Chemistry and Biology 56 Nomura, M., Higuchi, N., and Fujii, Y., Mass dependence of uranium isotope effects in the U(IV) – U(VI) exchange reaction, J. Am. Chem. Soc., 118, 9127– 9130, 1996. 57 Bigeleisen, J., Statistical mechanics of isotopic systems with small quantum corrections. I. General considerations and the rule of the geometric mean, J. Chem. Phys., 23, 2264– 2267, 1955. 58 Bigeleisen, J., The significance of the product and sum rules to isotope fractionation processes, In Proceedings of the Amsterdam Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North Holland Publishing Co., Amsterdam, pp. 121– 157, 1959. 59 Singh, G. and Wolfsberg, M., Calculation of isotopic partition function ratios by a perturbation theory technique, J. Chem. Phys., 62, 4165– 4180, 1975. 60 Bigeleisen, J., Quantum mechanical foundations of isotope chemistry, In Isotopes and Chemical Principles, Rock, P., Ed., American Chemical Society, Washington, pp. 1 – 28, 1975. 61 Bigeleisen, J. and Ishida, T., Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. V. Isotope chemistry and molecular structure. Simplified theory of end atom isotope effects, J. Am. Chem. Soc., 95, 6155– 6157, 1973. 62 Wolfsberg, M. and Stern, M. J., Validity of some approximation procedures used in the theoretical calculation of isotope effects, Pure Appl. Chem., 8, 225– 242, 1964. 63 Stern, M. J. and Wolfsberg, M., The absence of isotope effects in the absence of force constant changes, J. Chem. Phys., 45, 2618– 2630, 1966. 64 Bigeleisen, J., Lee, M. W., Ishida, T., and Bigeleisen, P. E., Approximation of isotopic reduced partition function ratios by the WINIMAX polynomial perturbation method (WIMPER), J. Chem. Phys., 68, 3501–3510, 1978. 65 Lee, M. W. and Bigeleisen, J., Generalized finite polynomial approximation (WINIMAX) to the reduced partition function of isotopic molecules, J. Chem. Phys., 68, 3496– 3500, 1978. 66 Nemeth, G., Gellai, B., and Jansco, G., Application of the minimax approximation to the reduced partition function of isotopic molecules, J. Chem. Phys., 54, 1701– 1708, 1971. 67 Ishida, T., Spindel, W., and Bigeleisen, J., Theoretical analysis of chemical isotope fractionation by orthogonal polynamial methods, In Isotopes and Chemical Processes, Advances in Chemistry Series, Vol. 89, Gould, R. F., Ed., American Chemical Society, Washington, DC, pp. 192– 247, 1969. 68 Bigeleisen, J., Ishida, T., and Lee, M. W., Correlation of the isotope chemistry of hydrogen, carbon, and oxygen with molecular forces by the WIMPER (2) method, J. Chem. Phys., 74, 1799– 1816, 1981. 69 Bigeleisen, J., Ishida, T., and Spindel, W., Correlation of isotope effects with molecular forces. I. The diatomic molecule, Proc. Nat. Acad. Sci. U.S.A., 53, 113– 119, 1970. 70 Ishida, T. and Bigeleisen, J., Isotope chemistry and molecular structure. Deviations from the first rule of the mean, J. Chem. Phys., 64, 4775–4789, 1979. 71 Wolfsberg, M., Massa, A. A., and Pyper, J. W., Effects of vibrational anharmonicity on the isotopic self-exchange equilibriums H2X þ D2X ¼ 2HDX, J. Chem. Phys., 53, 3138– 3146, 1970. 72 Skaron, S. A. and Wolfsberg, M., The calculation of isotopic partition functions ratios by a perturbation theory technique. 2. Dissection of the isotope effect, J. Am. Chem. Soc., 99, 5253– 5261, 1977. 73 Eyring, H. and Polanyi, M., Simple gas reactions, Z. physik. Chem., 12, 279– 311, 1931. 74 Marcellin, M. R., Contribution to the study of physical chemistry — kinetics. Part I, Ann. Phys., 3, 120– 184, 1915. 75 Pelzer, H. and Wigner, E., Velocity coefficient of interchange reactions, Z. physik. Chem., B15, 445– 471, 1932. 76 Eyring, H., Activated complex in chemical reactions, J. Chem. Phys., 3, 107– 115, 1935. 77 Evans, M. G. and Polanyi, M., Application of the transition-state method to the calculation of reaction velocities, especially in solution, Trans. Faraday Soc., 31, 875– 894, 1935. 78 Farkas, L. and Wigner, E., Calculation of the rate of elementary reactions of light and heavy hydrogen, Trans. Faraday Soc., 32, 708– 723, 1936. 79 Hirschfelder, J., Eyring, H., and Topley, B., Reactions involving hydrogen molecules and atoms, J. Chem. Phys., 4, 170– 177, 1936. 80 Yankwich, P. E. and Calvin, M., An effect of isotopic mass on the rate of a reaction involving the carbon– carbon bond, J. Chem. Phys., 17, 109– 110, 1949. 81 Willi, A. V., Isotopeneffekte bei Chemischen Reaktionen, Georg Thieme Verlag, Stuttgart, 1983. 82 Bigeleisen, J., The validity of the use of tracers to follow chemical reactions, Science, 110, 14 – 16, 1949.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

37

83 Bigeleisen, J., Isotope effect in the decarboxylation of labelled malonic acids, J. Chem. Phys., 17, 425– 426, 1949. 84 Bigeleisen, J. and Friedman, L., 13C isotope effects in the decarboxylation of labeled malonic acid, J. Chem. Phys., 17, 998– 999, 1949. 85 Lindsay, J. G., Bourns, A. N., and Thode, H. G., Influence of temperature on the intermolecular carbon13 isotope effect in the decarboxylation of normal malonic acid, Can. J. Chem., 30, 163– 165, 1952. 86 Roe, A. and Hellmann, M., Determination of an isotope effect in the decarboxylation of malonic-1C14 acid, J. Chem. Phys., 19, 660, 1951. 87 Yankwich, P. E. and Promislow, A. L., Intramolecular carbon isotope effect in the decarboxylation of liquid malonic acid near the melting point, J. Am. Chem. Soc., 76, 4648– 4651, 1954. 88 Yankwich, P. E., Promislow, A. L., and Nystrom, R. F., C14 and C13 intramolecular isotope effects in the decarboxylation of liquid malonic acid at 140.5, J. Am. Chem. Soc., 76, 5893– 5894, 1954. 89 Griggs, E. C. M., C14 intramolecular isotope effects of malonic and bromomalonic acids, Aust. J. Chem., 9, 252– 257, 1956. 90 Stern, M. J. and Wolfsberg, M., Theoretical considerations of carbon isotope effects in the decarboxylation of malonic acid, J. Chem. Phys., 39, 2776 –2777, 1963. 91 Wolfsberg, M., Isotope effects on reaction rates and the reaction coordinate, J. Chem. Phys., (1960), 21 – 22, 1960. 92 Jones, W. M., The relative rates of reaction of hydrogen and tritium hydride with chlorine, J. Chem. Phys., 19, 78– 85, 1951. 93 Wheeler, A., Topley, B., and Eyring, H., Absolute rates of reaction of hydrogen with the halogens, J. Chem. Phys., 4, 178– 187, 1936. 94 Magee, J. L., Directed valence in chemical reactions, J. Chem. Phys., 8, 677– 687, 1940. 95 Bigeleisen, J. and Wolfsberg, M., Semiempirical study of the H2Cl transition complex through the use of hydrogen isotope effects, J. Chem. Phys., 23, 1535–1539, 1955. 96 Sato, S., Potential energy surface of a system of three atoms, J. Chem. Phys., 23, 2465– 2466, 1955. 97 Bigeleisen, J., Klein, F. S., Weston, R. E. Jr., and Wolfsberg, M., Deuterium isotope effect in the reaction of hydrogen molecules with chlorine atoms and the potential energy of the H2Cl transition complex, J. Chem. Phys., 30, 1340– 1351, 1959. 98 Stern, M. J., Persky, A., and Klein, F. S., Force field and tunneling effects in the molecular hydrogenatomic chlorine reaction system. Determination from kinetic isotope effect measurements, J. Chem. Phys., 58, 5697–5706, 1973. 99 Weston, R. E. Jr., A survey of reactions involving H0 ,H00 , and chlorine atoms, J. Phys. Chem., 83, 61 – 68, 1979. 100 Allison, T. C., Lynch, G. C., Truhlar, D. G., and Gordon, M. S., An improved PES for the H2Cl system and its use for calculations of rate coefficients and kinetic isotope effects, J. Phys. Chem., 100, 13575– 13587, 1996. 101 Bian, W. and Werner, H-J., Global ab initio potential energy surfaces for the ClH2 reactive system, J. Chem. Phys., 112, 220– 229, 2000. 102 Mielke, S. L., Allison, T. C., Truhlar, D. G., and Schwenke, D. W., Quantum mechanical rate coefficients for the Cl þ H2 reaction, J. Phys. Chem., 100, 13588– 13593, 1996. 103 Manthe, U., Bian, W., and Werner, H-J., Quantum-mechanical calculation of the thermal rate constant for the H2 þ Cl ¼ H þ HCl reaction, Chem. Phys. Lett., 313, 647– 654, 1999. 104 Kumaran, S. S., Lim, K. P., and Michael, J. V., Thermal rate constants for the Cl þH2 and Cl þD2 reactions between 296 and 3000 K, J. Chem. Phys., 101, 9487– 9498, 1994. 105 Michael, J. V., Su, M-C., and Sutherland, J. W., New rate constants for D þ H2 and H þ D2 between approximately 1150 and 2100 K, J. Phys. Chem., 108, 432– 437, 2004. 106 Mielke, S. L., Garrett, B. C., and Peterson, K. A., The utility of many-body decompositions for the accurate basis set extrapolation of ab initio data, J. Chem. Phys., 111, 3806– 3811, 1999. Mielke, S. L., Garrett, B. C., and Peterson, K. A., A hierarchical family of global analytic Born – Oppenheimer potential energy surfaces for the H þ H2 reaction ranging in quality from double-zeta to the complete basis set limit, J. Phys. Chem. A., 106, 1557– 1566, 2002. 107 Mielke, S. L., Lynch, G. C., Truhlar, D. G., and Schwenke, D. W., Ab initio chemical kinetics: converged quantal reaction rate constants for the D þ H2 system, J. Phys. Chem., 98, 8000– 8008, 1994.

38

Isotope Effects in Chemistry and Biology 108 Mielke, S. L., Peterson, K. A., Schwenke, D. W., Garrett, B. C., Truhlar, D. G., Michael, J. V., Su, M-C., and Sutherland, J. W., H þ H2 thermal reaction: a convergence of theory and experiment, Phys. Rev. Lett., 91, 063201/1 – 063201/4, 2003. 109 Rabinovich, I. B., Influence of Isotopy on the Physicochemical Properties of Liquids, Consultants Bureau, New York, 1970. 110 Topley, B. and Eyring, H., Separation of the hydrogen isotopes by electrolysis. I, J. Chem. Phys., 2, 217– 230, 1934. 111 Van Hook, W., Condensed matter isotope effects, Isotope Effects in Chemisty and Biology, pp. 119– 152. 112 Bigeleisen, J., Vapor pressures of isotopic molecules, J. Chim. Phys., 60, 35 – 43, 1963. 113 Bigeleisen, J., Lee, M. W., and Mandel, F., Equilibrium isotope effects, Ann. Rev. Phys. Chem., 24, 407– 440, 1973. 114 Jansco, G., Isotope effects, In Handbook of Nuclear Chemistry, Vertes, A., Nagy, S., and Klencsar, Z., Eds., pp. 85 – 116, 2003. 115 Armstrong, G. T., Brickwedde, F. G., and Scott, R. B., Vapor pressure of the methanes, J. Res. Natl. Bur. Stand., 55, 39 – 52, 1955. 116 Grilly, E. R., The vapor pressures of hydrogen, deuterium, and tritium up to three atmospheres, J. Am. Chem. Soc., 73, 843– 846, 1951. 117 de Boer, J., Quantum theory of condensed permanent gases. I. The law of corresponding states, Physica, 14, 139– 148, 1948. 118 Herzfeld, K. F. and Teller, E., Vapor pressure of isotopes, Phys. Rev., 54, 912– 915, 1938. 119 Bigeleisen, J., Statistical mechanics of isotopic systems with small quantum corrections, Phys. Rev., 99A, 638, 1955. 120 Libby, W. F. and Barter, C. A., Vapor pressures of the tritium liquid hydrogens. Dependence of hydrogen vapor pressure on mass of the molecule, J. Chem. Phys., 10, 184– 186, 1942. 121 Bigeleisen, J. and Kerr, E. C., Vapor pressure of HT, J. Chem. Phys., 23, 2442– 2443, 1955. 122 Bigeleisen, J. and Kerr, E. C., Vapor – liquid equilibriums of dilute solutions of HT in e-H2 and DT in e-D2 from the triple points to the critical temperatures of the solutions, J. Chem. Phys., 39, 763– 768, 1963. 123 Babloyantz, A., Rotational effect on the thermodynamic properties of isotopes, Mol. Phys., 2, 39 – 48, 1959. 124 Bigeleisen, J., Quantum effects in liquid hydrogen, J. Chem. Phys., 39, 769– 777, 1963. 125 Keesom, W. H. and Haantjes, J., Vapor pressures of neon of different isotopic compositions, Physica, 2, 986– 999, 1935. 126 Giauque, W. F. and Egan, C. J., Carbon dioxide. The heat capacity and vapor pressure of the solid. The heat of sublimation. Thermodynamic and spectroscopic values of the entropy, J. Chem. Phys., 5, 45 – 54, 1937. 127 Johnston, H. L., Clarke, J. T., Rifkin, E. B., and Kerr, E. C., Condensed gas calorimetry. I. Heat capacities, latent heats and entropies of pure para-hydrogen from 12.7 to 20.3 K. Description of the condensed gas calorimeter in use in the Cryogenic Laboratory of the Ohio State University, J. Am. Chem. Soc., 72, 3933 –3938, 1950. 128 Roth, E. and Bigeleisen, J., Vapor pressures of the neon isotopes, J. Chem. Phys., 32, 612, 1960. 129 Bigeleisen, J. and Roth, E., Vapor pressure of the neon isotopes, J. Chem. Phys., 35, 68 – 77, 1961. 130 Boato, G., Scoles, G., and Vallauri, M. E., Vapor pressure of isotopic liquids. I. Ar, N, O, below the boiling-point, Nuovo Cimento, 14, 735– 747, 1959. 131 Boato, G., Casanova, G., Scoles, G., and Vallauri, M. E., Vapor Pressure of Isotopic Liquids III. Some corrections to previous papers, Nuovo Cimento, 20, 87 – 93, 1961. 132 Furukawa, G. T., Vapor pressures of natural neon and of the isotopes neon20 and neon22 from the triple point to the normal boiling point, Metrologia, 8, 11 – 27, 1972. 133 Clusius, K., Flubacher, P., Piesbergen, U., Schleich, K., and Sperandio, A., Results of lowtemperature research. XXVII. Comparison of the atomic heat, heat of fusion, and entropies of the condensed isotopes Ne20 and Ne22, Z. Naturforsch, 15a, 1 – 9, 1960.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

