J . Phys. Chem. 1991,95,7465-7471
7465
acids, such a linear relationship is only observed from C, to CB: (pK,, 5: 3.397 0.00546MW. In comparison to the case for the metal complexes, the slope B is very small for the acids. This m a y be related to the exceptionally small relative mass of the proton. The linear relationships between log /Ivalues and molecular weights for inorganic complexes is only observed when a strong ligating group is attached to a long substituent and changes of 4 2 the molecular weight of the substituent do not effect the met04 I al-ligand bond in an electronic sense. A related effect due to 0 100 200 isotope exchange is well known for acidity but the molecular weight dependence on molecular weight for other classes of chemical Figure 8. Plot of pK, (K,= acidity constant) of (A) monocarboxylic compounds was not realized until now. The observed effect cannot acids and (B)dicarboxylic acids vs molecular weight. be explained by a change in average vibrational energies, as simple calculations show that the influence would be too small to be plexes (see Table 111). Obviously, the values of A are different detected. However, the mass correlation may be a result of the for each metal-ligand system and hence should be related to entropy of the ligand solvation increasing with increasing size of intrinsic bond properties. The slopes B fall in a range 0.01-0.03 the ligand. This should be analogous to hydrophobic solvation and a correlation of B with the atomic weight of the metal ion that often accompanies biochemically important reactions. In the exists. case of M(Rldtc)2 the average radius of [R&]- increases as R On the basis of the above discussion, it could be expected that increases. Because of the $ / r dependence on the Born solvation similar relationships can be observed for the reciprocals of the expression this results in a decrease in solvation. As [R,dtc]- is logarithm of the acidity constants (or pK,'s) of organic a c i d ~ . ~ + ~ lless solvated, the equilibrium M(R2dfc)2 P M" + 2[R&tc]- will Figure 8 shows plot of pK, versus molecular weight for alkyl increasingly favor M(R2dtc)z since M(R,dtc), is uncharged. carboxylic acids RCOOH and alkyl dicarboxylic acids HOOAcknowledgment. Many of the mercury and lead compounds CRCOOH. A linear relationship is observed from C2to C9for used in this work were kindly supplied by D. R. Mann and A. F. free n-alkyl carboxylic acids: pK, = 4.313 f 0.004O8Mw (M, Hollenkamp. The financial contribution from the Deakin = molecular weight in the acids). In the case of the dicarboxylic University Research Committee that enabled F.S.to visit Deakin University and undertake this work is gratefully acknowledged. (30)Stability Constants of Metal-Ion Complexes, Part B, Organic Ligands; compiled by Perrin, D. D.; Permagon Press: Oxford, U.K., 1979. (31) Stability Constants of Metal-Ion Complexes; compiled by Sillen, L. (32)Ballinger, P.; Long, F. A. 1. Am. Chem. Soc. 1%9,81, 2347. G., Martell, A. E.;The Chemical Society: London, 1964, Special Publiciation (33) Kingerley, R.W.; LaMer, V. K. J . Am. Chem. Soe. 1941,63,3256.
+
nos. 17 and 25.
(34)Hulanicki, A. Talanta 1967,14, 1371.
Ion-Dipole Model Perturbation Theory Applied to Simple Electrolytes KwongYu Chan Department of Chemistry, University of Hong Kong, Pokfulam Road, Hong Kong (Received: January 11, 1991; In Final Form: April 10, 1991)
An application of the ion-dipole perturbation theory developed by Henderson, Blum, and Tani is attempted. Expressions for the activity coefficient and methods of predicting density changes are reported. These are tested against experimental data of chloride solutions. The serious disagreement shown is mainly due to failure in predicting the reference hard-sphere fluid properties.
Introduction Statistical mechanical theories for electrolytes have been developed according to two Hamiltonian models: the primitive model in which the solvent is a continuum and the ion-dipole model in which the solvent is modeled as dipolar hard spheres. For the primitive model, theories, simulations, and comparison with experimental data have been reported extensively in the past. A perturbation theory was first developed by Stell and Lebowitz.' Waisman and Lebowitz2 and Blum' have solved the mean spherical approximation (MSA) equations. The solution of the hypematted chain (HNC) integral equation and comparison with data were reported by Rasaiah and Friedman.' Card and Valleau5 first reported Monte Carlo simulations for the primitive model. ( I ) Stell, G.; Lebowitz, J. L. J . Chem. fhys. 1968, 48, 3706. (2) Waisman, E.;Lebowitz, J. L. J . Chem. fhys. 1972,56,3086,3093. (3) Mum, L. Theor. Chem. 1980, 5. 1. (4) Rasaiah, J. C.; Friedman, H. L. J. Chem. fhys. 1968,18,2742. (5) Card, D. N.;Valleau, J. P. J. Chem. fhys. 1970,12, 6232.
