INGMAR GRENTHE* Royal Institute of Technology,Stockholm, Sweden
HANS WANNER Swiss Federal Nuclear Safety Inspectorate,Villingen-HSK, Switzerland
Thermodynamic data always refer to a selected standard state. The defini-tion given by IUPAC [1] is adopted in the NEA-TDB project as outlined inthe corresponding guideline [2]. According to this definition, the standardstate for a solute B in a solution is a hypothetical solution, at the standardstate pressure, in which mB=mo=1mol � kg�1, and in which the activitycoefficient gB is unity. However, for many reactions, measurements cannotbe made accurately (or at all) in dilute solutions from which the necessaryextrapolation to the standard state would be simple. This is invariably thecase for reactions involving ions of high charge. Precise thermodynamicinformation for these systems can only be obtained in the presence of aninert electrolyte of sufficiently high concentration, ensuring that activityfactors are reasonably constant throughout the measurements. The objec-tives of this chapter are to describe and illustrate various methods ofextrapolation to zero ionic strength and to describe the methods preferred inthe NEA review along with recommended values of the auxiliary para-meters. By following these guidelines, the members of the NEA specialistteams performing the review will be assured of using the same theory con-sistently for the extrapolation to zero ionic strength.
*Retired.
By kind permission from the OECD NEA-TDB project, reprinted from Report TDB-2,
Guidelines for the extrapolation to zero ionic strength, OECD Nuclear Energy Agency, Data
Bank, F-91191 Gif-Sur Yvette, France. For latest revised version, see Chemical Thermo-
dynamics of Americium Silva, R. J., Bidoglio, G., Rand, M. H., Robouch, P. B., Wanner, H.,
and Puigdomenech, I., Eds., Elsevier/North-Holland 1995.
The activity factors of all the species participating in reactions inhigh ionic strength media must be estimated in order to reduce the ther-modynamic data obtained from the experiment to the standard state (I=0).These estimates are based on the use of extended Debye-Huckel equations,either in the form of ‘‘specific ion interaction methods’’ or the Daviesequation. However, the Davies equation (see section 6.2.4) should in gen-eral not be used at ionic strengths larger than 0.1mol � kg�1. The follow-ing forms of specific ion interaction methods have been elaborated in thepast:
1. The Brønsted-Guggenheim-Scatchard approach (abbreviated ‘‘B-G-Sequation’’ in this document), (see section 6.1).
2. The Pitzer and Brewer ‘‘B-method’’ (abbreviated ‘‘P-B’’ herein) (seesection 6.2.1).
3. The Pitzer virial coefficient method, see section 6.2.2.Methods 1 and 2 areequivalent and differ only in the form of the denominator in the Debye-Huckel term. Method 3 requires more parameters for the description ofthe activity factors. These parameters are not available in many cases.This is generally the case for complex formation reactions.
The method preferred in the NEA Thermochemical Data Base review is thespecific ion interaction model in the form of the Brønsted-Guggenheim-Scatchard approach.
One may sometimes have access to the parameters required for thePitzer approaches, e.g., for some hydrolysis equilibria and for some solu-bility product data, cf. Baes and Mesmer [3] and Pitzer [4]. In this case,the reviewer should perform a calculation using both the B-G-S and theP-B equations and the full virial coefficient methods and compare theresults.
6.1 THE SPECIFIC ION INTERACTIONEQUATIONS
6.1.1 Background
The Debye-Huckel term, which is the dominant term in the expressionfor the activity coefficients in dilute solution, accounts for electrostatic,nonspecific long-range interactions. At higher concentrations, short-range,nonelectrostatic interactions have to be taken into account. This is usuallydone by adding ionic strength dependent terms to the Debye-Huckel ex-pression. This method was first outlined by Brønsted [5,6], and elaborated
by Scatchard [7] and Guggenheim [8]. The two basic assumptions in thespecific ion interaction theory are:
Assumption 1: The activity coefficient gj of an ion j of charge zj in thesolution of ionic strength Im may be described by Eq. (6.1).
log 10 �j ¼ �z2j DþXk
"ð j;k;ImÞmk (6.1)
D is the Debye-Huckel term:
D ¼ AffiffiffiffiffiImp
1þ BajffiffiffiffiffiImp (6.2)
A and B are constants that are temperature dependent, and aj is theeffective diameter of the hydrated ion j. The values of A and B as a functionof temperature are listed in Table 6.1.
