Kenneth S. Pitzer
26
ground state is minimum. Thus the optical excitation of the solvated electron from the ground state to the excited state i s followed by the rearrangement of the solvent molecules. In the semicontinuum model12 the optical excitation is followed by the expansion of the cavity. On the other hand, in bhe configuration coordinate model, the optical excitation is followed by the change of the orientational polarization from Po to PI (see Figure 4). The change of the orientational polarization from Po to PI implies the decrease of the orientational polarization around the electron (see Figure 2). The decrease of the
orientational polarization implies the deorientation of the solvent molecules. Thus in the configuration coordinate model the optical excitation is followed by the deorientation of the solvent molecules. In the present calculation the effect of the electronic polarization is neglected. This effect will be taken into consideration in our later paper.
Acknowledgments. The authors wish to thank Professor T. Watanabe and Dr. M. Natori for helpful discussions.
Thermodytriarnics of Electrolytes. I . Theoretical Basis and General Equation Kenneth S. Pitter inorganic Materials Research Division of the Lawrence Berkeley Laboratory and Department of Chemistry, University of Caiifornia, Berkeiey, California 94720 (Received May 72, 7972) Publication cosfs assisted by The U.S. Atomic Energy Commission
A system of equations for the thermodynamic properties of electrolytes is developed OR the basis of theoretical insights from improved analysis of the Debye-Huckel model as well as recently published numerical calculations for more realistic models. The most important result is the recognition of an ionic strength dependence of the effect of short-range forces in binary interactions. By modifying the usual second virial coefficients to include this feature, one obtains a system of equations which are only slightly more complex than those of Guggenheim but yield agreement within experimental error to concentrations of several molal instead of 0.1 M . If one compares instead with the recently proposed equations of Scatchard, Rush, and Johnson, the present equations are very much simpler for mixed electrolytes (and somewhat simpler for single electrolytes) yet appear to yield comparable agreement with experimental results for both single electrolytes and mixtures.
The thermodynamic properties of aqueous electrolytes have been extensively investigated both experimentally and theoretically. The monographs of Harned and Owen1 and of Robinson and Stokes2 provide excellent summaries. While the detailed nature of these solutions is so complex that an ab inibio quantum-statistical theory is not feasible, the data appear to relate to few enough independent parameters 'to make relatively exact semiempirical representation possible. It is the present objective to develop equations which reproduce the measured properties substantially within experimental accuracy, which are compact and convenient in that only a very few parameters need be tabulated for each substance and the mathematical calculations are simple, which have appropriate form for mixed electrolytes as well as for solutions of a single solute, and whose parameters have physical meaning as far as possible. In 1960 Brewer and the writer3 selected as the best available system one proposed and applied to dilute solutions by Guggenkeid with modifications suggested by Scatchard5 for concentrated solutions. While this system was useful in providing a simple and compact summary of experimental data, it did not fully satisfy the other desired qualities. Recent theoretical advances of Friedman and collaborators"? and of Card and Valleaus provide imThe Journal of Physical Chemistry, Voi. 77, No. 2, 1973
portant insights and support for the greatly improved semiempirical treatment proposed below. Indeed it is interesting that a key idea can be obtained by introducing the Debye-Huckel model and distribution function into modern equations relating such functions to the osmotic pressure. But first we review the Guggenheim equations. (1) H. S. Harned and 8. 8.Owen, "The Physical Chemistry of Electrolytic Solutions," Reinhold, New York, N. Y., 1950. (2) R. A. Robinson and R. H, Stokes, "Electrolytic Solutions," Butterworths, London, 1959. (3) K. S. Pitzer and L. Brewer, revised edition of "Thermodynamics" by G. N. Lewis and M. Randall, McGraw-Hili, New York, N. Y., 1961. (4) (a) E. A. ,Guggenheim, Phil. Mag., [7] 18, 588 (1035); (b) E. A , . Guggenheim and J. C. Turgeon, Trans. Faraday Soc., 51, 747 (1955); (c) E. A. Guggenheim, "Applications of Statistical Mechanics, Clarendon Press, Oxford, 1966, pp 166-170; here Guggenhelm surprisingly abandons the limitation to 0.1 M for validity of his equations for the difference in ~pfrom a reference salt. A5 is obvious from Figure 1 or the figures and tables of ref 4, this is a crude approximation. (5) G. Scatchard. Chem. Rev., 19, 909 (1939); also in W. J. Hamer, Ed., "The Structure of Electrolytic Solutions." Wilev. New York. N. Y., 1959, p 9. (6) J. C. Rasaiah and H. L. Friedman, J. Chem. Phys., 48, 7242 (1968); 50, 3965 (1969). (7) P. S . Ramanathan and H. L. Friedman, J Chem. Phys.. 54, 1086 (1971). (8) (a) D: N. Card and J. P. Valleau, J. Chem. Plrys., 52, 6232 (1970); (b) J. C. Rasaiah, D. N. Card, and J. P. Valleau, J. Chem. Phys., 56, 248 (1972).
