Extract from MODELLING IN AQUATIC CHEMISTRY … · (OECD Publications, 1997, 724 pp., ISBN 92-64-15569-4) Scientific Editors: ... Andreas Jakob, Theo Karapiperis, An drey V. Plyasunov,

Chapter X

Temperature Corrections to

Thermodynamic data and Enthalpy

Calculations y

Ignasi PUIGDOMENECH � Joseph A� RARDOECD Nuclear Energy Agency Geosciences and Environmental TechnologiesLe Seine � Saint Germain Environmental Programs Directorate�� boulevard des Iles Lawrence Livermore National LaboratoryF�� Issy�les�Moulineaux �France Livermore� California �� USA

Andrey V� PLYASUNOV � Ingmar GRENTHEDepartment of Inorganic Chemistry Department of Inorganic ChemistryRoyal Institute of Technology Royal Institute of TechnologyS�� Stockholm �Sweden S�� Stockholm �Sweden

X�� Introduction

Recalculation of chemical equilibrium data from the reference temperature of �� K��C� to any desired temperature is made by using the relationships provided by thermodynamics� The procedures are straightforward provided that information is availablefor rH

�m or rS

�m at the reference temperature and for their temperature dependencies�

Complete information of this kind is rarely available for formation reactions of chemical complexes in aqueous solution and it is therefore necessary to rely on approximationmethods of various kinds� These methods will be described in some detail in this chapter�

y A substantial portion of this Chapter originates from an internal NEA technical report �TDB�� Permanent address� Department of Inorganic Chemistry� Royal Institute of Technology�

S�� Stockholm� Sweden�� Permanent address� Institute of Experimental Mineralogy� Chernogolovka� Moscow District� ��

Russia�

��

Extract from MODELLING IN AQUATIC CHEMISTRY (OECD Publications, 1997, 724 pp., ISBN 92-64-15569-4) Scientific Editors: Ingmar Grenthe and Ignasi Puigdomenech Contributors: Bert Allard, Steven A. Banwart, Jordi Bruno, James H. Ephraim, Rolf Grauer, Ingmar Grenthe, Jörg Hadermann, Wolfgang Hummel, Andreas Jakob, Theo Karapiperis, Andrey V. Plyasunov, Ignasi Puigdomenech, Joseph A. Rard, Surendra Saxena, Kastriot Spahiu Secretariat: OECD Nuclear Energy Agency Data Bank: M.C. Amaia Sandino and Ignasi Puigdomenech Original text processing and layout: OECD Nuclear Energy Agency Data Bank: Cecile Lotteau. © OECD 1997

Temperature Corrections to Thermodynamic Data and Enthalpy Calculations

Experimental thermodynamic information about the chemical species �chemical speciation� in a particular system forms the basis of thermodynamic databases� However experiments only provide information about the species that are present in noticeableamounts in the laboratory systems �and thus are detectable in the experiments�� A complication in the modelling of the properties of systems at di�erent temperatures is thepossibility of a change in speciation with temperature since even a temperature changeas small as from �� to �� K may result in the appearance of new species ��BAE�MES ��CIA�IUL�� The modelling in such situations can only be made using the general principles of chemistry as outlined in Chapter III� One may also have to make experimentaldeterminations in situations where one is reasonably con�dent that a change in speciationis of critical importance for the understanding of the system at higher temperatures�

There are some general considerations that can be used as guidelines�

� The dielectric constant of liquid water decreases strongly with increasing temperature �cf� Figure X�� on p�� hence complexes of low or zero charge are favouredat higher temperatures�

� Hydrolysis of metal ions increases with increasing temperature in keeping with thegeneral increase of acidity of water with increasing temperature�

� The relative amounts of polynuclear complexes and other complexes with highcharges decrease with increasing temperature ��PLY�GRE�� This e�ect can becorrelated with the decrease in the dielectric constant of the solvent water with increasing temperature �cf� Figure X�� Even surprisingly simple electrostatic modelsare able to describe this feature with fair accuracy� In the �nal section of this chapter we will describe the characteristic features of one such model and its predictiveproperties�

Thermodynamic data may also need to be corrected for ionic strength e�ects both at�� K and at other temperatures� This is not a straightforward problem at T �� K as discussed in Chapter IX�As will be seen from the following text the approximation methods used to describe

the temperature dependencies of chemical equilibria rely heavily on simple electrostaticmodels which treat the participating ionic species as pointcharges and the solvent as ahomogeneous dielectric continuum� both assumptions are oversimpli�cations�The extrapolation of experimental values of rG

�m�T � �or conversely equilibrium con

stants� to a reference temperature generally �� K is usually done by using variousmodi�cations of the socalled second and thirdlaw methods� The thirdlaw extrapolations require free energy functions and are generally the preferred method of calculationwhen long temperature extrapolations are required particularly where the reactants andproducts are pure phases for which experimental heat capacities or relative enthalpiesare available or can be accurately estimated� That is thirdlaw extrapolations shouldgenerally be used for equilibria between di�erent phases at high temperatures� When

��

Second�law extrapolations

extrapolations over relatively small temperature ranges are made then secondlaw extrapolations can be used for accurate calculations but this method requires experimentalor estimated heat capacities around the temperature of interest� Secondlaw extrapolations should generally be used for aqueous equilibria�

X�� Second�law extrapolations

The standard molar Gibbs energy change for a reaction at any given temperature is givenby

rG�m � rH

�m � TrS

�m� �X��

and the temperature dependence of the Gibbs energy is��rG

�m

�T

�p

� �rS�m�

The Gibbs�Helmholtz equation is��rG

�m�T

�T

�p

��rH

�m�T �

T �� X��

which has the integrated formy�

Z T

T�d

�rG

�m�T �

T

��

Z T

T�

�rH

�m�T �

T �

�dT� �X��

From the integration of this equation and from using the temperature derivatives of theenthalpy and entropy �

�rH�m

�T

�p

� rC�p�m� �X��

��rS

�m

�T

�p

�rC

�p�m

T� �X��

it is possible to write the temperature dependence of the Gibbs energy as a function ofthe entropy at the reference temperature �T� � �� K� as well as the heat capacityfunction� It is however also possible ��NOR�MUN Eq� �� and simpler to useEqs� �X�� X�� and �X�� to write

rG�m�T � � rH

�m�T��

Z T

T�rC

�p�mdT

� T

�rS

�m�T��

Z T

T�

rC�p�m

TdT

��

y T� stands for the reference temperature �� K��

��


This equation is usually recast in terms of only one thermodynamic function other thanGibbs energy and heat capacity e�g� if the choice is entropy

rG�m�T � � rG

�m�T�� T � T��rS

�m�T��

�Z T

T�rC

�p�mdT � T

Z T

T�

rC�p�m

TdT� �X��

Alternatively one may write the temperature dependence of the equilibrium constantas a function of the standard enthalpy and the standard heat capacity

log��K��T � � log��K

��T�� rH�m�T��

R ln��

��

T� �

T�

�

� �

RT ln��

Z T

T�rC

�p�mdT �

�

R ln��

Z T

T�

rC�p�m

TdT� �X��

where R is the gas constant �� J �K�� mol�� Either Eq� �X�� or �X�� may beused to calculate equilibrium constants at a temperature T if�

�� the equilibrium constant at �� K is known�

�� the temperature dependence of rC�p�m is known or the temperature interval is small

enough that rC�p�m can be assumed to be constant�

�� either rH�m�T�� or rS

�m�T�� are known�

Unfortunately experimentally derived heat capacity data have not been measured formost aqueous species and for many solid phases� Therefore in order to use Eqs� �X�� or�X�� one will have to make either approximations �as discussed below� or estimations�cf� Section X��

If the reaction of interest involves only species for which the pertinent coe�cients inthe following equation for the heat capacityy

C�p�m�T � � a� b T � c T � � j T � � dT�� e T�� k T��

� f lnT � g T lnT � hpT � i

�pT

�X��

y This equation is used as a general form for the temperature dependency of the heat capacity intemperature intervals that do not involve any phase transition� For any speci�c system� only a fewof these coe�cients will be required �frequently a� b and e�� Some of these terms should obviouslynot be used if the heat capacity equation is required to be valid as T � �� since they become in�niteat that limit�

��


are available �which is seldom the case� then the approximation methods given below arenot needed and the integrals in Eqs� �X�� and �X�� can be performed analytically andwill take the following form�Z T

T�rC

�p�mdT � a �T � T��

b

�

�T � � T �

�

�

�c

�

�T � � T �

�

��j

�

�T � � T �

�

��d ln

�T

T�

�

�e��

T� �

T�

�� k

�

�T�� T��

�

��f �T �� lnT �� T� �� lnT�� g

�

hT � �� lnT �� T �

� �� lnT��i

��h

�

�T �� T

��

�� i

�pT �

qT�

�� X��

Z T

T�

rC�p�m

TdT � a ln

�T

T�

��b �T � T��

�c

�

�T � � T �

�

��j

�

�T � � T �

�

�

�d��

T� �

T�

�� e

�

�T�� T��

�

�

� k�

�T�� T��

�

��f

�

h�lnT �� lnT��

i�g �T �� lnT �� T� �� lnT�� h

�pT �

qT�

�� i

��pT� �p

T�

�� X��

where a b c �� are the changes in the parameters a b c �� of Eq� �X�� with thereaction i�e�

a �Xi

�iai�

etc� and where �i are the stoichiometric coe�cients of the species i of the reaction�If equilibrium constants are known at several temperatures the parameters of Eq� �X��

�together with Eqs� �X�� and �X�� can be evaluated from the experimental data by aleastsquares procedure�It is sometimes convenient to use �apparent� standard partial molar Gibbs energies and

enthalpies cf� Refs� ��HEL�KIR ��HEL�DEL ��TRE�LEB� de�ned as

aG�m�i� T � � fG

�m�i� T�� G

�m�i� T ��G�

m�i� T�� X��

and

aH�m�i� T � � fH

�m�i� T��

Z T

T�C�p�m�i� T � dT� �X��

��


Using the relationship

S�m�i� T � � S�

m�i� T�� Z T

T�

C�p�m�i� T �

TdT� �X��

it is possible to rewrite Eq� �X��

rG�m�T � �

Xi

�iaH�m�i� T �� T

Xi

�iS�m�i� T � �X��

�Xi

�iaG�m�i� T � �X��

�Xi

�i

�fG

�m�i� T�� T � T��S

�m�i� T��

�Z T

T�C�p�m�i� T � dT � T

Z T

T�

C�p�m�i� T �

TdT

�� X��

If the heat capacity is expressed according to Eq� �X�� the integrals in Eq� �X�� havethe same form as Eqs� �X�� and �X�� except that in Eqs� �X�� and �X�� rC

�p�m

a b c etc� must be substituted for C�p�m a b c etc�

Eqs� �X�� through �X�� are especially useful when the assumption is made that forsome of the reactants �or products� the heat capacity does not vary with temperature whereas for the rest of the reactants �or products� Eq� �X�� applies�

X�� The hydrogen ion convention

The hydrogen ion convention states that the conventional standard partial molar Gibbsenergy of formation entropy and heat capacity of H� are all set equal to zero at alltemperatures� This allows values to be assigned for the thermodynamic properties ofeach ionic species participating in a reaction� Therefore it is possible to write

rG�m�T � �

Xi

�iG�m�i� T �� X��

where G�m�i� T � is the resulting standard state Gibbs energy of ion i based on the hydrogen

ion convention�As just stated the hydrogen ion convention involves the arbitrary assignment of the

standard ionic entropy of the aqueous hydrogen ion as being equal to zero at all temperatures i�e� S�

m�H�� aq� T � � �� Another assumption of this convention is that the

standard partial molar Gibbs energy of formation of the hydrogen ion is equal to zero atall temperatures fG

�m�H

�� aq� T � � �� However the entropy of formation of an ion isequal to the temperature derivative of its Gibbs energy of formation�

��fG

�m�i� T �

�T

�p

� �fS�m�i� T ��

��


Thus the assumption that fG�m�H

�� aq� T � � � also implies that fS�m�H

�� aq� T � � � and according to Eq� �X�� it also implies that fH

�m�H

�� aq� T � � ��The reaction for formation of the aqueous hydrogen ion is

�

�H��g� �� H� � e��

for which the entropy of formation is given by

rS�m�T � � fS

�m�H

�� aq� T �

� S�m�H

�� aq� T � � S�m�e

�� T �� S�m�H�� g� T �

In this case the standard partial molar entropy of formation of the hydrogen ion is identicalto the entropy of reaction� However the standard partial molar entropy of formation ofthe hydrogen ion and its standard partial molar ionic entropy are both set equal to zeroin the hydrogen ion convention fS

�m�H

�� aq� T � � S�m�H

�� aq� T � � �� Consequently toremain consistent with the standard entropy of H��g� the �aqueous electron� must beassigned an e�ective molar entropy of

S�m�e

�� T � ��

�S�m�H�� g� T �

At �� K the CODATA key values ��COX�WAG� at � bar pressure yield

S�m�e

�� T�� J �K�� mol��This value must be included when the entropy of an ionic species is being calculated fromthe entropies of the elements from which it is formed�As an example the ionic entropy of the divalent calcium ion will be calculated at

�� K from its entropy of formation� The formation reaction is

Ca�cr� �� Ca�� e��

and the corresponding entropy of formation is given by

fS�m�Ca

�� aq� �� K� � S�m�Ca

�� aq� �� K� � �S�m�e

�� K�

� S�m�Ca� cr� �� K�

The entropy of formation of the calcium ion can be calculated from its standard Gibbsfree energy of formation and the enthalpy of formation as given in the CODATA tablesto yield fS

�m�Ca

�� aq� �� K� � �� J �K�� mol�� Then S�m�Ca

�� aq� �� K� � S�m�Ca� cr� �� K�� S�

m�e�� K�

� fS�m�Ca

�� aq� �� K�

� �� J �K�� mol�� y

y This calculated value of S�m�Ca�� aq� � �� K� is identical to that given in the CODATA tables�

whereas the uncertainties are quite di�erent� This arises because the calculations involved di�erentthermodynamic cycles�

��


X�� Approximations

For many chemical reactions there is a lack of heat capacity functions for all or someof the species involved and therefore approximations �as described in this section� orestimations �cf� Section X�� must be made in order to use Eqs� �X�� X�� or �X��The method of choice will depend on the type of chemical reaction being considered�

X�� Constant enthalpy of reaction

The simplest assumption to be made is that the heat capacity change of reaction is zeroat all temperatures �i�e� the standard molar enthalpy of reaction does not vary withtemperature cf� Eqs� �X�� and �X�� In that case Eq� �X�� reduces to the integratedvan�t Ho� expressiony


��T�� rH

�m�T��

R ln��

��

T��

T

�� X��

For a temperature range �T � T�� equal or less than �� K the error introduced inlog��K

��T � by this simpli�cation will in most cases be well within its uncertainty limits�

Eq� �X�� is applicable to chemical reactions in a single phase or a multiphase systemat constant total pressure provided no further constraint is placed upon the system� Forcertain other cases for example when the temperature dependence of a solubility productis being studied the system is constrained to fall on the saturated solution curve andEq� �X�� must be modi�ed� See Section X��

X�� Constant heat capacity of reaction

Another approach which is often used in conjunction with Eq� �X�� when the extrapolation extends over a temperature range larger than about �� K is to assume that the heatcapacity of the reaction does not vary with temperature� In that case Eq� �X�� becomes


��T�� rH

�m�T��

R ln��

��

T��

T

�

�rC

�p�m

R ln��T��T �� ln�T�T�� X��

where as mentioned earlier the reference temperature is T� � �� K�

y By combining Eq� �X�� and the relation �rG�m�T � � �RT lnK��T �� it is possible to obtain� for a

given constant pressure p�

d lnK�p �T �

dT�

�rH�m�T �

RT �

which is called the van�t Ho� equation�

��


Using Eq� �X�� it is possible to rewrite this expression as follows �compare also withEqs� �X�� and �X��

rG�m�T � � rG

�m�T�� T � T��rS

�m�T��

� rC�p�m �T � T� � T ln�T�T�� X��

Use of Eqs� �X�� or �X�� to represent values of log��K�T � will give enthalpies ofreaction which are more reliable than those obtained assuming a �constant enthalpy ofreaction� Eq� �X�� but the rC

�p�m�T�� values obtained will be imprecise ��HEL�� For

most reactions the assumption made in Eqs� �X�� and �X�� will be appropriate fortemperatures in the range �� to �� K�As an example Eq� �X�� can be used to �t the hightemperature equilibriumconstants

reported by ��SER�NIK� for the reaction

CO�� UO��

�� UO�CO��aq�

which will be evaluated below as an isoelectric reaction� The resulting values arelog��K�T�� rH

