heat capacities liquids, solutions and vapours

3.7 Discussion of Results 49Chapter 4 Heat Capacities and Related Properties of Liquid Mixtures 54 Emmerich Wilhelm and Jean-Pierre E.. 7.4 Studies of Apparent Molar Heat Capacities andS

Heat Capacities

Liquids, Solutions and Vapours

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The role of energy in our society, whether it is from fossil fuels or an native source such as hydrogen, has been the topic of much discussionrecently An example of this is to be found in the volume entitled FutureEnergy: Improved, Sustainable and Clean Options for our Planet (Elsevier,Amsterdam 2008) edited by T M Letcher, published under the banner ofthe International Union of Pure and Applied Chemistry (IUPAC) and theInternational Association of Chemical Thermodynamics (IACT), which is anAssociate Organization of IUPAC Not surprisingly, there is also resurgence

alter-in the broad topic often termed thermochemistry that is covered by the remit

of IACT, and was previously the concern of the IUPAC Commission onThermodynamics These organizations take an active role in the definitionand maintenance of standards in the fields of thermochemistry This roleincludes, but is not limited to, the recommendation for calorimetric proce-dures, the selection and evaluation of reference standards for thermo-dynamic-measurement techniques of all types, the dissemination of evaluatedthermodynamic data of the fluid state, the standardization of nomenclatureand symbols in chemical thermodynamics as well as the surveillance of inter-national pressure and temperature scales and promotion of the subject withawards by IACT to young scientists (further information can be found atwww.IACTweb.org)

This book, entitled Heat Capacities: Liquids, Solutions and Vapours, is alsopublished under the auspices of both IUPAC and IACT It is another in alineage that started in 1956 with Experimental Thermochemistry Volume I(Interscience Publishers, Inc New York) edited by F D Rossini, who ishonoured at each biennial International Conference on Chemical Thermo-dynamics, sponsored by both IACT and IUPAC, with the delivery of thecoveted Rossini Lecture Volume II, edited by H A Skinner was published in

1962 (Interscience-Wiley, New York) An update of the material in thesevolumes was given in Combustion Calorimetry, edited by S Sunner and M.Ma˚nsson (Pergamon Press, Oxford, 1979) There is in addition a series ofvolumes on ‘‘Experimental Thermodynamics’’, for which there are now sevenvolumes in print and an eighth in preparation: (a) Volume I, Calorimetry ofNon-reacting Systems, edited by J P McCullough and D W Scott

v

(Butterworths, London, 1968); (b) Volume II, Experimental Thermodynamics ofNon-reacting Systems, edited by B LeNeindre and B Vodar, (Butterworths,London, 1975); (c) Volume III, Measurement of the Transport Properties ofFluids, edited by W A Wakeham, N Nagashima, and J V Sengers (BlackwellScience Publications, Oxford, 1991); (d) Volume IV, Solution Calorimetry,edited by K N Marsh and P A G O’Hare (Blackwell Science Publications,Oxford, 1994); (e) Volume V, Equations of State for Fluids and Fluid Mixtures,edited by J V Sengers, R.F Kayser, C.J Peters, and H.J White Jr (ElsevierScience, New York, 2000); (f) Volume VI, Measurement of the ThermodynamicProperties of Single Phases, edited by A R H Goodwin, K N Marsh, and W.

A Wakeham (Elsevier 2003); (g), Volume VII, Measurement of the dynamic Properties of Multiple Phases, edited by R D Weir and T W de Loos(Elsevier 2005); and (h) Volume VIII, Applied Thermodynamics, edited by A R

Thermo-H Goodwin, J V Sengers, and C J Peters (Royal Society of Chemistry,Cambridge, to be published 2010)

The current text has been produced with an international team of tinguished experimentalists who describe the current state of development ofheat capacity measurements It continues the theme started with volume (f)above to provide a framework for both academia and industry The latter oftenrequire measurements fit-for-purpose obtained from instruments with workingequations based on the principles of physics, without recourse to maintainingthe breath of expertise required to perform these measurements Naturally,many of the author team are IACT members and to all great appreciation andgratitude are owed for their willing and enthusiastic contributions Interna-tional cooperation in this venture fulfils one mission for IUPAC and all shouldnot forget the tireless editors who, faced with the task (common to all editors ofco-operative efforts) of constructing a coherent whole from the independentcontributions have achieved their goal The material contained in this volume is

dis-of considerable value, perhaps also providing guidance for the development dis-ofnew and more precise techniques