The term Baj in the denominator of the Debye-Huckel term has beenassigned a value of Baj=1.5, as proposed by Scatchard [9] and accepted byCiavatta [10]. This value has been found to minimize, for several species, theionic strength dependence of e( j,k,Im) between Im=0.5m and Im=3.5m. Itshould be mentioned that some authors have proposed different values for
Baj, ranging from Baj=1.0 [11] to Baj=1.6 [12]. However, the parameterBaj is empirical and as such correlated to the value of "ð j;k;ImÞ. Hence, thisvariety of values for Baj does not represent an uncertainty range, but ratherindicates that several different sets of Baj and "ð j;k;ImÞ may describe equallywell the experimental mean activity coefficients of a given electrolyte. Theion interaction coefficients listed later in the chapter in Tables 6.3–6.5 haveto be used with Baj=1.5.
The summation in Eq. (6.1) extends over all ions k present in solution.Their molality is denoted mk. The concentrations of the ions of the ionicmedium is often very much larger than those of the reacting species. Hence,the ionic medium ions will make the main contribution to the value oflog10 gj for the reacting ions. This fact often makes it possible to simplifythe summation
Pk "ð j;k;ImÞmk so that only ion interaction coefficients
between the participating ionic species and the ionic medium ions areincluded, as shown in Eqs. (6.5)–(6.9).
Assumption 2: The ion interaction coefficients "ð j;k;ImÞ are zero for ions of thesame charge sign and for uncharged species. The rationale behind this is thate, which describes specific short-range interactions, must be small for ions ofthe same charge, since they are usually far from one another due to elec-trostatic repulsion. This holds to a lesser extent also for uncharged species.
Equation (6.1) will allow fairly accurate estimates of the activitycoefficients in mixtures of electrolytes if the ion interaction coefficients areknown. Ion interaction coefficients for simple ions can be obtained fromtabulated data of mean activity coefficients of strong electrolytes or from thecorresponding osmotic coefficients. Ion interaction coefficients for com-plexes can either be estimated from the charge and size of the ion ordetermined experimentally from the variation of the equilibrium constantwith the ionic strength. Ion interaction coefficients are not strictly constantbut vary slightly with the ionic strength. The extent of this variation dependson the charge type and is small for 1:1, 1:2, and 2:1, electrolytes for mol-alities less than 3.5m. The concentration dependence of the ion interactioncoefficients can thus often be neglected. This point was emphasized byGuggenheim [8], who has presented a considerable amount of experimentalmaterial supporting this approach. The concentration dependence is largerfor electrolytes of higher charge. In order to accurately reproduce theiractivity coefficient data, concentration-dependent ion interaction coef-ficients have to be used: (see Pitzer and Brewer [14]; Baes and Mesmer [3];or Ciavatta [10]). By using a more elaborate virial expansion, Pitzerand coworkers [4, 15–21] have managed to describe measured activitycoefficients of a large number of electrolytes with high precision over a large
concentration range. Pitzer’s model generally contains three parameters, ormore, as compared to one in the specific ion interaction model. The use ofthe model requires the knowledge of all these parameters. The derivation ofPitzer coefficients for many metal complexes would require a very largeamount of additional experimental work, since no data of this type arecurrently available.
The way in which the activity coefficient corrections are performedaccording to the specific ion interaction model is illustrated below for ageneral case of a complex formation reaction. Charges are omitted forbrevity.
mMþ qLþ nH2O(1)ÐMmLq(OH)n þ nHþ (6.3)
The formation constant of MmLq(OH)p, *bn,q,m, determined in an ionicmedium (1:1 salt NX) of the ionic strength Im, is related to the corre-sponding value at zero ionic strength, *bon,q,m, by Eq. (6.4).
The subscript (n,q,m) denotes the complex ion MmLq(OH)n. If the concen-trations of N and X are much greater than the concentrations of M, L,MmLq(OH)n and H, only the molalities mN and mX have to be taken intoaccount for the calculation of the term
Pk "ð j;k;ImÞmk in Eq. (6.1). For
example, for the activity coefficient of the metal cation M, gM, Eq. (6.5) isobtained.
log10�M ¼ �z2M0:5091
ffiffiffiffiffiImp
1þ 1:5ffiffiffiffiffiImp þ "ðM;X;ImÞmX (6.5)
Under these conditions, Im&mX=mN. Substituting the log10 gj values inEq. (6.4) with the corresponding forms of Eq. (6.5) and rearranging leads to
as a reactant or product require a correction for the activity of water, aH2O.