Thermodynamics of Electrolytes
269
~ ~ ~ ~ e n h ~ ihard ~ -Equations " S ~ ~ i , c Since the Guggenheim4 equations have had considerable success, it is desirable to recall them at this point and to discuss the aspects that need improvement. The equations for osmotic and activity coefficients are
0.4
-m
o 0.3
f
B I 2s x
a_ -1s
0.2
0
I___
0
1.0
0.5
15
ml/2
Figure 1. Difference in osmotic lytes from that of RbN03.
Here the s u m over M and X cover all positive and negative ions, respt:ctively; A, is the usual Debye-Huckel coefficient with N , Avogadro's number and d, the density of the solvent; and I ie; a useful parameter which expresses the distance a t which the electrostatic energy for singly charged ions in the dielectric just equals thermal energy. The quantities are constants (at given T and D ) analogous to second virial coefficients which represent the net effect of various short-range forces between the M and X ions. It should be noted that Guggenheim followed Br~nsted'sg princEple of specific interaction and excluded terms related to short-range forces between ions of like sign. We shall not distinguish between direct interactions of solute species at short distances and changes in solvation with concentration since bolh phenomena influence the effective interionic potential of average force and thereby the second and higher virial coefficients. Likewise the distinction between molality and concentration will be ignored since tho statistical calculations based upon molecular models are used only to suggest appropriate forms for empirical use and to provide qualitative understanding. 'The advant,age of the molality in practical calculations, especially if the temperature varies, is overwhelming. It is also useful to consider the total excess Gibbs energy (for a solution containing n, kg of solvent) which is obtained by approprrate integration of the activity or osmotic coefficient.
r(n.) = (3/X3)[in( I -I-
-
c (x2/2)1
For a single 1-1electrolyte eq 1and 2 reduce to
+ m1/2)]+ 2 P ~ x m -(A,m1/2/3) u(rnlf2)+ P M x m
In y := --tA,ml/2/(1 cp
_- 1
E=
(8) (9)
Guggenheim and Turgeondb showed that eq 8 and 9 fitted, essentiallly within experimental error, the data for 1-1 electrolytes in water at 0" and at room temperature at concentrations up to 0.1 M . They acknowledge, however, and others have showin that substantial discrepancies arise at higher concentrations. In seeking equations that can be used at higher concentration one may examine separately
coefficient of several electro-
the first and second terms on the right side. The first term in eq 6 must remain a general function of ionic strength if these equations are to maintain their simplicity and utility for mixed electrolyte solutions. We shall return later to the question of an improved mathematical form of this first term. Next we note that, by taking differences between the properties of different electrolytes of the same type and at the same concentration, the first term cancels and we have (for 1-1 electrolytes) In Y M ~ Y In Y M'X' = 2 m ( 0 w y - i ( 3 h f y ) (10) PWX"- 'FM'X' = m ( @ M y- pMfxf) (11) These equations (10 and 11) still do not fit the experimental properties above 0.1 M with constant P's. Following a suggestion of Scatchard,5 Brewer and the writer3 considered @ to vary slowly with concentration and produced thereby a very compact tabulation of the experimental data for pure electrolyte solutions. If the P's are no longer constants, however, the relationships between eq 1, 2 , and 6 and between 8 and 9 are no longer valid. We shall return to the derivation of correct relationships corresponding to the variable @'s