�m�T�� kJ � mol�� and rC

�p�m�T��

�� J �K�� mol�� If the equilibrium constant at �� K is not a �tting parameter but instead is set equal to the recommended value of log��K

��T�� GRE�FUG� the following results are obtained� rH

�m�T�� kJ � mol�� and

rC�p�m�T�� J �K�� mol��

A comparison between the results from Eq� �X�� using both sets of �tted parameters�and the experimental data is shown in Figure X��The reaction enthalpy is found in this case to be zero within the experimental error�

The entropy of reaction is given �cf� Eq� �X�� by

rS�m�T��

rH�m�T��

T�� ln��R log��K

��T��

��rS�m� � �� rH

�m� � ��

��log��K��

which gives rS�m�T�� J � K�� mol�� when the equilibrium constant at

�� K is also �tted to Eq� �X�� or which gives rS�m�T�� J �K�� mol��

�with log��K��T�� xed at ��

Eqs� �X�� and �X�� are useful to calculate higher temperature equilibrium constantswhen average heat capacities for aqueous ions have been determined experimentally orestimated �cf� Section X��Alternatively partial molar heat capacities at �� K are sometimes considered to be

constant with temperature� This assumption is based on the fact that although valuesfor ionic heat capacities generally increase with temperature they usually also have amaximum around �� to �� K and then begin to decrease �see for example Figure �in Ref� ��PAT�SLO� and Figures �� to �� in Ref� ��HEL�KIR�� Therefore settingC�p�m�T � � C�

p�m�T�� may be a valid simpli�cation in the temperature range between ��and �� to �� K depending on the nature of the reaction ��HEL�� However using the

��


Figure X�� Equilibrium constants from ��SER�NIK� for Reaction �X�� CO��

UO��

�� UO�CO��aq� compared with the two di�erent least squares �ts to the �constantrC

�p�m� equation Eq� �X�� described in the text�

�

��

��

�

�

��

��

� ��

log��K�

T�K

��SER�NIK� �

� �

�

�

�

�

�

log��K��T�� adjusted

log��K��T��

heat capacity at the average temperatureC�p�m�

T�T�� may be an even better approximation

over some temperature intervals�Partial molar heat capacities at �� K for ions in aqueous solutions may be esti

mated from ionic entropies as discussed later with some of the equations given in Refs��CRI�COB� ��HEL�KIR ��SHO�HEL��Once heat capacities have been estimated they can then be used as follows ��HEL��

� If the chemical reaction involves only aqueous ionic species then the average heatcapacities may be combined into a single average heat capacity of reaction andEqs� �X�� or �X�� can be used� This method is used for example in Ref��LEM�TRE��

� If the chemical reaction includes phases for which heat capacity functions areavailable �i�e� expressions compatible with Eq� �X�� it is convenient to useEq� �X��

rG�m�T � �

Xi

�iaH�m�i� T �� T

�Xi

�iS�m�i� T �

�� X��

together with the following equations ��HEL� for the aqueous species �which areto be used instead of Eqs� �X�� and �X��

aH�m�i� T � � fH

�m�i� T�� C�

p�mjTT��i� �T � T�� X��

��


Table X�� Temperature contributions in Eq� �X��

t��C� T �K� �T � T � T� T ln�T�T�� T � T ln�T�T��K� �K� �K�

��

S�m�i� T � � S�

m�i� T�� C�p�mjTT��i� ln�T�T�� X��

while for the nonionic species Eqs� �X�� and �X�� can be used for which �asmentioned earlier� the integrals take the form of expressions like Eqs� �X�� and�X�� except that one must substitute rC

�p�m a b �� etc� for C

�p�m a b c

�� etc�

It may be interesting to note that if rS�m and rC

�p�m are comparable in magnitude

the term in Eq� �X�� with rC�p�m becomes small compared to rS

�mT over short

temperature ranges as shown in Table X��In many cases the rC

�p�m term can be neglected as it is probably smaller than the

error in the estimates of rS�m� This amounts to the assumption that rC

�p�m � � and

thus rH�m and rS

�m each have the same values for the reaction at any two di�erent

temperatures �Eq� �X�� This may be a good approximation for small temperatureranges�

X�� Isoelectric and isocoulombic reactions

For a reaction involving aqueous ionic species �but without oxidation�reduction� theenthalpy of reaction may be divided conceptually into two contributions� electrostatic andnonelectrostatic� The main part of the enthalpy of reaction is due to the electrostaticinteractions between the ionic species participating in the reaction and between theseionic species and the solvent� Figure X�� shows that the dielectric constant of waterdecreases signi�cantly with temperature reaching values which at T � �� K are similarto those of some organic solvents �acetone ethanol etc�� at room temperature� Because ofthis temperature dependence of the dielectric properties of water electrostatic interactionswill bring the largest contribution to the heat capacity of reaction ��GUR ��SHO�HEL��

��


Figure X�� Temperature dependence for the static dielectric constant of water calculatedaccording to the equation of Bradley and Pitzer ��BRA�PIT� at the standard pressure�� bar� for T �� K and at the steam saturated pressure ��KEE�KEY Eq� �� intheir Appendix� at T � �� K�

��

�

�

��

��

��

��

�

��

� ��

� for H�O�l�

T�K

steam saturation

�

�Isoelectric� reactions are de�ned as reactions in which a� the total amount of positivecharges among the reactants equals the sum of positive charges among the products andb� the same applies for negative charges among reactants and products� For example��

M�� H�O�l� �� MOH�� H�

is an isoelectric reaction while

M�� OH� �� MOH��

is not�In isoelectric reactions the electrostatic contributions to the temperature dependence

will balance out to a large extent and the heat capacities of reactions will be small andcan be assumed constant with temperature� For these reactions the �constant enthalpyof reaction� Eq� �X�� is generally a reliable approximation over a fairly large temperature interval �up to T � �� K�� Furthermore the �constant heat capacity of reaction�approximation Eq� �X�� is generally reliable up to T � �� K�� M�� R� and �AN� are used as general abbreviations for metal� rare earth �lanthanide�� and actinide

respectively�

��


All ionic species participating in isoelectric reactions often �but not necessarily� havethe same sign in the electrical charges ��BAE�MES� i�e� either all charged species havepositive or all have negative charges�The term �isocoulombic� ��LIN ��JAC�HEL� is used for isoelectric aqueous reactions

in which the magnitude of the electrical charge of each individual ionic species also isbalanced between reactants and products for example

M�OH�� HCO��

�� MCO�� OH

� �H�O�l��

For this kind of reaction rC�p�m � � and thus Eq� �X�� becomes an even better approx

imation of the experimental data� It should be noted that there is also an approximatecancellation of activity coe�cients in the expression for the equilibrium constant of thisreaction�It is possible to estimate the solvation contribution to the absolutey standard partial

molar ionic heat capacity with a continuum electrostatic model� For example using theBorn equation ��HEL�KIR their Eqs� �� and �� SHO�HEL their Eq� �� TAN�HEL their Appendix D�

C��absp�m�s�i� T � �

z�ire��i

NAe�

��T�

�

�� ln �

�T �

�p

�� ln �

�T

��

p

� �X��

�z�ire��i

NAe�

��TX�T �

where the subscript �s� stands for solvation and where �NAe�� m �

J � mol�� X�T � is a temperature function of the solvent�s dielectric constant which at�� K is equal to �� K�� TAN�HEL Table H�� and re� �units m� is thee�ective electrostatic radius which at T �� K is equal to the crystallographic radiusto which is added an empirical valencedependent constant �equal to zi � �� mfor cations and zero for anions ��TAN�HEL their Eq� ��The �rst hydrolysis step of the lanthanum�III� ion will be used as an example� The

reaction may be written either as an isoelectric reaction

La�� H�O�l� �� LaOH�� H�� X��

or as a complexation reaction involving the hydroxide ion

La�� OH� �� LaOH�� X��

The solvation contribution rC�p�m�s �in J � K�� mol�� to the heat capacity for the

isoelectric Reaction �X�� is according to Eq� �X��

rC�p�m�s�X�� T��

NAe�

��T�X�T��

��

��

��

��

� �� J �K�� mol��y �Absolute� ionic standard partial molar thermodynamic values are de�ned in Section X�� p��

��


where the crystallographic and e�ective radii for LaOH�� were taken equal to that of La��

�ri � �� m given in ��SHO�HEL Table �� and the e�ective radius for H� wastaken from ��TAN�HEL their Table �� The heat capacity change for Reaction �X��would then be estimated as

rC�p�m�X�� T�� rC

�p�m�s�X�� T�� C�

p�m�H�O� l� T��

� �� J �K�� mol��

For Reaction �X�� the ionic radius for OH� given in ��TAN�HEL their Table �� resultsin a solvent contribution of

rC�p�m�s�X�� T��

NAe�

��T�X�T��

��

��

��

��

� �� J �K�� mol��

and the heat capacity change for Reaction �X�� is therefore estimated to be �� J �K�� mol�� In this case the value of the estimated heat capacity change for Reaction �X�� is about one third of that for Reaction �X�� Similar results are obtainedfor the �rst hydrolysis step of a divalent cation like Fe�� or for a tetravalent cation likeU��Although the examples given above only take into account solvation contributions to

the heat capacity of reaction �furthermore estimated with an oversimpli�ed electrostaticmodel� they support the assertion that the approximation rC

�p�m � const� is more

appropriate for isoelectric reactions than for most other types of reactions� However forisocoulombic reactions rC

�p�m may be even closer to zero�

Many hydrolysis equilibria are isoelectric ��RUA�� As an example the data reportedby Nikolaeva ��NIK� �in the temperature range �� to �� K� for the reaction

H�O�l� � UO��

�� H� �UO�OH� �X��

are plotted in Figure X�� The values of log��K� are essentially a linear function of T��

in this temperature interval indicating that rC�p�m is small� Other examples with linear

or nearlylinear regions for acidbase equilibria are given by Lindsay ��LIN� Cobble etal� ��COB�MUR pp�� to �� and by Mesmer et al� ��MES�MAR ��MES�PAL��If a reaction of interest is neither isocoulombic nor isoelectric it may be converted into

an isocoulombic or isoelectric reaction by combination with an appropriate reaction forwhich accurate hightemperature equilibrium constants are known� This approach hasbeen widely used e�g� ��BAE�MES ��PHI�SIL ��JAC�HEL ��RUA ��IZA�CHR ��IZA�CHR� ��OSC�GIL ��OSC�IZA ��IZA�OSC ��CHE�GIL ��CHE�GIL� ��GIL�OSC��The data by Piroshkov and Nikolaeva ��PIR�NIK� and by Sergeyeva et al�

��SER�NIK� in the temperature range �� to �� K for the equilibrium

CO�� UO��

�� UO�CO��aq� �X��

��


Figure X�� Equilibrium constants from ��NIK� for Reaction �X�� H�O�l� � UO��

��H� � UO�OH� �� compared with results from the �constant rH

�m� equation

Eq� �X�� using log��K��T�� and rH

�m� �� kJ �mol��

�solid line��

��

��

��

��

��

��

��

log��K�

�� K�T

�

�

�

��

�

�

Figure X�� Equilibrium constants from ��SER�NIK ��PIR�NIK� for Reaction �X��CO��

� �UO��

�� UO�CO��aq��

��

��

�

�

��

��

��

log��K�

�� K�T

��PIR�NIK� �

��

�

�

�


��

�

�

�

�

�

��


are used as an example�It is readily seen in Figure X�� that Eq� �X�� cannot be used directly to describe the

experimental data since log��Keq is not linear in T�� However it is possible to combineEq� �X�� with Eq� �X��

CO��aq� � H�O�l� �� CO�� H�� X��

to obtain the isoelectric reaction

UO�� CO��aq� � H�O�l� �� UO�CO��aq� � �H

�� X��

This example is used in two ways� Firstly a discussion of the procedure to be taken toextrapolate hightemperature data to �� K will be given and secondly a descriptionwill be presented of how Reactions �X�� to �X�� may be used together with Eq� �X��to calculate equilibrium constants at high temperatures�

X�� Correlation of high�temperature equilibrium constants

In order to make a temperature extrapolation of the log��K��T � data like those avail

able from Refs� ��PIR�NIK ��SER�NIK� for Reaction �X�� one adds the wellknownlog��K

��X�� T � to log��K��X�� T �� The values of log��K

��X�� T � may be obtainedfrom Table � in Ref� ��PAT�SLO� and Table IV of Ref� ��PAT�BUS� cf� Table X��This converts the reaction to an isoelectric form�The values of the last row of Table X�� are added to the log��K

��T � for Reaction �X��reported in Refs� ��SER�NIK ��PIR�NIK�� The resulting values are plotted in Figure X�� It can be seen that the isoelectric approach �rC

�p�m � �� can be used successfully

to describe the experimental data using Eq� �X�� with log��K��T��

and rH�m � �� kJ �mol��

For data of higher quality over larger temperature intervals the �constant heat capacityof reaction� approximation Eq� �X�� should be used instead to obtain more reliablethermodynamic values at the reference temperature�

X�� Extrapolation of �� K data to higher temperatures

The following selected values for Reaction �X�� are reported in Table III�� of��GRE�FUG�� log��K

��X�� K� � �� rH�m�X�� K� �

�� kJ � mol�� From the data for auxiliary compounds the following results areobtained for Reaction �X�� at �� K� log��K

��X�� K� � �� and rH

�m�X�� K� � �� kJ � mol�� Therefore the following val

ues are found� log��K��X�� K� � �� and rH

�m�X�� K� �

�� kJ �mol�� Eq� �X�� is now used to extrapolate the equilibrium constantof Reaction �X�� to higher temperatures� Once this is done it is possible to obtain theequilibrium constants of Reaction �X�� at higher temperatures by subtracting the values

��


Figure X�� Equilibrium constants for Reaction �X�� UO�� CO��aq� � H�O�l� ��

UO�CO��aq��H� �obtained by combining results from Refs� ��PIR�NIK ��SER�NIK�for Reaction �X�� with the values in the last row of Table X�� The line represents aleast squares �t to the �constant rH

�m� equation Eq� �X��

��

��

��

��

��

��

log��K�

�� K�T

��PIR�NIK� �

�

�

�

�

�

�


�

�

�

�

�

�

Eq� �X��

Figure X�� Equilibrium constants for the reaction CO�� UO��

�� UO�CO��aq� from

Refs� ��SER�NIK ��PIR�NIK� compared with values obtained with the isoelectric procedure described in the text�

�

��

��

�

�

��

��

� ��

log��K�

T�K

��PIR�NIK� ��

��

�

�

��SER�NIK� ��

�

�

�

�

�

��GRE�FUG� �see text�

��


Table X�� Experimental equilibrium constantsa for the ionisation of carbonic acid fromRefs� ��PAT�SLO� and ��PAT�BUS��

log��K�

��C ��C ��C ��C ��C ��C ��C ��C�� K �� K �� K �� K �� K �� K �� K �� K

HCO�� H� � CO��

� �

��

CO��aq� � H�O�l� �� HCO�� H��

��

CO��aq� � H�O�l� �� CO�� H��

��

a Values at in�nite dilution as extrapolated by the authors�

for Reaction �X�� at the same temperature �cf� Table X�� The results are shown inFigure X�� compared with the available literature data for Reaction �X��The practical importance of this method is the fact that the approximation

rC�p�m�X�� K� � � can be used� The values of the individual standard partial

molar heat capacities of the reacting species are thus not required� For neutral complexesor molecular solutes this is very important because the available methods to estimatetheir standard partial molar heat capacities are less well developed than for electrolytes�However if a value of rC

�p�m�� K� is available more accurate predictions of

equilibrium constants at higher temperatures are obtained by using the �constant heatcapacity of reaction� approximation Eq� �X�� The assumption rC

�p�m�� K� � �

should only be used if heat capacities cannot be estimated�The isoelectric method is however limited to reactions that are either isoelectric in

themselves or which can be converted to such reactions� For reactions including specieswith electrical charges greater than �� this is certainly a problem because it is less probable that literature data can be found for an additional reaction which may be combinedto obtain an isoelectric reaction �in the same way as Reaction �X�� was used above��

X�� Calculation of the enthalpy of solution from temperature dependence of solubility

Consider the dissolution of a hydrated salt to form a saturated solution

M��A�� xH�O�s� �� Mz� � ��A

z� � xH�O�sln�

��


for which the thermodynamic solubility product is given by�

K�s �T � � �

��

�� m�

s ��a

xw�

Here aw is the activity of water in a saturated solution� � the mean molal activitycoe�cient of the solute for a saturated solution� � � �� z� and z� the charges onthe cation and anion respectively� and ms is the molality of the saturated solution�The starting point for our calculations in the di�erential form of the van�t Ho� equation