The activity of water in an electrolyte mixture can be calculated as
log 10aH2O ¼���kmk
lnð10Þ � 55:51(6.10)
where F is the osmotic coefficient of the mixture and the summation extendsover all ions k with molality mk present in the solution. In the presence ofan ionic medium NX in dominant concentration, Eq. (6.10) can be simpli-fied by neglecting the contributions of all minor species, i.e., the reactingions. Hence, for a 1:1 electrolyte of ionic strength Im&mNX, Eq. (6.10)becomes
log 10aH2O ¼�2mNX�
lnð10Þ � 55:51(6.11)
Values of osmotic coefficients for single electrolytes have been compiledby various authors, e.g., Robinson and Stokes [22]. The activity of watercan also be calculated from the known activity coefficients of the dissolvedspecies.
In the presence of an ionic medium NX of a concentration much largerthan those of the reacting ions, the osmotic coefficient can be calculatedaccording to Eq. (6.12) [23].
1� � ¼ A lnð10Þjzþz�jmNX � (1:5)3 [1þ 1:5
ffiffiffiffiffiffiffiffiffiffimNX
p � 2 log 10(1þ 1:5ffiffiffiffiffiffiffiffiffiffimNX
p)
� 1
1þ 1:5ffiffiffiffiffiffiffiffiffiffimNXp ]þ 1
ln (10Þ "(N;X)mNX (6.12)
The activity of water is obtained by inserting Eq. (6.12) into Eq. (6.11). Itshould be mentioned that in mixed electrolytes with several componentsat high concentrations, it is necessary to use Pitzer’s equation to calculatethe activity of water. On the other hand, aH2O is near constant (and =1)in most experimental studies of equilibria in dilute aqueous solutions,where an ionic medium is used in large excess with respect to the reactants.The ionic medium electrolyte thus determines the osmotic coefficient of thesolvent.
In natural waters the situation is similar; the ionic strength of mostsurface waters is so low that the activity of H2O(l) can be set equal to unity.A correction may be necessary in the case of seawater, where a sufficientlygood approximation for the osmotic coefficient may be obtained by con-sidering NaCl as the dominant electrolyte.
In more complex solutions of high ionic strengths with more than oneelectrolyte at significant concentrations, e.g., (Na+, Mg2+, Ca2+) (Cl�,SO4
2�), Pitzer’s equation may be used to estimate the osmotic coefficient; thenecessary interaction coefficients are known for most systems of geochem-ical interest.
Note that in all ion interaction approaches, the equation for meanactivity coefficients can be split up to give equations for conventional singleion activity coefficients in mixtures, e.g., Eq. (6.1). The latter are strictlyvalid only when used in combinations that yield electroneutrality. Thus,while estimating medium effects on standard potentials, a combination ofredox equilibria with H+ +e�)* 1
2H2(g) is necessary (see Example 3).
6.1.2 Estimation of Ion Interaction Coefficients
6.1.2.1 Estimation from Mean Activity Coefficient Data
Example 1: The ion interaction coefficient "ðHþ;Cl�) can be obtained frompublished values of g± ,HCl vs. mHCl.
2 log 10� � ;HCl ¼ log 10�þ;Hþ þ log 10��;Cl�
¼ �Dþ "ðHþ;Cl�ÞmCl� �Dþ "ðCl�;HþÞmHþ
log 10� � ;HCl ¼ �Dþ "ðHþ;Cl�ÞmHCl
By plotting log10g± ,HCl+D vs. mHCl, a straight line with the slope "ðHþ;Cl�Þis obtained. The degree of linearity should in itself indicate the range ofvalidity of the specific ion interaction approach. Osmotic coefficient datacan be treated in an analogous way.
6.1.2.2 Estimations Based on Experimental Values ofEquilibrium Constants at Different Ionic Strengths
Example 2: Equilibrium constants are given in Table 6.2 for the reaction
UO2þ2 þ Cl� Ð UO2Cl
þ (6.13)
The following formula is deducted from Eq. (6.6) for the extrapolation toI=0:
log 10�1 þ 4D ¼ log 10�o1 ��"Im (6.14)
The linear regression is done as described in the NEA Guidelines forthe Assignment of Uncertainties [24]. The following results are obtained:
The experimental data are depicted in Fig. 6.1, where the area between thedashed lines represents the uncertainty range that is obtained by using theresults in log10b1
o and De and correcting back to I= 0.