Anthony R H Goodwin

Chairman

International Association of Chemical Thermodynamics

Many of the most significant developments in physical chemistry and cal engineering during the last century have been based on contributions che-mical thermodynamics has provided The continuously increasing number ofarticles containing experimental data on thermodynamic properties and onphase equilibria, as well as on new experimental techniques and advances intheory and computer simulation, demonstrate the unabated growth of thisfield Most noteworthy is the accelerating thrust in biophysical chemistrytowards achieving a broader, quantitative thermodynamic basis of the physi-cochemical phenomena involved in biological processes Heat capacitiesbelong to the most important thermodynamic/thermophysical properties,playing a central role in the pure sciences as well as in chemical engineeringand industrial applications In this monograph the reader will find 22 con-tributions dealing with heat capacity, from various angles, of (mostly) liquidsand gases/vapours, both pure and mixed It may thus be regarded as a nat-ural complement to the volumes on Experimental Thermodynamics (Volume

chemi-I, 1968, through Volume VIchemi-I, 2003) with their quite general coverage of thefield, in that the focus is now on a single property Thermodynamics is a the-oretical discipline in close contact with experiment, and the 22 chapters are,

in fact, testimony to it However, they do not cover the entire subject.Amongst other topics, chemically reactive systems are not included Weintend to fill the gaps at a later point of time

This book has its origins in committee meetings of the International ciation of Chemical Thermodynamics It is a project produced under theauspices of the International Union of Pure and Applied Chemistry (IUPAC)

Asso-In true IUPAC image, the authors, which represent some of the most importantnames in their respective fields, come from many countries around the world,including: Austria, Belarus, Belgium, Canada, Czech Republic, France,Germany, Israel, Italy, Japan, Poland, South Africa, United Kingdom, and theUnited States of America

Heat Capacities: Liquids, Solutions and Vapours

Edited by Emmerich Wilhelm and Trevor M Letcher

r The Royal Society of Chemistry 2010

Published by the Royal Society of Chemistry, www.rsc.org

vii

Two features are of paramount importance in monographs like this one: thetimeliness of the topic and the coverage and critical evaluation of the pertinentpublications In fact, this book highlights the underlying theory, some of themost important experimental techniques, modelling and computer simulation,

as well as significant and new results related to heat capacity, thereby scoring its importance; and as shown by the contributions, the authors haveendeavoured to cover the relevant literature up to about 2008 Overwhelmingly,the contributing authors have adhered to the nomenclature/symbols suggested

under-by the Green Book of IUPAC (3rd edition, 2007) The few differences are eitherdue to their desire to present a more concise, unequivocal notation, or tocompliance with usage accepted by the scientific communities working in theirrespective specialised fields We, as editors, did not interfere if exact definationswere supplied, though in a few instances we added explanatory footnotes at therelevant places Such as approach is in accord with the spirit of the Green Book

as expressed so admirably by Martin Quack in his Historical Introduction on

p XII of its 3rd edition: It is not the aim to present a list of recommendations inform of commandments Rather we have always followed the principle that thismanual should help the user in what may be called ‘‘good practice of scientificlanguage’’

One of the objectives of the book is to bring together research from disparatedisciplines which have a bearing on HEAT CAPACITIES Links between thesechapters, we believe, could lead to new ways of solving problems and looking atnew and also old HEAT CAPACITIES related issues Underlying this philo-sophy is our inherent belief that a book is still an important vehicle for thedissemination of knowledge

This book is meant for researchers in chemical thermodynamics, either fromacademia or from (applied) chemical engineering, to show the progress recentlyachieved Its success rests with the 35 authors We would like to thank all ofthem for their cooperation; and we would also like to thank the publishers, theRoyal Society of Chemistry, whose representatives were helpful and patient inproducing this monograph on heat capacity