39

134 Bernardes, N., Theory of solid neon, argon, krypton, and xenon at 0 K, Phys. Rev., 112, 1534– 1539, 1958. 135 Bigeleisen, J. and Ribnikar, S. V., Structural effects in the vapor pressures of isotopic molecules. O18 and N15 substitution in N2O, J. Chem. Phys., 35, 1297– 1305, 1961. 136 Bigeleisen, J., Ribnikar, S. V., Slobodan, V., and Van Hook, W., Molecular geometry and the vapor pressure of isotopic molecules. Equivalent isomers cis-gem-, and trans-dideuterioethylenes, J. Chem. Phys., 38, 489–496, 1963. 137 Bigeleisen, J., Stern, M. J., and Van Hook, W., Molecular geometry and the vapor pressure of isotopic molecules. C2H3D and C12H2 – C13H2, J. Chem. Phys., 38, 497–505, 1963. 138 Wolfsberg, M., Isotope effects on intermolecular interactions and isotopic vapor pressure differences, J. Chim. Phys., 60, 15 – 22, 1963. 139 Stern, M. J., Van Hook, W., and Wolfsberg, M., Isotope effects on internal frequencies in the condensed phase resulting from interactions with the hindered translations and rotations. The vapor pressures of the isotopic ethylenes, J. Chem. Phys., 39, 3179– 3196, 1963. 140 Bigeleisen, J., Brooks, F. P., Ishida, T., and Ribnikar, S. V., Cryostat for thermal measurements between 2 to 300 K, Rev. Sci. Instrum., 39, 353–356, 1968. 141 Van Hook, W., Vapor pressures of the deuterated ethanes, J. Chem. Phys., 44, 234– 251, 1966. 142 Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., Vapor pressures of the isotopic ethylenes. V. Solid and liquid ethylene-d1, ethylene-d2 (cis, trans, and gem), ethylene-d3, and ethylene-d4, J. Chem. Phys., 66, 1689– 1700, 1977. 143 Clusius, K., Schleich, K., and Vogelmann, M., Low-temperature research. XL. The vapor pressures of Ar36 and Ar40 between the melting and boiling points, Helv. Chim. Acta, 46, 1705– 1714, 1963. 144 Boato, G., Casanova, G., and Vallauri, M. E., Vapor pressure of isotopic liquids II. Ne and Ar above the boiling point, Nuovo Cimento, 16, 505– 520, 1960. 145 Lee, M. W., Fuks, S., and Bigeleisen, J., Vapor pressures of argon-36 and argon-40 intermolecular forces in solid and liquid argon, J. Chem. Phys., 53, 4066– 4075, 1970. 146 Phillips, J. T., Linderstrom-Lang, C. U., and Bigeleisen, J., Liquid– vapor argon isotope fractionation from the triple point to the critical point. Mean Laplacian of the intermolecular potential in liquid argon, J. Chem. Phys., 56, 5053– 5062, 1972. 147 Singh, R. R. and Van Hook, W., Excess free energy in solutions of isotopic isomers. I. Monatomic species. II. Polyatomic species, J. Chem. Phys., 86, 2969– 2975, 1987. 148 Prigogine, I., Bellemans, A., and Mathot, V., Molecular Theory of Solutions, North Holland Publishing Co., Amsterdam, 1957. 149 Rebelo, L. P. N., Dias, F. A., Lopes, J. N. C., and Calado, J. C. G., Nunes da Ponte M. Evidence for nonideality in the fundamental liquid mixture (36Ar þ 40Ar), J. Chem. Phys., 113, 8706– 8716, 2000. 150 Lee, M. W. and Bigeleisen, J., Calculation of the mean force constants of the rare gases and the rectilinear law of mean force, J. Chem. Phys., 67, 5634– 5638, 1977. 151 Lopes, J. N. C., Padua, A. A. H., Rebelo, L. P. N., and Bigeleisen, J., Calculation of VPIE in the rare gases and their mixtures using an integral equation theory, J. Chem. Phys., 118, 5028–5037, 2003. 152 Klein, M. L., Blizard, W., and Goldman, V. V., Calculation of the vapor-pressure ratio of the isotopes of solid neon and argon, J. Chem. Phys., 52, 1633–1635, 1970. 153 Lee, M. W., Calculation of the lattice modes of the isotopic carbon dioxide molecules and their reduced partition function ratios, J. Chem. Phys., 62, 2094– 2097, 1975. 154 Bilkadi, Z., Lee, M. W., and Bigeleisen, J., Phase equilibrium isotope effects in molecular solids and liquids. Vapor pressures of the isotopic carbon dioxide molecules, J. Chem. Phys., 62, 2087– 2093, 1975. 155 Yato, Y., Lee, M. W., and Bigeleisen, J., Phase equilibrium isotope effects in molecular solids and liquids. Vapor pressures of the isotopic nitrous oxide molecules, J. Chem. Phys., 63, 1555– 1563, 1975. 156 Popowicz, A. M., Lu, T. H., and Bigeleisen, J., Temperature dependence of the liquid– vapor isotopic fractionation factors in methane-d3-methane and fluoromethane-d3-fluoromethane, Z. Naturforsch, 46, 60 – 68, 1991.

2

Enrichment of Isotopes Takanobu Ishida and Yasuhiko Fujii

CONTENTS I. II.

III.

IV.

V.

VI.

Overview ............................................................................................................................ 42 A. Separation Factor, Material Balance, and Cascade of Separation Stages ................ 42 Enrichment Processes......................................................................................................... 44 A. Enrichment Processes Based on Steady State Phenomena of Reversible Processes.............................................................................................. 44 1. Distillation ........................................................................................................... 50 2. Chemical Exchange............................................................................................. 50 3. Gas Centrifugation .............................................................................................. 53 B. Enrichment Processes Based on Nonsteady State Phenomena of Reversible Processes.............................................................................................. 53 C. Enrichment Based on Irreversible Processes............................................................. 53 1. Laser Isotope Separation ..................................................................................... 53 2. Gaseous Diffusion ............................................................................................... 54 3. Thermal Diffusion ............................................................................................... 55 4. Electrolysis .......................................................................................................... 55 5. Electromagnetic Method: Calutron..................................................................... 56 Separation Cascade ............................................................................................................ 56 A. Ideal Cascade: Thermodynamic Efficiency and No-Mixing..................................... 56 B. Product-End Refluxer................................................................................................. 58 C. McCabe –Thiele Diagram for Square Cascade ......................................................... 61 1. Case of Total Reflux ........................................................................................... 63 2. Case of Minimum Reflux Ratio.......................................................................... 63 D. Separative Capacity for Close-Separation, Ideal Cascade ........................................ 64 E. HETP (Height Equivalent of Theoretical Plate) ....................................................... 65 Startup of Isotope Enrichment Cascade ............................................................................ 66 A. Time-Dependence of Enrichment Profile along the Length of Cascade during Startup.......................................................................................... 66 B. Rate of Attainment of Steady-State Profile vs. Holdups .......................................... 67 Empirical Determination of HETP and Separation Factor a............................................ 67 A. By Use of Analytic Solution of Material Balance Equation under Transient Condition.................................................................................................... 67 B. From Graphical Solution of Material Balance Equation under the Condition of Zero Time-Dependence at All Stages............................................ 69 Miscellaneous Other Considerations ................................................................................. 69 A. Possible Needs of Chemical Waste Disposal............................................................ 70 B. Possibility of Failure to Achieve a High Target Enrichment ................................... 70 C. Possible Explosion of Working Material .................................................................. 71 D. Consideration of Supply for the Feed ....................................................................... 72

41

42

Isotope Effects in Chemistry and Biology

VII.

Enrichment by Nonsteady State Phenomena Involving Reversible Process .................... 72 A. Ion Exchange Isotope Separation .............................................................................. 72 B. Chromatographic Isotope Separation......................................................................... 74 C. Nonsteady-State Enrichment...................................................................................... 75 1. Enrichment Profile............................................................................................... 75 2. HETP ................................................................................................................... 77 D. Isotope Separation by Ion Exchange ......................................................................... 78 1. Boron Isotope Separation .................................................................................... 78 2. Nitrogen Isotope Separation................................................................................ 79 3. Uranium Isotope Separation................................................................................ 81 VIII. Concluding Remarks............................................................................................................... 82 Acknowledgments .......................................................................................................................... 83 References....................................................................................................................................... 83

I. OVERVIEW Similarity in the chemical and physical properties of isotopomers is the basis of isotopic tracer techniques, while isotope effect studies take advantage of the slight differences, and isotope separation works against the similarities. Isotope separation is thus one of the practical applications of the theories of isotope effects (IE), which suggest which features of molecules would cause an effective separation of isotopes. The theoretical basis of isotope effects is presented by Jacob Bigeleisen1 in the present publication and in other excellent reviews, of which Refs. 2– 18 are classic examples. Regarding the distinction between the terms isotopomers and isotopologues, “isotopomer” will be used exclusively in this chapter, because the discussions of this chapter will be equally applicable for both. The distinction becomes important only when discussion involves mechanics of interactions of molecular structures, molecular forces, and external forces.

A. SEPARATION FACTOR, M ATERIAL B ALANCE, AND C ASCADE OF S EPARATION S TAGES Because one pass of isotopic material through an isotope separation unit would not achieve a sufficiently high enrichment, many such units must be interconnected one after another to effect a practically useful joint multiplicative result. Such a system of interconnected units is called a cascade of separation stages. A separating unit for any process can be envisioned as a black box (Figure 2.1) that, in the simplest scheme, separates a single feed stream into a product stream which is somewhat enriched in the desired isotope, and a waste stream, which is somewhat depleted in that component. In Figure 2.1, F, P, and W are the total molar flows of the molecules containing the desired isotope in the feed, the product, and the waste streams, respectively, and x represents the mole fraction of the desired isotope or, more precisely, the average atom fraction of the desired isotope in the form of exchangeable state in each stream. For each stage, material conservation requires that the total amount of material fed into the stage be equal to the amounts leaving in the forms of product and waste streams (Figure 2.1) and that the quantities of the desired isotope in the two exit streams be equal to the moles of the isotope that have entered the stage: Total molecular balance: F ¼PþW

ð2:1Þ

Fxf ¼ Pyp þ Wzw

ð2:2Þ

Total isotopic balance:

Enrichment of Isotopes

43 Depleted Stream W moles Atom fraction = x w

Feed Stream

Separation Stage

F moles Atom fraction = x f P moles Atom fraction = x p Enriched Stream

FIGURE 2.1 An isotope separation stage.