in a later section but here point out in Figure 1 the qualitative nature of the dependence of fl on concentration. The notable (and initially surprising) feature is the substantial change in p at low concentration in contrast to its relative constancy at higher concentration. While the change in @ below 0.1 M is not large, there is no reason to doubt that the curves maintain their slope in that region. Thus an improved theory which adequately accounts for these effects at higher concentration will presumably also represent an improvement below 0.1 M . Recently Scatchardlo and coworkers'j have extended and elaborated the Guggenheim equations in several ways. First the Debye-Huckel term in eq E is subdivided into a series of terms with different coefficients of 1112 in T corresponding to different distances of closest approach for the solute components. A p p r o ~ ~ derivatives ia~~ then (9) J N Brqnsted, Kgl Dan Vidensk Selsk Mat Fys Medd 4, (4) (1921), J Amer Chem ~ O C44,877 , (1922), 45,2898 (1923) (10) G Scatchard, J Amer CkLem Soc , 83,2636 (1961) (11) G Scatchard, R M R u b , and J S Johnson J Phys C h e m , 74, 3786 (1970) The Journal of Physical Ghernisrry, Vol. 77, No. 2, 1973
Kenneth S. Pitzer
270
yield correct but very complex formulas for the osmotic and activity coefficients. Secondly, the Br$nsted principle of specific interaction is abandoned and terms are introduced for the short-range interaction of ions of like sign. Finally, arrays of third and fourth virial coefficients are added. On this basis, Lietzke and Stoughton12 were able to represent accurately the osmotic coefficients of 20 pure electrolytes; also several systems of mixed electrolytes have been treated.l[)$J~ This system allows enough terms to be included to represent experimental data accurately; but the equakions are very complicated and there seems little promise OC simple physical interpretation of the parameters. Consequently, it seems worthwhile to seek simpler equations with fewer and more meaningful parameters. Also noteworthy is the work of Reilly, Wood, and Robinson13 which ih directed primarily to the process of mixing of pure electrolytes. Their general pattern of equations is similar in many aspects to that presented hereafter, but these authors did not apply their equations to the numerical expression of the properties of single electrolytes and thus did not deal with one of the principal subjects of this paper. Hard-Core Effects in Debye-HCekel Theory Traditional Debye-Huckel theory of electrolyte solutions1.2 recognizes the distance of closest approach a in the calculativn 3f the electrostatic energy of the distribution of ions hut ignores the kinetic effect of the hard core on the osmotic pressure or other properties. As Kirkwood,14 among others noted, this hard core effect cannot be treated rigorously by the traditional charging process methods of cdculating free energy. Van Rysselberghe and Eisenberg15 did add approximate hard-core terms to the traditional Debye-Hhckel formulas, but the feature of particular interest was lost in that approximation. Recently it has been shown6e16 that there are several equations which relate the intermolecular potential and the radial distribution function (also called the pair correlation function) to thermodynamic properties. Each equation will give the same set of thermodynamic results if the distribution function is exact, but different results arise from approximate distribution functions. The most convenient equation for a hard-core potential is the so-called “pressure” equation which yields the pressure of a pure fluid or the osinotic pressure of a solution.