Emmerich Wilhelm

Institute of Physical Chemistry, University of Wien

Wien (Vienna), Austria

Trevor M Letcher

University of KwaZulu-Natal

Durban, South Africa

Chapter 2 Calorimetric Methods for Measuring Heat Capacities of

Lee D Hansen and Donald J Russell

Chapter 3 An Analysis of Conductive Heat Losses in a Flow Calorimeter

J David Raal

3.3 Mathematical Formulation for a Model Flow

Heat Capacities: Liquids, Solutions and Vapours

Edited by Emmerich Wilhelm and Trevor M Letcher

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ix

3.7 Discussion of Results 49

Chapter 4 Heat Capacities and Related Properties of Liquid Mixtures 54

Emmerich Wilhelm and Jean-Pierre E Grolier

Andrew W Hakin and Mohammad M.H Bhuiyan

7.4 Studies of Apparent Molar Heat Capacities and

Standard Partial Molar Heat Capacities of Aqueous

Chapter 8 Scanning Transitiometry and its Use to Determine Heat

Chapter 10 Speed-of-Sound Measurements and Heat Capacities of

Toshiharu Takagi and Emmerich Wilhelm

10.2 Apparatus for the Speed-of-Sound Measurement in

10.3 Temperature and Pressure Dependences of the

10.4 Speed of Sound and Thermodynamic Properties 228

Chapter 11 Heat Capacities and Brillouin Scattering in Liquids 238

Emmerich Wilhelm and Augustinus Asenbaum

xiContents

11.4 Selected Results and Discussion 248

Chapter 12 Photothermal Techniques for Heat Capacities 264

Jan Thoen and Christ Glorieux

13.3 Design and Operational Implementation 29213.4 Phase Transition Studies in Binary and Ternary

13.5 Phase Transition Studies in Liquid Crystals 299

Mikhail Anisimov and Jan Thoen

15.3 Experimental Heat Capacity 335

15.5 Quantitative Thermal Analysis of Polymeric

Werner W Streicher and George I Makhatadze

16.3 Experimental Determination of Partial Heat

M Marinelli, F Mercuri and U Zammit

17.1 Brief Introduction to Liquid Crystals 36717.2 The Nematic–Isotropic Phase Transition 37017.3 The Smectic A–Nematic Phase Transition 37317.4 The Smectic A–Smectic C Phase Transition 37617.5 The Smectic A–Hexatic B Phase Transition 37817.6 Phase Transitions in Restricted Volumes 381

Chapter 18 Heat Capacities and Phase Transitions for the Dynamic

Chemical Systems: Conformers, Tautomers, Plastic

Gennady Kabo, Eugene Paulechka and Michael Frenkel

18.2 Practical Calculations of Heat Capacity for

Gases and Liquids by Statistical Thermodynamic

Acknowledgment 418

Chapter 19 The Estimation of Heat Capacities of Pure Liquids 421

Milan Za´bransky´, Zdenˇka Kolska´, Vlastimil Ru˚zicˇka

and Anatol Malijevsky´

Chapter 20 Computer Simulation Studies of Heat Capacity Effects

Dietmar Paschek, Ralf Ludwig and Jo¨rg Holzmann

21.3 Thermodynamics, Data Reduction, Data

Once the fabric is woven it may be embellished at will.

Nero Wolfe in The Golden Spiders, by Rex Stout, Bantam edition, New York,

NY, 1955

1.1 Introduction

Heat capacities belong to the most important thermophysical properties ofmatter: they are intimately related to the temperature dependence of funda-mental thermodynamic functions; they may be determined in the laboratorywith great accuracy; and they are of key importance for linking thermo-dynamics with microscopic fluid structure and dynamics, as evidenced by thecontributions to this book They are thus indispensable in physical chemistry aswell as in chemical engineering For instance, as a classical example, considerthe standard entropies of liquids at T¼ 298.15 K They are evaluated fromexperimental heat capacities at constant pressure from low temperatures to298.15 K and entropies of phase changes in between (assuming applicability of

Heat Capacities: Liquids, Solutions and Vapours

Edited by Emmerich Wilhelm and Trevor M Letcher

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1

the third law of thermodynamics) The measured heat capacity of an organiccompound can usually be extrapolated to 0 K by fitting a Debye heat capacityfunction to the experimental values at, say, 10 K.

The nature and the size of this monograph’s topic make it impractical to coverthe entire subject in one volume As indicated in the title, the focus will be on heatcapacities of chemically non-reacting liquids, solutions and vapours/gases(though polymers and liquid crystals are also covered) The individual specialisedchapters have been written by internationally renowned thermodynamicists/thermophysicists active in the respective fields Because of their topical diversity,

in this introductory chapter I shall try to summarise concisely the major aspects

of the thermodynamic formalism relevant to fluid systems, to clarify, perhaps,some points occasionally obscured, to indicate some ramifications into neigh-bouring disciplines, and to point out a few less familiar yet potentially interestingproblems The omission of any topic is not to be taken as a measure of itsimportance, but is predominantly a consequence of space limitations

Calorimetric determinations of heat capacities of liquids have a long tion, and many distinguished scientists have contributed to this subject Onecan only marvel about the careful work of some of the early researchers, such asEucken and Nernst,1,2who developed precursors of modern, adiabatic calori-meters The adiabatic method for heat capacity measurements at low tem-peratures was pioneered by Cohen and Moesveld,3and Lange,4and becamewidely used Indeed, during the following decades, many alternative designs ofincreasing sophistication have been devised and successfully used A selection

tradi-of adiabatic calorimeters which were described in the literature up to about