The degree of separation achieved by one passage through such a unit is expressed by the singlestage (or elementary) separation factor, a, defined as the abundance ratio of the desired isotope in the product stream divided by the abundance ratio in the waste stream: Elementary separation factor:

a;

xp =ð1 2 xp Þ xw =ð1 2 xw Þ

ð2:3Þ

If a is unity, there is no separation.1 With exceptions for hydrogen isotope exchanges and laser isotope separation (LIS) and a few other notable exceptions, the order of magnitude of ða 2 1Þ of all isotope separation processes are several percent at the most. The separations for which the magnitude of a 2 1 is very small compared to unity is called the close separation, for which ln a < a 2 1: The isotope separation processes that have been studied have some merits and some disadvantages, and evaluating them for a specific isotope depends on several factors. Two most important factors are: (a) whether isotopes of light mass (e.g., deuterium), intermediate mass (e.g., nitrogen), or heavy mass (e.g., uranium) elements are to be separated, and (b) whether the quantities needed are grams per day (as in research) or multiple tons per year (as for use in power reactors). The choice depends on properties of the element, the degree of enrichment needed, and the scale and continuity of the demand; even for a given isotope, there is usually no one best method. For large scale applications, availability of feed materials, capital costs, environmental cleanliness (vide infra) and energy requirements may be the overriding considerations, while for laboratory needs simplicity of operation and/or versatility may be most important. For a large-scale production of a highly enriched isotope, the net content of the desired isotope in the feed becomes a very important issue: the number of moles of the desired isotope produced by a plant must be contained in the feed material. Since the natural abundance of desired isotope is usually very low, a large quantity of the feed material for the isotope plant must be abundantly available. The problems of feed supply and considerations regarding the processes that do not provide high throughputs or require large energy inputs or chemical refluxing will be discussed in Sections III and VI of the present chapter.

44

Isotope Effects in Chemistry and Biology

II. ENRICHMENT PROCESSES Most isotope enrichment processes fall into one of the following three categories: (1) The steady state phenomena of reversible processes: examples: distillation, chemical exchange, gaseous centrifugation. (2) The nonsteady state phenomena of reversible processes: examples: ion exchange separation of isotopes of heavy elements. (3) Irreversible processes: examples: laser isotope separation, gaseous diffusion, thermal diffusion, electromagnetic separation, electrolysis, ionic migration. Examples of applications of these processes are tabulated in Table 2.1. The separation factors listed in Table 2.1 are for the ambient conditions as quoted in the references cited in the last column of Table 2.1.

A. ENRICHMENT P ROCESSES B ASED ON S TEADY S TATE P HENOMENA OF R EVERSIBLE P ROCESSES Distillation can be regarded as an exchange process between liquid and its vapor. Except for the associated refluxer, the same theory of separation cascade can be used for both processes. In a distillation column a liquid flows downward and a gas stream flows upward, and a refluxer converts the liquid into the gas at the bottom and the gas is converted into liquid at the top. Isotope exchange takes place between the liquid and the gas in the column. When the isotope effect is normal, that is, when the vapor pressure of the heavier isotopomer is lower than that of the lighter isotopomer, the heavier isotope enriches in the liquid. In relatively rare cases of inverse isotope effects at the temperature of operation, in which the direction of net migration of isotopes between two phases are reversed, the heavier isotope moves toward the gas. This occurs when an increase in the effect of intramolecular forces on the reduced partition function ratio associated with vaporization outweighs the effect of intermolecular forces in the liquid upon the reduced partition function ratio (Ref. 1 and later in this section). The equilibrium constant K of an exchange reaction AX0 þ BX ¼ AX þ BX0

ð2:4Þ

is K¼

QðAXÞQðBX0 Þ QðAX0 ÞQðBXÞ

ð2:5Þ

where AX0 refers to a lighter isotopomer and AX refers to the heavier isotopomer, A and B are the polyatomic entities of two chemical species to which the isotopes X0 and X are attached, and Q is the molar partition function. For the ideal gas, Q ¼ qN0 =N0 !; where q is the molecular partition function and N0 is the Avogadro number. When isotopic atoms distribute themselves between two chemical species, the preference of an isotope for associating themselves with one chemical species rather than the other is determined by the chemical environments provided by the two species competing for the isotopic atom in question. The chemical environments for a gaseous molecule are provided by the intramolecular (vibrational) forces, while those for a condensed phase or in a solution are provided by the intermolecular forces as well as the intramolecular vibrational forces.1

6

Lithium Li ¼ 7.56% 7 Li ¼ 92.44%

Hydrogen H ¼ 99.985% D ¼ 0.015%

Element

Water (‘)

Water (‘) vs. dihydrogen (g) Water (‘) vs. dihydrogen (g) Water (‘) vs. dihydrogen (g) Water (‘) þ dihydrogen, and electrlysis of water Water Ammonia (‘) vs. dihydrogen (g) Water (‘) vs. hydrogen sulfide (g)

Electrolysis

Chemical exchange

Chemical exchange Chemical exchange Chemical exchange þ Electrolysis Chemical exchange þ Distillation Chemical exchange

Chemical exchange

LiCl (fused) Li-amalgam vs. LiBr in DMF Li-amalgam vs. Liþ (C2H5OH soln) Li- amalgam vs. LiCl (in DMF, THF, DMSO, etc.)

Ion-migration

Chemical exchange Chemical exchange Chemical exchange

LiCl(aq), LiOH(aq)

Water (‘)

Distillation

Electrolysis

Dihydrogen (g)

Working Substances

Distillation

Method

TABLE 2.1 Examples of Isotope Enrichment

Self-contained heavy water production by bithermal process For a, see Table 2.2 Self-contained heavy water production in bithermal process Both isotopes; laboratory scale a ¼ 1:055 6 Li enrichment; laboratory scale. a ¼ 1:0075 6 Li enrichment. a ¼ 1:05 6 Li enrichment. a ¼ 1:025 6 Li enrichment. Lab scale a ¼ 1:04 , 1:058 @ T ¼ 2 8 , 858C

Production of D2O: final enrichment a ¼ 3.61 @ Ttriple;1.81@ nbp Production D2O, final enrichment a ¼ 1.120 @ Ttriple;1.026 @ nbp Production: final enrichment of output from other process, or partial enrichment for input of other processes. a < 3 , 10 (See text) Self-contained heavy water production by bithermal (dual temperature) process Development of hyrophobic Pt-based catalyst Production: India, Argentina, and others The CECE process: production of heavy water. For a, see Table 2.2 Detritiation

Remarks

29 29 30

26–28

24,25

8,9,12

8,9,12

23

22

21

8,9,11,12

8,9,20

continued

8,9,11– 13,20

8,9,11– 13,59

Reference

Enrichment of Isotopes 45

Nitrogen 14 N ¼ 99.63% 15 N ¼ 0.37%

Carbon 12 C ¼ 98.89% 13 C ¼ 1.11%

Boron 10 B ¼ 19.61% 11 B ¼ 80.39%

Element

Method

Thermal diffusion

N2(g) þ 14N15N(g) þ 14N15N(g)

(NHþ 4 ) vs. NH3(g)

Chemical exchange 15

HNO3 (aq) vs. NO (g)

Chemical exchange

Distillation

HCN(g) vs. CN2 (aq) Bicarbonate(aq) vs. CO2(g) Carbamate (in organic solution) vs. CO2(g) NO

Li(s) vs. Li (glymes, propylene carbonate) Li(s) vs. LiCl(aq) BF3(g) vs. BF3(CH3)2O(‘) BF3(g) vs. BF3 –Et2O(‘) BF3(g) vs. BF3-ethers (‘) BF3 CO

þ

Working Substances

Chemical exchange Chemical exchange Chemical exchange

Chemical exchange Exchange distillation Exchange distillation Exchange distillation Distillation Distillation

Chemical exchange

TABLE 2.1 Continued

Enriched 15N production For a, see Table 2.4 Enriched 15N production: Nitrox process, a ¼ 1:055 (258C) Laboratory scale. Column overall separation < 1.22 , 1.79 Laboratory scale, Clusius –Dickel Column

Labolatory scale. a < 1:0075 , 1:02

32 33–35 36 37,38 39 8,9,40

K ¼ 1.046 ^ 0.013 @ 296.6K Mainly 10B: laboratory scale. a ¼ 1:016 Pilot scale. a ¼ 1.026 @ 738C Laboratory scale. a ¼ 1:016 Laboratory scale for 10B. a ¼ 1:0075 Laboratory scale for 12C and 13 C a ¼ 1:011 for (12/13) a ¼ 1:008 for (16/18) Laboratory scale a ¼ 1:026

47

44

45,70,71

43

41 53 42

31

Reference

K ¼ 1.030 ^ 0.005 @ 296.6K

Remarks

46 Isotope Effects in Chemistry and Biology

CO NO Water O2 Water (‘) vs. CO2 (g) Sulfite(aq) vs. SO2(g)

UF6(g) UF6(g) UF6(g) UCl4(g) U atoms or UF6(g)

Distillation

Distillation

Distillation þ Electrolysis Thermal diffusion Chemical exchange Chemical exchange

Gas diffusion

Centrifuge

Thermal diffusion

Electro-magnetic

Laser

Production of enriched 235U. For a and other data, see Ref. 8 Production of enriched 235U. For a and other data, see Ref. 8 Laboratory scale. For a and other data, see Ref. 8 Production of 235U (1945–1946) For all data, see Ref. 8 Production (Discontinued in U.S. and France) a ¼ 6:20 , 6:24

Laboratory scale. a ¼ 1:044 Pilot scale for 34S a34 ¼ 1:012

By-product of CO-distillation for 13C a ¼ 1:0038 By-product of NO distillation for 15N. For a, see Table 2.4 Laboratory scale, for 18O and 17O

8,9,60–65

8,9,16–18,57,69

8,9,15–18

8,9,14–18,58,66–68

8,9,14–18

8,9,48–50 51,52 53,54 55,56

43

40

Ttr ; Triple point; nbp, Normal boiling point of the liquid. The separation factors without any indication of temperatures are the values reported by the authors of the Reference, which is mostly at ambient temperatures.

Sulfur32S ¼ 95.0% 34 S ¼ 4.22% Uranium

Oxygen 16 O ¼ 99.76% 17 O ¼ 0.037% 18 O ¼ 0.204%

Enrichment of Isotopes 47

48

Isotope Effects in Chemistry and Biology

In the Boltzmann distribution, relative attractiveness of a quantum state of a molecule is expressed by the molecular partition function q¼

AllX states

e21j =kT

ð2:6Þ

j

where the relative weight of a state is the negative exponential of the reduced energy, the energy relative to the thermal energy, kT. The greater the q-value or Q-value, the greater the effective capacity of the molecule or the molar ensemble of the molecules. Equation 2.6 simply states that K is the ratio of the relative capacities of AX compared to AX0 and the relative capacities of BX compared to BX0 . If AX provides higher molecular forces than BX, it follows that 1,

QBX Q , AX QBX0 QAX0

ð2:7Þ

and, therefore ð2:8Þ

K.1 The heavier isotope X enriches in AX rather than in BX. Classically1 QAX Q0AX0

!

s0 ¼ s Cl

Isotopic Yatoms i

mi m0i

!3=2

ð2:9Þ

where s and s0 are the symmetry numbers of the isotopomers, AX and AX0 , respectively, and mi and m0i are the masses of the isotopic atoms. All Q atomic masses in the isotopomer molecules except the isotopic atom(s) cancel out for the product in Equation 2.9. The equilibrium constant, Kclassical ; according to the classical statistical mechanics is, from Equation 2.9 QAX QAX0 QBX QBX0

KCl ¼

Cl

¼

Cl

s0 s s0 s

AX=AX0

ð2:10Þ

BX=BX0

Thus, KCl is independent of temperature and it is only the ratio of the symmetry number ratios. Quantum mechanically1 Qqm < ðQTranslation ÞCl ðQRotation ÞCl ðQVibration Þqm

ð2:11Þ

and Qqm QCl

AX

¼

AX 2uj =2 Y e =ð1 2 e2uj =2 Þ ð1=uj Þ j

ð2:12Þ

where ðQqm =QCl ÞAX is the reduced partition function of the isotopomer AX, and Kqm ¼

QAX QAX0 QBX QBX0

qm

qm

¼

Qqm QCl Qqm QCl

AX

AX0

Qqm QCl Qqm QCl

BX0

BX

QAX QBX0 QAX0 QBX

Cl

¼

s f s0 s f s0

AX=AX0 BX=BX0

KCl

ð2:13Þ

Enrichment of Isotopes

49

where

s f s0

AX=AX0

Qqm QCl Qqm QCl

;

AX

ð2:14Þ

AX0

in which ðs=s0 Þf of the AX/AX0 pair is the isotopic ratio (AX/AX0 ) of Bigeleisen – Mayer’s2 reduced partition function ratio (RPFR), Qqm =QCl : The separation factor for the exchange reaction Equation 2.4 is the ratio of QðAX=AX0 Þqm over and above its finite classical limit QðAX=AX0 ÞCl in comparison to QðBX=BX0 Þqm over and above its classical limit QðBX=BX0 ÞCl ; that is s 0 0 f ðAX=AX Þ Kqm s a¼ s ¼ KCl f ðBX=BX0 Þ s0 ln a ¼ ln

s AX f s0 AX0

2 ln

s s0

Qqm QCl

2ln

Qqm QCl

¼ ln

AX

ð2:15Þ

BX BX0 AX0

2 ln

Qqm QCl

2ln BX

Qqm QCl

BX0

ð2:16Þ

For an ideal gas, each reduced partition function (RPF), Qqm =QCl ; consists of contributions of intramolecular motions, while for a condensed phase molecule each RPF consists of contributions of inter- and intra-molecular motions. The intermolecular motions such as translation and libration and low-frequency vibrations such as a bond-torsion and a wagging contribute a 1=T 2 terms of the form

1 h 24 k

2

W1 T2

Low-energy motions X

ðnj02 2 n2j Þ;

or /

X j

ðuj02 2 u2j Þ

to the RPF, where nj 0 and nj are the frequencies (sec21) for the lighter and heavier isotopomers, and u ; hn=kT: The coefficient W1 is the modulating coefficient for the first order term generated by the orthogonal polynomial approximation19 of the reduced partition function, Equation 2.14. The contributions to the RPF of the higher energy-motions such as the bond-stretching and most of the bond-angle-bending vibrations are, at ordinary temperatures, of the form of the zero-point energy approximation of RPF, that is

1 h 1 2 k T

High-energy motions X

ðn 0j 2 nj Þ;

or /

X j

ðu0j 2 uj Þ

The motions that contribute the terms of the form u=2 at low temperatures tend to move toward the terms of the form u2 =24 at higher temperatures, so that effective contributions at intermediate temperatures may take other forms of temperature-dependence. The words, high-energy and low-energy are only relative to the thermal energy at the temperature of the operating system in question.