�cf� Section X�� footnote on p��

d lnK�s �T �

dT�

solH�m

RT ��

where solH�m is the enthalpy change that occurs when one mole of the hydrated solid is

dissolved to form an in�nitely dilute solution� Taking this derivative gives

d lnK�s

dT� �

d lnms

dT� �

d ln �dT

� xd ln awdT

�

where these derivatives are constrained to fall along the saturated solution molality curveat a constant pressure greater than or equal to the saturation vapour pressure of thesaturated solution at the highest temperature considered� �At temperatures below about�� K the di�erences between values of these derivatives taken at constant pressure andthose taken at the saturation vapour pressure of the solution will be insigni�cant comparedto experimental error��Since � and aw are being constrained to fall along the saturated solution molality

curve they are functions of both saturation molality and of temperature� Thus

d ln �dT

�

�� ln ��T

�ms�p

�

�� ln ��ms

�T�p

�dms

dT

�

and�

d ln awdT

�

�� ln aw�T

�ms�p

�

�� ln aw�ms

�T�p

�dms

dT

��

Combining the last four equations and using the relationship

d lnms

dT�

�dms

dT

��

ms

�

then gives�

solH�m � RT �d lnK

�s �T �

dT

� RT �

��

ms� �

�� ln ��ms

�T�p

� x

�� ln aw�ms

�T�p

��dms

dT

�

�

�� ln ��T

�ms�p

� x

�� ln aw�T

�ms�p

� ��

��


The last term can be recast using well known expressions for these temperature derivatives�

�

�� ln ��T

�ms�p

� x

�� ln aw�T

�ms�p

� � �

RT ��L� � xL��

where L� and L� are the relative partial molar enthalpies of solute and solvent in thesaturated solution respectively relative to in�nite dilutiony� The enthalpy of solution toform a saturated solution is related to the enthalpy of solution to form an in�nitely dilutesolution by�

solHm�sat� � solH�m � ��L� � xL��

One additional simpli�cation can be made� The GibbsDuhem equation for a binarysolution can be cast into the form��

� ln aw�ms

�T�p

� ��ms

nw

�� ln a��ms

�T�p

where a� is the activity of the solute in the saturated solution and nw is the number ofmoles of water in � kg of water� Taking this derivative yields�

�� ln a��ms

�T�p

� �

�� ln ��ms

�T�p

��

ms�

The �nal expression for the enthalpy of solution then becomes�

solHm�sat� � �RT �� xms

nw

�� ms

�

�� ln ��ms

�T�p

��dms

dT

��

y L� and L� denote the excess �relative to the standard state of in�nite dilution� or relative partialmolar enthalpy of the solvent �water� and the solute� respectively� The total enthalpy of a solutionis expressed as a function of its composition and the partial molar enthalpies of its constituents�

H�solution� � n�H� � n�H�

where ni and Hi are the amount and the partial molar enthalpy of a substance� The excess enthalpyis then

H�solution� � n�H�� n�H

�� L � n�L� � n�L�

and Li � Hi �H�i � These relative partial molar enthalpies are related to the activities of the solute

and solvent as follows�

L� � �RT�

�� ln aw�T

�p�m

� and L� � ��RT�

�� ln��T

�p�m

�

Values of Li are usually calculated from experimental measurements of enthalpies of either solutionor dilution�

��


By setting � � � this equation also becomes valid for a nonelectrolyte and by settingx � � for a solid anhydrous electrolyte or nonelectrolyte�

We note that the temperature dependence of solubilities �i�e� of ms� gives solHm

for the formation of a saturated solution and not for an in�nitely dilute solution as isgenerally �and erroneously� assumed� These two types of solution enthalpies will di�ervery little for sparingly soluble solutes but their di�erences can be substantial for moresoluble electrolytes� Some numerical calculations of solHm�sat� were given byWilliamson��WIL� and Brice ��BRI�� Williamson also gave the �rst systematic presentation ofsolubility equations for hydrated and nonhydrated electrolytes and nonelectrolytes�

X�� Alternative heat capacity expressions for aqueous species

As an alternative to the general heat capacity temperature function given in Eq� �X�� Clarke and Glew ��CLA�GLE� proposed a Taylor series expansion for the temperaturedependence of the heat capacity� This approach was used subsequently by Phillips andSilvester ��PHI�SIL�� Clarke and Glew�s equations do not o�er any special advantageover Eq� �X�� and the Taylor series expansion requires more parameters than modelsdescribed below which are based either on the density of the solvent or on the Bornequation� However as pointed out by Clarke and Glew it is not justi�able to set lowerorder temperature derivatives equal to zero in these expansions while retaining higherorder ones �e�g� do not set c � � in Eq� �X�� if the jT � term is retained��

Electrostatic models can be used to predict electrolyte behaviour at high temperatureswith a lesser number of parameters� The model that is perhaps most widely used amonggeochemists is that of Helgeson and coworkers �e�g� ��SHO�HEL�� Some of their equations are presented below�

X�� DQUANT Equation

The DQUANT equation was proposed by Helgeson ��HEL� and it is of historical interest because it has been used by several researchers for example by Haas and Fisher��HAA�FIS� Helgeson�s group ��JAC�HEL� Smith Popp and Norman ��SMI�POP� etc� although the authors of Refs� ��HAA�FIS ��SMI�POP� used additional terms forthe nonelectrostatic contributions to the heat capacity� Furthermore the EQ�� geochemical computer program package ��JAC�WOL� uses the DQUANT equation to calculate hightemperature equilibrium constants of dissociation for neutral inorganic complexes�

The name of �DQUANT� appears to have its origin in the name of a computer program which was used earlier at the Laboratory of Theoretical Geochemistry Universityof California Berkeley�

Assuming that the temperature dependence of the heat capacity change of a dissociationreaction is proportional to the temperature dependence of the electrostatic contribution Helgeson ��HEL his Eqs� �� and �� obtained the expression

��


log��K��T � �

rS�m�T��

ln��RT

�T� � �

�

�� exp

�exp�b� aT �� c�

T � T��

��

� rH�m�T��

ln��RT� �X��

which is consistent with the following expressions for the heat capacity change of thedissociation reaction and the dielectric constant �relative permittivity� for water�

rC�p�m�T � �

TrS�m�T��

��exp

�exp�b� aT �� c�

T � T��

��h

�� exp�b� aT �� exp�b� aT �i

�X��

��T � � �� exp�� exp�b� aT �� T

�

��

where �� b � �� a � �� K�� K c � exp�b � aT�� ac�� and � � a� � ��It should be noted that Eqs� �X�� and �X�� have been superseded by subsequent

models of Helgeson and coworkers described in next Section which generally yield morereliable model �ts�Helgeson ��HEL ��HEL� claimed agreement of Eq� �X�� with experimental values

for most reactions in the temperature range �� to �� or to �� K with the uppertemperature limit depending on the reaction� The errors at �� K were of the order of � to� � of log��K

��T � ��HEL p�� but increased with temperature� Note however thatfor some dissociation reactions whose rH

�m�T�� and rS

�m�T�� and�or the heat capacity

of dissociation are positive the use of Eqs� �X�� and �X�� is not recommended ��HEL pp��Eq� �X�� is of interest because it does not require any knowledge of the heat capacity

change of a reaction� For neutral inorganic species in aqueous solution except for afew simple dissolved gases there are no known methods to estimate the standard molarheat capacities� Therefore Eq� �X�� is of special interest to estimate hightemperatureequilibrium constants for dissociation of neutral species�As an example Figure X�� compares experimental results ��SER�NIK� for Reac

tion �X��

CO�� UO��

�� UO�CO��aq� �X��

with calculated values of log��K��T � using Eq� �X�� and selected reaction parameters

from ��GRE�FUG��

X�� The revised HelgesonKirkhamFlowers model

The electrostatic heat capacity model used by Helgeson et al� which was brie�y describedin Section X�� has evolved into a set of equations of state for the standard partial molar

��


Figure X�� Comparison of equilibrium constants from ��SER�NIK� for Reaction �X�� CO��

� � UO��

�� UO�CO��aq� with calculated values using the DQUANT equation Eq� �X�� and the following selected reaction values ��GRE�FUG�� log��K�T�� and rS

�m�T�� J �K�� mol��

�

��

��

�

�

��

��

��

��

� ��

log��K�

T�K


� �

�

�

�

�

�

DQUANT Eq�

properties of aqueous species ��TAN�HEL ��SHO�HEL ��SHO�HEL ��SHO�HEL��These equations of state which constitute the �Revised HelgesonKirkhamFlowers�model ��Revised HKF� model� allow predictions to be made to �� K and to � kbar�Neither this equation nor the DQUANT equation are expected to be reliable near the critical �point� of water �� K and �� bar ��LEV�KAM ��HAA�GAL ��SAU�WAG��According to this model the standard partial molar heat capacity for an aqueous ion i

is given by

C�p�m�i� � c� �

c��T � ��

��T

�T � ��

a��p � p�� a� ln

�� p

�� p�

��

� �TX � �TY

��

�T

�p

� T��

��

��

�T �

�p

�

where p� � � bar is the standard state pressure the pressure p has units of bar � is thedielectric constant �relative permittivity� of H�O�l� which is temperature and pressuredependent �see for example Figure X�� or ��HEL�KIR Table �� c� c� a� and a� aretemperature and pressure independent parameters and are speci�c to each ion i and X Y and � are temperature and pressure dependent functions �cf� our Eq� �X�� for X Eq� �X�� below for � and Ref� ��SHO�HEL Eq� �� for Y � which were tabulatedby Tanger and Helgeson ��TAN�HEL�� Equations for the partial derivatives of � withregard to temperature and pressure are also given there ��TAN�HEL Appendix B��

��


Pressure e�ects will be neglected here for the following reasons� The saturated pressureof steam at �� K is �� bar ��HEL�KIR Table �� The e�ect that this pressure willhave on equilibrium constants will depend on the reactants and products involved and onthe partial molar volume change of reaction� For example the e�ect that a pressure of�� bar has on the chemical potential of H�O�l� is about �� kJ �mol�� STU�MOR� which at �� K would change an equilibrium constant involving one water molecule by�� log�� units� This may be accounted for if the �apparent� Gibbs energy of H�O�l�as calculated from Table III� of Ref� ��COX�WAG� �see also Eqs� �X�� and �X�� isused in the calculations of the temperature e�ects on equilibrium constants� In general the pressure e�ect on an equilibrium constant may be estimated assuming that the molarvolume of reaction is independent of pressure ��HEL p�� STU�MOR pp��

log��K��p� � log��K

��p�� rV�m�p � ��

RT��ln��

For a partial molar volume change of reaction of �� dm� �mol�� this equation estimatesthe pressure e�ect at �� K and �� bar as �� on log��K��Since the major temperature range of interest for the modelling of aqueous systems is

�� to �� K and in order to simplify the equations of the �Revised HKF� model pressuree�ects on temperature corrections will be neglected here� The reader interested in evenhigher temperatures �and therefore higher pressures� is referred to the original publications��TAN�HEL ��SHO�HEL ��SHO�HEL ��SHO�HEL� ��SHO�SAS ��SVE�SHO��By neglecting the pressure e�ects the following equation is obtained� The apparent

standard partial molar Gibbs energy of an aqueous ion i cf� Eqs� �X�� and �X�� isgiven by the expression

aG�m�i� T � � fG

�m�i� T�� S�

m�i� T��T � T�� c�

�T� � T � T ln

�T

T�

��

� c�

��

T � ��

T� � �� T

��

�

� T

��ln

�T��T � ��T �T��

��

� ��T ��

��

�� T��

��

��

�� T � T��T��Y �T�� X��

where c� and c� are the nonsolvation parameters speci�c to each aqueous ion� the dielectric constant �relative permittivity� of H�O�l� � is temperature and pressure dependent��HEL�KIR Table �� at �� K and � bar �� cf� Table X�� and Y �T�� hasthe value �� K�� TAN�HEL Table H��The temperature �and pressure� dependent function ��T � is de�ned for ionic aqueous

species as ��TAN�HEL Eq� �B��

��i� T � �NAe

�

��

�z�i

ri � jzij�kz � g�T �� zi�� g�T �

�� X��

��


where �NAe�� is equal to ��m �J �mol�� ri is the crystallographic ionic

radius kz is a charge dependent constant �equal to zero for anions and �� m forcations� and g�T � is a non saltspeci�c function which accounts for the dependence of thee�ective electrostatic ionic radius on temperature and pressure �at the steam saturatedpressure g is zero for T �� K ��TAN�HEL Table H� ��SHO�OEL Table �� AsEq� �X�� shows the Revised HKF model uses an e�ective radius for the electrostaticinteractions between the dissolved species and the solvent�

re��i�T � � ri � jzij�kz � g�T �� X��

For neutral species the model function � cf� Eq� �X�� is assumed to be independentof temperature and pressure and thus becomes a model parameter� Correlations at�� K with standard partial molar entropies ��SHO�HEL� their Eqs� �� and ��give for volatile neutral nonpolar aqueous species �noble and diatomic elemental gases�

��i� � ��S�m�i� T��

and for neutral polar aqueous species �H�S�aq� CO��aq� SiO��aq� etc��

��i� � �� S�m�i� T�� X��

where � is in units of J �mol��The c� and c� parameters for Eq� �X�� of this model are temperature and pressure

independent and are correlated at �� K with the standard partial molar ionic heatcapacity as follows ��SHO�HEL Eqs� �� and �� TAN�HEL p�� Eqs� �� c� and ��

c� � �� C�p�m�i� T��

c� � C�p�m�i� T�� c�

��

T� � ��

� �� i� T��

where c� and c� are in units of J �K�� mol�� and J �K �mol�� respectively�The crystallographic ionic radius ri in Eq� �X�� is correlated to the standard partial

molar ionic entropy ��SHO�HEL Eq� ��

ri �� z�iS�m�i� T�� az

� jzijkz� �X��

where az is a charge dependent regression constant �equal to �� and �� J�K��mol��for mono di and trivalent ions respectively and az � ��jzij for cations or anions withzi � � ��SHO�HEL Eq� ��If the chemical reaction only involves aqueous species the calculations can be done with

Eqs� �X�� and �X�� If the chemical reaction includes phases for which heat capacityfunctions �of type Eq� �X�� are available �e�g� solid phases or H�O�l�� it is convenientto use Eq� �X�� for aqueous species and Eqs� �X�� and �X�� for the solid phases�

��


As mentioned earlier the integrals in Eqs� �X�� and �X�� may have expressions likeEqs� �X�� and �X�� except that one must disregard the delta signs on the right handside of Eqs� �X�� and �X�� For water the apparent Gibbs energy should be calculatedfrom values given in the CODATA tables ��COX�WAG��In order to avoid computational errors it is advantageous to use a computer pro

gram to do the calculations described here� The computer program SUPCRT is availablefrom Helgeson�s laboratory at Berkeley ��JOH�OEL� and on the GEOPIG home page�http��zonvark�wustl�edu�geopig�� That program also includes a mineral and aqueousspecies data base�As it stands Eq� �X�� contains �ve parameters for each aqueous species �standard

partial molar ionic entropy and standard Gibbs energy of formation as well as c� c� and ri�� When Eqs� �X�� and �X�� are included in the model only three parametersremain in Eq� �X�� the standard partial molar entropy the standard partial molarGibbs energy of formation and the standard partial molar heat capacity��

Tremaine Sway and Barbero ��TRE�SWA� and Apps and Neil ��APP�NEI� havereanalysed some experimental data using the HelgesonKirkhamFlowers equations�As an example of the application of Eq� �X�� to predict hightemperature equilibrium

constants the calcite dissolution reaction is used�

CaCO��cr� �� Ca�� CO�� X��

The data for the aqueous ions are taken from CODATA ��COX�WAG� except for thestandard partial molar ionic heat capacities which are not given in the CODATA publication and are taken instead from Ref� ��DES�VIS�� For calcite the value for the entropyand the heat capacity function of CODATA ��GAR�PAR� are used whereas the standardGibbs energy of formation is adjusted to �� kJ�mol�� in order to force the logarithmof the equilibrium constant at �� K to be log��K

��X�� as recommendedby Plummer and Busenberg ��PLU�BUS�� The predicted temperature dependence ofthe equilibrium constant for calcite dissolution is shown in Figure X�� together with experimental values from Ref� ��PLU�BUS�� Large di�erences in calculated values are ingeneral obtained at T � �� K between the �constant rC