1970 is provided by references 5 through 19 For details, the interested readershould consult the classic IUPAC monograph edited by McCullough andScott,20 or the more recent ones edited by Marsh and O’Hare,21 and byGoodwin, Marsh and Wakeham,22 or the monograph on calorimetry byHemminger and Ho¨hne.23

More specialised reviews have been prepared by Lakshmikumar and Gopal,24Wadso¨25 and Gmelin.26 A monograph focusing on differential scanningcalorimetry has been presented by Ho¨hne, Hemminger and Flammersheim.27

To date, the most widely used instruments for measuring heat capacities ofliquids and liquid mixtures are based on the differential flow calorimeterdesigned by Picker,28,29which was commercialised by Setaram Because of theabsence of a vapour space, differential flow calorimeters are particularly useful.They may be fairly easily modified to be used at elevated temperatures andpressures, including the critical region The first instrument of this type wasconstructed by Smith-Magowan and Wood,30 with improved versions beingdue to White et al.,31and Carter and Wood.32 However, comparison of heatcapacities measured by different types of flow calorimeters and differentialthermopile conduction calorimeters shows small differences in measured heatcapacities, which are attributed to conductive and convective heat losses.Conductive heat losses, the principal problem in flow calorimetric heat capacitymeasurements on liquids, have recently been analysed by Hei and Raal33for afive-zone model calorimeter

Because of the importance of heat capacity data of liquids in chemicalthermodynamics and chemical engineering, numerous critical data compila-tions have been published – starting at the end of the nineteenth century withBerthelot’s Thermochimie,34and including such well-known publications as theInternational Critical Tables,35 Timmermans’ Physicochemical Constants ofPure Organic Compounds,36 Landolt-Bo¨rnstein,37 and Daubert and Danner’sPhysical and Thermodynamic Properties of Pure Chemicals: Data Compilation,DIPPRsDatabase.38 The most recent and the most comprehensive compila-tion of critically evaluated heat capacity data of pure liquids is the monograph

on Heat Capacities of Liquids: Volumes I and II Critical Review and mended Values, authored by Za´bransky´ et al (1996),39 with Supplement I of

Recom-2001.40This monograph also contains a valuable survey of calorimetric niques for determining heat capacities of liquids, and useful comments onterminology and criteria for the classification of calorimeters

tech-As concerns heat capacity data of mixtures, the situation is somewhat lesssatisfactory Critically selected excess molar heat capacities at constant pressure

of binary liquid organic mixtures have been included in the International DATASeries, SELECTED DATA ON MIXTURES, Series A,41 and the DortmundData Bank (DDB)contains a large number of data sets on heat capacities ofmixtures/excess heat capacities.42 However, a monograph devoted to a rea-sonably comprehensive compilation of heat capacity data of liquid mixtures,though highly desirable, is not available

For more than a century, experimental studies of real-gas behaviour atlow or moderate densities, have held a prominent position in physical chem-istry They were motivated, and still are, either by the need to solve practicalproblems – such as those encountered in the reduction of vapour–liquidequilibrium data – or by their usefulness as valuable sources of information

on intermolecular interactions in both pure gases/vapours and gaseous tures In this context, perfect-gas (ideal-gas) state heat capacities are ofcentral importance, say, in the calculation of property changes of single-phase,constant-composition fluids for any arbitrary change of state They may bedetermined by vapour-flow calorimetry, or by speed-of-sound measurements.The statistical–mechanical calculation of perfect-gas state heat capacities(they are 1-body properties which do not depend on molecular interactions)has reached a high level of sophistication, with obvious great practicaladvantages For instance, the calculations readily allow extension of experi-mental data to temperature ranges currently inaccessible to measurement.Data compilations of heat capacities of pure substances in the perfect-gas(ideal-gas) state may be found in Selected Values of Physical and Thermo-dynamic Properties of Hydrocarbons and Related Compounds,43 Landolt-Bo¨rnstein,37 in Stull, Westrum and Sinke’s The Chemical Thermodynamics ofOrganic Compounds,44 in the TRC Thermodynamic Tables,45 in the book byFrenkel et al.,46 and in the NIST-JANAF Thermochemical Tables.47 Oneshould always keep in mind, however, that only comparison of experi-mental with calculated values leads to better approximations and/or newconcepts

mix-3Heat Capacities: Introduction, Concepts and Selected Applications

1.2 Thermodynamics: Fundamentals and Applications

To set the scene for this monograph, a few selected basic thermodynamicrelations will be summarised below For further aspects and details the inter-ested reader should consult a textbook close to his/her taste, perhaps one ofthose listed in references 48 through 58