50

Isotope Effects in Chemistry and Biology

1. Distillation Examples of distillation systems are found in Table 2.1. Except for the hydrogen isotopes, the separation factor, a, of distillation is generally small primarily due to two reasons. First, the separation factor for isotopic distillation is s Low-energy f X 1 h 2 W1 motions s0 condensed ln a < ln ðnj02 2 n2j Þcondensed ¼ 2 s 24 k T f s0 vapor 1 ½dðZPEÞcondensed 2 dðZPEÞvapor þ 2

ð2:17Þ

where dðZPEÞ ; ðZPEÞ0 2 ðZPEÞ; which represents the isotope shift in the zero-point energy. And, dðZPEÞcondensed 2 dðZPEÞvapor ¼ ½ðZPEÞ0condensed 2 ðZPEÞ0vapor 2 ½ðZPEÞcondensed 2 ðZPEÞvapor , which is the isotopic difference in the ZPE-shift upon condensation of the vapor. This term is usually negative and tends to partially cancel the positive first term of Equation 2.17. Secondly, the distillation is usually carried out near the normal boiling point of the liquid for practical reasons and, T being in the denominator in Equation 2.17, these relatively high temperatures for the distillate used in the isotope separation reduces both terms of Equation 2.17. For heavy water production, the deuterium enrichment system is built as a parasitic plant adjacent to a chemical plant, e.g., a plant consisting of a hydroelectric water-electrolysis plant producing large quantities of dihydrogen, plus a dihydrogen distillation plant that uses the hydroelectric dihydrogen as its feed. Another distillation process that is notable for yielding a relatively high a for distillation of an isotope outside heavy water is that of nitric oxide for 15N production. The separation factor is high (Ref. 39 and Table 2.1 of this chapter) because, unlike other distillations, NO in its liquid state exists primarily as a dimer (NO)2, in which two NO molecules are bound to each other by a weak bond between approximate mid-points of the two NO molecules.76 2. Chemical Exchange Two chemicals flow countercurrently. The heavier isotope usually enriches in the stream of the chemical in which stronger molecular forces are exerted on the isotopic atom than in the other chemical.1 For an exchange reaction AX0 ð‘; or aqÞ þ BXðgÞ ¼ AXð‘; or aqÞ þ BX0 ðgÞ 1 h ln a < 24 k þ

2

1 T2

Low-energy motions X

2

AX;AX0

1 h 16 2 k T4

AX;AX0

0

2 2 0 2 nj;AX Þ ðnj;AX

High-energy X 6 motions

ð2:4aÞ

ðnj;AX0 2 nj;AX Þ 2

High-energy motions X BX;BX0

3

7 ðnj;BX0 2 nj;BX Þ7 5

ð2:17aÞ

Because the chemical exchange processes take advantage of relatively large differences in the isotopic differences in the vibrational ZPE’s between two chemical species, the separation factors of chemical exchange processes are usually greater than those of distillation at comparable temperatures. Because a is large, the chemical exchange requires a smaller number of separation stages, and needs shorter process columns and requires lower flow rates of the material through the stages. The chemical process, however, usually needs chemical refluxing of enriched product

Enrichment of Isotopes

51

(cf. Section III.A). Feed of a third chemical to the refluxer is needed and by-products of the reflux chemical reaction must be disposed of. The amounts of the refluxing chemical and the by-products increase proportionately to the amount of refluxing required, but the latter increases very rapidly with the amount of enriched isotope withdrawn and with the level of enrichment of the product (cf.: Section III.C). In the early days of isotope chemistry, i.e., in the early 1930s, H.C. Urey and his students (and, later, the students’ collaborators) investigated many chemical exchange reactions that formed the bases of the contemporary isotope exchange processes. They have been included in Table 2.1. For the chemical exchange reactions given in Table 2.1, the compound in which the heavier isotope enriches is listed first under the column “working substances”, and the compound depleted in the heavy isotope is listed second, except for the processes for uranium. Some of the exchanging species listed in Table 2.1 may appear to be violating the rule that the heavier isotope enriches in the species that provides a stronger molecular force around the isotopic atom, but all of these cases in fact exemplify the subtlety of the rule. For example, take the exchange distillation involving BF3(g) vs. BF3 anisole (‘). (Do not be bothered by the word “distillation” in “exchange distillation,” because the word simply refers to a special type of product-end reflux and has nothing to do with the direction of isotope enrichment. The word will be explained in Section III.B). In the reaction, Donor 11 BF3 ð‘Þ þ 10 BFðgÞ ¼ Donor 10 BF3 ð‘Þ þ 11 BF3 ðgÞ 11

B enriches in the gaseous BF3, while the common chemist’s intuition anticipates that 11B will go to the BF3-donor complex. The key for the apparently unusual behavior of boron is that the fluorine atom is so small. When the BF3 molecule is formed with fluorine, one of the three electron pairs on one of the fluorine atoms is repulsed from the F-atom and occupies one of two vacant orbitals of boron perpendicular to the B – F bond, thus forming a p bond. The resonance energy of BF2· · ·F has been calculated to be 48 kcal/mol. When the Lewis base forms a coordination bond with BF3, the coordination energy is smaller than the resonance energy of the p bond.37,38 The chemical exchange process is of major importance for the isotopes of light elements, especially for heavy water production. Table 2.2 is an excerpt of a tabulation by Benedict, Pigford, and Levi8 on the separation factors of hydrogen exchange. From the viewpoint of the separation factor and its temperature-dependence, the exchange between water and dihydrogen is the best of all. (For the importance of T-dependence of a, see below.) However, this exchange reaction between liquid water and dihydrogen gas requires a catalyst, unlike others listed in Table 2.2. Even platinum catalysts readily lose their activity for cleaving the hydrogen– hydrogen bond of

TABLE 2.2 Separation Factors of H/D Exchange Reactions Involving Watera Separation Factor a

Ratio

Exchange Reaction

a/K

08C

258C

508C

1008C

1258C

2008C

a25/a125

H2O(‘Þ þ NH2D(g) ¼ HDO þ NH3 H2O(‘Þ þ HDS(g) ¼ HDO þ H2S H2O(‘Þ þ DCl(g) ¼ HDO þ HCl H2O(‘) þ HD(g) ¼ HDO þ H2

3/2 1 1/2 1

1.02 2.60 2.87 4.53

1.00 2.37 2.51 3.81

1.00 2.19 — 3.30

0.99 1.94 — 2.65

0.99 1.84 1.88 2.43

0.99 1.64 — 1.99

1.01 1.24 1.34 1.57

a

Excerpt from Table 13.17 of Benedict, Pigford, and Levi (Ref. 8).

52

Isotope Effects in Chemistry and Biology

dihydrogen, because the platinum surfaces are too hydrophilic: the surface is quickly covered by water molecules after the surface is exposed to water leaving no room for dihydrogen. The problem was solved by an invention of “hydrophobic platinum catalysts”21,22 in which platinum is deposited on porous polytetrafluoroethylene or on high surface area carbon bonded to a variety of column packings using Teflon. All water (‘) — dihydrogen exchange processes listed in Table 2.1, including the process for detritiation, are dependent on hydrophobic platinum catalysts. Removal of tritium from heavy water is an isotope separation process and detritiation is, for instance, needed to keep the level of tritium in the heavy water coolant of nuclear power reactors under a safety limit. The problem associated with the needs for chemical reflux becomes acute in large-scale isotope plants such as the ones for heavy water production. It has been solved by an ingenious method called the dual-temperature (or, bithermal) process. The principle of the bithermal process is schematically illustrated by Figure 2.2. Figure 2.2a is the scheme of an ordinary exchange process. In Figure 2.2b which is a block diagram of bithermal process, the cold tower operates like the exchange column of Figure 2.2a in which deuterium is enriched in the water stream but, instead of the chemical product-refluxer of (A), the hot tower provides an enriched hydrogen sulfide, because a lower separation factor at the higher temperature (cf.: Table 2.2) for the reaction, H2O(‘) þ HDS(g) ¼ HDO(‘) þ H2S(g), generates a requisite highly enriched hydrogen sulfide. Product refluxing is thus done physically rather than chemically. The effective separation factor of a dualtemperature process8 is

aeff ¼

aC aH

ð2:18Þ

where aC and aH are the separation factors at the temperatures of the cold and hot towers, respectively. Table 2.2 illustrates that the water –hydrogen system is best with regard to the high value of aeff : See Ref. 8 for a detailed analysis of the dual-temperature principle.

Feed (Natural water)

Feed

Depleted Hydrogen Sulfide

Depleted (Waste)

(Natural water)

Depleted Hydrogen Sulfide

Cold Tower 25 °C a = 2.37

Exchange Column

Product

Hot Tower

Enriched Hydrogen Sulfide

Product (Heavy Water)

125 °C a = 1.84

Refluxing Chemical Product Refluxer

(a)

D2S

(Heavy Water)

Chemical Waste

FIGURE 2.2 Dual temperature process.

(b)

Waste Depleted Water

Depleted Hydrogen Sulfide

Enrichment of Isotopes

53

3. Gas Centrifugation8,9,15,16,18,66 – 68 The gas centrifuge is based on the dependence of equilibrium population of gaseous molecules under the influence of position-dependent centrifugal force. When a gas of molar mass M is placed in a uniform gravitational field g, the partial pressure at altitude h is, according to the Boltzmann distribution law, pðhÞ ¼ pð0Þexpð2Mgh=RTÞ: Similarly, when a gas of mass M and density r is placed in a centrifuge of radius r rotating with an angular velocity v, the pressure gradient of the gas at distance r from the axis is dp=dr ¼ M rv2 r ¼ ðMp=RTÞv2 r; the ratio of the partial pressure at the axis ð p0 Þ to that at the periphery ð pÞ is p0 =p ¼ expðM v2 r 2 =2RTÞ; and the separation factor between the molar masses M 0 and M is "

ðM 0 2 MÞv2 r 2 a ¼ exp 2RT

#

The separation factor thus depends on the square of the peripheral velocity and on the difference in the masses. This last dependence is an important advantage of the centrifugal method in comparison with other isotope separation processes, especially of the isotopes of heavy elements, because in most other methods a depends on ðM 0 2 MÞ=M 0 M rather than on ðM 0 2 MÞ: When gaseous uranium hexafluoride spins in a centrifugal cylinder at a high rotational speed, the heavier 238UF6 molecules move preferentially toward the periphery. The gas enclosed is subject to centrifugal acceleration thousands of times greater than the gravity of the earth. At a peripheral speed of above 500 m/sec, the 235UF6 content at the center of the cylinder could be as much as 18% greater than at the periphery. A system of rotating baffles and stationary scoops induces longitudinal countercurrent gas flow with light gas flowing upward near the axis and heavier gas flowing downward near the periphery. Only seven stages would be needed in the ideal cascade to produce 3% 235U product and waste at 0.25%. The separation factor being dependent on high spin speed, the centrifuge process depends on high strength cylinder material and design of the bearings.

B. ENRICHMENT P ROCESSES B ASED ON N ONSTEADY S TATE P HENOMENA OF R EVERSIBLE P ROCESSES The ion-exchange separation of uranium isotopes is an example of the processes in this category. In the column, a solution of uranium compounds flows down a stationary column packed with ion exchangers, and the adsorption bands of uranium isotope migrate down and, in the process, isotope fractionation takes place (Ref. 1 and Section VII in the present chapter). Except for the Asahi plant, ion-exchange methods have not left the environment of laboratories. Besides accomplishing a separation of uranium isotopes, however, it has led to the discovery of a new type of nonBorn – Oppenheimer isotope effect, the effects due to differences in the shape and size of the isotopic atoms of uranium (and other heavy elements), which cause isotopic irregularities of the extranuclear electronic energies, resulting in reversals of orders of magnitudes of separation factor vs. mass number.