Example 3: When using the specific ion interaction theory, the relationshipbetween the normal potential of the redox couple UO2
2+/U4+ in a mediumof ionic strength Im and the corresponding quantity at I=0 should becalculated in the following way. The reaction in the galvanic cell
Pt;H2jHþkUO2þ2 ;U4þ;HþjPt (6.15)
is
UO2þ2 þH2(g)þ 2Hþ Ð U4þ þ 2H2O (6.16)
Table 6.2 The Preparation of the Experimental
Equilibrium Constants for the Extrapolation to I = 0 with
the Specific Ion Interaction Method, According to
Eq. (6.13)
Im log10b1(exp)a log10b1,m
b log10b1,m + 4D
0.1 �0.17 ± 0.10 �0.174 0.264 ± 0.100
0.2 �0.25 ± 0.10 �0.254 0.292 ± 0.100
0.26 �0.35 ± 0.04 �0.357 0.230 ± 0.040
0.31 �0.39 ± 0.04 �0.397 0.220 ± 0.040
0.41 �0.41 ± 0.04 �0.420 0.246 ± 0.040
0.51 �0.32 ± 0.10 �0.331 0.371 ± 0.100
0.57 �0.42 ± 0.04 �0.432 0.288 ± 0.040
0.67 �0.34 ± 0.04 �0.354 0.395 ± 0.040
0.89 �0.42 ± 0.04 �0.438 0.357 ± 0.400
1.05 �0.31 ± 0.10 �0.331 0.491 ± 0.100
1.05 �0.277 ± 0.260 �0.298 0.525 ± 0.260
1.61 �0.24 ± 0.10 �0.272 0.618 ± 0.100
2.21 �0.15 ± 0.10 �0.193 0.744 ± 0.100
2.21 �0.12 ± 0.10 �0.163 0.774 ± 0.100
2.82 �0.06 ± 0.10 �0.021 0.860 ± 0.100
3.5 0.04 ± 0.10 �0.021 0.974 ± 0.100
aEquilibrium constants for Eq. (6.13) with assigned uncertainties,
corrected to 258C where necessary.bEquilibrium constants corrected from molarity to molality units,
according to the procedure described in Ref. 2.
Note: The Linear Regression of this Set of Data is shown in
The relationship between the equilibrium constant and the standard poten-tial is
lnK ¼ nF
RTE (6.23)
lnKo ¼ nF
RTEo (6.24)
E is the standard potential in a medium of ionic strength I, Eo is the cor-responding quantity at I=0, and n is the number of transferred electrons inthe reaction considered. Combining Eqs. (6.22)–(6.24) and rearranging themleads to Eq. (6.25).
E � 10DRT lnð10Þ
nF
� �¼ Eo ��"mClO�4
RT lnð10ÞnF
� �(6.25)
For n=2 in the present example and T=298.15K, Eq. (6.25) becomes
6.1.3 On the Magnitude of Ion InteractionCoefficients
Ciavatta [10] made a compilation of ion interaction coefficients for a largenumber of electrolytes. Similar data for complexation reactions of variouskinds were reported by Spahiu [25] and Ferri, et al. [26]. These and someother data have been collected and are listed in Tables 6.3–6.6. It is obviousfrom the data in these tables that the charge of an ion is of great impor-tance for the magnitude of the ion interaction coefficient. Ions of the samecharge type have similar ion interaction coefficients with a given counte-rion. Based on the tabulated data, it is judged possible to estimate, with anerror of at most±0.1 in e, ion interaction coefficients for cases where thereare insufficient experimental data for an extrapolation to I=0. The errormade by this approximation is estimated to be±0.1 in De in most cases,based on comparison with De values of various reactions of the same chargetype.