�p�m� equation Eq� �X�� and

the simpli�ed revised HKF model Eq� �X�� and Eqs� �X�� to �X��The �tting capabilities of the revised Helgeson Kirkham and Flowers model are very

high because of the large number of parameters for each dissolved species� With thesimpli�cation introduced by Eqs� �X�� and �X�� the number of parameters is reduced but still one must avoid over�tting experimental data� Only good quality data in a broadtemperature range should be used to determine the model parameters� This is due to thefact that ionic heat capacities generally show gentle maxima at about �� to �� K andhave a steep decrease as the temperature approaches the critical temperature of water� �� K� ��LEV�KAM ��HAA�GAL ��SAU�WAG�� The heat capacities for ions are� Note added in press� Sverjensky� Shock and Helgeson ��SVE�SHO� have recently published sev�

eral examples of this technique� Furthermore they present several correlation strategies to estimatethermodynamic properties of aqueous complexes �mostly inorganic��

��


Figure X�� Comparison of experimental equilibrium constants for calcite dissolution��PLU�BUS� Reaction �X�� with those predicted with the simpli�ed �RevisedHelgesonKirkhamFlowers model� described in the text �continuous line�� The predictions with the �constant rC

�p�m� equation Eq� �X�� using rS

�m � �� J �K�� mol��

and rC�p�m � �� J �K�� mol�� are shown as a dotted line�

��

��

��

��

��

��

��

� ��

log��K�

T�K

��

usually not determined reliably by a leastsquares representation of equilibrium constantsin the range �� to �� K�As an example of the uncertainties that might be found in �tting S�

m or C�p�m to experi

mental values of equilibrium constants the data on the solubility of zincite are used� Theequilibrium constants for the reaction

ZnO�cr� � H� �� ZnOH� �X��

are taken from Khodakovskiy and Yelkin ��KHO�YEL their Table �� For zincite thestandard entropy and the heat capacity function are taken from Kubaschewski et al��KUB�ALC� and the standard Gibbs energy of formation from the US NBS tables��WAG�EVA�� For ZnOH� the standard ionic partial molar Gibbs energy is set to thevalue required for achieving an equilibrium constant at �� K of log��K

� � �� which is the value recommended in Ref� ��KHO�YEL�� A leastsquares �t to the simpli�ed �Revised HKF model� on the four experimental values of log��K

��X�� yieldsS�m�ZnOH

�� aq� T�� J �K�� mol�� and C�p�m�ZnOH

�� aq� T�� J �K�� mol�� and is shown in Figure X�� The large uncertainties are due to the correlation between S�

m�T�� and C�p�m�T�� A more precise value of C

�p�m�T�� is obtained by a

leastsquares �t if the value of S�m�ZnOH

�� aq� T�� is �xed to �� J �K�� mol�� the valueobtained above� namely C�

p�m�ZnOH�� aq� T�� J � K�� mol�� Figure X��

��


Figure X�� Comparison of experimental equilibrium constants ��KHO�YEL� for Reaction �X�� ZnO�cr� � H� �� ZnOH� with a leastsquares �t to the simpli�ed �RevisedHKF model�� The dotted lines re�ect the e�ect of an uncertainty of �� J �K�� mol��in C�

p�m�ZnOH�� aq� T��

��

��

��

��

� ��

log��K�

T�K

�

�

�

�

shows that data at higher temperatures are needed in order to constrain the leastsquares�ts su�ciently to obtain unambiguous thermodynamic parameters for ZnOH��An example where experimental data up to �� K are available ��PAT�SLO� is the

reaction

CO��aq� � H�O�l� �� HCO�� H

�� X��

The thermodynamic data are taken from CODATA ��COX�WAG� except for thestandard partial molar heat capacities of the bicarbonate ion which is taken fromRef� ��DES�VIS� and of CO��aq� which is �tted by leastsquares to the experimental equilibrium constants of ��PAT�SLO�� The calculation yields C�

p�m�CO� aq T�� J � K�� mol�� and the model results are shown in Figure X��For comparison results obtained with the DQUANT expression Eq� �X�� and the�constant rC

�p�m� Eq� �X�� with a heat capacity of reaction of �� J � K�� mol��

are also displayed� The approximation expressed by the DQUANT equation Eq� �X�� resulted in errors of up to �� in log��K��T ��A better �t of the data could be achieved by using the isocoulombic reaction�

CO��aq� � OH� �� HCO�

�

and the �constant rC�p�m� approximation Eq� �X�� Reaction �X�� has been used

for illustrative purposes only�

��


Figure X�� Comparison of experimental equilibrium constants for the hydrolysis ofCO��aq� ��PAT�SLO� Reaction �X�� with those obtained from �tting C�

p�m�CO� aq T�� with the simpli�ed �Revised HKF model�� Results obtained using the DQUANTequation Eq� �X�� and the �constant rC

�p�m� equation Eq� �X�� with rH

�m�T��

�� kJ �mol�� rS�m�T�� J �K�� mol�� and rC

�p�m � �� J �K�� mol��

are also displayed for comparison�

��

� ��

� ��

��

��

��

��

� ��

log��K�

T�K

simpli�ed Rev� HKF modelconstant �rC

�p�m

DQUANT� PAT�SLO� e

e

ee e

ee

e

e

e

e

e

e

e

X�� The RyzhenkoBryzgalin model

Ryzhenko and Bryzgalin ��RYZ ��BRY�RAF ��RYZ�BRY� and Bryzgalin ��BRY ��BRY� describe the temperature dependence of mononuclear complex formation reactions using a simple electrostatic model where the Gibbs energy of reaction rG

�m�T� p�

is described as a sum of two contributions �an idea �rst proposed in detail by Gurney��GUR Chapter ��

rG�m�T� p� � rG

�m�nonel �rG

�m�electr�T� p� �X��

where rG�m�nonel is a temperature and pressure independent nonelectrostatic contribu

tion to the Gibbs energy and rG�m�electr�T� p� is a temperature and pressure dependent

electrostatic contribution given by the following Coulombtype equation�

rG�m�electr�T� p� � �jZcZaje�

re�

NAe�

��

�

��T� p��X��

where e is the elementary charge and NA is the Avogadro constant �NAe��

�� m � J �mol�� cf� Table II�� T� p� is the relative permittivity of thesolvent �in this case the dielectric constant of water�� re� is an e�ective bond distance

��


Table X�� Polarisabilities of ions in aqueous solutions from Ref� ��NIK p�� alsoreported in ��RYZ�SHA ��RYZ�BRY�� We note however that there are inconsistenciesbetween the values listed in this table and the data found in Refs� ��RIC ��SYR�DYA� apparently because the latter two references contain electronic ionic polarisabilities of ionseither in vacuo or in crystal lattices�

Anion �� Cation �� Cation � � ��

�m� �m� �m�

OH� �� H�� Zn��

F� �� Li� �� Cd�� Cl� �� Na� �� Hg�� Br� �� K� �� Mn�� I� �� Rb� �� Mn�� CN� �� Cs� �� Mn�� NO�

� �� Mg�� Fe�� HCOO� �� Ca�� Fe�� SiO��

� �� Sr�� Co��

SO�� Ba�� Ni��

CO�� Pb�� Al��

CrO�� Cu�� Y��

La��

which in most cases is approximately equal to the sum of radii of the central ion andligand and it is independent of the total number of ligands in the complex� jZcZaje� is an�e�ective charge� which is a function of the formal charges of the anion and the cation�Za and Zc respectively� and of the number of ligands and geometry of the complex�The model can only be applied to monodentate ligands but protonation equilibria forpolybasic acids can be described by considering complexes where the central ion is theanion and the ligands are protons ��RYZ�BRY��Bryzgalin and Rafal�skiy ��BRY�RAF� give the following equation to calculate jZcZaje�

for a number of di�erent coordination geometries�

jZcZaje� � jZcZajL�QZ�a �

�aZ�cL

� r�e�� aZc��Q

� r�e��X��

where L is the total number of ligands in the mononuclear complex� Q is a stereochemicalfactor Q � ��L��L�� and �a is the polarisability of the ligand �cf� Table X�� The�rst term on the right hand side of Eq� �X�� takes into account the attraction betweenthe central ion and each of the ligands� the second term includes mutual repulsions between

��


ligands� the third term deals with the attractive interactions between the central ion andthe induced dipoles in the ligands� and the last term considers the mutual repulsion amongthe dipoles induced in the ligands ��BRY�RAF�� For �� complexes L � � Q � � andthe last term of Eq� �X�� is modi�ed so that the e�ective charge is instead given by��BRY��

jZcZaje� � jZcZaj� �aZ�c

� r�e��cZ

�a

� r�e��X��

where �c is the polarisability of the central ion �cf� Table X��Electrostatic models are based on incompressible spherical ions� The thermodynamic

values obtained by such models require the conversion from the molar volume of the idealgas to that of a solution at � mol � �kgH�O�� This correction is adsorbed by adjustableparameters in the HelgesonKirkhamFlowers model which is described in Section X��The RyzhenkoBryzgalin model yields thermodynamic data in volumetric concentration

units ��PRU� i�e� on the molar concentration scale� The conversion of equilibriumconstants from molar to molal units is straightforward cf� Eq� �II�� on page �� wherethe factor for the conversion of molality to molarity � at in�nite dilution becomes equalto the density of H�O�l� ��T� p� in kg � dm�� and �at in�nite dilution� this conversionfactor is � � at T �� K� Taking this units conversion into account it is possible toobtain from Eqs� �X�� to �X�� an expression for the formation equilibrium constantsof mononuclear complexes�

log��K��T� p� �

T�Tlog��K

��T�� p��

�jZaZcje�re�

NAe�

��RT ln��

��

��T� p��

��T�� p��

�

�Xi

�i log�� T� p� �X��

provided that re� is assumed to be independent of temperature and pressure� In thisequation it is seen that the model requires only two parameters� the equilibrium constantat one temperature and an e�ective bond distance re��For the standard partial molar entropy of a reaction involving the formation of mononu

clear complexes the following expression is obtained�

rS�m�T� p� � �

��rG

�m

�T

�p

� � jZcZaje�re�

NAe�

��T� p��

��T� p�

�T

�p

�Xi

�iR ln ��T� p�

�Xi

�iRT�T �T� p� �X��

where �T �T� p� � ��V �m� ��V

�m��T �p � �� T �p � � �� ln ��T �p is the coe�

cient of thermal expansion of water cf� Table X�� here V �m represents the molar volume

��


of pure liquid water in cm� �mol�� V �m�T� p� � ��T� p�� At �� K and � bar

the contribution of the second term is negligible and one can then write�

rS�m�T�� p

��

�� jZcZaje�

re��

Xi

�i

�J �K�� mol�� X��

where re� is in units of m� Similarly for the standard partial molar heat capacity ofa reaction involving the formation of mononuclear complexes the following equation isobtained�

rC�p�m�T� p� � T

��rS

�m

�T

�p

� �T jZaZcje�re�

NAe�

��T� p��

��T� p�

�T �

�p

� �

��T� p�

��T� p�

�T

��

p

�A

� �Xi

�iRT�T �T� p� �Xi

�iRT�

��T �T� p�

�T

�p

�X��

At �� K and � bar the contribution of the second term is negligible and the followingequation can be used�

rC�p�m�T�� p

��

�� jZcZaje�

re��

Xi

�i

�J �K�� mol�� X��

The uncertainty in this expression is �� as estimated from the uncertainty in thederivatives ��T �p and ��

��T ��p�� The expression for the standard partial molarvolume of a reaction involving the formation of mononuclear complexes is�

rV�m�T� p� �

��rG

�m

�p

�T

�jZcZaje�re�

NAe�

��T� p��

��T� p�

�p

�T

�Xi

�iRTkT �X��

where kT �T� p� � � ��V �m� ��V

�m��p�T � �� ln ��p�T is the coe�cient of isothermal com

pressibility of pure water� At �� K and � bar the following equation may be used�

rV�m�T�� p

�� jZcZaje�

re��

Xi

�i

�cm� �mol��

The uncertainty in rV�m�T�� p

�� should be �� from the estimated uncertainty in��p�T ��

��


The values of the temperature and pressure derivatives of the dielectric constant ofwater which are needed in Eqs� �X�� to �X�� can be calculated from the equations of Refs� ��JOH�NOR ��SHO�OEL� or from the equations of Archer and Wang��ARC�WAN� together with an equation of state for water to calculate the densityof liquid water ��T� p� �such equations of state may be found for example in Refs��HAA�GAL ��KES�SEN ��SAU�WAG ��WAG�PRU�� All the data needed to usethe RyzhenkoBryzgalin model for aqueous solutions from �� K to �� K are givenin Table X��The RyzhenkoBryzgalin model considers thermodynamic quantities at zero ionic

strength hence it is in general necessary to recalculate the experimental data to I � ��This can be done for example by using the speci�c ion interaction theory as describedin Chapter IX�As mentioned above the RyzhenkoBryzgalin model is a two parameter model requir

ing only K�T�� p�� and re�� If the value of rS�m for the reaction is known at �� K �or

may be estimated cf� Section X�� p�� the required value of the distance parameterre� may be calculated with Eq� �X�� However in general it is possible to estimate the�distance� parameter re� from the sum of crystallographic radii of the central ion and theligand� This allows the calculation of preliminary values of log��K

��T� p� even when allthermodynamic properties of reaction are lacking except for log��K

��T�� p��

X�� Example the mononuclear Al ��OH� system

This section describes the procedure used and indicates the accuracy of the method�Further details on this example are given in Ref� ��PLY�GRE��In the system Al��OH� the following mononuclear complexes have been established

��BAE�MES�� Al�� AlOH�� Al�OH�� Al�OH��aq� and Al�OH�� Standard values

of the equilibrium constants of formation of these complexes log�� i at �� K have

been selected by Plyasunov and Grenthe ��PLY�GRE� from recent literature studies�� and �� for i� � to � respectively� Thesecond step is to �nd re� for this system� Several estimations of this parameter are givenin ��PLY�GRE�� Preliminary calculations readily show that a reasonable agreement withthe available experimental data can be achieved with re� � �� m� Theresults obtained with Eq� �X�� are compared with the literature equilibrium constantsfor the �rst and last hydrolysis steps �for which more accurate data have been obtainedexperimentally� in Figures X�� and X�� In addition calculated and experimentalequilibrium constants for the stepwise reaction�

AlOH�� OH� �� Al�OH��

are compared in Figure X��The calculations indicate that the simple electrostatic model of Ryzhenko and Bryz

galin is able to describe at least semiquantitatively the temperature dependence of theformation constants for the mononuclear hydroxy complexes of aluminium up to �� or

��


Table X�� Temperature and pressure variation of the density and dielectric constant �relative permittivity� of liquid water at the standard pressure �p� � �bar� for T � ��Kand at the vapour saturation pressure at T � ��K as well as some other properties of liquid water at the reference temperature T� and at the standard pressure p� namely� the coe�cient of thermal expansion and its temperature derivative the coe�cient of isothermal compressibility the �rst and second temperature derivatives and thepressure derivative of the dielectric constant� These values have been calculated with theequation of state of Kestin et al� ��KES�SEN� and the equation of Archer and Wang��ARC�WAN� for the dielectric constant of water�

t T p � �

��C �K �bar �g � cm��

��

Values at T� � �� K and p� � � bar�

�T � �� K��

��T��T p� � �� K��

kT � �� bar��

��T p� �� K��

��T �p� � �� K��

��pT� � �� bar��

��


Figure X�� Comparison of experimental equilibrium constants for the reaction of formation of AlOH�� with those obtained on the basis of the RyzhenkoBryzgalin model�Adapted from Ref� ��PLY�GRE��

��

��

��

��

��

��

��

��

� ��

log��K�

T�K

��NIK�TOL� �

��

��

��VOL�PAV�

� �COU�MIC� �

��

��VER� �

��

�

��CAS� �

��

��PAL�WES� u

u uu

uu

�PAL�WES��BOU�KNA� e

e

e

e

calc�calc��calc��

�� K using the values of stability constants at �� K with only one �tting parameter re� which has the same value for all mononuclear complexes formed in the system� Athigher temperatures the calculated values of K� di�er systematically from the equilibriumconstants determined experimentally�

The model is put to a more rigorous test by trying to predict rS�m rC

�p�m and rV

�m

for the formation reactions with Eqs� �X�� to �X�� and the same size parameter asabove �re� � �� m�� Precise experimental values for rS