C. ENRICHMENT B ASED ON I RREVERSIBLE P ROCESSES 1. Laser Isotope Separation8,9,15 – 18,60 – 65 Figure 2.3 is a cross-sectional view depicting an atomic vapor laser isotope separation (AVLIS) module. It takes advantage of small differences in absorption spectra of isotopic species. By using sufficiently monochromatic light of an appropriate wavelength, a particular isotopic species is preferentially excited to an upper energy level. Referring to Figure 2.3, atoms of uranium heated in a high vacuum chamber vaporize and flow upward through a region between a pair of negatively charged plates and get illuminated by a laser having an appropriate energy to excite 235U but not

54

Isotope Effects in Chemistry and Biology Tails Collection Surface

Depleted Uranium Flow

Laser - illuminated area

Electromagnetic (Plasma) ion Extraction Structure Ion deflector plates

Product Collector plates

electron bea m rgy ne

Uranium vapor flow

hig h

e

Magnetic Field

U

Water - cold crucible

FIGURE 2.3 Cross-sectional view of an atomic vapor laser isotope separation module for (Redrawn based on the references listed under Ref. 8.)

235

U enrichment.

238

U. Then, at least one other laser beam having energies sufficient to ionize the excited 235U but insufficient to ionize unexcited 238U is used, and 235U ions are collected by a bank of negatively charged plates. Since even the largest energy difference between the atoms of 235U and 238U corresponds to a difference of the order of 1 in 50,000, the excitation laser must be tuned more closely than this. Lasers generate pulses of photons at a fixed repetition rate. The lasers for the LIS must have a sufficiently rapid repetition rate to catch all the 235U atoms passing between the collection plates leading to ionization of the 235U atoms. Thus, LIS requires a high-power, rapid-repetition, finely tunable laser. The method has been used successfully for laboratory scale separation of isotopes as heavy as uranium and, among others, those as light as hydrogen, boron, chlorine, sulfur, and bromine. 2. Gaseous Diffusion8,9,15 – 18 Although this process is usually called gaseous diffusion, the process is based on the physical phenomenon of molecular effusion. The difference is that effusion is a phenomenon of individual molecules passing through a tiny hole on a wall of a vessel just large enough to allow such molecular passage without permitting continuous mass flow of the gas. The rate of passage of two gaseous isotopomers is inversely proportional to the square root of their molecular mass (Graham’s law). For example, sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffi mð238 UF6 Þ 352 a¼ ¼ 1:00429 ¼ 235 349 mð UF6 Þ The unit stage of a gaseous diffusion cascade consists of a compressor, a cooling chamber, and a converter containing a porous diffusion barrier. See Ref. 8 for detailed description of theory

Enrichment of Isotopes

55

and equipment of the U.S. plant that began its operation in 1945 and others built in European countries since then. 3. Thermal Diffusion51,52 When a mixture of fluid (usually a gas mixture) is subjected to a temperature gradient, a small diffusion current is induced, with one component transported in the direction of heat flow and the other component in the opposite direction. The phenomenon is called thermal diffusion. The theory of the thermal diffusion phenomenon is detailed by Hirschfelder, Curtiss, and Bird.46 The basic design of thermal diffusion apparatus that multiplies the separation effect of thermal diffusion was first developed by Clusius and Dickel.51,52 The Clusius – Dickel column is a vertical tube externally cooled and internally (along the tube axis) heated, and a (gaseous isotopic) mixture is confined in a narrow annular space between the two walls. When a temperature difference is maintained between the two walls, not only is the lighter isotope concentrated along the hotter inner wall while the heavier isotope is concentrated along the cooler outer wall, but also a convection current is induced so that the lighter-isotope-enriched gas near the axis flows upward and the heavier-isotope-enriched gas near the outer wall flows downward. The convection flow multiplies the effect of the thermal diffusion. A Clusius –Dickel-type column with a narrow annular space (e.g., a few millimeters wide) can produce an height equivalent of theoretical plate (HETP) (cf. Section III.E) of the order of a centimeter, so that a single Clusius –Dickel column can yield thousands of stages. The design of the thermal diffusion column is simple and has no moving parts and is extremely convenient for a small-scale operation, and the same Clusius– Dickel column can be easily converted for other isotope separation purposes. However, the throughput (i.e., the magnitude of interstage flows) are small, and the energy consumption is high. 4. Electrolysis8,9,11 – 13 When a portion of an aqueous electrolyte solution, e.g., KOH(aq), is electrolyzed, the deuterium content of the residual water becomes higher than the D-content of the feed water and higher than the D-abundance of dihydrogen generated at the cathode. If electrolysis is continued, the residual water becomes more and more enriched in deuterium. If desired, the richer gas may be burned, and the water condensed and returned to an appropriate stage of the cascade of electrolytic cells. This is the basis of the electrolytic production of heavy water. All of the heavy water produced until 1943 was made by electrolysis. The largest of the plants in this period were: The Norsk Hydro Co at Rjukan, Norway (1.5 metric tons per year), which used cells equipped with steel cathodes and steel diaphragms between the cathodes and anodes; the Trail, British Columbia, plant, which used a chemical exchange process to an enrichment of a few percent in deuterium (6 metric tons of 2.37% D2O), which was then further enriched by electrolysis to 99.7%; Manhattan District plant at Morgantown, W. Va, which used cells equipped with steel cathodes and diaphragms. The Trail BC plant was initially an electrolytic production plant. It was later converted to an electrolysis and water – dihydrogen exchange scheme by SAM scientists. At Savannah River, S. Carolina, batch electrolysis was used for a final enrichment from 90 to 99.87% D. Other electrolysis plants are at Ems, Switzerland, and Nangal, India. The separation factor ranged between about 3 and about 13: a ¼ 5:3 (, 608C) at Rjukan, a ¼ 3 (60 , 708C) for the primary plant at Trail, and a ¼ 8 (238C) for the secondary plant at Trail, and a ¼ 6:0 , 8:2 (408C) at Morgantown. The separation factor is influenced by the cell design and operating conditions that are known to increase the irreversibility of the electrolysis process; for example, a is lower at higher temperature, at lower voltage, and in the presence of diaphragms, and with poisoning of the surfaces of the electrodes. Also, electrode materials that have catalytic properties for the reactions, 2H ! H2 and H2(g) þ HDO(‘) ¼ HD(g) þ H2O(‘),

56

Isotope Effects in Chemistry and Biology

reduce the separation factor. It is plausible that the overvoltage for the dihydrogen generation at the cathode is higher for the dideuterium generation than for dihydrogen generation and that this is the cause of H/D separation in electrolysis. (The lowest electrode voltage usable for the enrichment purposes is about 1.7 V, which is to be compared with V ¼ 1:23 V for the reversible electrolysis of water.) 5. Electromagnetic Method: Calutron57,69 The process is essentially that of a huge-scale mass spectrometer and dates back to World War II Manhattan District Project for enriching 235U in kilogram quantities. Since the end of the War the electromagnetic separators, called Calutron for California University Cyclotron because they were originally developed at the University of California by E.O. Lawrence, have been used for separating an amazing variety of isotopes. The Calutron was used for separating relatively small quantities of high enrichment isotopes of practically every element, including the target isotopes for use in the production of a variety of radio-pharmaceuticals by bombardment in a particleaccelerator. Unless the target material is isotopically pure, bombardment might produce radioisotopes of undesirable properties not fit, for instance, for injection in human patients. A workshop organized by the National Research Council, Washington, DC, issued a report,72 a wish-list (as of early eighties) of the research communities in various scientific fields for the need of enriched isotopes that could be separated especially by taking advantage of the versatility of Calutron.

III. SEPARATION CASCADE This section primarily deals with the cascade theory of steady state of cascade of reversible processes.

A. IDEAL C ASCADE : T HERMODYNAMIC E FFICIENCY AND N O -MIXING To achieve a useful enrichment, it is necessary to construct a stack of separation stages in which the product stream from a lower stage is fed to the next higher stage, whose product is in turn fed to the stage above, and so on. Here, the “lower stages” are the stages containing less enriched material than those in the “higher stages.” Such a stack of interconnected stages is called a separation cascade. Thermodynamically, the most efficient cascade is the one in which the streams containing different isotopic enrichments are not allowed to mix: such mixing would achieve an entropy of mixing, a process that will counteract the very purpose of isotope separation. At every stage of a cascade called an ideal cascade, two streams, i.e., the product stream from the stage below and the waste stream from the stage above converging into the stage in question, have the same isotopic enrichment. The ideal cascade that satisfies Equation 2.1 to Equation 2.3 at every stage necessarily have flow rates that must change with stage numbers as depicted in Figure 2.4.66 The interstage flows are at their highest at the feed point, because the flow of the low isotopic abundance material at this point must supply, at least, all the requirement for the target isotopes in the stages nearer the more enriched product end. Stages below the feed point in Figure 2.4, called the stripping section of the cascade, work to squeeze out some of the residual isotope left out by the stages above the feed, called the enriching section. The stripping section is useful especially in cases where the feed material is valuable. The feed material must be conserved in the most practical way where it has already been enriched to some extent and/or when the natural resources of the feed material are limited, e.g., uranium. The ideal cascade calls for a smooth decrease in the flow rates at every stage from the feed point to the product end (Figure 2.4). The ideal cascade is the most efficient cascade but such

Enrichment of Isotopes

57 Product (Enriched)

} A Stage Separating Units

} A Stage

Feed Natural Abundance

Tails (Waste) (Depleted)

FIGURE 2.4 Schematic representation of relative flow rates of an ideal cascade. The width of each stage is meant to be proportional to the interstage flow rate in the stage in a cascade of discrete separating units such as centrifuges in which each rectangle represents a discrete unit. For a continuous equipment such as a packed column, the number of the rectangles in the horizontal direction in the figure is proportional to the crosssectional flow rate. (Adapted from Ref. 66; “The Gas Centrifuge,” by Donald R. Olander, Copyright Scientific American, August, 1978.)

a smooth scale-down of flow rates requires changes of design of stage equipment from stage to stage, a highly expensive endeavor. Such a cascade is thus feasible only for a large-scale plant for enriching highly valuable products such as enriched uranium. For the isotopes of light elements, say, for those approximately up to sulfur in the periodic table, a cascade in which the interstage flow rates are constant throughout is more practical. It is called the square cascade. Sometimes, as a compromise between an ideal cascade and a square cascade, two or more straight columns of different diameters are interconnected, one on top of another, that crudely simulate the shape of an ideal cascade. This arrangement is called the tapered cascade. These cascades still require an often expensive product refluxer (vide infra) at every junction of the columns of different diameters. Of course, one loses the ideality, but the square design offers much simplified equipment and lower capital costs than an ideal cascade.

58

Isotope Effects in Chemistry and Biology

B. PRODUCT-END R EFLUXER Let us suppose that the plant is for enriching an isotope I1 of an element and that the plant works on two chemical streams A and B in a square cascade of an equilibrium process. Further, the Isotope I1 concentrates in the A-stream, which flows down, and an Isotope I2 concentrates in the B-stream, which flows up. The feed for every stage, i, is a mixture of the enriched stream from the stage “above” (i.e., stage number i 2 1), and the I1-depleted stream from the stage “below” (stage number i þ 1). The net feed to stage i is an average composition of the two streams. Here again, the word “below” is meant to be the next higher stage containing the isotope I1 in a higher enrichment, and the word “above” is meant to be the next lower stage containing the isotope in a lower enrichment. This common usage of the words, “above” and “below,” has come from the facts that (a) the target isotope is usually heavier than the other isotope, e.g., deuterium and tritium in hydrogen, 13C and 14C in carbon, 15N in nitrogen, and 17O and 18O in oxygen, and (b) the heavier isotope usually enriches in a molecule (i.e., in the A-stream in the present notation) that exerts higher molecular forces on the isotopic atom in question than the other molecule does,1 and (c) such a molecule is usually in a liquid state or in a liquid solution that physically flows downward in the cascade column, while the other molecule is usually gaseous that flows upward in the column (Figure 2.5). That is, the stage “below” is the higher stage, enrichment-wise, and the stage “above” is the lower stage, enrichment-wise. As one follows further and further downward into the more and more enriched stages, one finally comes to the “product” end where the enrichment has become sufficiently high to be called your product. But then a question arises as to what is the source of the highly enriched material B for the next-to-the-product-end stage, the stage that is one stage away from the satisfactorily high enrichment, i.e., the stage number n in Figure 2.5? Because you are the one producing the high enrichment product, you cannot expect to find from elsewhere a source of a sufficiently highly enriched stream of the material B to feed this highest stage. The only solution to this bootstrapping problem is to sacrifice a portion of your own high-enrichment product (still in the form of Chemical A) and convert it into Chemical B. Such a device is called the product-end refluxer. For example, in a process called Nitrox process for the enrichment of 15N, the isotope exchange reaction is H14 NO3 ðaqÞ þ 15 NOðgÞ ¼ H15 NO3 ðaqÞ þ 14 NOðgÞ In a packed column for this process, nitric acid (of about 10 M) flows down the column countercurrently against a gas stream of nitric oxide. Nitrogen-15 enriches in nitric acid with a separation factor a ¼ 1:055 (258C). At the bottom of the column nitric acid can be made to contain a highly enriched (, 99.9%) 15N. During the total reflux period, 100% (vide infra) of the nitric acid is reduced to nitric oxide using sulfur dioxide 3SO2 ðgÞ þ 2HNO3 ðaqÞ þ 2H2 Oð‘Þ ¼ 3H2 SO4 ð‘Þ þ 2NOðgÞ When the reflux conversion is complete, the nitric oxide thus produced is as highly enriched as the product nitric acid and does not contain any higher nitrogen oxides or nitrogen oxyacids. The gas then flows up the exchange column as it gradually gets depleted by the transfer of 15N to the nitric acid stream. It sounds like we have gone to great lengths in which 15N has been transported back and forth between the two chemicals without making any gain, but we actually do gain. One should remember that isotope separation takes advantage of a slight difference for our purpose just as we do in isotope effect research. Effects of the tiny differences will become clear in the sections that follow. Before leaving this section, it is extremely important to emphasize that the refluxing process should not tolerate an incomplete chemical conversion and any physical leaks around