The P-B equation is very similar to the B-G-S equation. The expression forthe activity coefficient of an ion i of charge zi takes the form
log 10�i ¼�z2i 0:5107
ffiffiffiffiffiImp
1þ ffiffiffiffiffiImp þ
Xj
B(i; j)mj (6.28)
Table 6.3 (continued )
j k!# Cl� ClO4
� NO3�
Al3 + 0.33 ± 0.02 - -
Fe3 + 0.56 ± 0.03 0.42 ± 0.08
Cr3 + 0.30 ± 0.03 - 0.27 ± 0.02
La3 + 0.22 ± 0.02 0.47 ± 0.03 -
La3 + ?Lu3 + - 0.47? 0.52h -
UOH3 + - 0.48 ± 0.08j -
UF3 + - 0.48 ± 0.08e -
UCl3 + - 0.59 ± 0.10j -
UBr3 + - 0.52 ± 0.10e -
UNO3þ3 - 0.62 ± 0.08j -
Be2OH3 + - 0.50 ± 0.05k -
Pu4 + - 1.03 ± 0.05f -
Np4 + - 0.82 ± 0.05 -
U4 + - 0.76 ± 0.06e -
Th4 + 0.25 ± 0.03 - 0.11 ± 0.02
aSource: Refs. [10,24] unless specified. Uncertainties are 95% confidence level.bIon interaction coefficients can be described more accurately with an ionic strength-dependent
function, listed in Table 6.5.cFerri et al. [26].dEvaluated in NEA-TDB review on uranium thermodynamics [37].eEstimated in NEA-TDB review on uranium thermodynamics [37].fRiglet et al. [36], where the following assumptions were made: "ðNp3þ ;ClO�4 Þ � "ðPu3þ ;ClO�4 Þ ¼ 0:49 as
for other ðM3þ;ClO�4 Þ interactions, and "ðNpO2þ2;ClO�4 Þ � "ðPuO2þ
2;ClO�4 Þ � "ðUO2þ
2;ClO�4 Þ ¼ 0:46.
gHedlund [35].hTaken from Spahiu [25].iThe coefficients "ðMnþ;Cl�Þ and "ðMnþ ;NO�3 Þ reported by Ciavatta [10] were evaluated without taking
chloride and nitrate complexation into account. See section 6.3.jEvaluated in NEA-TDB review on uranium thermodynamics [37] using "ðU4þ ;ClO�4 Þ ¼ ð0:76� 0:06Þ.kTaken from Bruno [38], where the following assumptions were made: "ðBe3þ ;ClO�4 Þ ¼ 0:30 as for other
"ðM2þ ;ClO�4 Þ; "ðBe2þ;Cl�Þ ¼ 0:17 as for other "ðM2þ ;Cl�Þ, and "ðBe2þ;NO�3 Þ ¼ 0:17 as for other "ðM2þ ;NO�3 Þ.
aSource: Refs. [10,34] unless specified. Uncertainties are 95% confidence level.bIon interaction coefficient can be described more accurately with an ionic strength-dependent
function, listed in Table 6.5.cEvaluated in NEA-TDB review on uranium thermodynamics [37].
where the summation over j covers all anions for the case that i is a cationand vice versa. Tables of B(I, j) are given by Pitzer and Brewer [14] and byBaes and Mesmer [3]. The Debye-Huckel term [see Eq. (6.2)] is differentfrom that in the B-G-S equation. Apart from a slightly different value for A,the factor Baj has been chosen equal to 1.0 in the P-B equation compared to1.5 in the B-G-S equation. The B-G-S equation is preferred to the P-Bequation in the critical evaluations of the NEA-TDB Project for the reasonsgiven in section 6.1.1.
6.2.2 The Pitzer Equations
The following text is only intended to provide the reader with a brief outlineof the Pitzer method. This approach consists of the development of an ex-plicit function relating the ion interaction coefficient to the ionic strength andthe addition of a third virial coefficient to Eq. (6.1). For the solution ofa single electrolyte MX, the activity coefficient may be expressed by Eq.(6.29) [15]:
ln� ¼ jzMzXjf� þm(2�M�X)
�B�MX þm2 2ð�M�XÞ3=2
�
!C�MX (6.29)
where nM and nX are the numbers of M and X ions in the formula unitand zM and zX their charges. m is the molality of the solution and n=nM+ nX. In aqueous solutions at 258C and 105 Pa, the following relationsare given [16]:
f � ¼ �0:392ffiffiffiffiffiImp
1þ 1:2ffiffiffiffiffiImp þ 1:667 lnð1þ 1:2
ffiffiffiffiffiIm
pÞ
� �(6.30)
B�MX ¼ 2�
ð0ÞMX þ
�ð1ÞMX
2Im(1� (1þ 2
ffiffiffiffiffiIm
p� 2Im)e
�2 ffiffiffiffiImp ) (6.31)
C�MX ¼ 3
2C�
MX (6.32)
where f g is the Debye-Huckel term extended to include osmotic effects, theparameters b(0)MX and b(1)MX define the second virial coefficient (correspondingto e in the B-G-S equation), and Cf
MX defines the third virial coefficient.Im=1
2
Pmizi
2 is the ionic strength in molal units. In the case of 2:2 elec-trolytes, one could add to the second virial coefficient terms of the sameform. Pitzer’s equations have been extended to cover electrolyte mixtures[17], including terms allowing for the interactions of ions of the same chargesign. Equation (6.33) is an extension of Eq. (6.29) for a anions and c cations
[4]. Here a and a0 cover all anions, c and c0 cover all cations, and(P
mz)=P
cmczc=P
ama|za|.