Convenient starting points are the fundamental property equations (alsocalled the Gibbs equations) of a single-phase PVT system, either open or closed,where P denotes the pressure, V is the molar volume and T is the thermo-dynamic temperature No electric, magnetic or gravitational fields are con-sidered in such a simple system For a multicomponent system, where the totalamount of substance is given by n¼P

i

ni, with nibeing the amount of stance of component i, the fundamental property equation in the energyrepresentationis

in composition occurring as a result of chemical reactions (reactive system) orboth Corresponding to Equations (1) and (2), the primary functions (or car-dinal functions, or Euler equations) are

In both the energy and entropy representations the extensive quantities arethe mathematically independent variables, while the intensive parameters arederived, which situation does not conform to experimental practice The choice

of nS and nV as independent variables in the fundamental property equation inthe energy representation is not convenient, and Equation (4) suggests thedefinition of useful alternative energy-based primary functions The appro-priate method for generating them without loss of information is the Legendretransformation These additional equivalent primary functions are the molarenthalpy

dðnGÞ ¼ ðnSÞdT þ ðnVÞdP þX

i

The four fundamental equations presented so far are equivalent; however, each

is associated with a different set of canonical variables{nS, nV, ni}, {nS, P, ni},{T, nV, ni} and {T, P, ni}

A primary function which arises naturally in statistical mechanics is thegrand canonical potential It is the Legendre transform when simultaneouslythe entropy is replaced by the temperature and the amount of substance by thechemical potential:

with the canonical variables{T, nV, mi}

The complete Legendre transform vanishes identically for any system Thecomplete transform of the internal energy replaces all extensive canonicalvariables by their conjugate intensive variables, thus yielding the null-function

0¼ U  TS þ PV X

i

as final alternative primary function in the energy representation This property

of the complete Legendre transform gives rise to

of the system and shows that they are not independent of each other

While the extensive parameters of a simple phase are independent of eachother, the conjugate intensive parameters are not, as shown above For a givenphase, the number of intensive parameters which may be varied independently

is known as the number of thermodynamic degrees of freedom

Treating the sumP

i

mini as a single term, the total number of equivalent mary functions and therefore the total number of equivalent fundamental prop-erty equations for a thermodynamic system is 2k Thus for nU¼ nU(nS, nV, n)

there are but eight distinct equivalent primary functions [nU, Equation (4), plusseven alternatives] and eight distinct forms of the fundamental equation [d(nU),Equation (1), plus seven alternatives] Of the seven Legendre transforms of theinternal energy, five have been treated above (including the null-function) Theremaining two, X¼ U P

which quantity is of central importance in mixture/solution thermodynamics.For a homogeneous fluid of constant composition, the following four energy-based fundamental property relations apply:

P, V, S and the energy-based functions U, H, F, G In view of the definitions of

Fand G and Equation (29), the Gibbs–Helmholtz equations

7Heat Capacities: Introduction, Concepts and Selected Applications

H ¼ G  T @G=@Tð ÞP ð31Þ

are obtained

A Legendre transformation of the primary function in the entropy sentation, Equation (5), resulting in the replacement of one or more extensivevariables by the conjugate intensive variable(s) 1/T, P/T and mi/T, defines aMassieu–Planck function For instance, the molar Massieu function is

Another second-order transform is the molar Kramer function

At this juncture it is convenient to introduce, by definition, a few auxiliaryquantities commonly known as the mechanical and the isentropic coefficients.Specifically, these are the isobaric expansivity

aP¼ V1ð@V=@TÞP ¼ r1ð@r=@TÞP ð49Þthe isothermal compressibilityw

bT ¼ V1ð@V=@PÞT ¼ r1ð@r=@PÞT ð50Þthe isochoric pressure coefficient

and the isentropic compressibilityw(often loosely called adiabatic compressibility)

bS¼ V1ð@V=@PÞS ¼ r1ð@r=@PÞS ð52Þwhere r¼ M/V is the density and M is the molar mass

where R is the gas constant

At low temperatures, where gV of liquids is large, direct calorimetric mination of CV of liquids is difficult (it becomes more practicable near the

deter-w

In this chapter the isothermal compressibility is represented by the symbol b T and not by k T as was recently recommended by IUPAC Similarly, the isentropic compressibility is represented by the symbol b and not by k

critical point, where gV is much smaller) Thus most of the isochoric heatcapacity data for liquids reported in the literature have been obtained indirectlythrough the use of Equations (55) and (56), that is to say from experimentalmolar isobaric heat capacities, isobaric expansivities and ultrasonic speeds.However, see for instance reference 59 Since also