Enrichment of Isotopes

59

L moles / hr Point A

G moles / hr

yf

xf =x1

yf

x1

Stage 1 Point B

y1

x2

Stage 2 y2

x3

Stage 3 y3

x4

yi −1

xi

Envelope E

Point C

Stage i yi

xi +1

Stage i+1 yi +1

xi +2

yn−1

xn

Stage n Point P

yn = y p

xp

yp

xp

Product-end Refluxer

FIGURE 2.5 Schematics of a square cascade. The mole fractions xi and yi are in isotope exchange equilibrium with each other in the streams leaving the stage-i; xiþ1 and yi are the mole fractions of the desired isotope in the streams at the same horizontal levels along the cascade, i.e., xiþ1 enters the stage i and yi leaves the stage i. The Points A, B, C, and P correspond to the Points A, B, C, and P in Figure 2.7.

the refluxer: incomplete conversion will be a potential source of parasitic isotope fractionation and difficulty in achieving the target enrichment, and leaks will offset the material balance and be a major cause of inability to reach the target level of enrichment. After a successful buildup of isotope concentration profile along the length of the separation column (vide infra) in the startup phase of the operation, one would start withdrawing the enriched product from the product end, always with a small and controlled rate of withdrawal. Even with extreme care, the withdrawal will inevitably lower the enrichment level just achieved prior to the beginning of the product withdrawal, as will be explained later in the present chapter. The higher the withdrawal rate, the more depressed

60

Isotope Effects in Chemistry and Biology

the enrichment of the withdrawn product will become. If there is a leak from around the product refluxer, the enrichment will never reach an expected level, because your cascade cannot distinguish between a purposeful withdrawal for the extraction of the product and an unintentional withdrawal called a leak. The cascade system responds to a leak in the same manner as to an intentional withdrawal. There is another chemical engineering consideration concerning the choices of a type of refluxer. The issue becomes a major one when considering the cost of large-scale enrichment. There are three schemes of refluxing method commonly used in the reversible isotope separation processes, as depicted in Figure 2.6. Distillation requires only addition or removal of heat for the product and waste end refluxing. Chemical exchange needs a chemical to convert the enriched product, e.g., from HNO3 to NO (Figure 2.6) using SO2. The refluxer also produces a chemical waste, e.g., H2SO4 (at a concentration about 10 M and saturated with sulfur dioxide) in the case of the Nitrox process. Exchange distillation (Figure 2.6) is an excellent compromise that takes advantage of a relatively high a of chemical exchange method and a clean and inexpensive thermal refluxing of distillation. In the example of exchange distillation shown in Figure 2.6, 13C/12C are exchanged between CO2 and a nonaqueous solution of carbamate, RR0 – NCOOH, and 13C is enriched in the carbamate. In its product refluxer, heat is applied to decompose the enriched carbamate to carbon dioxide, which is sent back up the separation column, and the amine in an organic solvent is pumped to the top of the column, where it is recombined with CO2 emerging from the low-enrichment end of the column. The chemical exchange processes usually provide a higher single stage separation factor than distillation processes operating at the same temperature, but the distillations require only thermal reflux. However, the chemical exchange needs a chemical to effect the refluxing conversion and produces chemical wastes. This is one of the most significant issues especially in a large-scale production, because the amount of reflux reaction needed is easily many

Feed NO

Depleted CO2 Heat

Condenser

Boiler

Enriched CO2

Amine CO2

Thermal Decomposer

Heat DISTILLATION

Heat EXCHANGE DISTILLATION

FIGURE 2.6 Comparison of reflux methods.

H2O, O2 Depleted HNO3

Depleted NO Feed HNO3

Packed Column

Packed Column

Carbamate

Waste Refluxer

CO2

Carbamate

Enriched NO

Heat

Thermal Recombiner

Feed CO2

Packed Column

Depleted NO

HNO3

Enriched HNO3

NO

Product Refluxer

SO2 CHEMICAL EXCHANGE

Waste H2SO4

Enrichment of Isotopes

61

thousands of times the number of moles of enriched product withdrawn (cf.: Sections III.C and VI.A).

C. MC CABE – THIELE D IAGRAM FOR S QUARE C ASCADE Although mathematical theory and design formula that describe the square and ideal cascades have been derived8,9 from Equation 2.1 to Equation 2.3, the principle generated from Equation 2.1 to Equation 2.3 can be best illustrated using a McCabe – Thiele diagram. For the flow scheme of Figure 2.5, which shows a two-isotope system involving two countercurrent streams, a McCabe – Thiele diagram, Figure 2.7, is constructed using two lines in the x ¼ ½0; 1 , y ¼ ½0; 1 space, where x and y represent the mole fractions of the desired isotopes in streams g and ‘; respectively. For simplicity in this discussion of the diagram the stripping section and the refluxers will not be included. The ‘-stream having a molar flow rate L (cf. Figure 2.5) and a composition of the feed stage yf enters Stage 1 and meets the g-stream having a molar flow rate G and a composition x2 : After an isotopic exchange equilibrium is reached in Stage 1, it produces the atom fraction y1 in the ‘-stream and atom fraction xf ( ¼ x1 ) in the g-stream. The stream compositions are (xf ; yf ), i.e., Point A of Figure 2.5 and Figure 2.7, just above Stage 1 and (x2 ; y1 ), which is Point B, just below Stage 1. Similarly, the column composition at Point C is (x3 ; y2 ). The line A –P is the locus of the points representing the compositions of the working fluids on the same horizontal planes between the successive stages (cf.: Figure 2.5). The line is straight, because the isotope balance in the Envelope E in Figure 2.5 is yf L þ xiþ1 G ¼ yi L þ xf G

yi ¼ yf þ

or

G ðx 2 xf Þ L iþ1

ð2:19Þ

where ðxf ; yf Þ is a fixed point and G and L are constant in a square cascade. The straight line, A through P, in Figure 2.7 is called the operating line. 1 yp

P

y

Equilibrium Line

Operating Line y2

C

y1

B

yf

A x2

0

x3

0

1 xf

X

xp

FIGURE 2.7 Basic McCabe– Thiele diagram for a square cascade. See Figure 2.5 for the flow scheme notations.

62

Isotope Effects in Chemistry and Biology

The points representing the equilibrium composition in Figure 2.7 are given by an equivalent of Equation 2.3 which, in terms of the notation of Figure 2.5 and Figure 2.7, y 12y a¼ x 12x

ð2:20Þ

or y¼

ax 1 þ ða 2 1Þx

ð2:21Þ

This line is called the equilibrium line. The line is concave upward when a . 1: As has been shown above, one separation stage corresponds to a step consisting of a vertical line and a horizontal line drawn between the equilibrium line and the operating line. The diagram illustrated by Figure 2.7 is called the McCabe – Thiele diagram. The operating line ends at its highenrichment end on the main diagonal in the McCabe – Thiele diagram, which corresponds to a perfect product-end refluxing; “perfect” in the sense that the refluxer does not change the isotopic abundance during the reflux process and does not have any leaks. The number of stages needed for a given separation can be graphically determined by counting the number of steps between the operating and equilibrium lines between Points A and P of the McCabe – Thiele diagram. As is evident from Equation 2.21, the smaller the magnitude of la 2 1l; the closer the equilibrium line is to the operating line, and the greater the number of steps required to accomplish a given separation. Mathematically, the number of stages required for a given set of terminal conditions and flow rates can be obtained from the material balance and equilibrium equations. For instance, for the ideal cascade without stripping section, the overall separation S is xp 1 2 xp S¼ xw 1 2 xw

ð2:22Þ

where xp and xw are the mole fractions in the product and waste streams, respectively. Also, S ¼ an

ð2:23Þ

where n is the total number of stages in the cascade. In the case of close separation in which a is very small compared to unity, i.e., lal ! 1; we have

an < eða21Þn ¼ e1n

ð2:24Þ

1;a21

ð2:25Þ

where

Here, a is the overall separation of a stage between two streams leaving the stage. This a is the same as the a used by Benedict,1 Bigeleisen,4 and many others, which is the “heads-to-tails” separation factor. This is in contrast to the a used by Karl Cohen, which is “heads-to-feed” ratio. Because of the eminence of Cohen’s book,14 there has sometimes been minor confusion among the latter day researchers in the use of the word, separation factor, despite the general trend that favors the heads-to-tails terminology.

Enrichment of Isotopes

63

When yp ð¼ xp ) and yf are given, the slope of the operating line of the square cascade is (cf.: Figure 2.5 and Figure 2.7) Slope ¼

yp 2 yf 1 G ¼ ¼12 R L yp 2 xf

ð2:26Þ

where R ¼ Reflux ratio ;

L P

ð2:27Þ

1. Case of Total Reflux For a given rate of product enrichment, i.e., when Point P is fixed, fewer stages are required when the slope of the operating line is steeper (cf.: Figure 2.7). The steepest slope is the case in which the feed point is on the main diagonal of the McCabe – Thiele diagram. This corresponds to Slope ¼ 1 and R ! 1; called the case of total reflux: The entire product is refluxed, and there would be no product withdrawn. Then nmin ¼

lnbxp =ð1 2 xp Þc 2 ln½xw =ð1 2 xw Þ ln a

ð2:28Þ

2. Case of Minimum Reflux Ratio Another extreme case corresponds to the smallest slope of the operating line, that is, the case of minimum reflux ratio. For the square cascade it occurs when the feed point A (Figure 2.7) is on the equilibrium line. Mathematically, from the equilibrium condition, Equation 2.21 y2x¼

yða 2 1Þð1 2 yÞ y þ að1 2 yÞ

and the material balance between the feed point and the product point yf L ¼ Pxp þ ðL 2 PÞÞxf Combining the two equations Rmin ¼

L P

min

¼

yp b1 þ ða 2 1Þð1 2 yf Þc 2 yf ða 2 1Þyf ð1 2 yf Þ

ð2:29Þ

A cascade operating on the minimum reflux ratio will produce a (fictitious) maximum product rate but will require infinite number of stages, since it would take an infinite number of steps to move the operation point away from the equilibrium line. Table 2.3 illustrates examples of the number of stages and the need for large amounts of reflux reaction in the Nitrox process. Note that the ratio of the amount of the enriched product to that of the reflux chemicals is huge. For an ideal cascade in which the interstage flow rates change from one stage to the next, the slope of the operating line changes from stage to stage, so that the condition for the minimum slope must be applied on the individual segments of the operating line. As the reflux ratio decreases, say, from that of the total reflux to smaller values, the increase in the enrichment between the stages decreases, and the difference in the enrichment between the successive stages decreases toward an eventual zero. In this situation there would be no enhancement of enrichment and the reflux ratio

64

Isotope Effects in Chemistry and Biology

TABLE 2.3 An Example of Parameters of Two-Section Tapered Cascade: Case of Nitrox Process Producing 100 kg 15N per year (5 18.26 mol 15N per day) Flow Rates (mol N/day)f

Enrichment

Number of Stagesd

Reflux Ratio

HNO3

NO

Reflux

SO2

90.0%a —

92 ฀(84) 137 ฀(125)

11.470 ฀(10.428) 411.6 ฀(343)

94,004 7,538

93,985 7,520

86,465 7,520

99.0%b —

92 ฀(84) 188 ฀(171)

11.388 ฀(10.352) 456.0 ฀(380)

103,428 8,349

103,410 8,331

95,079 8,331

99.9%c —

90 ฀(82) 246 ฀(224)

11.165 ฀(10.150) 468.6 (426)

104,370 8,580

104,350 8,561

95,790 8,561

129,698 11,280 140,978 142,619 12,497 155,116 143,685 12,842 156,527

e

Product

18.26 18.26 18.26 18.26 18.26 18.26

a

Atom fraction of depleted stream out of cascade ¼ 0.00037. A stripping section has been included in these calculations but not shown in the table. Taper point atom fraction ¼ 0.05. k1 ¼ 1:10; k2 ¼ 1:20: b Atom fraction of depleted stream out of cascade ¼ 0.00037. A stripping section has been included in these calculations but not shown in the table. Taper point atom fraction ¼ 0.05. k1 ¼ 1:10; k2 ¼ 1:20: c Atom fraction of depleted stream out of cascade ¼ 0.00037. A stripping section has been included in these calculations but not shown in the table. Taper point atom fraction ¼ 0.045, k1 ¼ 1:10; k2 ¼ 1:10: d In each enrichment entry the first number (e.g., 92) is the number of stages used in the calculation. The number in the parentheses (e.g., 84) in each entry is the minimum number of stages. The first and second lines for each enrichment entry are for the Section 1 and Section 2 of the two-section enrichment cascade, respectively. e The reflux ratio used in the Section 1 calculation (e.g., 11.470) is k1 times the minimum reflux ratio, which is enclosed in the parentheses (e.g., 10.428). The reflux ratio used in the Section 2 calculation (e.g., 411.6) is k2 times the minimum reflux ratio, which is enclosed in the parentheses (e.g., 343). f All flow rates are in moles of N per day, except for SO2, which is 1.5 times the rate of product-end reflux.

is at its minimum. The minimum reflux ratio for the ideal cascade is necessarily a function of stage number8 Liþ1 P

min

¼

ðyp 2 yi Þ½1 þ ða 2 1Þð1 2 yi Þ ða 2 1Þyi ð1 2 yi Þ

ð2:30Þ

In both square and ideal cascades the minimum reflux ratio and thus the flow rates at any stage in the cascade are inversely proportional to ða 2 1Þ: Therefore, the number of separation units, e.g., the compression –cooling – diffusion units at a given stage in a gaseous diffusion plant, is inversely proportional to ða 2 1Þ: Because the minimum number of stages, nmin ; is also inversely proportional to ða 2 1Þ; the minimum total size of the plant and thus the initial construction cost is inversely proportional to ða 2 1Þ2 .