ln�MX¼ jzMzXj f �þ2�M�
Xa
ma BMaþX
mz� �
CMaþ �X�M
�Xa
� �
þ2�X�
Xc
mc BcXþX
mz� �
CcXþ�M�X�Mc
� �
þXc
Xa
mcma jzMzXjB 0caþ1
�ð2�MzMCcaþ�M Mca
þ�X caXÞ� �
þ1
2
Xc
Xc0
mcmc0�X� cc0XþjzMzXj�0cc0
� �
þ1
2
Xa
Xa0
mama0�M� Maa0 þ jzMzXj�0aa0
� �(6.33)
In Eq. (6.33) f g is the same as in Eq. (6.29); CMX, BMX, and its derivativewith ionic strength, B0MX, have the forms
BMX ¼ �ð0ÞMX þ�ð1Þ
2Im(1� (1þ 2
ffiffiffiffiffiIm
p)e�2
ffiffiffiffiImp
) (6.34)
B0MX ¼�ð1ÞMX
2I2m(� 1þ (1þ 2
ffiffiffiffiffiIm
pþ 2Im)e
�2 ffiffiffiffiImp ) (6.35)
CMX ¼ C�MX
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffijzMzXj
p (6.36)
where b(0), b(1), Cf are the same as for pure electrolytes. In Eq. (6.33) the yterms summarize the interactions between ions of the same charge sign thatare independent of the common ion in a ternary mixture, and the c termsaccount for the modifying influence of the common ion on these interac-tions. Pitzer [4] points out that higher order electrostatic terms (beyondthe Debye-Huckel approximation) become important in cases of unsym-metrical mixing, especially if one of the ions has a charge of three or higher.Higher order electrostatic effects, on the other hand, were found to beunimportant for cases of symmetrical mixing and for pure unsymmetricalelectrolytes.
Based on the cluster integral method [27], Pitzer [19] divided the dif-ference terms yMN into two parts
at 258C and 105 Pa and J(x) is an integral that canbe approximated as
JðxÞ ¼ � 1
6x2ðln xÞe�10x2 þ
X6k¼1
Ckx�k
!�1(6.40)
The first term is in fact important only at very low ionic strengths; thus forcases used in equilibrium analysis one has
JðxÞ ¼ 4:118
xþ 7:247
x2þ 4:408
x3þ 1:837
x4þ 0:251
x5þ 0:0164
x6
� ��1(6.41)
The syMN parameters can be evaluated from data on mixtures of electrolytesby calculating the differences between the experimental value of gexp and thevalue calculated with the appropriate values for all pure electrolyte termsand Ey terms but zero values for sy and c terms. For the activity coefficientof MX in a MX-NX mixture one has Eq. (6.42) and equivalent expressionsfor other cases.
�
2�MmN� ln � ¼ s�MN þ 1
2mX þ zM
zX
��������mM
� � MNX
� �(6.42)
This is the equation of a straight line with intercept syMN and slope cMNX,when the left side of Eq. (6.42) is plotted against a function of the com-position 1
2ðmX þ j zMzX jmMÞ.
The Pitzer equation for single electrolytes with the value of parameterscollected in several publication may be used as a compact source of activitycoefficient data. From the values go� jk thus obtained, one may calculate, forexample, "ðj;k;ImÞ values using Eq. (6.1).
For the estimation of medium effects on solubility equilibria in mix-tures of electrolytes involving ions of charge less than three, one may neglecty and c terms in Eq. (6.33).
In equilibrium analysis studies carried out in the presence of an inertsalt (medium salt NX) and small (trace) concentrations of the reactants,only the terms involving mNX have to be considered in Eq. (6.33), whilethose involving mi
trace can be neglected. Nevertheless, as the main difficulty
there still remains the accurate estimation of the parameters of single elec-trolyte b(0), b(1), and Cf for species such as metal ion complexes.
