All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, including photocopying and recording without the written permission of the copyright holder, application for which should be addressed to the Publishers. Such written permission must also be obtained before any part of this publication is stored in a retrieval system of any nature. This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price given by the Publishers in their current price list. First published 1984 © Butterworth & Co (Publishers) Ltd 1984 British Library Cataloguing in Publication Data Tyrrell, H. J. V. Diffusion in liquids.—(Butterworths monographs in chemistry) 1. Heat—Transmission 2. Fluids I. Title II. Harris, K. R. 536'.2 QC320.2 ISBN 0-408-17591-5 Library of Congress Cataloging in Publication Data Tyrrell, H. J. V. (Henry John Valentine) Diffusion in liquids. (Butterworths monographs in chemistry) Includes index. 1. Diffusion. 2. Liquids. I. Harris, K. R. II. Title. III. Series: Butterworth's monographs in chemistry. QD543.T95 1984 541.3'4 83-7711 ISBN 0-408-17591-5 Filmset in Monophoto Times by Mid-County Press, London Printed in Great Britain at the Cambridge University Press Butterworths Monographs in Chemistry Diffusion in Liquids A theoretical and experimental study H. J. V. Tyrrell, MA, DSc Professor of Physical Chemistry, Chelsea College, University of London, England K. R. Harris, BSc, PhD Lecturer in Chemistry, Department of Applied Chemistry, Royal Melbourne Institute of Technology, Victoria, Australia Butterworths London Boston Durban Singapore Sydney Toronto Wellington Preface This book was first conceived as a revision of Diffusion and Heat Flow in Liquids which was published in 1961 and was concerned with diffusion, thermal diffusion and thermal conduction in liquid systems. It soon became evident that inclusion of all the important advances of the last two decades would make a new edition along the lines of the first inordinately long. We were therefore faced with a choice between maintaining the original breadth of coverage at a more superficial level than before and restricting the scope without a fundamental change of style. The aim of the first edition had been the provision of an account of theory and practice sufficiently complete to give a non-specialist an adequate base for the understanding of the original literature, and to give, in addition, a critical review of the then current state of knowledge in the field. On reflection, it seemed more useful to preserve this aim and to restrict the scope of this book almost entirely to a treatment of translational and rotational diffusion in isothermal liquid systems. The first four chapters are concerned with establishing the foundations of the theory of transport processes, starting with the classic phenomenological descriptions which define transport coefficients. These can also be defined, by use of the techniques of non-equilibrium thermodynamics, in terms of mobility (Onsager) coefficients or of resistance (friction) coefficients, thereby clarifying both the nature of the driving force for diffusion and the minimum number of independent coefficients needed to describe transport processes. This is particularly useful for diffusion in multicomponent mixtures. Finally, non- equilibrium statistical thermodynamics provides, at least in principle, me- thods for calculating transport properties from molecular properties. The intention in these chapters has been to provide a unified treatment of these topics which is adequate for the understanding of modern methods of interpreting the experimental data on translational and rotational diffusion reviewed in Chapters 6-8. These data can be obtained by use of a wide range of techniques and Chapter 5 is devoted to an account of many of these. Translational diffusion coefficients have been measured for well over a century though it was not until the late 1940s and the advent of optical techniques based on the Gouy and Rayleigh interference phenomena that really reliable data for binary systems became available. These, and other optical techniques, are still of importance iv Preface v in the study of interdiffusion, especially since the advent of laser sources, and are discussed in some detail, as are some other long-established and reliable methods. In recent years new methods, each with their own theoretical base, have come into use. For interdiffusion studies, chromatographic peak broadening or Taylor dispersion has become a method of considerable importance, while the advent of photon correlation techniques has made light- scattering methods particularly useful for diffusion studies on macromo- lecules, especially those of biological origin. In both cases the speed with which such measurements can be made has contributed greatly to their acceptance. For the measurement of intradiffusion (self-diffusion) coefficients the nuclear magnetic resonance method has been refined to a remarkable extent, but to understand the principles of both this and light-scattering techniques requires an exposition of the basic physics involved. The same applies to all the methods available for the study of rotational diffusion where many traps exist for the unwary. Consequently, Chapter 5 is long and, in parts, quite complicated; we hope, however, that the rather detailed discussions we have given will help the increasing number of non-specialists who use these methods to have an adequate understanding