Equations (54), (56) and (59) may be used for the indirect determination ofisothermal compressibilities from densities, isobaric expansivities, ultrasonicspeeds and molar isobaric heat capacities All these quantities may now be reli-ably and accurately measured, whence the indirect method for determining theisothermal compressibility of liquids has become an attractive alternative to thedirectmethod of applying hydrostatic pressure and measuring the correspondingvolume change For the difference between bTand bSone obtains, for instance,

ð@U

@VÞT ¼ P þ Tð@P=@TÞV¼ P þ TgV ð61Þð@H=@PÞT ¼ V  Tð@V=@TÞP¼ V  TVaP ð62Þthese equations lead to

When T and V are selected as independent variables,

where mJT is the Joule–Thomson coefficient All three quantities CP, (qH/qP)T

and mJTmay be measured by flow calorimetry.60,61(qH/qP)T is also known asthe isothermal Joule–Thomson coefficient, and frequently given the symbol j.For ideal gases TaP ¼ 1 and thus mJT¼ 0 For real gases, the temperature Ti(atthe inversion pressure Pi) where TiaP ¼ 1 is called the inversion temperature Atthat point the isenthalpic exhibits a maximum: for initial pressures PoPi, mJT4

0, and the temperature of the gas always decreases on throttling; for initialpressures P 4 Pi, mJTo0, and the temperature of the gas always increases onthrottling The maxima of the enthalpics form a locus known as the inversioncurve of the gas There exists a maximum inversion temperature at P¼ 0 Forpressures above the maximum inversion pressure, mJTis always negative.Because of Equation (67) one obtains, for instance, for the isentropic com-pression or expansion of a gas

Since aP of gases is always positive, the temperature always increases withisentropic compression and decreases with isentropic expansion.

In principle, the exact methods of classical thermodynamics are the mostgeneral and powerful predictive tools for the calculation of property changes ofsingle-phase, constant-composition fluids for any arbitrary change of state, say,from (T1,P1) to (T2,P2) For a pure fluid, the corresponding changes of molarenthalpy DH H2– H1and molar entropy DS  S2– S1are, respectively,

, where M

is the molar value of any extensive thermodynamic property of the fluid at(T,P), and Mpg is the molar value of the property when the fluid is in theperfect-gas state at the same T and P Given any volume-explicit equation ofstate, these residual functions may be calculated from

13Heat Capacities: Introduction, Concepts and Selected Applications

quantities are frequently used The differences between these two sets are theconfigurational properties of the perfect gas, and for U and CVthey vanish.

In actual practice, this approach would be severely limited by the availability

of reliable data for pure fluids and mixtures The experimental determination ofsuch data is time-consuming and not simple, and does not impart the glamourassociated with, say, spectroscopic studies Fortunately, statistical–mechanicalcalculations for CpgP are quite dependable for many substances, and so aregroup-contribution theories, for instance the techniques based on the work byBenson and co-workers.62,63

The search for generalised correlations applicable to residual functions hasoccupied scientists and engineers for quite some time The most successful onesare based on versions of generalised corresponding-states theory, which isgrounded in experiment as well as statistical mechanics The three-parametercorresponding states correlations, pioneered by Kenneth Pitzer and co-work-ers,64–67 have been capable to predict satisfactorily the PVT behaviour ofnormal, nonassociating fluids They showed that the compression factors ofnormal fluids may be satisfactorily expressed as

Z¼ Zð0ÞðTr; PrÞ þ oZð1ÞðTr; PrÞ ð79Þwhere

o 1  log10ðPs;rÞTr¼0:7 ð80Þ

is Pitzer’s acentric factor, Tr¼ T/Tcis the reduced temperature, Pr¼ P/Pcisthe reduced pressure, Ps,r¼ Ps/Pcis the reduced vapour pressure, here eval-uated at Tr¼ 0.7, Psis the vapour pressure of the substance, Tcis the criticaltemperature of the substance and Pcis its critical pressure In fact, this method

is a thermodynamic perturbation approach where the Taylor series is truncatedafter the term linear in o The generalised Z(0) function is the simple-fluidcontribution and applies to spherical molecules like argon and krypton, whoseacentric factors are essentially zero The generalised Z(1) function (deviationfunction) is determined through analysis of high-precision PVT data of selectednormal fluids where oa0 One of the best of the generalised Pitzer-type cor-responding-states correlations for Z(0), Z(1)and the derived residual functions isdue to Lee and Kesler.68