Equations for single ion activity coefficients [4], osmotic coefficients[17], and other thermodynamic quantities [28], as well as applications indifferent cases (e.g., H2SO4 and H3PO4 solutions) have been given by Pitzerand coworkers [4,20].
From Tables 6.3 and 6.4 it seems that the size and charge correlationscan be extended to complex ions. This observation is very important becauseit indicates a possibility to estimate the ion interaction coefficients forcomplexes by using such correlations. It is, of course, always preferable touse experimental ion interaction coefficient data. However, the efforts nee-ded to obtain these data for complexes will be so great that it is unlikely thatthey will be available for more than a few complex species. It is even lesslikely that one will have data for the Pitzer parameters for these species.Hence, the specific ion interaction approach may have a practical advantageover the inherently more precise Pitzer approach.
6.2.3 The Equations Used by Baes and Mesmer
Baes and Mesmer [3] use the function F(Im) proposed by Pitzer to expressthe ionic strength dependence of the ion interaction coefficient BMX inGuggenheim’s equations. For a single electrolyte
log 10�o�MX ¼ �jzMzXj 0:511
ffiffiffiffiffiImp
1þ ffiffiffiffiffiImp þ 2�M�X
�BMXmMX (6.43)
where
BMX ¼ B1MX þ (BoMX � B1MX)F(Im) (6.44)
FðImÞ ¼ 1� ð1þ 2ffiffiffiffiffiImp � 2ImÞe�2
ffiffiffiffiImp
4Im; Fð0Þ ¼ 1;Fð1Þ ¼ 0:
(6.45)
The Pitzer function linearizes the dependence of the ion interaction coeffi-cient on the ionic strength quite well, even in the cases of 4:1 and 5:1electrolytes, where constant e(M,X) values [see Eq. (6.1)] are not obtained athigh ionic strengths. The parameters Bo
MX and B?MX can be determined from
a single electrolyte activity coefficient datum by calculating first BMX. (Notethat BMX=e( j,k), since the Debye-Huckel term in Eq. (6.43) does not havethe factor 1.5 in the denominator.) By plotting BMX(Im) values against F(Im),B?
MX is obtained as the intercept while BoMX is obtained from the slope of
the straight line [see Eq. (6.44)]. The equation for a mixture is similar to
Eq. (6.43) and BMX=0 if M and X are of the same charge sign. In the caseof equilibrium constant measurements, DB values are expressed by equa-tions similar to Eq. (6.6).
The corresponding DBo and DB? can be obtained together with the bo
where b(Im,n) and ðaH2OÞn refer to the values of b and aH2O at ionic strengthIm,n. From the values obtained for DBo and DB? and equations similar toEq. (6.9), one may estimate the unknown Bo
MX and B?MX values.
6.2.4 The Davies Equation
The Davies equation [29] has been used extensively to calculate activitycoefficients of electrolytes at fairly low ionic strengths.
The Davies equation for the activity coefficient of an ion i of charge ziis, at 258C:
log10�i ¼ �0:5102z2iffiffiffiffiffiImp
1þ ffiffiffiffiffiImp � 0:3Im
� �(6.47)
The equation has no theoretical foundation but is found to work fairly wellup to ionic strengths of 0.1mol � kg�1. It should not be used at higher ionicstrengths. The Davies equation has a form similar to the B-G-S equation butwith ion interaction coefficients equal to 0.153zi
2, i.e., 0.15, 0.61, and 1.38 forions of charge 1, 2, and 3, respectively. These values do not agree very wellwith the tabulated e values.
6.3 ION INTERACTION COEFFICIENTS ANDEQUILIBRIUM CONSTANTS FOR ION PAIRS
Two alternative methods can be used to describe the ionic medium depen-dence of equilibrium constants:
� One method takes into account the individual characteristics of the ionicmedia by using a medium-dependent expression for the activity coeffi-cients of the species involved in the equilibrium reactions. The mediumdependence is described by virial or ion interaction coefficients as used inthe Pitzer equations and in the specific ion interaction model.
� The other method uses an extended Debye-Huckel expression wherethe activity coefficients of reactants and products depend only on the
ionic charge and the ionic strength, but it accounts for the medium-specific properties by introducing ionic pairing between the medium ionsand the species involved in the equilibrium reactions. Earlier, thisapproach has been used extensively in marine chemistry (See Refs. [30–32, 39]).