D. SEPARATIVE C APACITY FOR C LOSE -SEPARATION, I DEAL C ASCADE The sum of the total flow rates of the “heads” (enriched) streams, J, and the sum of the total flow rates of the “tails” (depleted) streams, K, for a close-separation ideal cascade is8,14 JþK ¼

8 D ða 2 1Þ2

ð2:31Þ

Enrichment of Isotopes

65

where D is called the separative capacity or separative duty and defined by D ; Wð2xw 2 1Þln

xp xw xf 2 Fð2xf 2 1Þln þ Pð2xp 2 1Þln 1 2 xp 1 2 xw 1 2 xf

ð2:32Þ

in which xw ; xp ; and xf are the isotopic atom fractions in the waste, product and feed streams for the cascade, respectively, and W, P, and F are the molar flow rates for the waste, product, and feed streams of the cascade, respectively. Note that they are for the whole plant and not for an individual stage. Equation 2.31 is obtained by summing up the material balance relations on all (i.e., feed, heads, and tails) streams of all stages. The factor, 8=ða 2 1Þ2 ; is a measure of the relative ease or difficulty of the separative process or the quality of the separation phenomenon in question, while the separative capacity, D, is a measure of the quantity of the actual separation being done. Many important characteristics of a plant are proportional to the separative capacity. When predicting an effect of changing a design and operating conditions, the most dependable, first hand, prediction can be made on the basis of changes in the separative capacity.

E. HETP (HEIGHT E QUIVALENT OF T HEORETICAL P LATE) If a distillation column consists of a series of discrete bubble-cup plates and if the cup-plate stage were designed to let the liquid and vapor streams entering it reside for a sufficiently long period of time so that the streams would reach an equilibrium with each other before they leave the stage, then each of such stages will be equivalent to a fully bona-fide separation stage. In a packed column for distillation or isotope exchange system, one can imagine a (hopefully short) vertical section of the column to be designated as an equivalent of one stage in which an isotope exchange equilibrium is attained. Whatever the actual two-phase kinetics is, it is conceivable to expect that an equilibrium would be reached sooner or later so that after such a period the vertical section of the column becomes an equivalent of one stage. The two-way streams continue to flow while they exchange toward the equilibrium, so that, the slower the two-phase exchange kinetics is, the higher the vertical thickness of the column it would take to achieve one equilibrium. This is the height equivalent of theoretical plate (HETP). The HETP multiplied by the number of stages equals the height of the separation cascade: Height of cascade ¼ ðHETPÞn

ð2:33Þ

The HETP of a packed column is influenced by the rate of isotope exchange, the flow rates per cross-sectional area and nature of the packing. The rate of exchange depends on the chemical kinetics and the rates of diffusion of the isotope-exchanging chemicals through two fluid layers, especially through the liquid. The primary aim of the packing is to provide an increased interphase area and a thin liquid layer without losing its own wet surface area while maintaining a smallest possible holdup. The higher the column height, the higher the pressure drop through the column, and the higher will be the holdup of the desired isotope by the column, causing among other things a higher capital cost of construction and longer startup transition period longer (cf.: Section IV of present chapter). This period is not only a waste of time because it does not produce the enriched isotope but also because, in case of an operational accident that requires a shut-down of the cascade, all efforts and costs expended up to that point will become wasted. And, the longer the startup period, the more probable that an accident may occur. Depending on the scale of production, holdups, and target enrichment, the transition period may be days, weeks, or months (cf.: Section IV of present chapter).

66

Isotope Effects in Chemistry and Biology

IV. STARTUP OF ISOTOPE ENRICHMENT CASCADE A. TIME -DEPENDENCE OF E NRICHMENT P ROFILE OF C ASCADE DURING S TARTUP

ALONG THE

L ENGTH

We will consider a startup of a simple case of a square, close separation cascade involving two isotopes and two streams, one stream of chemical ‘ and another of g. The plant’s feed point may be taken as Stage 0 and the abundance of the desired isotope at the feed point is kept constant (usually at the natural abundance) at all times. The stage number, s, increases from 1 to n, n being the highest stage which may be identified with the product-end refluxer. The desired isotope is assumed to enrich in the stream of ‘: The magnitude of the holdup, H moles of the desired isotope, in the refluxer depends on the design of the reflux process. Similarly, the holdup, h moles of the desired isotope per stage per unit flow, or the average process time per stage, also depends on the design of the separation process and the stage component such as the column packing material. Prior to a startup of the process, the separation column is flooded with the substance ‘ of the isotopic composition of the feed. Thus, at time t ¼ 0; the mole fraction of the desired isotope, N, which corresponds to the notations x and y in Figure 2.5 and Figure 2.7, is N0 throughout the cascade, where N0 is the enrichment of the feed material. The cascade operation is then started by initiating the input flow at the feed stage and the total reflux operation at the product-end. As the stream of the material ‘ flows toward the higher stages countercurrently against the stream of the material g, the target isotope moves toward the ‘-stream and the other isotope moves from the ‘-stream into the g-stream in the isotope exchange. When the front end of the ‘-stream first reaches the product end, however, the enrichment in that stream at that time will not be anywhere near the target enrichment level, because the level in every stage has just started with the very low enrichment of the feed material. As the ‘-steam comes into the product-end refluxer, it is (supposedly) 100% converted to the material of the g-stream in the refluxer, which then begins its journey up toward the plant feed point. This g-stream is somewhat more enriched than the original flooding material and provides a source of somewhat higher levels of enrichment in the ‘-stream flowing down. As plant operation under total reflux continues, the isotope concentration profile along the column slowly builds up. At any time during the startup period, the enrichment becomes higher with the higher stage gaining more in the enrichment than the lower stage. Thus, N is a function of the stage number, s, and the time, t, during the startup period: At t ¼ 0;

Nðs; t ¼ 0Þ ¼ N0

At s ¼ 0 ðat feed pointÞ;

½all s

Nð0; tÞ ¼ N0

At s ¼ n ðat product endÞ; P ¼ Product withdrawal rate ¼ 0

ð2:34aÞ ½all t ½all t , t1

ð2:34bÞ ð2:34cÞ

where t1 is the time it takes for the cascade to asymptotically attain a satisfactorily high enrichment profile, at which time the withdrawal of the enriched product may begin. Since the enrichment at every stage in the cascade will drop when the product is withdrawn (cf.: Section III.D, this chapter), the total reflux operation must be continued for a while longer even after the enrichment at the refluxer has reached a satisfactory level, in order to provide a buffer against such a drop. The time-dependent material balance between the product stage and the sth stage in a square, close separation cascade with a constant flow rate L,14 leads to a partial differential equation:

l

›Nðs; tÞ ›2 Nðs; tÞ › 2 1 {CNðs; tÞ þ Nðs; tÞ½1 2 Nðs; tÞ } ¼ 2 ›t ›s ›s

ð2:35Þ

Enrichment of Isotopes

67

where 1;a21

ð2:25Þ

l ; 2h

ð2:36Þ

2P 1L

ð2:37Þ

and

C;

in which P is the production rate at the product-end, and L the total interstage flow rate at stage s. L is independent of s at all times in a square cascade and of t during the total reflux operation. When the mole fraction of the target isotope is small compared to unity as in most of the isotope separation processes, Equation 2.35 becomes linear:

l

›N ›2 N ›N ¼ 2 1ð1 þ CÞ ›t ›s ›s2

ð2:38Þ

The solution of Equation 2.38 is in the form N 2 N0 ¼ 1 2 A1 e2B1 t 2 A2 e2B2 t 2 · · · N1 2 N0

ð2:39Þ

where N1 ¼ e1n is the overall separation at steady state, t ; t=ln2 is the reduced time, or the time measured in the units of average process time ðlnÞ divided by the number of stages ðnÞ; and ðN 2 N0 Þ=ðN1 2 N0 Þ represents the fractional equilibrium attainment. Cohen14 tabulated A1 and B1 for the cases of K=ln ¼ H=hLN ¼ 0.1 (0.1) 0.5 and 1n between 0.1 (0.1) 1.2, where K ; 2H=L in which H is the product-refluxer holdup in moles. Wieck and Ishida73 extended the solution of Equation 2.38 to include the second term of the solution, Equation 2.39.

B. RATE OF ATTAINMENT OF S TEADY-STATE P ROFILE VS. H OLDUPS As a more direct application of Equation 2.39, calculated reduced time, t0:95 ; the time required for the achievement of 95% of the steady-state enrichment has been plotted in Figure 2.8. It clearly shows an increasing trend of the transient time with increasing holdup in the product refluxer, K=ln; and the overall separation, 1n: The actual time it takes is ln2 t0:95 : Figure 2.9 illustrates an approach of the overall separation toward a steady state in a laboratory scale experiment74 on an exchange of nitrogen isotopes between liquid N2O3 – N2O4 mixture and their vapor phase under pressured conditions. The HETP for this system is 1.07 cm, while for the Nitrox process under comparable conditions70,71 (with 10 M HNO3 at 1.6 ml cm22 min through a 2.5 cm dia £ 95 cm long column packed with Helipak #3013) is HETP ¼ 2.8 cm.

V. EMPIRICAL DETERMINATION OF HETP AND SEPARATION FACTOR a A. BY U SE OF A NALYTIC S OLUTION OF M ATERIAL B ALANCE E QUATION UNDER T RANSIENT C ONDITION The method is an application of Equation 2.39 assuming first that the approximation up to the first transient term, A1 e2B1 t ; is sufficient. A plot of ln½ðN1 2 NÞ=ðN1 2 N0 Þ against t would be a straight line whose intercept and slope are ln A1 and 2B1 ; respectively. Both A1 and B1 are functions of 1 and n. This information, combined with experimental data on 1nð¼ ln N1 Þ leads to the first approximation for 1 and n. The second transient term of Equation 2.39 may then be added

68

Isotope Effects in Chemistry and Biology 10 9 8 7 6

l = 0.9 kn

5

t 0.95

4 0.5 0.4 0.3 0.2

3

2

0.1 0.0

1

0

1.0 en

2.0

FIGURE 2.8 The 95% equilibrium-attainment time, t0:95 : See the text for the notations; K=ln ¼ Ratio of product refluxer holdup and total separative stage holdup, 1n ¼ Overall separation of the cascade. (Reprinted with permission from Ref. 73. Courtesy of Marcel Dekker Inc.)

Overall Separation, S(t)

4.0

3.0

2.0

1.0

0

24

48

72 96 Time (hours)

120

144

FIGURE 2.9 Overall separation as a function of time in a laboratory experiment under the following conditions. Points up to 96 h were taken under total reflux. Points thereafter were observed at the product withdrawal rate, P ¼ 0.207 m mol N/min. (Reprinted from Ref. 74 by courtesy of Marcel Dekker Inc.) Exchange system: Exchange of 15N and 14N between liquid N2O3 – N2O4 mixture and their vapor phase under pressured condition. 15 NOðgÞ þ 14 N2 O3 ð‘Þ ¼ 14 NOðgÞ þ 15 N14 NO3 ð‘Þ Liquid composition: A mixture of the major component, N2O3(‘), and the minor component, N2O4(‘). Gaseous component: major component: NO(g). Minor components: NO, N2O3, NO2, N2O4. Product-end reflux: reduction of NOx by SO2(g), similar to the reflux for the Nitrox process. Other operating conditions: T ¼ þ 15.08C, total pressure ¼ 4.08 atm, liquid flow rate ¼ 0.48 ml/cm2 min, liquid molar flow rate ¼ 17.4 mmol Nitrogen per min, column packing ¼ Podbielniak SS Helipak No. 3013. Parameters obtained: aeffective ¼ 1:030; HETP ¼ 1.07. Minimum required parameters: nmin ¼ 345; min molar flow rate ¼ 9.31 mmol N/cm2 min, min column height ¼ 369 cm, min column cross-section ¼ 24.8 cm2.

Enrichment of Isotopes

69

to ðN1 2 NÞ=ðN1 2 N0 Þ to correct, especially, for possible deviations from a straight line for small t. The natural logarithm of the resulting quantity would then be plotted against t, and a second approximation for 1 and n would be obtained accordingly. Then, the HETP is equal to the total height of all separation stages divided by the number of stages, n.