of their advantages and limitations. Any authors attempting to cover such a field profit greatly from the candid criticisms of successive drafts by independently minded friends. Among these we would like to thank Dr P. J. Dunlop of the University of Adelaide, and Dr R. Mills of the Diffusion Research Unit at the Australian National University, Canberra, both of whom read a late draft and enriched the final version by their comments. An even greater influence on the final shape of the book was exerted by Dr D. G. Miller of the Lawrence Livermore National Laboratory who commented extensively and very acutely on two successive drafts. His detailed knowledge of the history, the theory, and the practice of diffusion measurements has helped us greatly. In the (to us) less familiar field of rotational diffusion we had useful discussions with Professor D. Phillips of the Royal Institution, London, and with Dr P. Quinn of the Department of Biochemistry at Chelsea College. We are, of course entirely responsible for any errors of fact or of understanding which may remain. H.J.V.T. K.R.H. List of symbols A Cross-sectional area Parameter defined in equations (3.62), (3.63) Translational-rotational coupling constant for diffusion Aj Coefficients in series expansion, equation (5.52) A Instrumental constant, equation (5.160) m A Translational-rotational coupling constant for viscosity n A Affinity of rth reaction r A(r,j>; {N}) Fluctuating force in Langevin equation t ^(r,p) Time-independent part of Hamiltonian A Surface area s s/ Spin-echo amplitude, equation (5.340) a Thermal diffusivity (thermometric conductivity) Thickness of diffusion cell, equation (5.59) a Thermodynamic activity of component i i a State parameter, equation (2.46) k B Parameter defined by equations (3.62), (3.63) Parameter defined in equation (6.106) Parameter defined in equation (6.137) B* Thermodynamic factor [1 +(dln fi/d In Xi) T] p defined in c mole fraction terms B, B™ Thermodynamic factors defined similarly in terms of t molar and molal concentration scales respectively B Peak-to-peak amplitude in saturation transfer EPR m measurements B Magnetic field strength, equation (5.314) b Optical distance between the centre of a diffusion cell and the image plane, equation (5.59) Savart plate shear distance, equation (5.159) C Maximum deflection from geometric optics in Gouy fringe t pattern C Parameter characteristic of rotating disc electrode, w equation (5.221) viii List of symbols ix c Molar density (total number of moles per unit volume) c Molar concentration of component i t c Specific heat capacity at constant pressure p D Translational diffusion coefficient: distinguished by superscripts v, m, x, s, c, for volume-, mass-, mole-, solvent- and cell-fixed reference frames, respectively D Interdiffusion (mutual diffusion coefficient) for binary 12 mixture D™ Limiting interdiffusion coefficient at infinite dilution of 2 solute D\ Modified solvent-fixed diffusion coefficient defined by 2 equation (3.26) D Integral diffusion coefficient D[, D\ Intrinsic diffusion coefficients in binary mixture, equation (3.115) D Rotational diffusion coefficient (also D and R ± Daf Intradiffusion coefficient for component / D Apparent diffusion coefficient, p. 197 D Diffusion coefficient calculated from the rth moment and rh the height h of a Gaussian refractive index gradient plot D o Area-rth moment diffusion coefficient r D Reduced rth moment diffusion coefficient, e.g. D , rm 2m equation (5.111) Dj* Thermodynamically corrected interdiffusion coefficient 2 @ Member of an un symmetric set of diffusion coefficients jk describing translational diffusion in a multicomponent mixture Sty Member of a similar but symmetric set, equation (3.49) £(r,t) Incident field in light-scattering experiments, equation f (5.266) E Total scattered field, equation (5.267); E* is the conjugate s of E s E Electric field strength e Electrical charge on particle i t F(t) Time-dependent part of perturbation term in Hamiltonian, equation (1.46) F Generalized Onsager force, equation (2.52) k F Effective electric field, equation (5.357) F Faraday constant / Distribution function for particle velocities, equation (2.14) f Phase shift due to 180° pulse, equation (5.346) ft Activity coefficient of component i on mole fraction scale x List of symbols fij Velocity correlation coefficient (Hertz) f(Z) Parameter defined by equation (5.117), used in the interpretation of free diffusion data f{co) Normalized distribution function of Larmor frequencies, equation (5.325) G( 1 ) Normalization factor, equation (5.369) G( 2(T)) Field autocorrelation function, equation (5.274) G (T) Intensity autocorrelation function, equation (5.277) AG* Molar Gibbs function change on formation of a transition state G(t) Magnetic field gradient integrated over time interval, equation (5.345) G(r,p;t) Microscopic state variable, equation (1.51) g Gr2avitational acceleration ga (dS/dadaj), equation (2.52) kj k g(\x2) ) Normalized field autocorrelation function, equation (5.275) # (T) Normalized intensity autocorrelation function, equation (5.278) g$ Pair distribution function g Magnetic field gradient, Section 5.13 H N-particle Hamiltonian function, equation (1.20) N H(Z*) Experimental parameter used in interpretation of X Rayleigh fringe patterns AH Molar enthalpy change on formation of transition state h Molar enthalpy h Specific enthalpy I Time-averaged light scattering