It can be shown that the virial type of activity coefficient equations and theionic pairing model are equivalent, provided that the ionic pairing is weak.In these cases, it is in general difficult to distinguish between complex for-mation and activity coefficient variations unless independent experimentalevidence for complex formation is available, e.g., from spectroscopic data,as is the case for the weak uranium(VI) chloride complexes. It should benoted that the ion interaction coefficients evaluated and tabulated by Cia-vatta [10] were obtained from experimental mean activity coefficient datawithout taking into account complex formation. However, it is known thatmany of the metal ions listed by Ciavatta form weak complexes withchloride and nitrate ions. This fact is reflected by ion interaction coefficientsthat are smaller than those for the noncomplexing perchlorate ion (seeTable 6.3). This review takes chloride and nitrate complex formation intoaccount when these ions are part of the ionic medium and uses the value ofthe ion interaction coefficient "ðMnþ;ClO�4 Þ for "ðMnþ;Cl�Þ and "ðMnþ;NO�3 Þ. Inthis way, the medium dependence of the activity coefficients is describedwith a combination of a specific ion interaction model and an ion pairingmodel. It is evident that the use of NEA-recommended data with ionicstrength correction models that differ from those used in the evaluationprocedure can lead to inconsistencies in the results of the speciation calcu-lations.
It should be mentioned that complex formation may also occurbetween negatively charged complexes and the cation of the ionic medium.An example is the stabilization of the complex ion UO2(CO3)3
5� at high ionicstrength.
6.4 TABLES OF ION INTERACTIONCOEFFICIENTS
Tables 6.3–6.5 contain the selected specific ion interaction coefficients usedin this review, according to the specific ion interaction model described insection 6.1. Table 6.3 contains cation interaction coefficients with Cl�,ClO4
�, and NO3�; Table 6.4 anion interaction coefficients with Li +, with
Na+ or NH4+ , and with K+ . The coefficients have the units of kg �mol�1
and are valid for 298.15K. The species are ordered by charge and appear,within each charge class, in standard order of arrangement [33].
In some cases, the ionic interaction can be better described byassuming ion interaction coefficients as functions of the ionic strength ratherthan as constants. Ciavatta [10] proposed the use of Eq. (6.48) for caseswhere the uncertainties in Tables 6.3 and 6.4 are±0.03 or greater.
" ¼ "1 þ "2 log 10Im (6.48)
For these cases, and when the uncertainty can be improved as compared tothe use of a constant e, the values e1 and e2 given in Table 6.5 should be used.
It should be noted that ion interaction coefficients tabulated inTables 6.3–6.5 may also involve ion pairing effects, as described in section6.3. In direct comparisons of ion interaction coefficients, or when estimatesare made by analogy, this aspect must be taken into account.
6.5 CONCLUSION
The specific ion interaction approach is simple to use and gives a fairly goodestimate of activity factors. By using size/charge correlations, it seems pos-sible to estimate unknown ion interaction coefficients. The specific ioninteraction model has therefore been adopted as a standard procedure in theNEA Thermochemical Data Base review for the extrapolation and correc-tion of equilibrium data to the infinite dilution standard state. For moredetails on methods for calculating activity coefficients and the ionic medium/ionic strength dependence of equilibrium constants, the reader is referred toRef. 40, Chapter IX.
REFERENCES
1. Lafitte, M. J. Chem. Thermodyn., 1982, 14, 805.
2. Wanner, H. Standards and conventions for TDB publications, TDB-5.1, Rev. 1,
OECD Nuclear Energy Agency, Data Bank, Gif-sur-Yvette, France, 1989.
3. Baes, C. F., Jr.; Mesmer, R. F. The Hydrolysis of Cations; John Wiley and Sons,
New York, 1976.
4. Pitzer, K. S. Theory: Ion interaction approach. In Activity Coefficients in
Electrolyte Solutions, Pytkowicz, R. M., Ed., Vol. I, CRC Press, Boca Raton,
Florida, 1979; 157–208.
5. Brønsted, J. M. Studies of solubility: IV. The principle of specific interaction of
ions. J. Am. Chem. Soc., 1922, 44, 877–898.
6. Brønsted, J. M. Calculation of the osmotic activity functions of univalent salts.
J. Am. Chem. Soc., 1922, 44, 938–948.
7. Scatchard, G. Concentrated solutions of strong electrolytes, Chem. Rev., 1936,
19, 309–327.
8. Guggenheim, E. A. Applications of Statistical Mechanics; Oxford, Clarendon