�m rC

�p�m

and rV�m at �� K and � bar are only available for one of the aluminium hydrolysis

reactions�

Al�� OH� �� Al�OH��

Table X�� shows that the calculated values of rS�m rC

�p�m and rV

�m are only in

qualitative agreement with the experimental determinations� This re�ects the limitationsof this simple electrostatic model for example the assumption that rG

�m�nonel is indepen

dent of T and p� More sophisticated models e�g� the revised HelgesonKirkhamFlowersmodel consider the nonelectrostatic term to be a function of temperature and pressure�

It should be noted that in order to improve the predictive capabilities of this simpleelectrostatic model Ryzhenko ��RYZ� suggested the following empirical temperature and

��


Figure X�� Comparison of experimental equilibrium constants for the reaction of formation of Al�OH�� from Al�� and OH� with those obtained on the basis of the RyzhenkoBryzgalin model� See Figure X�� for symbols� Adapted from Ref� ��PLY�GRE��

��

��

��

��

��

��

��

� ��

log��K�

T�K

��

��

�

�u

uu u

e

e

e

Figure X�� Comparison of experimental equilibriumconstants for the reaction AlOH��OH� �� Al�OH�� with those obtained on the basis of the RyzhenkoBryzgalin model�See Figure X�� for symbols� Adapted from Ref� ��PLY�GRE��

��

��

��

��

��

��

��

��

� ��

log��K�

T�K

�

��

��

u

uu u

ee

e

��


Table X�� The calculated and experimental values of rS�m rC

�p�m and rV

�m for re

action Al�� OH� �� Al�OH�� at �� K and � bar� Adapted with revisions fromRef� ��PLY�GRE�� The experimental values were derived in Ref� ��PLY�GRE� fromexperimental results presented in the literature sources�

Property Calculated Experimental Literature sourcevalue value

�rS�m �� a� �� PAL�WES�

� J �K�� mol�� CHE�XU� �COX�WAG�

�rC�p�m �� a� �� HOV�HEP�

� J �K�� mol�� PAL�WES� �� PAL�WES�

�rV�m �cm� �mol�� b� �� HOV�HEP�

�a� The � term is based on the uncertainty in re� ��b� The � term has been estimated from the uncertainty in ��p�T � cf�

Eq� �X��

pressure dependence of re��

re��T� p� � re��T�� p��

�V �m�H�O� l� T� p�

V �m�H�O� l� T�� p��

��

�X��

where values for the molar volume of pure water are needed both at the T and p of interest and at the reference temperature T� � �� K and the standard pressure p� � � bar� Inlater publications Ryzhenko and Bryzgalin ��RYZ�BRY ��BRY ��RYZ�BRY ��BRY�used a somewhat di�erent function�

re��T� p� � re��T�� p��

��T�� p��

��T� p�

��T�p��

�X��

Eqs� �X�� and �X�� are perhaps indirect attempts to take into account both di�erencesin concentration units �molar to molal as discussed above� as well as physical phenomenaoccurring when the molar volume of water increases� Eqs� �X�� and �X�� predict agradual increase of re� with temperature typically in the range �� to �� mat �� K� It should be noted that in the revised HelgesonKirkhamFlowers model cf�Eq� �X�� the e�ective radius decreases with temperature because the function g�T � hasnegative values ��TAN�HEL Table H� ��SHO�OEL Table �� From Eqs� �X�� and�X�� it follows that if re� is temperature and pressure dependent then jZcZaje� will alsobe a function of T and p and Eqs� �X�� and �X�� etc� must be rewritten accordingly�

��


Figure X�� Comparison of experimental equilibrium constants �squares� ��PAL�DRU� circles� ��PAL�HYD�� for reaction� Fe�� CH�COO� �� FeCH�COO� with those obtained on the basis of the RyzhenkoBryzgalin model�

��

��

��

��

��

��

��

��

� ��

log��K�

T�K

�

�

�

�

�

�

e

e

e

e

e

e

See for example Ref� ��PLY�GRE� for re� given by Eq� �X�� Nevertheless in boththe RyzhenkoBryzgalin and in the revised HelgesonKirkhamFlowers models the e�ectsof the variation in re� with increasing temperature are essentially negligible at T � �� K�For simplicity the examples in this chapter have been worked out with Eqs� �X�� to�X�� that is assuming that re� is independent of temperature and pressure� Plyasunovand Grenthe ��PLY�GRE� presented the results of the RyzhenkoBryzgalin model on theAl�III�OH� system using Eq� �X�� to describe the T and p dependence of re��

X�� Example the stability of acetate complexes of Fe ��

Palmer et al� ��PAL�DRU ��PAL�HYD� have used a potentiometric method to studythe stability of acetate complexes of Fe�II� in aqueous solutions at temperatures between�� and �� K �at steam saturated pressures above �� K�� Palmer and coworkers obtained equilibrium constants for the formation of FeCH�COO� and Fe�CH�COO��aq� and observed indications of the presence of Fe�CH�COO�

�� at high concentrations of ac

etate� The experimental equilibriumconstants ��PAL�DRU ��PAL�HYD� are comparedin Figures X�� and X�� with those obtained with the RyzhenkoBryzgalin model usingre� � �� m and a polarisability of zero for the acetate ion�

��


Figure X�� Comparison of experimental equilibrium constants ��PAL�DRU� for reaction� Fe�� CH�COO� �� Fe�CH�COO��aq� with those obtained on the basis of theRyzhenkoBryzgalin model�

��

��

��

��

��

��

��

� ��

log��K�

T�K

�

�

�

�

�

�

X�� The density or �complete equilibrium constant� model

The density model is based on the experimental observation that in aqueous systems thelogarithm of the equilibrium constants of many reactions at isothermal conditions arelinear functions of the logarithm of the density of the solvent over large p and T ranges��FRA ��FRA�� The theoretical basis for this has been discussed by many authors anda comprehensive review of the literature is given by Anderson et al� ��AND�CAS�� Theconclusion from these discussions is that the origin of the linear relationship is unknown cf� ��AND�CAS pp�� The model postulates a direct proportionality betweenlog��K

��T� p� for reactions involving aqueous species and log�� T� p� where ��T� p� isthe density of pure water� This proportionality was discovered by Franck ��FRA ��FRA�during his conductimetric investigations of the electrolytic properties of KCl and otherelectrolytes at temperatures between �� to �� K at a wide range of pressures� Later Marshal ��MAR ��MAR� con�rmed this observation and formulated the concept of�complete equilibrium constant� on the basis of determinations of dissociation constantsof many salts and acids at temperatures above �� K� This concept is outlined by thefollowing relation�

log��K��T� p� � log��K

��T � � k�T � log�� T� p� �X��

where K��T� p� stands for the conventional equilibrium constant in molar concentrationunits which depends both on temperature and pressure� K ��T � is the �complete equi

��


librium constant� which is assumed to depend on temperature only� k�T � is a functionwhich Marshal ��MAR ��MAR� considered to represent the change in hydration numbers between the products and the reactants� and ��T� p� is the density of pure water� Itmust be pointed out that Eq� �X�� has no rigorous thermodynamic basis� However itprovides a remarkably good correlation valid over a very wide range of state parameters�It describes the dissociation constants for HCl ��FRA�MAR� HBr ��QUI�MAR�� NaCl��QUI�MAR� NaBr ��QUI�MAR�� NaI ��DUN�MAR� and NH�OH ��QUI�MAR��in the temperature range �� to �� K and a water density of �� or �� to ��g � cm�� as well as the pressure dependence of log��K

��T� p� for the dissociation of anumber of electrolytes up to � kbar at room temperature ��MAR� etc� Eq� �X�� alsodescribes the thermodynamic solubility product for salts for instance that of CaSO��s�at temperatures between �� and �� K and pressures up to �� bar ��MAR��The following type of equation was used by Marshall and Franck ��MAR�FRA� to

describe the molal ion product of water as a function of p and T and the same expressioncan also be used for other chemical equilibria�

log��K��T� p� �

�A�

B

T�

C

T ��

D

T �

�

��E �

F

T�

G

T �

�log�� T� p� �X��

Eq� �X�� is a formulation issued by the International Association for the Propertiesof Steam ��MAR�FRA� which describes the dissociation constant of pure water up to�� K and �� kbar practically within the experimental uncertainties in the whole T prange �at least at densities above �� g � cm��The density model is a useful empirical generalization of a large number of experimental

data on the thermodynamic behaviour of solutes at high temperatures and pressures�The full set of coe�cients of Eq� �X�� can be determined only if experimental data

on log��K��T� p� are available in a wide range of temperature and pressure �or solvent

density�� Anderson et al� ��AND�CAS� cited Mesmer for the use of a simpli�ed form ofEq� �X��

lnK��T� p� � a� �a�T�a�Tln ��T� p� �X��

where a� a� and a� are independent of T and p� This equation is very useful for correlations up to �� or �� K depending on the system� Standard thermodynamic relationshipsallow the derivation of the parameters a� a� and a� as follows ��AND�CAS��

a� � lnK��T�� p��

rH�m�T�� p

��

RT�� rC

�p�m�T�� p

��T �T�� p��

RT� ��T��T �p�

a� � �rH�m�T�� p

��

R��T��T �T�� p�� ln ��T�� p��rC

�p�m�T�� p

��

RT� ��T��T �p�

a� � � rC�p�m�T�� p

��

RT� ��T��T �p�� rV

�m�T�� p

��

RkT �T�� p��

��


Figure X�� The temperature dependence of parameter k�T � of Eq� �X�� between�� and �� K from experimental studies on the dissociation equilibria of H�O�l��MAR�FRA� and NH�OH�aq� ��MES�MAR��

��

��

��

��

��

k�T �

�� K�T

H�O�l�

NH�OH�aq�

� �MAR�FRA�� MES�MAR�

where �T and kT are the coe�cient of thermal expansion of water and the coe�cient ofisothermal compressibility of water respectively and the subscript �p�� in the derivative��T��T �p� indicates that the derivative is taken at T� and p�� The values for �T kT and ��T��T �p� needed to calculate a� a� and a� are given in Table X�� Values forthe density of water � which are needed in Eq� �X�� may be calculated at the steamsaturated pressure and for any temperature below the critical point with the equationgiven by Wagner and Pruss ��WAG�PRU��

A comparison of Eqs� �X�� and Eq� �X�� shows that the latter equation assumes thesimplest possible temperature dependence for log��K

��T � and a simple T�� dependenceof k�T �� This simpli�cation reduces the temperature range of applicability of Eq� �X�� because the true temperature dependence of k�T � might be more complicated for exampleas in Eq� �X�� Among the most reliable expressions available for k�T � are those forthe dissociation of H�O and NH�OH� These expressions are shown in Figure X�� whichshows that k�T � can be considered to be approximately linear functions of T�� only upto about �� K� Anderson et al� ��AND�CAS� estimated the upper temperature limit ofapplicability of Eq� �X�� to be �� K�An important feature of the �density� model in the AndersonCastetSchottMesmer

modi�cation ��AND�CAS� is the possibility of using experimentally determined thermodynamic quantities like rH

�m�T�� p

�� rC�p�m�T�� p

�� or alternatively rS�m�T�� p

��

��


Figure X�� Comparison of experimental equilibriumconstant for the reaction CO��aq��H�O�l�� HCO�

� �H� ��PAT�SLO� with those calculated on the basis of the �density�

model and the constant rC�p�mmethod� See Eq� �X�� Section X�� and Figure X��

��

� ��

� ��

��

��

��

��

� ��

log��K�

T�K

constant �rC�p�m Eq�

�density� model� PAT�SLO� e

e

ee e

e

e

e

e

e

e

e

e

e

rV�m�T�� p

�� etc�� together with the value of log��K��T�� p�� to predict values of

log��K��T� p�� This calculation also requires the numerical values of ��T� p� �T �T�� p��

kT �T�� p�� and ��T��T �p� for pure water at saturation water vapour pressure whichare well known� All the data needed to use the �density� model of Anderson et al��AND�CAS� are given in Table X�� p� ��

Unlike the electrostatic model of Ryzhenko and Bryzgalin which is valid only for formation �or dissociation� reactions involving exclusively aqueous species the �density�model can be used for any reaction involving aqueous species including reactions wheresolid phases participate �see ��AND�CAS� for further details� and the �density� modelappears to be as general as the revised HelgesonKirkhamFlowers model described inSection X�� However relatively fewer aqueous systems have been analysed with the�density� model�

We will demonstrate the accuracy of the method by using experimental data for the�rst deprotonation constant of CO��aq� from ��PAT�SLO� cf� Figure X��

From the experimental values at �� K and � bar �log��K��T�� p��

rH�m�T�� p

�� kJ �mol�� rC�p�m�T�� p

�� J � K�� mol��we obtain the following values of the parameters a� to a� in Eq� �X�� a� � �� a� � �� K a� � �� K� The calculated values of log��K� �the solid curve� andthe experimental values are shown in Figure X�� The maximal deviations are ��logarithmic units at �� K�

��

Third�law method

X�� Third�law method

The secondlaw method of extrapolation described in previous sections is frequently used and it is well suited for extrapolation of Gibbs energy data when only a small temperatureinterval is involved� However if a temperature extrapolation over a large temperaturerange is required and the original Gibbs energy data are for a restricted temperatureinterval then the secondlaw extrapolation method can be very sensitive to extrapolationerrors� This arises because the integral on the right hand side of Eq� �X�� dependson rH

�m�T �� For experimental Gibbs energy data over a short temperature range the

derived values rH�m�T � and rS

�m�T � can be very inaccurate �even if rG

�m�T � is fairly

accurately known� since they depend on the intercept and slope of


�m�T �� TrS

�m�T �� X��

This issue has been discussed in many textbooks� For example according to the revised version of Lewis and Randall�s Thermodynamics ��LEW�RAN p�� Oftentemperaturedependent errors are di�cult to eliminate from the equilibrium measurements and while the resulting equilibrium constants or free energies of reaction are approximately correct the temperature coe�cient and the corresponding heat of reactionfrom the second law � � �may be greatly in error� The thirdlaw method will also yield theheat of reaction when F � values have been determined over too small a temperatureinterval to determine the temperature coe�cient accurately��The preferred method to obtain fH

�m�T�� of a compound or ion is by direct use of

calorimetric measurementsy� For example both Ca�cr� and CaO�cr� readily dissolve inaqueous solutions of strong mineral acids and these solH

�m values can be combined to

yield fH�m�CaO cr T�� However for some systems this type of measurement is not

possible� That approach cannot be used for Tc�cr� and TcO��cr� Ru�cr� and RuO��cr� Pd�cr� and PdO�cr� etc� and other oxides which are only very slightly soluble in aqueoussolutions of mineral acids and where the metals are even less reactive� Direct combustionof the metal with O��g� under pressure can sometimes be used instead to yield directcalorimetric values of fH

�m�T�� but the desired oxide is not always obtained or the

reaction may be incomplete or yield more than one product� Combustion of Tc�cr� yieldsmainly Tc�O��cr� for example so it will not yield data for TcO��cr��An alternative way of obtaining thermodynamic data is by use of Gibbs energy mea

surements at very high temperatures� For example a reaction such as

MO��cr� �� M�cr� � O��g�

can be studied by means of oxygen decomposition pressure measurements or solid stateemf measurements� �This reaction does not occur for some metaloxide systems e�g� inthe case of TcO��cr� since it sublimes rather than decomposes at high temperatures��In many cases it may be necessary to use temperatures of �� K or higher to obtain a

y T� stands for the reference temperature �� K��

��


signi�cant vapour pressure of oxygen from decomposition of a MO� and �� to �� K orhigher to obtain su�cient solid state di�usion required for a solid state emf measurement�A thirdlaw extrapolation of such rG

�m�T � data to �� K will in general be much

more reliable than direct use of the secondlaw method�The thirdlaw method makes use of the free energy functions for reactants and products

G�m�T ��H�

m�T��

T�

which have much smaller variations with temperature than G�m�T �� Thus this method

can generally be used for numerical or graphical interpolations or extrapolations with ahigher degree of precision than direct calculations with either G�

m�T � or H�m�T ��

The thirdlaw method equation can be written in the general form

r

�G�

m�T ��H�m�T��

T

��

Xproducts

�G�

m�T ��H�m�T��

T

�

� Xreactants

�G�

m�T ��H�m�T��

T

�

�rG

�m�T �

T� rH

�m�T��

T� �X��

For this equation to be used the value of rG�m�T � must be known from lowtemperature

calorimetric data or from hightemperature Gibbs energy measurements and the valueof rH

�m�T�� needs to be obtained� Once rH

�m�T�� has been determined it can be used

to derive fH�m�T�� of one of the reactants or products� Two di�erent cases will now be

considered�

X�� Evaluation from high and low�temperature calorimetric data

If both lowtemperature �heat capacity� and hightemperature �relative enthalpies as fromdrop calorimetry� thermal data are available then the calculation of rH