intensity, equation (5.273) s J(v) Optical spectrum, equation (5.276) I(co) Polarized light scattering intensity yu J||(0> J±(0 Scattered fluorescent light intensity / Nuclear spin quantum number J Ionic strength, defined as \ £ z cf t I Ionic strength fraction of ion species i t I Elect1ri2c/ current, equation (1.3) (-D i Unit vector J Total fringe shift in integral fringe pattern J Scalar spin-spin coupling constant x Heat flux density, equation (1.1) J Molar flux density of component i: distinguished by f superscripts indicating the reference frame used List of symbols xi Flux density of component i in mass units relative to a mass-fixed frame i[ Intrinsic flux density of component /c, equation (3.115) Jv Vacancy flux density, equation (3.125) Jx Molar flux density of component i in a system containing a labelled component, equations (4.30), (4.31) f Volume integral of flux density, equation (1.72) j Interference fringe number in integral fringe pattern j Electric current density ]j(r) Mean momentum density, equation (1.69) K Integration constant, equation (1.31) Parameter with dimensions of diffusion constant defined by equation (5.248) Kt Equilibrium constant, equation (2.34) K Scattering wave vector, equation (5.268) k Coefficient of thermal conductivity, equation (1.1) k Boltzmann constant kp Ratio of density at atmospheric pressure to that at the experimental pressure, equation (5.37) /ca, betc, kr Reaction rate constants, cf. equations (2.33), (6.101) k Wave vectors, equation (5.268) Lij 9L*7- Mobility coefficients, cf. equations (1.17), (2.63) Mobility coefficients in solvent-fixed frame, equation (3.82) L0 Optical path length, equation (5.115) L Angular momentum / Length of conducting bar, equation (1.4) Thickness of diaphragm in a diaphragm cell Ijj Mobility coefficient defined in a mole-fixed reference frame lt j Mobility coefficient defined in a mole-fixed reference frame for isotopically labelled systems, equations (4.30), (4.31) M Magnetization of sample Mr rth semi-moment of distribution curve, equation (5.68) M Molecular weight or molar mass m{ Mass of particle, equation (1.20) m Molality m2 Mass of solute injected in pulse mr rth moment of distribution curve, cf. equations (5.101), (5.297) 7VA Avogadro number NL Amount of labelled material remaining in a capillary, equation (5.310) Nx Dimensionless group defined by equation (5.45) xii List of symbols n Refractive index n Stokes-Einstein number, equation (7.20) tt^r,^) Number density, equations (1.28), (6.31) n Unit vector normal to surface element dS, equation (5.331) P Parameter defined in equation (3.65) Parameter defined in equation (5.260) P(r) Probability function, equation (1.32) P(r, t) Dielectric polarizability, equation (5.266) p Pressure p Momentum vector of particle i f Q Parameter defined by equation (3.65) Quadrupole moment of the nucleus q Symbol for heat Local electric field gradient, equation (5.374) R Perturbation of Hamiltonian function Gas constant Resistance between two electrodes Radius of capillary tube, equation (5.45) Ratio of outer to inner distance parameters for the square well potential Re Reynold's number, equation (5.256) R Refractive index increment for component i in a T multicomponent mixture 0t Radius of gyration, equation (6.130) r Radial coordinate in a circular section tube r(t) Polarization anisotropy, equation (5.368) i-; Position vector of particle i s Entropy Sc Schmidt number, equation (5.257) S' Specific entropy flux into volume element, equation (2.138) AS* Entropy of formation of a transition state se Diameter of a circular sheet s Time interval Parameters defined in equation (6.146) et seq. S Specific entropy T Temperature (Kelvin) T Critical temperature c T T NMR relaxation times 192 t Time Numerical parameter in modified Stokes-Einstein equation List of symbols xiii Transport number of ion i Hittorf transport number 'Cell-fixed' transport number U Internal energy U Average longitudinal velocity of liquid in a capillary, equation (5.239) u Specific internal energy u Quantity defined by equation (5.135) m u Mobility of ion species j ; uj Intrinsic mobility of species j, equation (3.115) uf Mobility of species j defined in terms of its intradiffusion coefficient u(r) Longitudinal velocity of liquid in a capillary at a distance r from the axis V Volume V Volume element occupied by an ensemble, equation (4.155) 0 Volume occupied by close-packed hard-spheres ViVi — Tj) Pair interaction energy between particles i and j v Molar volume v Specific volume v Reaction velocity vR Velocity of particle i f y Arbitrary velocity of reference frame, equation (2.163) vj Velocity of species j in the intrinsic reference frame W Flow rate of mercury at a dropping mercury electrode, equation (5.212) iV Rotation frequency of rotating disc electrode w Mass fraction of component i t X Generalized force, equation (1.16) £ External force per unit mass acting on component j, equation (2.121) Kj Force per mole on species j in systems containing a labelled component, equations (4.30), (4.31) Y + Extensive thermodynamic property Y Ratio SY/SV where V is a volume element Yj Lateral fringe displacement, equation (5.58) Yj Force on unit mass defined by equation (3.53) y Activity coefficient of component i on molarity scale t yj Displacement of fringe j in interference pattern y Coordinate in experimental refractive gradient curve, equation (5.64) Z Reduced spatial coordinate, equation (5.69)