Where n is the osmotic pressure, c, and c I , are concentrations of species i arid .I, c is the total solute concentration 2 c L ,urJis the interniolecular potential, and gzJ is the radial distribution function. The sums cover all solute species. We introduce ihe pot entia1 u2/=
w,
rCa
uLJ= x,zJe2/Dr, r 2 a
and obtain
n - chT
1-
~ c , c l z , z I g1,(r)4nrdr
= (e2/BD,l I
(134 (13b)
+
J
Where the second term, involving g(a), the radial distribution function at the outer surface of contact, constitutes The Journal of Physfcal Chemistry. Yo/ 77, No. 2, 7973
the kinetic effect of the hard core. Note that we have assumed the same core size a for all species of ions but different charges z, and z J in accordance with Debye-Huckel theory. The usual Debye-Hiickel distance 1 / is~defined K~
= (4ae2/DkT)~
I Z ~ ~ C ,
(15)
and the function qJr) = [z,zJe2/DkT(1+ ~ a ) ] ( e - x ( r - @ ’ / r ) (16)
In the Debye-Huckel derivation the radial distribution of ions around a given ion is obtained from Boltzmann’s law with the interaction energy given by the product of the charge on one icn and the average electrical potential surrounding the other ion. Summation over the various ions yields a charge density which, after approximation by its linear term, is substituted into the Poisson equation. The solution of the Poisson equation yields an explicit result for the electrical potential. The quantity qLI(r)defined in eq 16 is the resulting exponent in the Boltzmann factor for the radial distribution function, which is g L I ( r ) = exp[-q,,(r)I. The charge density around a given ion t now becomes
In the expanded expression the first term vanishes because the solution is electrically neutral. Since qcl contains the factor z J , the second or linear term is a sum over cJz,2 and does not vanish; indeed, it yields the charge distribution in Debye-Huckel theory. The third term involves a sum over cJz13 which vanishes for symmetrical electrolytes and is relatively small in other cases. Thus for symmetrical electrolytes there is no inconsistency with the linearized Poisson-Boltzmann solution in retaining the third term, but it has no effect on the charge density and therefore no effect on the conventional Debye-Huckel results for thermodynamic properties. Now, however, we wish to include the kinetic effect of the hard core by the use of eq 14 which involves the sum over c,c,g,,(a). In discussion of this sum we again consider the expanded expression for the radial distribution function gJr) = 1 - q L I ( r+ ) Y2qZj2(r) ~
I
.
(17)
for r L a; g = 0 for r < a, of course. In this case the sum for the first term is finite while that for the second term is zero. But for the hard-core effect the third term again yields a finite contribution. Thus for symmetrical electrolytes this third term can be included without any inconsistency with the Debye-Huckel derivation, and its inclusion should yield an improved approximation for the hard-core effect. Even for unsymmetrical electrolytes the effect of the third term 0x1 the charge density is small; hence the “inconsistency” of its retention is small and it may be desirable to include its contribution t o the hard core-effect. (12) M. H. Lietzke and R . W. Stoughton, J. Phys. Chem., fie, 508 (1962). (13) (a) P. J. Reilly and R. H. Wood, J. Phys. Chem., 73, 4292 (1969); (b) P. J. Reiliy, R. H. Wood, and R. A. Robinson, ibid., 75, 1305 (1971). (14) J. c. Kirkwood, Chem. Rev., 19, 275 (1936); J, C , Kirkwood and J. C. Poirier, J. Phys. Chem., 58, 591 (1954). (15) P. Van Rysselberghe and S, Eisenberg, J . Amer. Chem. SOC.,61, 3030 (1939); 62, 451 (1940). (16) T. L. Hill, “Statistical Mechanics,” McGraw-Hill, New York, N. Y . , 1956.