An alternative to the direct experimental route to high-pressure PVT dataand CP(T,P) is to measure the thermodynamic speed of ultrasound v0 as afunction of P and T (at constant composition), and to combine these results, inthe spirit of Equations (50), (54) and (60) with data at ordinary pressure, say

P1¼ 105Pa, i.e r(T,P1) and CP(T,P1) For a pure liquid, upon integration atconstant temperature, one obtains69,70

The first integral is evaluated directly by fitting the ultrasonic speed data withsuitable polynomials, and for the second integral several successive integrationalgorithms have been devised The simplicity, rapidity and precision of thismethod makes it highly attractive for the determination of the density, isobaricexpansivity, isothermal compressibility, isobaric heat capacity and isochoricheat capacity of liquids at high pressures Details may be found in the appro-priate chapters of this book, and in the original literature.

From experimentally determined heat capacities of liquids, relatively simplemodels have been used to extract information on the type of motion executed

by molecules in the liquid state In general, they are based on the separability ofcontributions due to translation, rotation, vibration and so forth Though none

of them is completely satisfactory, they have provided eminently useful insightsand thereby furthered theoretical advances Following the early work ofEucken,71 Bernal,72 Eyring,73 Stavely,74 Moelwyn-Hughes,75 Kohler,18,76Bondi77and their collaborators, one may resolve the molar heat capacity CVofsimple, nonassociated liquids into the following contributions:78,79

CV¼ Ctrþ Crotþ Cintþ Cor ð82ÞThe translational (tr) contribution arises from the motion of the moleculesunder the influence of all molecules (translational movement within theirrespective free volumes), the rotational (rot) contribution arises from rotation

or libration of the molecules as a whole, the internal (int) contribution arisesfrom internal degrees of freedom, and the orientational (or) contribution, fordipolar substances, results from the change of the dipole–dipole orientationalenergy with temperature Cint can be subdivided into a part stemming fromvibrations (Cvib) which usually are not appreciably influenced by densitychanges (i.e by changes from the liquid to the perfect-gas state), and anotherpart, Cconf, resulting from internal rotations (conformational equilibria), whichdoes depend on density Preferably, all these contributions to CVare discussed

in terms of residual quantities in (T,V)-space.78,79The residual molar isochoricheat capacity of a pure liquid is defined by

CrV CVðT; VÞ  CpgVðTÞ ð83ÞFor liquids composed of fairly rigid molecules, such as tetrachloromethane,benzene or toluene, to an excellent approximation CrintE 0, whence

repre-15Heat Capacities: Introduction, Concepts and Selected Applications

The resolution of the variation of CV of pure liquids along the orthobariccurve (subscript s), i.e for states (T, Ps), into the contributions due to theincrease of volume and to the increase of temperature, respectively, is a highlyinteresting problem.78–80It is important to realise that due to the close packing

of molecules in a liquid, even a rather small change of the average volumeavailable for their motion may have a considerable impact on the moleculardynamics: volume effects may become more important in influencing molecularmotion in the liquid state than temperature changes Since

right-of (q2P/qT2)Vare not plentiful Available data70,81indicate that it is small andnegative for organic liquids, that is to say, CVdecreases with increasing volume.Alternatively, one may use18,78,79

0.60 J K1cm3 These results indicate a substantial contribution of (qCV/qV)TVas to the change of CV along the orthobaric curve as well as to thecorresponding change of CrV

Equation (56) is a suitable starting point for a discussion of the temperaturedependence of k CP/CVof a liquid along the orthobaric curve:

Usually, the second term in parenthesis on the right-hand side of Equation (88)

is positive and the third term is negative; the fourth term may contributepositively or negatively Thus k may increase or decrease with temperature.The importance of the heat capacity in the perfect-gas state has been stressedrepeatedly Flow calorimetry is a commonly used method for measuring CPofgases and vapours,84 and allows straightforward extrapolation to zero pres-sure85to obtain CpgP The virial equation in pressure

ZPV

RT ¼ 1 þ B0Pþ C0P2þ ð89Þwhere B0 is the corresponding second virial coefficient and C0 the third virialcoefficient, may be used to calculate the residual heat capacity of a pure fluidaccording to

The thermodynamic speed of ultrasound (below any dispersion region) isrelated to the equation of state, and hence to the virial coefficients For a realgas, v2may thus be expressed as a virial series in molar density 1/V,92i.e.