�m�T�� is straight

forward� The middle terms of Eq� �X�� G�

mT �H�

mT�T

are then known and they can becombined with each experimental rG

�m�T � point at each temperature to yield a value

of rH�m�T�� If there is no drift in the calculated values of rH

�m�T�� within the tem

perature range over which rG�m�T � was measured then the thermodynamic data are

internally consistent and the average of these rH�m�T�� should be reliable� We now give

a speci�c example of this type of calculation�Consider the simple case of the calculation of subH

�m�T�� from hightemperature sub

limation data of a pure metal to form its monoatomic vapour� The reaction is thus

M�cr� �� M�g��

and rG�m�T � becomes subG

�m�T �� For the solid phase

Gm�cr� T � � G�m�cr� T ��

��

Third�law method

and for the vapour phase

Gm�g� T � � G�m�g� T � �RT ln f�

where f is the fugacity of M�g�� Vapour pressures of metals are generally quite low attemperatures used for sublimation measurements so the fugacity can be equated to thevapour pressure

Gm�g� T � � G�m�g� T � �RT ln p�

For the sublimation process

subGm�T � � Gm�g� T ��Gm�cr� T �

� G�m�g� T � �RT ln p �G�

m�cr� T �

� �G�m�g� T ��G�

m�cr� T �� RT ln p

� subG�m�T � �RT ln p�

Vapour pressure measurements are measurements of an equilibrium property sosubGm�T � � � at each T � Thus for a closed system

subG�m�T � � G�

m�g� T ��G�m�cr� T �

� �RT ln p�For this case the thirdlaw extrapolation Eq� �X�� becomes

subH�m�T�� T

�G�

m�cr� T ��H�m�cr� T��

T� G�

m�g� T ��H�m�g� T��

T

�

�RT lnp�

The T values were not cancelled in the �rst term of the right hand side of this equation since free energy functions will be used to interpolate thermal data�As a numerical example we will reanalyse ruthenium vapour pressure measurements

reported by Carrera et al� ��CAR�WAL�� They were measured from �� to �� Kby use of weight loss recorded with a microbalance �Langmuir method�� Values of�G�

m�T ��H�m�T�� both for Ru�cr� and Ru�g� are taken from Hultgren et al� ��HUL�DES��

thermodynamic properties for the vapour phase in that report were obtained from statistical thermodynamic calculations� Values of �G�

m�cr� T ��H�m�cr� T�� G

�m�g� T ��H�

m�g� T��and

P�T are summarised in Table X�� Here

Pdenotes

X� �G�

m�cr� T ��H�m�cr� T�� G�

m�g� T ��H�m�g� T��

Carrera et al� ��CAR�WAL� reported a large number of vapour pressures but we willreanalyse only a few representative values� The results of these calculations are given inTable X�� pressures in atm are converted to bar and values of

P�T at each temperature T

are obtained graphically from a plot ofP�T against T � There is no trend in the calculated

��


Table X�� Free energy functions needed for the evaluation of sublimation data of ruthenium� From Ref� ��HUL�DES��

T G�m�cr� T �H�

m�cr� T� G�m�g� T �H�

m�g� T�P�T

�K � kJ �mol�� kJ �mol�� kJ �K��mol��

� ��

Table X�� Ruthenium vapour pressure data ��CAR�WAL� and standard sublimationenthalpies calculated with the thirdlaw extrapolation method�

T p �R ln pP�T �subH

�m�T�

�K �bar �kJ �K��mol�� kJ �K��

�mol�� kJ �mol��

��

values of subH�m�T�� with the temperature of measurement� They yield an average value

of subH�m�T�� kJ � mol�� Carrera et al� ��CAR�WAL� reported a

thirdlaw standard enthalpy of sublimation of subH�m�T�� kJ � mol��

based on an analysis of all �� of their most reliable vapour pressures� Our value ofsubH

�m�T�� kJ �mol�� from analysis of a �ve point subset of their vapour

pressures is in good agreement�

Since subH�m�T�� fH

�m�g T�� fH

�m�cr T�� and fH

�m�cr T�� we then

have a value of fH�m�Ru g T�� This can be combined with the calculated statistical

thermodynamic entropy of Ru�g� to yield a calculated value of fG�m�Ru g T��

��

Third�law method

X�� Evaluation from high�temperature data

There are some systems for which lowtemperature heat capacities �and thus entropies�have not been measured but for which hightemperature thermal and Gibbs energy results are available� These can be analysed by a variant of the thirdlaw method to yieldapproximate values of rH

�m�T�� and rS

�m�T��

Our fundamental equation for the thirdlaw analysis Eq� �X�� can be rearrangedinto the form

rG�m�T �

T� rH

�m�T��

T� �rS

�m�T�� rS

�m�T ��rS

�m�T��

�rH

�m�T ��rH

�m�T��

T�

where rG�m�T � is known and both �rS

�m�T � � rS

�m�T�� and �rH

�m�T � � rH

�m�T��

can be calculated from hightemperature thermal �relative enthalpy� results� This equation has two unknowns rS

�m�T�� and rH

�m�T�� One approach is to use a leastsquares

�t to obtain values of rS�m�T�� and of rH

�m�T�� from all of the experimental tempera

tures� This would yield greater uncertainties than in the case discussed in Section X��As an example we will reanalyse the same vapour pressure data for ruthenium that

were used in the preceding section� Rearrangement of the last equation for the case ofsublimation yields�

subH�m�T��

T�subS

�m�T��

subG�m�T �

T��subH

�m�T ��subH

�m�T��

T

�

� �subS�m�T ��subS

�m�T��

� �R ln p�X ��

where�

X� � �

�subH

�m�T ��subH

�m�T��

T

�� subS

�m�T ��subS

�m�T��

Table X�� contains values ofP� at round values of the temperature which were cal

culated from the tabulated critically assessed thermodynamic values of Hultgren et al��HUL�DES��Table X�� contains the results obtained by reanalysis of the �ve selected vapour pres

sures� Values ofP� at each experimental temperature were obtained by graphical inter

polation from a plot ofP� against T �

By using a linear leastsquares analysis of the values in Table X�� we obtain�

subH�m�T��

T�subS

�m�T��

�� T

� ��

with a correlation coe�cient of �� Thus this calculation yields subS�m�T��

�� J�K�� mol�� and subH�m�T�� kJ�mol�� This value

��


Table X�� Thermodynamic values for the reanalysis of sublimation data for ruthenium�

T ��subH�m�T ��subH

�m�T��T ��subS

�m�T ��subS

�m�T��

P�

�K � J �K��mol�� J �K��

�mol�� J �K��mol��

� ��

Table X�� Recalculation of ruthenium vapour pressures ��CAR�WAL� by the thirdlawextrapolation method without �xing the entropies of Ru�cr� and Ru�g� at T� � ��K�

T �R ln pP�

�R ln p�P�

�K � J �K��mol�� J �K��

�mol�� J �K��mol��

��

of subH�m�T�� is in very good agreement with the value obtained by a standard third

law extrapolation �cf� Section X�� kJ � mol�� but is considerably moreuncertain� The larger uncertainty arises because two parameters are now being determined by the same �ve data points whereas in the standard thirdlaw extrapolation onlysubH

�m�T�� was determined� It is possible to reduce these large uncertainties somewhat

by using more of the published vapour pressures in the calculation� However the samebasic conclusion will still be reached i�e� that the third law calculation of a standardenthalpy of reaction will always yield signi�cantly larger uncertainties when no reliablevalues of the standard entropies are available than when the calculation is constrainedusing the standard entropies�

For the examples just given involving the sublimation of rutheniummetal both types ofthirdlaw extrapolations give values of subH

�m�T�� in good agreement� However consider

a case where either the vapour pressures or hightemperature calorimetric data have a

��

Third�law method

temperaturedependent systematic error �or if the calorimetric �data� were estimatedand these estimated values were systematically in error�� An analysis of such data by thestandard thirdlaw method as described in Section X�� would yield values of rH

�m�T��

that vary with the temperature of the measurements� It would thus be obvious that therewas an error in one or more of the input values for the calculations�In contrast if the calculations were done as described in the present section then the

errors due to certain types of systematic errors could be adsorbed into the coe�cientsof the linear �t� Consequently both subH

�m�T�� and subS

�m�T�� would be in error but

it would not be obvious to the person doing the calculations� The standard thirdlawmethod should be considered more trustworthy in most cases and it should be used whensu�cient calorimetric data are available�

X�� A brief comparison of enthalpies derived from the second and third�law methods

The starting point for a secondlaw calculation of the standard enthalpy of reaction fromGibbs energy of reaction data is


�m�T �� TrS

�m�T �� X��

where rH�m�T � is extracted from a linear or higherorder �t of the Gibbs energy of

reaction data as a function of temperature� If a linear �t is appropriate then the assessedvalue of rH

�m�T � refers to the mean temperature of the measurements Tav� The standard

enthalpy of reaction is then calculated from the integration of Eq� �X�� from T� to themean temperature of the hightemperature Gibbs energy of reaction measurements�

rH�m�T�� rH

�m�Tav��

Z Tav

T�rC

�p�m�T � dT

Carrera et al� ��CAR�WAL� have compared values of subH�m�T�� derived from second

and thirdlaw analyses of their vapour pressures� There is no point in repeating thosecalculations and we merely cite their results� They performed �ve series of measurementsand a separate analysis of each data set gave thirdlaw values of subH

�m�T�� ranging from

�� to �� kJ � mol�� In contrast their analysis by the secondlawmethod of these same �ve data sets gave values of subH

�m�T�� ranging from ��

to �� kJ � mol�� Quite clearly the thirdlaw based values are more preciseand more consistent than the secondlaw based values�Although thirdlaw based values of rH

�m�T�� are usually more precise and consistent

than secondlaw based values this better consistency is not in itself a proof that theoriginal Gibbs energy measurementswere completely accurate since there are certain typesof systematic errors that do not a�ect these consistency checks� Agreement of secondlawand thirdlaw based values of rH

�m�T�� within their �experimental� precision is usually

a better indication that the hightemperature Gibbs energy data and the correspondingcalorimetric data are of high quality�

��


X�� Estimation methods

Experimental or estimated values of C�p�m and S�

m are needed in order to calculate hightemperature equilibrium constants either with Eqs� �X�� or �X�� or with Eq� �X��which assumes a temperature independent heat capacity of reaction�� The same situationapplies to the electrostatic models described in Section X��Therefore estimation methods for C�

p�m�T � and S�m�T � will be described here� For a

broader presentation of thermodynamic estimation techniques the reader is referred tothe discussions in references ��NOR�MUN Section �� KUB�ALC Chapter �� LEW�RAN pp�� and ��LAT Appendix III��Recently heat capacities have been measured for several aqueous electrolytes and non

electrolytes to about �� K� See the article by Wood et al� ��WOO�CAR� for referencesto many of the original studies� Since these measurements do not yet include the majorityof aqueous electrolytes and nonelectrolytes estimated values are still required for mostapplications�

X�� Estimation methods for heat capacities

X�� Heat capacity estimations for solid phases

Kopp�s rule of additivity ��NOR�MUN Eq� �� may be used to estimate the heatcapacity of a solid phase as the sum of the molar heat capacities of the elements present�

C�p�m �

Xi

�iC�p�m�i��

This rule is only valid for elements which are also solid in their standard state� However e�ective contributions can be used for elements that are not solid in the standard stateand adjustments can be made to the entropies of the solid light elements to improve theaccuracy of the estimated values� Sturtevant ��STU pp�� reported the followinge�ective atomic contributions �in units of J �K�� mol�� for the light elements� C ��H �� B �� Si �� O �� F �� P �� S �� and �� for the heavier atoms�Kubaschewski Alcock and Spencer ��KUB�ALC� reported a technique similar to that

of Latimer�s method for standard entropies �cf� Section X�� p�� to estimate heatcapacities of solids at �� K by adding the contributions from the cationic and anionic groups present in a solid phase� Cationic and anionic contributions are listed in��KUB�ALC their Tables IX and X�� Coe�cients for temperature functions of the type

C�p�m�T � � a� b T � c T��

may be estimated by their approach if the melting temperature of the solid phase is known��KUB�ALC their Eqs� �� and ��Parameter estimates for the same type of heat capacity function for oxide minerals were

made by Helgeson et al� ��HEL�DEL Eqs� �� and �� using the sum of the heatcapacities for the constituent oxides�

��

Estimation methods

For estimations on chalcogenides the discussion given by Mills should be consulted��MIL Section �� For a discussion of the chalcogenides of rare earths and actinides see Ref� ��MIL Section ��a��

X�� Heat capacity estimations for aqueous species

For standard partial molar heat capacities of aqueous ions the estimation methods aremainly of two types�

� Correlations between C�p�m�T�� and standard partial molar ionic entropies at

�� K� The method of Criss and Cobble ��CRI�COB ��CRI�COB�� will be described because of its historical interest� The equations of Helgeson and coworkers��HEL�KIR ��SHO�HEL ��SHO�SAS ��SVE�SHO� are modern correlations ofthis kind�

� Electrostatic models of ion hydration� These types of C�p�m�T � estimations are de

scribed as alternative temperature functions for the heat capacity �Section X��

The �rst type of estimation methods �correlations between C�p�m�T�� and ionic entropies

at �� K� are limited to a few ion types which do not yet include metal ligand complexes and which sometimes exclude neutral aqueous species� Therefore these methodsare not always useful unless further assumptions are made� For example Baes and Mesmer ��BAE�MES� assigned partial molar heat capacities for a mononuclear hydrolysisproduct equal to that of another metal cation having the same ionic radius as that of theunhydrolysed cation of interest and having the same charge as the hydrolysis product ofinterest� Another approximation for metalligand complexes which was used by Lemireand Tremaine ��LEM�TRE� is to use the correlation parameters for simple cations andanions�

X�� Criss and Cobble s method

The method developed by Criss and Cobble ��CRI�COB ��CRI�COB�� is of interestbecause it was widely used during the ��s and there are many publications whichhave used it� It has however several associated problems which are discussed below andnowadays it has been largely superseded by the correlation equations of Helgeson andcoworkers described in Section X��Criss and Cobble ��CRI�COB ��CRI�COB�� observed that by assigning a speci�c

value to the standard partial molar ionic entropy of H� at each temperature for simpleions a linear correspondence could be obtained between the standard partial molar ionicentropies at �� K and at other temperatures

S��absm �i� T � � a�T � � b�T �S��abs

m �i� T�� X��

where a�T � and b�T � are temperature dependent parameters which di�er for di�erent ion types �i�e� for simple cations simple anions oxyanions and acid oxyanions�

��


and S��absm �i� T � are �absolute� partial molar entropies �referred to a speci�c value for

S��absm �H�� aq� T �� At �� K the optimum value of S��abs

m �H�� aq� T�� was found to beequal to �� J �K�� mol�� CRI�COB ��CRI�COB�� and therefore the relationshipbetween �absolute� and conventional entropies at �� K is

S�m�i� T�� S��abs

m �i� T�� ziS��absm �H�� aq� T��

� S��absm �i� T�� zi� �X��

where zi is the charge of the ion i�A related type of correlation was given by Couture and Laidler several years earlier

��COU�LAI� in which S�m�T�� was related to the mass of the ion and a term in the

inverse of the ionic radius �of the same basic form as the Born solvation term�� In thatstudy S��abs

m �H�� aq� T�� J � K�� mol�� which is similar to Criss and Cobble�svalue ��CRI�COB ��CRI�COB��Criss and Cobble ��CRI�COB� found that one set of a�T � and b�T � was appropriate for

simple cations another for simple anions a third set for oxyanions and a fourth for theiroxyacids� They tabulated values of a�T � b�T � and S��abs

m �H�� aq� T � at �� and �� K�The practical importance of Eq� �X�� is that ionic partial molar heat capacities can

be estimated from the ionic partial molar entropies of these ions averaged between twotemperatures� Criss and Cobble ��CRI�COB�� calculated �cf� our Eq� �X��

C��absp�m jTT��i� �

S��absm �i� T �� S��abs

m �i� T��

ln�T�T�� X��

in which case the �constant rC�p�m� equations Eqs� �X�� and �X�� may be used to

calculate equilibrium constants at higher temperatures� By substituting Eq� �X�� intoEq� �X�� Criss and Cobble obtained average �absolute� heat capacities�

C��absp�m jTT��i� �

a�T �� b�T ��S��absp�m �i� T��

ln�T�T��X��

� ��T � � ��T �S��absp�m �i� T��

where

C��absp�m �i� T � � C�

p�m�i� T � � ziC��absp�m �H�� aq� T � �X��

with

C��absp�m �H�� aq� T�� J �K�� mol��

Two extra digits were retained from the conversion of �� cal �K�� mol�� to this value�An alternative procedure is to estimate values for conventional standard partial molar

ionic entropies at higher temperatures �Eqs� �X�� and �X�� and to do a leastsquares�t on these values to the constant heat capacity equation�

S�m�i� T � � S�

m�i� T�� C�p�mjTT��i� ln�T�T�� X��

��

Estimation methods

Table X�� Heat capacity parameters for Eq� �X�� Units of A are J � K�� mol�� whereas B is unitless�

Species A B

Cations �� Anions including OH� �� Oxyanions �� Acid oxyanions ��

This method was used for example by Lemire and Tremaine ��LEM�TRE��Similarly to Eq� �X�� Criss and Cobble found a linear correlation between �absolute�

standard partial molar ionic heat capacities and �absolute� standard partial molar ionicentropies at �� K

C��absp�m �i� T�� A�BS��abs

m �i� T��

� A�B�S�m�i� T�� zi��

and using Eq� �X��

C�p�m�i� T�� A�BS�

m�i� T�� zi��B � �� X��

The last equation relates conventional standard partial molar ionic heat capacities toconventional standard partial molar ionic entropies and to the electrical charge of aqueousionic species� Criss and Cobble ��CRI�COB�� give the A and B parameters listed in ourTable X��For temperatures above �� K Criss and Cobble ��CRI�COB�� noted that a�T � and

b�T � in Eqs� �X�� and �X�� could each be assumed to be linear functions of thetemperature y�

a�T � � a� � b�T

b�T � � a� � b�T�

However these equations imply that the standard partial molar ionic heat capacities arealso approximately linear functions of the temperature at T � �� K ��CRI�COB� ��LEW ��TAY� which is in direct disagreement with experimental evidence which shows

y Tremaine et al� ��TRE�MAS� note that the value of a�T � at �� K for simple anions including OH�

should be �� cal �mol��K�� instead of the value of �� cal �mol��K�� given in ��CRI�COB�which is presumably a misprint�

��


that they go through a maximumas a function of temperature and decrease asymptoticallytowards �� at the critical point of water �see for example Figure � in Ref� ��PAT�SLO��For this reason the a�T � and b�T � parameters of Criss and Cobble are not recommendedfor use above �� K�

Note that Criss and Cobble ��CRI�COB�� recommended these linear equations for�� K T �� K and Taylor ��TAY� for T �� K�

Criss and Cobble�s equation our Eq� �X�� has been criticised by Shock and Helgeson��SHO�HEL� since it does not agree very well with some more recent experimental datafor M�� and some M�� especially for aqueous ions whose S�

m�T�� are less than about�� J �K�� mol��

X�� Isocoulombic method

For neutral aqueous species and metal complexes the isocoulombic approach �Section X�� might be used to estimate a value for their C�

p�m�i�� For example for theestimation of the heat capacity for AmCO�

� the isoelectric reaction

Am�� HCO��

�� AmCO�� H

�

can be combined with the reactions

Am�OH�� H� �� Am�� H�O�l�

and

H�O�l� �� H� �OH�

to yield the isocoulombic reaction

Am�OH�� HCO��

�� AmCO�� OH

� �H�O�l��

If the assumption is made that rC�p�m � � for this last reaction then

� � C�p�m�AmCO

�� aq� T�� C�

p�m�OH�� aq� T�� C�

p�m�H�O� l� T��

� C�p�m�HCO

�� aq� T�� C�

p�m�Am�OH�� aq� T��

As the heat capacities of H�O�l� OH� and HCO�� have been determined experimentally

and as the heat capacity for the hydrolysis product Am�OH�� can be estimated withCriss and Cobble�s method our Eq� �X�� using the parameters for cations and anestimated entropy for Am�OH�� the heat capacity for AmCO

�� can then be estimated

directly without a need to estimate its ionic entropy or to make further assumptions�

��

Estimation methods

X�� Other correlation methods

There are not many alternatives to Eq� �X�� which was derived from equations proposedby Criss and Cobble ��CRI�COB�� Helgeson and coworkers presented correlations forsimilar ions with equal charge cf� ��HEL�KIR Eq� �� and ��SHO�HEL Eq� ��

C�p�m�i� T�� a� bS�

m�i� T�� X��

where the parameters a and b are tabulated by Shock and Helgeson ��SHO�HEL�� Thesevalues should only be used with the same S�

m�i� T�� used by Shock and Helgeson to derivethem�It seems rather strange to us that the parameters of Shock and Helgeson ��SHO�HEL�

for the light and heavy tripositive rare earth ions should di�er so greatly� These largedi�erences may be a computational artifact because C�

p�m�T�� was poorly determined atthat time for those M�� ions� We note that Shock and Helgeson ��SHO�HEL� derivedvalues for the standard partial molar ionic heat capacities for the aqueous tripositive rareearth ions in the range �� to �� J � K�� mol�� with values for the light rareearths generally being more negative than those for the heavy� However a reanalysis byRard ��RAR� of all the available heat capacities for aqueous rare earth chlorides andperchlorates gave values for the ionic heat capacities C�

p�m�T�� that ranged from �� to�� J�K��mol�� for the lighter rare earth ions and from�� to �� J�K��mol�� for theheavier rare earths� In addition Xiao and Tremaine ��XIA�TRE� reported experimentalvalues of C�

p�m�T�� J �K�� mol�� for La�� and �� J �K�� mol�� for Gd�� which di�er signi�cantly from Shock and Helgeson�s values of �� and �� J � K�� mol�� respectively� Di�erences at some other temperatures are even larger� Thus thevalues of C�

p�m�T�� reported by Shock and Helgeson are probably too negative by about�� to �� J � K�� mol�� or more� Consequently Eq� �X�� should not be used for M��

ions��

Modi�ed equations of this type were proposed by Khodakovskiy ��KHO�

C�p�m�i� T�� a� djzij � �

�S�m�i� T��

and

S�m�i� T � �

a�T � T��

T�� d�T � T��

T�jzij

� �� T � T��S�m�i� T��

for conventional standard state properties� He found that one set of coe�cients workedwell for cations a di�erent set worked well for nonoxygenated anions �mainly halides�and unionised dissolved gases �Ar Kr N� O� H� H�S� and a third set worked well foroxyacids and their oxyanions� These equations have the advantage of working for somedissolved gases also� Khodakovskiy�s recommended values of a and d for each of thesethree cases are given in Table X�� Note added in press� Recently Shock et al� ��SHO�SAS� have re�evaluated the parameters for

Eq� �X�� excluding rare earth cations�

��


Table X�� Parameters for Khodakovskiy�s equations ��KHO�� Units of a and d areJ �K�� mol��

Cases a d

Cations �� Non�oxygenated anions

and dissolved gases �� Oxyacids and oxyanions ��

Unfortunately these various correlations do not yet include inorganic complexes and therefore extra assumptions are needed to use them for reactions that involve metalcomplexes� Some preliminary work in this area was done by Cobble ��COB�� for halide cyanide and a few other complexes by using �hydration corrected� entropies�

Another method proposed is a heat capacity correlation among redox couples��JAC�HEL��

C�p�m�oxd��T �� C�

p�m�red��T �

zoxd�� zred��

C�p�m�oxd��T �� C�

p�m�red��T �

zoxd�� zred��

If the heat capacity temperature functions for a redox couple are known �say Fe��Fe��and C�

p�m�T � for a member of another couple is known then this method allows the estimation of the unknown temperature variation for the heat capacity of the other memberof the second redox couple� However the reliability of this estimation method has beentested for only a few aqueous systems�

X�� Heat capacity estimation methods for reactions in aqueous solutions

The RyzhenkoBryzgalin model which is described in Section X�� may be used to estimate rC

�p�m cf� Eq� �X�� and for �isocoulombic� reactions rC

�p�m may be estimated

to be equal to zero as discussed in Sections X�� and X��

Smith et al� ��SMI�POP� have proposed a set of average values of rC�p�m that may be

used for the estimation of this quantity for proton dissociation reactions of acid oxyanions�

Sverjensky ��SVE� has proposed correlations between the rC�p�m for metal complexa

tion reactions with halide and hydroxide ions and the value of C�p�m for the metal cation�

�

� Note added in press� Recently Sverjensky� Shock and Helgeson ��SVE�SHO� have proposed correla�tions for �rC

�p�m for complex formation reactions between divalent cations and monovalent ligands�

��

Estimation methods

X�� Entropy estimation methods

X�� Entropy estimation methods for solid phases

For ionic compounds Latimer�s method ��LAT� involves estimating entropies by addingempirically estimated ionic contributions� Based on later experimental data Naumov etal� ��NAU�RYZ Tables I� and I�� prepared a revised table of parameters to be usedwith Latimer�s method�

Langmuir ��LAN� described improved parameters for estimating entropy values forsolid compounds containing the UO��

� moiety� In his procedure the contributions ofUO��

� to the entropy of UO� �UO��OH�� and schoepite were used to estimate thecontribution of UO��

� to the entropies of other uranium compounds�

Although Latimer ��LAT� and Naumov et al� ��NAU�RYZ� suggested a value of�� J � K�� mol�� for the entropy of each water of hydration in a crystalline solidhydrate Langmuir ��LAN� considered the value of �� J �K�� mol�� the value for theentropy of ice I from the compilation of Robie and Waldbaum ��ROB�WAL� to be moreappropriate�

Ionic contributions to the entropy also have slightly di�erent values for the di�erentsources of parameters for Latimer�s method� Thus the entropies in the literature calculated by Latimer�s method may vary signi�cantly depending on the exact set of parameters used in the estimation� Table X�� presents the parameter values selected for theNEA�s uranium review ��GRE�FUG��

Also for some compounds especially binary solids better entropy values may be estimated by comparison with values for closely related solids than by Latimer�s method� Forchalcogenides the reader is referred to the discussion given by Mills ��MIL Section ��

Helgeson and coworkers ��HEL�DEL� gave entropy estimation techniques foroxide minerals using the sum of the entropies for the constituent oxides��HEL�DEL their Eqs� �� and ��

For some elements and ions including most of the rare earths and actinides the metalsand ions have magnetic contributions to their entropies that depend mainly on theirground state electronic con�gurations� The S�

mag values range from �� to �� J �K�� mol�� for various rare earth and actinide ionsy M�� M�� and M�� at �� K� Valuesfor the total entropy S�

m�T�� vary by �� J � K�� mol�� for RCl��cr� across the series for R�O� by �� J � K�� mol�� J � K�� mol�� per rare earth ion� and for R�� by�� J � K�� mol�� SPE�RAR ��RAR�� It is clear that S�

mag is a signi�cant fractionof the total variation of S�

m�T�� across the rare earth series and it must be taken intoconsideration for any accurate scheme for estimating S�

m�T�� At �� K there are threedi�erent structural types of RCl��cr� and three of R�O��cr� for the rare earth series sothere are also structural factors that a�ect series trends�

For lanthanides and actinides and their cations the �f or �f electrons are shieldedby their outer electrons from the electric �elds of their neighbouring ions so their or

y �M�� R� and �AN� are used as general abbreviations for metal� rare earth �lanthanum and thelanthanides�� and actinide respectively�

��


Table X�� Contributions to entropies of solids � J � K�� mol�� mainly fromRefs� ��NAU�RYZ� and ��GRE�FUG�� The values in parentheses were not directly basedon experimental values but were estimated by Naumov Ryzhenko and Khodakovsky��NAU�RYZ��

Anion Average cation charge

��

OH� �� O�� F� �� Cl� �� Br� �� I� �� IO�

� �� CO��

� �� NO�

� �� SO��

� �� a

SO��

PO�� a

PO�� a

HPO��

H�O ��UO��

�b ��

Uc ��

�a� Estimated by ��GRE�FUG��b� Treated as a dipositive ion for the purpose of

selecting anion contributions��c� For uranium compounds not containing

UVIO��

��

Estimation methods

bital angular momentum is not quenched and RussellSaunders coupling applies� thusthe J quantum number remains a valid measure of total angular momentum� If this istrue the degeneracies of the lanthanide and actinide cations are not altered signi�cantlyby their ionic environments and thus the J retain approximately their freeion values�Consequently the magnetic �electronic� entropies are then fully developed by room temperature and take the values S�

mag � R ln��J � �� For a few cases �e�g� Sm�� and Eu��S�mag also contains small contributions from lowlying excited electronic levels that canbecome thermally occupied at room temperature� Hinchey and Cobble ��HIN�COB��noted that excited level term contributions add an extra �� J � K�� mol�� to S�

mag�T��for Sm�� and �� J �K�� mol�� for Eu�� Table X�� contains a listing of S�

mag for variousM�� M�� and M��The electronic levels contribute to the total heat capacities through the Schottky heat

capacity term CSch�T � see Ref� ��WES�� It is mathematically related to the Einsteinheat capacity function�For most lanthanide and actinide ions the maximum in CSch�T � appears at a tem

perature around �� to �� K and this contribution to C�p�m�T � becomes fairly small at

temperatures near �� K� For systems with lowlying excited electronic levels that canbecome thermally occupied this is no longer true� For example CSch�T � for EuCl��cr�is essentially zero up to about �� K it increases to slightly greater than R up to �� K and then it very slowly decreases at higher temperatures ��WES�� Hinchey and Cobble��HIN�COB�� noted that CSch�T�� J �K�� mol�� for Sm�� and �� J �K�� mol��for Eu�� Schottky contributions to the heat capacities are still signi�cant at T� for somerare earth sesquioxides ��WES��Westrum ��WES� has shown that for a particular type of solid containing a tripositive

lanthanide ion �RCl� R�O� R�OH�� R�S�� values of the �lattice entropy� S�lat�T��

S�m�T�� S�

mag show a very smooth variation with the molar volume for an isostructuralseries of compounds with the same anion� These molar volumes can be calculated fromcrystallographic unit cell parameters� Note that each series of RX� or R�X� �where �X�represents either a halogen a chalcogen or an OHgroup� undergoes one or more structuralchanges in going from light to heavy rare earths except for a few cases like R�OH�� thesmooth variation of S�

lat�T�� with the molar volume applies only for compounds of thesame stoichiometry with a common structure�When S�

mag is added to interpolated values of S�lat the resulting S

�m�T�� are generally

highly accurate frequently to within �� J �K�� mol�� of experimental values for the rareearth compounds� The uncertainties in estimated values may be much greater for manyactinide compounds where available thermodynamic data are much less complete�

X�� Entropy estimation methods for aqueous species

Several methods are available to estimate entropies of aqueous species and entropies ofreaction at �� K� These methods use correlations between ionic entropies and a combination of crystallographic radii molar volumes and mass electrical charge etc�For some aqueous ions including most of the rare earths and actinides there are mag

��


Table X�� Magnetic �electronic� contributions to entropies at �� K�

f or �f Ground S�mag

b Ions

con�gu� state a � J �K��mol��

ration

f� �S� �� La�� Ce�� Th�� Pa�� U��UO�� Ac��

f� �F�� Ce�� Th�� Pa�� U�� Np�� Pu��

f� �H� � �� Pr�� U�� Np�� Pu�� Pa��

f� �I�� Nd�� U�� Np�� Pu��

f� �I� � �� Pm�� Np�� Pu��

f� �H�� Pu�� Am��

f� �H�� Sm��

f� �F� �� Sm�� Am�� Cm��

f� �F� �� Eu��

f� �S�� Eu�� Gd�� Tb�� Am�� Cm�� Bk��

f� �F� �� Tb�� Bk�� Cf��

f� �H�� Dy�� Cf��

f�� I� �� Dy�� Ho�� Es��

f�� I�� Er�� Es�� Fm��

f�� H� �� Tm�� Fm��

f�� F�� Tm�� Yb�� Md��

f�� S� �� Yb�� Lu�� No��

� �S� �� Y��

�a Term symbols were taken from Refs� ��HIN�COB� ��HIN�COB�� EDE�GOF��

�b The S�mag were calculated from S�

mag � R ln��J � �� Values for S�mag for Sm�� and

Eu�� contain contributions from thermally�populated higher electronic states� Similarcontributions may be present for some of the actinide ions�

��

Estimation methods

netic contributions to their entropies that depend mainly on the electronic con�gurationof their ground state �cf� Section X�� p��Hinchey and Cobble ��HIN�COB�� calculated the ionic entropies of the aqueous tri

positive rare earth ions �lanthanides and yttrium� from available thermodynamic data forhydrated rare earth chlorides� These data include lowtemperature heat capacities for thecrystals and enthalpies and Gibbs free energies of solution� They found that �S�

m� S�mag�

was a linear function of r�� within the scatter of the then available values� Here r is thecrystal radius for rare earth ions for a CNy of � and S�

mag is the magnetic contribution described in Section X�� and listed in Table X�� However they had to estimate Gibbsfree energies of solution for seven of the hydrated salts and entropies for six of them� Inaddition some of the experimental enthalpies of solution later proved to be inaccurate�Spedding Rard and Habenschuss ��SPE�RAR� recalculated these ionic entropies andincluded more complete and accurate data that were published after Hinchey and Cobble�s report� Spedding Rard and Habenschuss ��SPE�RAR� found that �S�

m�T�� S�mag�

actually had a tilted S shape as a function of r�� and this S shape correlated fairly wellwith the overall hydration of the rare earth ions�Powell and Latimer ��POW�LAT� found that �S�

m� ��R lnM� was a linear function of

r�� for aqueous ions of various stoichiometries whereas Couture and Laidler ��COU�LAI�found that �S��abs

m �T�� R lnM� was a linear function of z��Nradj� for oxyanions where

M is the mass of the oxyanion under consideration N the number of coordinated oxygenatoms present in this anion and radj the covalent radius of the central atom plus thevan der Waals radius of an oxygen� Here S��abs

m �T�� S�m�T�� jzj� changed the

conventional scale to an �absolute� scale by using �� J � K�� mol�� for the �absoluteentropy� of H�� Related equations are those of Cobble ��COB ��COB�� and �withoutthe mass term� Cobble ��COB�� Powell ��POW� Helgeson et al� ��HEL Eq� �� SHO�HEL Eq� �� HEL�KIR Eq� �� JAC�HEL Eqs� �� and �� and also Ruaya ��RUA Eq� �� Sassani and Shock ��SAS�SHO� and Shock et al��SHO�SAS��The importance of Powell and Latimer�s ��POW�LAT� and Couture and Laidler�s

��COU�LAI� work is that they showed that entropies of very di�erent types of ionscould be correlated using ionic mass charge type and simple structural features� The��R lnM term is derived from the statistical thermodynamic calculation of the absoluteentropy of a gas phase ion� An entropy of an aqueous ion can be considered as the sumof its gas phase entropy and the entropy of hydration of that ion by liquid water� Theentropy of hydration should mainly depend on the sign and charge of the ion and theionic radius� The presence of the �

�R lnM is then justi�ed by assuming this is a gas phasefeature that is not lost when the ion becomes hydrated in aqueous solutions�The same type of mass dependence was included by David ��DAV ��DAV�� in his

comprehensive analysis of ionic entropies of the aqueous tripositive lanthanide and actinideions� David ��DAV his Fig� �� found that the adjusted entropy

S�adj�T�� S�

m�T�� S�mag � �

�R lnM

y CN is used as an abbreviation for �coordination number��

��


was a simple and symmetrical function of the CN � � tripositive rare earth crystalradii� This curve was Sshaped in agreement with the �ndings of Spedding Rard andHabenschuss ��SPE�RAR� and that shape was understandable in terms of changes oftotal hydration of the rare earth ions as a function of the ionic radius� David also notedthat the hydrated radii of actinide ions AN�� were essentially proportional to those forrare earths R�� and thus the curve for rare earths can be used to estimate S�

adj�T��for AN�� with a fairly high degree of con�dence� An experimental value of S�

adj�T�� isavailable only for Pu�� among the actinides� although it falls slightly o� the curve basedupon values for the for R�� it does agree with its interpolated position on the S�

adj�T��curve for AN�� within its uncertainty limits� These estimated values of S�

adj�T�� for AN��

were then used by David for calculation of his S�m�T�� values�

David ��DAV ��DAV�� also estimated S�m�T�� values for all possible R

�� AN�� R�� and AN�� The dipositive ion values were based on S�

adj�T�� for Ca�� and Sr�� with CN �

� and those for the tetrapositive ions were based on Ce�� and Th�� with CN � �� Sincethere are insu�cient data to establish series trends as a function of ionic radii in thesecases David�s estimated entropies have much greater uncertainties than for the tripositiveR�� and AN�� ions�Powell and Latimer ��POW�LAT� found that the entropies of a variety of non

electrolytes in aqueous solution could be represented by the equation

S�m�T��

�R lnM � S�int � �� Vm�

where Vm is the molar volume of the nonelectrolyte in cm� � mol�� for its pure liquidstate and S�

int is the �internal� contribution to the entropy as calculated for an idealgas using statistical thermodynamic methods for rotational vibrational and electroniccontributions� In general nonpolar inorganic gases �inert gases halogens O�� a fewpolar gases �H�O HF NO N�O COS� and some saturated alkanes �CH� C�H�� werewell represented by this equation with deviations of only � to � J�K�� mol�� However anumber of other polar gases and liquids �H�S CO SO� CO� NH� CH�OH C�H�OH� andalso N� showed deviations of �� to �� J �K�� mol�� Cobble ��COB�� gave an extendedversion of this equation which is valid for small and medium sized organic molecules butit fails for larger molecules� A more general and accurate correlation for nonelectrolytesis certainly needed�An alternative to the Powell and Latimer equation has been proposed by Laidler

��LAI� Eqs� �X�� and �X�� below� The relative merits of this and Powell and Latimertype of equations have been discussed by Scott and Hughus ��SCO�HUG� and Laidler��LAI�� The following two empirical equations have been proposed for the estimation ofthe partial molar entropy of ions in aqueous solution

S��absm �i� T��

�

�R lnMi � �� z

�i

ru�i�X��

and

S��absm �i� T��

�

�R lnMi � �� zi

r�e��i�X��

��

Estimation methods

where ru is Pauling�s univalent radius ��PAU Table �� converted from units of A tometre and re� is an e�ective ion radius� The �ordinary� and absolute entropy scalesdi�er by the assigned values of the entropy of H� which are �� and �� J �K�� mol�� respectively� Hence these two entropies for an ion i of charge zi are related by the equation

S��absm �i� T�� S�

m�i� T�� zi �X��

The equations above are able to describe the absolute ion entropies within �� to ��J � K�� mol�� From the discussion given in ��SCO�HUG� and ��LAI� it is clearthat the theoretical foundations of both the Powell and Latimer and the Laidler type ofequations are rather weak� However they provide some guidelines for entropy estimations�For aqueous complex ions of uranium Langmuir ��LAN Figure �� gave a simple cor

relation between S�m�i� T�� and the ionic charge zi� The parameters for this correlation

were obtained by �tting data to a �th degree polynomial of zi �see also ��LAN�HER Fig� �� for Th�IV� complexes��For halide and hydroxide complexes Helgeson gave two equations that relate the en

tropy of a complex with either crystallographic radii only ��HEL�KIR Eq� �� forneutral complexes or e�ective electrostatic radii ��SHO�HEL Eq� �� for ionic species� or in combination with the stoichiometry of the complex see Ref� ��JAC�HEL Eqs� ��and �� and also ��RUA Eq� ��

The following methods are available to estimate standard reaction entropies�

� For hydrolysis reactions Baes and Mesmer gave correlations for the reaction entropyfor the �rst mononuclear hydrolysis product ��BAE�MES their Eq� �� for thepolynuclear products �their Eq� �� and for stepwise mononuclear hydrolysis �theirEq� ��

� Entropies of dissociation for a group of aqueous metal complexes �halide etc�� maybe estimated with equations similar to that given by Helgeson for charged andneutral metal chloride complexes see Ref� ��HEL Eq� �� and Ref� ��HEL�KIR Eq� �� for neutral complexes��

� Electrostatic models of complex formation were compared by Langmuir��LAN�� The simplest model ��LAN Eq� �� was used for making crude estimations of the entropy of formation of sulphate and �uoride complexes cf� ��LAN Figures �� and ��

� The RyzhenkoBryzgalin model as described in Section X�� cf� Eq� �X�� Sverjensky ��SVE� has proposed correlations between the rS

�m for metal complex

ation reactions with halide ions and the values of S�m for the metal cation and the

ligand�� Note added in press� Sverjensky� Shock and Helgeson ��SVE�SHO� have recently proposed correla�

tions for �rS�m for complex formation reactions involving monovalent ligands as well as SO��

� andCO��

� �

��


X�� Examples

Criss and Cobble�s publications ��CRI�COB ��CRI�COB�� appeared before the heatcapacity for Co�� and Th�� had been determined experimentally and therefore these ionsare suitable to do test calculations with Eqs� �X�� and �X�� For Co�� the standardpartial molar entropy given by NBS �� J �K�� mol�� WAG�EVA� is used here�

Reference C�p�m�Co

�� aq �� K�� J �K�� mol��

��SPI�SIN� �experimental� �� to ��Eq� �X�� Eq� �X��

For Th�� the NBS tables ��WAG�EVA� give an entropy of �� J � K�� mol��For this example Eq� �X�� is not applicable since there are insu�cient data for M��

ions to determine values of a and b� The estimation with Eq� �X�� gives an ionic heatcapacity for Th�� of �� J � K�� mol�� as compared with the experimental value of�� J �K�� mol�� MOR�MCC��These results give some indication on the accuracy expected from these kinds of esti

mations�A more practical example is to predict hightemperature equilibrium constants for a

reaction such as

U�� H�O�l� �� UOH�� H�� X��

The entropy and heat capacity for water at �� K are taken from CODATA��COX�WAG� and the corresponding values for U�� from the NEA�TDB review��GRE�FUG� while the values for H� are zero by the hydrogen ion convention� Therefore only the standard partial molar entropy and heat capacity at �� K for UOH��

will be estimated here�Langmuir�s equation ��LAN p�� gives an estimate of �� J � K�� mol�� for the

standard partial molar ionic entropy of uranium species with an electrical charge of ��In contrast Shock and Helgeson�s equation which ignores magnetic contributions to theentropy ��SHO�HEL Eq� �� using a ionic radius of ��m for UOH�� equalto that of U�� gives an estimate of �� J �K�� mol��In order to use Eq� �� in Ref� ��BAE�MES� the U�O distance of

�� m given in ��BAE�MES Table �� may be used� The logarithm of theequilibrium constant is also needed and the recommended value in the NEA�TDB uranium report will be used log��K

�� X�� K� � �� GRE�FUG�� This

��

Estimation methods

results in an entropy of reaction of �� J � K�� mol�� Finally adding the entropyvalues for H�O�l� and U�� the value of �� J �K�� mol�� is obtained for the estimateof S�

m�UOH�� aq� �� K��

Thus the estimated values for the standard partial molar entropy of UOH�� are�

Reference�Method S�m�UOH

�� aq� T�� J �K�� mol��

��LAN p�� BAE�MES their Eq� �� SHO�HEL their Eq� ��

To estimate the standard partial molar heat capacity for UOH�� with Criss and Cobble�sequation our Eq� �X�� the standard partial molar ionic entropy is needed� The valueof �� J �K�� mol�� will be used as a �selected estimation�� The derived heat capacityis then �� J �K�� mol�� The change in heat capacity of Reaction �X�� is thereforeestimated at �� J �K�� mol�� This small calculated heat capacity of reaction agreeswith our discussion on isoelectric reactions �cf� Sections X�� p��A comparison between experimental values for the equilibrium constant of Reac

tion �X�� NIK� and the calculated hightemperature values with Eq� �X�� i�e� constant rC

�p�m� is given in Figure X��

The solubility reaction

Am�OH��cr� � �H� �� Am�� H�O�l� �X��

is used as an example for an equilibrium involving a solid phase� As before the propertiesof water are taken from CODATA ��COX�WAG�� For the entropy of Am�OH��cr� themethod of Latimer ��LAT� will be used with the following entropy contributions� Am�� J �K�� mol�� estimated from ��LAT Appendix III Figure �� OH� with �� cation�� J �K�� mol�� NAU�RYZ Table I�� The magnetic contribution to S�

m is zero inthis case� Therefore S�

m�Am�OH�� cr T�� J �K�� mol��For the estimation of the heat capacity of Am�OH��cr� the method described in

Chapter � of Ref� ��KUB�ALC� will be used� The contribution to the heat capacity forAm�� is estimated as �� J �K�� mol�� from the trend between Th and U of Table IX inRef� ��KUB�ALC�� and that of OH� in a solid compound is given as �� J�K�� mol��KUB�ALC Table X�� The estimated value is therefore C�

p�m�Am�OH�� cr� T�� J�K�� mol��The standard partial molar entropy of Am�� was estimated as �� J�K��mol��

by Fuger and Oetting ��FUG�OET�� Using this value the standard partial molarheat capacity estimated by Criss and Cobble�s equation our Eq� �X�� is thenC�p�m�Am

�� aq� T�� J �K�� mol�� Shock and Helgeson�s equation our Eq� �X��

��


Figure X�� Equilibrium constants from Nikolaeva ��NIK� for Reaction �X�� U�� H�O�l�� UOH��H� compared with the calculated values using the �constant rC

�p�m�

equation Eq� �X�� with log��K��T�� and the estimated values �see text� of

rC�p�m � �� J �K�� mol�� and rS

�m�T�� J �K�� mol��

��

��

��

��

��

��

��

��

��

� ��

log��K�

T�K

�� NIK� e

e

e

e

e

e

e

Eq� �X��

��SHO�HEL� with the correlation parameters for the heavy rare earths would result inC�p�m�Am

�� aq�T�� J �K�� mol�� instead� However this value has no experimental or theoretical basis and the values of C�

p�m�R�� aq� T�� for rare earth ions derived by

Shock and Helgeson ��SHO�HEL� seem to be much too negative �cf� the comments onC�p�m estimation methods for aqueous species in p�� Section X�� consequentlythe heat capacity obtained with the method of Criss and Cobble will be used here�Therefore for Reaction �X�� the following values are estimated�

rS�m�T�� J �K�� mol��

rC�p�m�T�� J �K�� mol��

X�� Concluding remarks

Secondlaw extrapolation procedures must be used with caution in the absence of experimental heat capacities� When �tting hightemperature equilibrium constants morethan one equation should be tested �for example both the �constant rC

�p�m� equation

Eq� �X�� and the Revised HelgesonKirkhamFlowers model Eq� �X�� and the resulting reaction properties obtained at �� K with di�erent methods �e�g� entropyand heat capacity� should be compared as should their uncertainties and the di�erencebetween the calculated and experimental equilibrium constants at all temperatures�

��

Acknowledgements

When extrapolating equilibrium constants to higher temperatures from lower temperature data several methods should also be tested and the results compared with eachother �cf� Figures X�� to X�� Comparison of the results from these alternative methodswill give a good estimate of the magnitude of the extrapolation error� This uncertainty which di�ers for di�erent methods of extrapolation gives increased uncertainty to thethermodynamic reaction properties at higher temperatures compared to �� K�A similar attitude should be adopted with estimated thermodynamic properties which

are used to make temperature extrapolations� If possible several estimation methodsshould be compared� It is unfortunate that there are only a few estimation methods ofC�p�m�i for aqueous species all of which have a limited �eld of application� This necessitates the use of less reliable methods like the DQUANT equation Eq� �X�� and theisocoulombic approach �cf� Section X�� The measurement of aqueous solution heatcapacities should be given a high priority�

X�� Acknowledgements

Thanks are due to Kenneth Jackson �Lawrence Livermore National Laboratory� for hismany suggestions and computer calculations which allowed us to check the numericalresults of some computer programs used in this work�We are grateful to Hans Wanner �Swiss Federal Nuclear Safety Inspectorate� for many

corrections and improvements to the manuscript� Robert Silva �Lawrence LivermoreNational Laboratory� is acknowledged for suggestions and comments� B�N� Ryzhenko�Vernadsky Institute of Geochemistry and Analytical Chemistry� has kindly providedphotocopies of a few pages of Ref� ��NIK�� We thank Kaname Miyahara �PNC Japan�for pointing out a few misprints� We are specially grateful to Peter Tremaine �MemorialUniversity Newfoundland� for his critical review of our draft manuscript�Major support for the contributions of Joseph A� Rard was from the O�ce of Basic

Energy Sciences Division of Geosciences and the O�ce of Environmental Restorationand Waste Management �EM �� of the U� S� Department of Energy� His contributionwas performed under the auspices of the U� S� Department of Energy by the LawrenceLivermore National Laboratory under contract No� W��ENG��

��

Documents

Extract from MODELLING IN AQUATIC CHEMISTRY … · (OECD Publications, 1997, 724 pp., ISBN 92-64-15569-4) Scientific Editors: ... Andreas Jakob, Theo Karapiperis, An drey V. Plyasunov,