Thermodynamics of Electrolytes
271
Since the tneory is only approximate, however, there is no reason why. the exponential expressions for g,, should not be used, if desired, and Card and Valleausa have found good agreement between such distribution functions and their Monte Carlo calculations for 1-1 electrolytes up to about 1Ad. For our puiposes i t is preferable to use the three-term expansion of cq 17 since this allows simpler mathematical expression of the results while still providing a good approximation whencmr the general Debye-Huckel solution i s valid. The osmotic coefficient is found, on this basis, to be p
-
1
la ~
ckT
-K J
I
==
24nc(l C Ka>
3-
--’-) 48n (
K4a
)] (18)
I all ions have charges & z this reduces to
Now K = (4nl)ll%zcl’2 and I = e2/DkT as defined earlier. In eq 18 and 19 the first term on the right arises from the electrostatic energy and the second term from the hard core; each has interesting implications. This result for the electrostatic effects differs from the traditional Debye-Huchel formula although it reduces to the same limiting law when ~a can be neglected as compared to unit). Such a difference is not unexpected since the distribution functiou is approximate except in the limit of small K. In Figure 2 we compare the alternate therrnodynaimrc expressions based on the Debye-Huckel distribution f mctaon with the theoretically exact but numerically ~ ~ ~ ~ ~ Monte ~ r s Carlo o m statistical e treatments of the Game molecular model ( a = 4.25 A, aqueous solution at ‘25”) The new expression, eq 19, agrees, within computational. ~ ~ c e r t a ~with n ~ ythe , Monte Carlo results for c o n c e n t i ~ a t ~up o ~t~o I = 0.5 M . Substitution of the conventional term for the electrostatic effects causes a serious discrepancy. It is also interesting that this theoretical curve of eq 19 (or the Monte Carlo results) fits the experimental data for HBr very well. In view of the approximations of the molecular model, hOWe~er,we shall regard both expressions for the thermodynamic effect of the charge distribution as candidates for adoption on the basis of success in fitting experimental data. O f greater interest is the hard-core term in eq 18 and 19. The first term within the brackets, 2xa3/3, is independent of concentration and was considered by Van Rysselberghe and Eisexzberg.15 The second term within the brackets, howevei, does depend, through K , on the ionic strength of thc solution. Furthermore, this expression in brackets varies with concentration in just the same manner as was shown in Figure 1 with an initially rapid decrease from the value at zero concentration to a smaller arid relatively consta~ntvalue at high concentration. This decrease in the second virial coefficient with ionic strength arises from a similar decrease in the average radial distribution function at contact. Thus it is important to note that the inore rigorous statistical treatments by the Monte C‘ario method yields a similar behavior of the radial ~ ~ s t r i ~ 9lunction ~ t i ~ n at contact. Furthermore, the aPrisnnthaa and Friedman? on a model with a soft repulsive potential show a similar decrease
-0. I
Figure 2.
u .o
0
0.5 -p2
I
Osmotic coefficient for the Debye-Huckel model
(a = 4.25 A, aqueous solution at 25’): the solid circles are calculated values by the Monte Carlo method,6athe solid curve is from
the present theory with eq 19, the dashed curve differs from the solid curve by substituting the conventional D-H expression for the electrostatic energy for the corresponding term in eq 19, and the triangles are experimental values for HBr. with increase in ionic strength in the average radial distribution function at a comparable radius. The Debye-Huckel distribution function was also inserted in the so-called “compressibility” equation of statistical mechanics.6 The result is much more complex, mathematically, although similar in qualitative behavior. Hence it does not add anything useful for our present purposes. While the Debye-Huckel treatment does not allow the consideration of different ionic radii within the same solution, we can compare the properties of different solutions, each of a single solute with a particular radius sum. We consider the difference in osmotic coefficient for symmetrical electrolytes with radius sums at and a2 and ion charge z. “Fz
- cp1 - 2rr c
4-