B. FROM G RAPHICAL S OLUTION OF M ATERIAL B ALANCE E QUATION UNDER THE C ONDITION OF Z ERO T IME -D EPENDENCE AT A LL S TAGES Another method for emipirical determination of a and HETP that has been used by many researchers in laboratory scales is based on a formula obtained by integrating Equation 2.38 over the stages, s, under the condition of steady state, that is, when ›N=›t ¼ 0: The method does not rely on the tables of coefficients such as these14,73 for the parameters A1 ; A2 ; B1 , and B2 (cf. Equation 2.39). The integration yields P 1 L ln n¼ P P 1þ 1þ L L1 2 P Sp L1 12

ð2:40Þ

where n ¼ number of theoretical plates, P ¼ product withdrawal rate (moles of desired isotope per unit time), L ¼ total interstage flow rate of the ‘ and g streams (moles per unit time), N0 ¼ mole fraction of the target isotope in the feed, Sp ¼ Np =N0 ¼ overall separation when the production rate is P moles per unit time. Set P=L1 ; g

ð2:41Þ

and rewrite Equation 2.40. Then, 2

13

0

6 B 1 þ g C7 1 C7 ¼ B exp6 4n1@ P A5 1þg 12 2g L Sp and en1 ¼ S0 ¼ the separation at total reflux. Thus, 2

S0

1þg P 12 L

¼

1þg 2g Sp

ð2:42Þ

Given the experimental values of P, L, S0 ; and Sp ; each side of Equation 2.42 is a function of g. Let the left-hand side of Equation 2.42 be FðgÞ; and let the right-hand side be GðgÞ: Then, the value of g that satisfies Equation 2.42 is obtained as an intersect of the plot of FðgÞ vs. g and the plot of GðgÞ vs. g. The corresponding 1 is equal to P=gL; and n is obtained from S0 ¼ ð1 þ 1Þn :

VI. MISCELLANEOUS OTHER CONSIDERATIONS The single stage separation factor as a function of temperature, pressure, etc. and the HETP are not the only factors that one should take into consideration for design of an isotope enrichment plant and evaluation of experimental results. There are numerous “other” considerations, which depend on the particular isotope separation in a particular production scale. In the following, four examples will be discussed to illustrate varieties of “other” consideration.

70

Isotope Effects in Chemistry and Biology

A. POSSIBLE N EEDS OF C HEMICAL WASTE D ISPOSAL As illustrated in Table 2.3, the refluxer of the Nitrox process that produces 100 kg of 99.9% enriched 15N per year (18.26 mol 15N per day) needs 156,527 mol of sulfur dioxide per day and produces the equivalent number of moles of sulfuric acid per day. This sulfuric acid is produced at about 10 M and saturated with sulfur dioxide and it is not marketable as is. It must be treated as an industrial waste: a large-scale Nitrox plant must be run either with a large waste-disposal cost or with a sulfur-resource recycling system attached to it.

B. POSSIBILITY OF FAILURE TO ACHIEVE A H IGH TARGET E NRICHMENT One of the first things to suspect when one becomes aware that an expected target enrichment is not being attained after a prolonged total reflux period is a possible leak around the high-enrichment end of the plant such as the product refluxer, as previously explained. However, under special circumstances, there could be other more fundamental reasons for an apparent difficulty in reaching a design enrichment level. For example, the cryogenic distillation of nitric oxide cannot attain a high 15N enrichment by a single-pass distillation process, because nitrogen and oxygen isotopes in NO do not undergo exchanges at cryogenic temperatures, although they do at ambient temperatures. At ambient temperatures, the isotopes 14N, 15N, 16O, 17O, and 18O are distributed in accordance with the completely random distribution; when the isotopes are randomly distributed, i.e., when the distribution of isotopomers is in the classical limit of the Boltzmann distribution, the mole fractions of the isotopomers are the terms of the binomial expansion of the equality such as ½ð1 2 xÞ þ x ½ð1 2 y1 Þ þ y1 þ y2 ¼ 1

ð2:43Þ

where x, y1 ; y2 are the atom fractions of 15N, 17O, and 18O, respectively, (cf. Refs. 43,76; and Table 2.4). The isotopes are locked in this distribution in the feed material, which is cooled and distilled without further isotope exchange. As the distillation continues, the less volatile isotopomers become enriched near the boiler (the product-end refluxer) and the more volatile isotopomers get enriched near the top of the column in accordance with the relative volatilities (Table 2.4). The separation factors here are approximately equal to the vapor presure ratios.1

TABLE 2.4 Mole Fraction of the Isotopomers of Nitric Oxide at Room Temperature and Separation Factors of Distillation at 121K Separation Factor Isotopic NO 14–16 14–17 14–18 15–16 15–17 15–18 a

Mole Fractiona

At Natural Abundance

McInteerb

Bigeleisenc

ð1 2 xÞð1 2 y1 2 y2 Þ ð1 2 xÞy1 ð1 2 xÞy2 xð1 2 y1 2 y2 Þ xy1 xy2

0.99390 0.00037 0.00203 0.00369 0.00000137 0.00000755

1.000 1.019 1.027 1.037 1.046 1.064

1.000 1.019 1.028 1.040 1.049 1.069

x ¼ Natural abundance of 15N ¼ 0.37%; y1 ¼ Natural abundance of 17O ¼ 0.037%; y2 ¼ Natural abundance of O ¼ 0.204%; For the mole fraction formula, see Equation 2.43. b Fitted to column operation data at 121K on the basis of an assumption of equal difference in ða 2 1Þ for the oxygen series Ref. 39. c From measurements of isotopic vapor pressures at 120K. Ref. 72. 18

Enrichment of Isotopes

71 Product end

Feed point for Column 2

100 90 80 14–16

Mole Percent

70 60 15–16

50 40

14–18

30 15–18

20 10

14–17

0

15–17

0

100 200 Stage number

300

FIGURE 2.10 Isotopomer distribution profile along the second column of a two-section tapered cascade for the cryogenic distillation of nitric oxide at 121K under 1 atm. Composition of the feed NO ¼ Random distribution as in Table 2.4. (Reprinted with permission from Ref. 43. Copyright (1965) American Chemical Society).

However, as illustrated in Figure 2.10 (Ref. 43), the mole fractions of the isotopomers that were present in the feed in low concentrations, e.g., xy2 ¼ 0:00000755 for 15 –18, cannot build up a high concentration profile near the refluxer because, having started with low abundance, the buildup of 15– 18 is blocked by the isotopomers such as 14– 18 that are available in abundance. A compromise solution to circumvent the problem is (i) withdrawing the enriched product in a manner as if it is the final product, warming it up to the ambient temperature outside the cyrogenic column, accumulating it, and using it as the feed for a separate distillation column, or (ii) running the distillation column with an isotope exchange catalyst.

C. POSSIBLE E XPLOSION OF W ORKING M ATERIAL For example, in the cryogenic distillation of nitric oxide, solid deposits may accumulate in the parts of the distillation column causing clog-up of the gas flow, which may lead to an explosion. The solid deposit could be the higher oxides of nitrogen, all of which have higher melting points than nitric oxide. The most prominent culprit is the disproportionation of NO, 3NO ! NO2 þ N2O. The rate of this disproportionation does not significantly change with temperature.77 The accumulation of the higher oxides with time at ambient temperature is illustrated in Table 2.5. A careful purification of initial feed material, nitric oxide, as soon prior to the feeding as possible, is of primary importance. If the purified NO and enriched NO must be held in storage tanks, it is highly recommended that the storage pressure be kept as low as practical. The partial pressure of NO below 5 atm is advisable.78

72

Isotope Effects in Chemistry and Biology

TABLE 2.5 Calculated Accumulation of NO2 and N2O through Disproportionation of NO at 298K Mole Fractions of NO2 and N2O Time (h)

P0 5 50

P0 5 10

P0 5 5

P0 5 2

P0 5 1

24 48 72 240 30 days 1 year

0.0010 0.0019 0.0029 0.0094 0.0264 0.1689

0 0 0.0001 0.0004 0.0011 0.0124

0 0 0 0.0001 0.0003 0.0032

0 0 0 0 0 0.0001

0 0 0 0 0 0.0001

P0 ¼ Initial pressure of pure NO in atm.

D. CONSIDERATION OF S UPPLY FOR

THE

F EED

The feed material for an isotope separation plant is a chemical that is usually not available inexpensively and in abundance, and the desired isotope contained in the typical feed is usually a minor component of the feed. Therefore, a large supply of the feed material is needed especially for a large-scale plant. For instance, if the atom fraction of the target isotope in a feed is, say, 0.0001 (i.e., 0.01%), the absolutely minimum amount of the feed required to produce 1 mol of 100% enriched target isotope is 10,000 mol of the feed material, which is without any regard to the separation factors and other cascade realities such as the target isotopes lost with the deleted (waste) stream. A common solution of the problem is that of a parasitic isotope separation plant, parasitic in the sense of setting up the isotope plant adjacent to a chemicals plant that produces the feed chemical in a sufficiently large quantity. Use a product of an appropriate chemicals plant as the feed for the isotope plant, extract the desired isotope to produce an enriched isotope, and return the isotope-depleted chemical to the chemicals plant. The isotopically depleted chemical will pass as the ordinary chemical, and the amount not returned by the isotope plant will be a small fraction of the amount borrowed.

VII. ENRICHMENT BY NONSTEADY STATE PHENOMENA INVOLVING REVERSIBLE PROCESS A. ION E XCHANGE I SOTOPE S EPARATION The isotope effect in ion exchange process was first observed by Taylor and Urey. They found that the isotopic abundance ratios of lithium, potassium, and nitrogen were changed when lithium, potassium, and ammonium ions were eluted from inorganic ion exchanger “zeolite” packed in a stainless steel pipe.79 Isotope effects of lithium were intensively studied by Lee, Begun, and Drury by using organic cation exchange resin.80 – 82 They determined single stage separation factors of lithium isotopes between the phases of an aqueous solution and synthetic cation exchange resin at different experimental conditions of temperature, concentration of eluent, crosslinking and functional groups of resin, etc. Ion exchange was applied for nitrogen isotope separation by Spedding et al. They were successful in enrichment of 15N and obtained highly enriched 15N starting from natural abundance of 0.366% by ion exchange chromatography.83 Since then several works have been carried out on nitrogen isotope separation by cation exchange resins. The observed single stage separation factor of the NH3 – NHþ 4 system is in the range 1.020 , 1.025. The experimentally

Enrichment of Isotopes

73

determined isotope separation factor arises from the equilibrium constant of the following isotopic exchange reaction 14

NH4 – R þ 15 NH3 H2 O ¼ 15 NH4 – R þ 14 NH3 H2 O

ð2:R:1Þ

where R represents a cation exchange resin. In general, isotope exchange in the ion exchange resin system is expressed as M – R þ M0 L ¼ M0 – R þ ML

ð2:R:2Þ

where M and M0 represent heavy and light isotope ion, respectively, and L is the ligand used for the elution of the isotopic species. In many cases of isotopic metal ions, light isotopes are enriched in the resin phase, while the heavy isotopes are enriched in the complex species in the aqueous phase. Recently studied examples are copper, zinc, vanadium, and gadolinium.84 – 86 The cations of these elements make strong complexes with organic acids and chelating reagents, such as malate and EDTA, while they are in the form of hydrated ions in the resin phase. As expected from the theory of isotope effects, the heavy isotopes are fractionated in the complex species in the solution phase. In the case of lithium isotopes, M is Liþ ion, L is a water molecule or a group of water molecules and ML represents fully hydrated Liþ ion in the aqueous phase. Due to the dehydration in ion exchange resin, light isotope 6Li is enriched in the resin phase and 7Li is enriched in the aqueous phase. On the other hand in the case of the above-mentioned NH3 –NHþ 4 system, the direction of the fractionation of nitrogen isotopes becomes opposite; the heavy isotope is enriched in the resin phase due to the formation of NHþ 4 , for which the RPFR in the resin phase is greater than that of NH3 in the aqueous phase. This tendency of heavy isotope enrichment in the ion exchange resin also occurs among the heavy alkali metals such as Rb: heavy isotope 87Rb is fractionated in the cation exchange resin, while light isotope 85Rb is fractionated in the aqueous solution. Isotope effects in pure ion exchange are closely related to the hydration states of the ions. Isotope effects provide a tool for the study of inorganic solution chemistry. The separation factor of the system is defined by the following equation and the deviation from unity is expressed in terms of 1,

a ¼ ½M0 – R ½ML =½M – R ½M0 L ¼ 1 þ 1

ð2:44Þ

The separation factor is experimentally determined by measuring the isotopic abundance ratios in each phase as

a ¼ {½M0 =½M }re ={½M0 =½M }aq ¼ {½M =½M0 }aq ={½M =½M0 }re

ð2:45Þ

where subscripts re and aq represent the resin and aqueous solution, respectively. The isotopic fractionation in an ion exchange system is depicted in Figure 2.11. It should be noted that the pure AX + BY = BX + AY K>1

Isotope Exchange Equilibrium Ion Exchange Equilibrium

Isotopic Fractionation

MY

MX [A] [B] resin Resin Phase

FIGURE 2.11 Isotopic fractionation in ion exchange resin system.