CpgV ¼ 1 þ R

For constant-composition fluids, the acoustic virial coefficients Bac, Cac, are functions of temperature only They are, of course, rigorously related to theordinary (PVT) virial coefficients For instance,

Since pressure is the preferred experimental parameter, one may also write avirial expansion for v2 in powers of the pressure with corresponding virialcoefficients B0ac, C0ac, The coefficients of the density and pressure expan-sion are interrelated; for example

All this valuable thermophysical information can then be used to obtainreliable second virial coefficients over large temperature ranges For a fluid withspherically symmetric pair potential energy u(r),

Inver-While a discussion of experimental acoustical methods is way outside thescope of this introductory chapter, the following comment is indicated Forgases/vapours at low to moderate pressures not too close to saturation, thehighest experimental precision, when measuring v0, is obtained through use of aspherical resonator, a technique which was pioneered by Moldover, Mehl andco-workers.95,96

So far, the focus was on homogeneous constant-composition fluids, of whichpure fluids are special cases I will now briefly consider the case where a pureliquid is in equilibrium with its vapour Such a situation is encountered, forinstance, in adiabatic calorimetry, where the calorimeter vessel is incompletelyfilled with liquid in order to accommodate the thermal expansion of the sample(usually, the vapour space volume is comparatively small) One has now aclosed two-phase single-component system The heat capacity of such a system

is closely related to CLs, i.e the molar heat capacity of a liquid in equilibriumwith an infinitesimal amount of vapour (as before, the saturation condition isindicated by the subscript s) For a detailed analysis see Hoge,97Rowlinsonand Swinton,56and Wilhelm.98

19Heat Capacities: Introduction, Concepts and Selected Applications

The molar heat capacity at saturation of the substance in the equilibriumphase p (denoting either the liquid, p¼ L, or the vapour, p ¼ V) is given by

The differences between the first four quantities are generally much smaller thanbetween CLVand (qUL/qT)s.

While the general equations apply also to the saturated vapour (p¼ V), theinequality does not Since aVPVVis always large, for saturated vapours the dif-ference CVs–CVPis always significant [see Equation (104)] In fact, for vapours ofsubstances with small molecules, such as argon, carbon dioxide, ammonia andwater (steam), aVPVVmay be large enough to make CVs even negative Finally wenote that the difference between the saturation heat capacities in the vapourphase and the liquid phase may be expressed as98

where DvapHdenotes the molar enthalpy of vaporisation, and DvapV VV VL

is the volume change on vaporisation In deriving these equations, use wasmade of the exact Clapeyron equation

gs¼ DvapH

and the exact Planck equation.99

There are, of course, many additional details and fascinating topics, inparticular when mixtures and solutions are considered, which fact is amplyevidenced by the contributions to this monograph Enjoy!

1.3 Concluding Remarks

Calorimetry and PVT measurements are the most fundamental and also theoldest experimental disciplines of physical chemistry Although simple inprinciple, enormous effort and ingenuity has gone into designing the vastarray of apparatus now at our disposal In this introductory chapter, I did notcover design of experiments beyond the bare rudiments – the reader is referred

to the relevant articles and books quoted, and to the chapters of this bookfocusing on this aspect Let it suffice to say that the advances in instrumentationduring the last decades have greatly facilitated the high-precision determination

of caloric and PVT properties of fluids over large ranges of temperature andpressure At the same time cross-fertilisation with other disciplines, notablywith ultrasonics and hypersonics, and with biophysics, is becoming increasingly

21Heat Capacities: Introduction, Concepts and Selected Applications

important, as is the close connection to equation-of-state research and, ofcourse, chemical engineering.56,61,79,98,100–103The discussion presented here and

in the chapters to follow may perhaps best be characterised by a statement due

to Gilbert Newton Lewis (1875–1946) on the practical philosophy of scientificresearch:

The scientist is a practical man and his are practical aims He does not seek theultimate but the proximate He does not speak of the last analysis but rather of thenext approximation On the whole, he is satisfied with his work, for whilescience may never be wholly right it certainly is never wholly wrong; and it seems

to be improving from decade to decade

By necessity, this introductory chapter is limited to a few topics, the selection

of which was also influenced by my current interests In conclusion, I hope tohave:

 formulated concisely some important aspects of the thermodynamicformalism needed in this area of research;

 discussed and made transparent some key aspects of experiments;

 shown how to apply and to appropriately extend well-known concepts toperhaps less familiar, yet potentially important, problems;

 stimulated some colleagues to enter this fascinating and important field ofresearch

Success in any of these points would be most rewarding

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25Heat Capacities: Introduction, Concepts and Selected Applications