Here the first term inside the braces arises from the electrostatic effects and the second and third terms from the hard-core effects ( i e . , the same order as in eq 19). The quantity ((p2 - cpl)/c corresponds to the quantity ((p2 cpl)/m which was shown in Figure 1 and would be a constant if the Guggenheim equations were valid. It is apparent from eq 20 that it will yield the same qualitative behavior as was shown in Figure l; specifically, the second term is constant, but both the first and third terms decrease from a finite value a t zero concentration to very small values as K increases. The principal consequence of this section is to suggest that the properties of electrolyte solutions can be expressed by an “electrostatic” term plus a virial coefficient series in which the virial coefficients may be functions of the ionic strength of the solution. More particularly it is suggested that the second virial coefficient will vary with ionic strength in the pattern indicated by Figure 1 or eq 18 and 19. An alternate and probably improved form is also proposed for the “electrostatic” term. The suggestion of a second virial coefficient which is a The Journal of Physical Chemistry, Vo;. 77, No. 2, 1973
Kenneth S. Pitzer
272
function of the ionic $strength is, of course, implicit in the much more complex theory of Mayer.17 The preceding argument leads more simply to this same type of expression and thereby should help to clarify the reasons for this ionic strength dependence as well as suggesting simple mathematical forms. We turn now to the choice of equations wbich wi 11 provide improved empirical representation of the properties of electrolytes. While the discussion up to this point should help make the following choices plausible, the final adoption of any particular equation will be based on its empirical effectiveness. Let 11s now s~t u p a system of equations for the thermodynamic properties of pure or mixed electrolytes in rather general form We will later choose the exact form for certain functions ky comparison with experimental data. The total excess Gihbs energy for a solution containing n, kg of solvent and s,,n,, . . . moles of' solute species i,j, . . . is taken to be
(21) Mere f(1)is a function of ionic strength (also temperature and solvent properties) expressing the effect of the longrange electrostatic forces; f ( I ) may have the form of the first term of eq 6 or the similar form which can be obtained by integration Qf eq 18; XEl(Z) is a function of ionic strength with the qualitative behavior indicated by Figure 1 on the second term 011 the right in eq 18 or 19. The effect of short-range forces between species i and j is, of course, the basis for A,](l). We also include a term for triple ion interaction which may be significant at high concentration but igncre any dependence of p l l R on ionic strength. We assume that the X and p matrices are symmetric, i e . , A,, = Allr etc. The equation 2.0
A, = Y3Ay = Y3(2nNodw/1000)1'213'2
(47b) (48)
A,, is the usual Debye-Huckel constant for the osmotic coefficient and all three forms reduce to the usual limiting law a t small I. The first form, DMQ for "DebyeHuckel-osmotic," is that derived above, eq 18 and 19, by use of the Debye-Huckel radial distribution function in the "pressure" equation of statistical mechanics which yields the osmotic pressure directly. The second form, DHC for "Debye-Huckel-charging," is the familiar result obtained by initially calculating the activity coefficient from a charging process and subsequent conversion by the Gibbs-Duhem equation to the osmotic coefficient. The third form is one recommended by Glueckaufl8 which goes smoothly from the limiting law into a cube-root dependence on I at high concentration. The parameter b is to be optimized separately for each form. Two forms were tested for the second virial coefficient
Br' = p ( 0 ) + [P'"/(l
+ ,P'Z)Z]
(49)
Brr*= pi01 - Pi11 exp( - - & / 2 )
(50)
The first form is suggested directly by eq 18 and 19 while the second is considered because it is an especially simple form having the same general properties, namely; (a) finite value a t zero ionic strength, (b) rapid change linear in 11'2 at low ionic strength, and (c) smosth approach to a constant value at high ionic strength. Although in principle all three parameters a , p i * ) , and may be adjusted for each solute, it was hoped that a: might remain the same for broad classes of electrolytes and that there would (18) E. Glueckauf, Proc. Roy. SOC.,Sei. A, 310,444 (1969) The Journal of Physical Chemistry, Vol. 77, No. 2, 1973
Kenneth S. Pitzer
274
TABLE Parameters tor Thermodynamic Functions for Dilute Aqueous Electrolytes at _ I -
Electrolyte
HCI LiCl
0.1802 0.1575 0.0781 0.0460 0.0320 0.0059 ---0.0143 -0.0663 0.5789 0.4869 0.4162 0.1 186 0.0428
NaCl KCI CSCl
NaN03 NH4N03
RbN03
Ca(C10.4z MgClz
CaC12 Na2CrO4 Na2S04 ab
= 1.20,
08
= 2.00,
103
@!I I
pc01
0.2753 0.2811 0.2659 0.2186 0.0273 0.1714 0.1045 -0.0623 2.5883 2.1062 2.2324 1.8765 1.3491
0.6 0.8
0.7 0.4 1.5 0.7 1.5 1.9 1.1 3.3 1.6 2.1 1.6
C'O = 0, range 0-2 M
TABLE I I: Par