0022-3654/91/2095-7465$02.50/0
The MSA has been applied by Planche and Renod to correlate data of several electrolyte systems. The ion-dipole model is the simplest nonprimitive model. Only a few theories and simulation results are available in recent years. Both the MSA'** and the HNC9 equations have been solved. Henderson and colleaguesI0 reported a perturbation theory of ion-dipole mixtures. Recently, computer simulations for the ion-dipole system have been reported."-13 The structural (6)Planche, H.; Renon, H.J. fhys. Chem. 1981,85, 3924. (7) Blum, L. Chem. fhys. Lctt. 1974,26,200.Blum, L.J. Chem. fhys. 1974.61,2129. (8) Adelman, S.A.; Deutch, J. M.J . Chem. fhys. 1974,60,3935. (9)Levcsque. D.; Weis, J. J.; Patey, G. J. Chem. fhys. 1980, 72, 1887. (IO) Henderson, D.; Blum, L.; Tani, A. In ACS Symposium Series No. 300, Chao, K. C., Robinson, R. L., Eds.;American Chemical Society: Washington, DC, 1986. (1 1) Chan, K. Y.; Gubbins, K.E.;Henderson, D.; Blum, L. Mol. fhys. 1989,66,299. (12) Caillol, J. M.;Levesque, D.; Weis, J. J. Mol. fhys. 1990, 69,199. Q 1991 American Chemical Society
7466 The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 properties reported are very different from those of the primitive model. A hybrid of the perturbation theory and the MSA theory gives fair agreement with the simulation results.” The H N C theory gives good agreement for the energies but cannot have convergent solutions for realistically high values of charges. Comparison of these ion-dipole theories with experimental data, however, has not been reported. It is necessary to assess the accuracy of the Hamiltonian and to determine whether the more realistic ion-dipole model gives any improvement over the primitive model. In a previous paper,“ the perturbation theory for the primitive model is compared with some experimental data. The activity of an ion can be described by an expression composed of a Coulombic attraction part and a hard-sphere repulsion part. While the theory can fit the data well, the size effect of the ions cannot be explained. This abnormality may be due to the unrealistic treatment of the solvent in the primitive model. It is therefore important to compare the perturbation theory of iondipole mixtures with experimental data to see if a more realistic Hamiltonian will agree better with experimental data. This paper will report a preliminary comparison.
Chan and h!,, are the total pair and triplet correlation functions of hard spheres at the density PI. Using the analytical expressions of the structural properties from the Percus-Yevick equation, Larsen et a1.I6gave expressions for the various integrals in eq 5. The series converges very slowly, and a Pad6 approximation can be formed to speed up the convergence:
where nl, dl, and d2 are constants related to the various integrals in eq 5. Expressions of n,, dl, and d2are given in Larsen et a1.l6 together with details of their evaluation, The primitive model Hamiltonian of eq 4 oversimplifies the effect of the solvent. The simplest nonprimitive model is to treat the solvent as dipolar hard spheres. For the ion-dipole model, the Hamiltonian can be defined as u++(r)= u-.. =
-LA+-
=
e2z1z2 r
Perturbation Theory of Ion-Dipole Mixtures Reviews and discussion of perturbation theories15 are available in the literature. In general, the Helmholtz free energy is expanded into a series of succesive higher order terms of a certain parameter starting from the equation: A - A,, = -NkT In ( Q / Q o ) (1) where A and A. are the free energy of the system and reference system, Q and Qoare the configuration partition function of the system and the reference system, k is the Boltzmann constant, N is the number of particles, and T i s the absolute temperature. The configuration partition function depends on the Hamiltonian of the system: (2) where the integration is over the whole coordinate phase space. The Hamiltonian for the primitive model of electrolytes is
4 = Eu/j(r/j) i oij
(4)
where u,) is the pair interaction energy between i and j , zi, zj are the charge valencies of i and j , rr, is the distance between i and j , e is the charge of an electron, D is the dielectric constant of the medium, and u/= uj = u,, is the case of the restricted primitive model. Perturbation theory of the restricted primitive model (RPM) has been solved by Stell and Leb0witz.I Their result is
where q is the diameter of the ion, Ud is the diameter of the dipole, (Tid = O S ( q Ud), and I,, I,’, I,’!, I, Id, I&, and [Md are the various integrals involving two-body correlations, three-body correlations, intermolecular distance, and angles evaluated at total density of ions and dipoles. The numerical expressions of these integrals are given in Henderson et a1.I0 The inverse Debye length in unit dielectric medium is defined as
+
and the reduced dipole moment is where where i0N
is the inverse Debye length, pi is the number density of the ion i, NI is the total number of ions, pi is the total number densit of ions, AHS is the Helmhotz free energy of hard spheres, and hJI 2 K
(13) Eggebrecht. J.; Ozler, P. J . Chem. Phys. 1990, 93, 2004. (14) Chan. K. Y. J . Phvs. Chem. 1990. 94. 8412. (l5j Henderson, D. Annu. Rev. Phys. Chem. 1974,25,461. Gubbins, K. E.Fluid Phase Eguilib. 1983, 13, 35.
x = -Ni =
N
number of ions total number of particles
(12)
The original series contains some divergent integrals. Equation 9 is obtained after a resummation. The coefficients are chosen to reproduce the linearized MSA solution due to Blum.’ In the (16) Larsen, B.; Rasaiah, J . C.; Stell, G. Mol. Phys. 1977, 33, 987. (17) Chan, K. Y. Ph.D. Dissertation, Cornell University, 1988.
The Journal of Physical Chemistry, Vol. 95, NO.19, 1991 7467
-
Ion-Dipole Model Perturbation Theory limit of y 0 for which case no dipoles are present, the StellLebowitz result of eq 5 is recovered with D = 1 . Equation 9 is a double expansion in terms of and y and represents the ion-ion, ion-dipole, and dipolcdipole interactions. Some of the terms containing K, may be transformed to K by the expansion of the dielectric constant of a dipolar fluid:I0 D = 1 3y + 3y2 ... (13)
+
Differentiating eq 15 and using
aK” = ! ( E ) api
Now K instead of q, appears. The dipolar effect is present explicitly only in the y2 and y3 terms for dipole-dipole interaction. The ion-dipole interaction is present implicitly through D in various terms. This equation has the same leading term as the primitive model expansion of ~5 except that p is now the density of ions plus dipoles and A - A is divided by the total number of particles. Convergence of both eqs 9 and 14 is slow, and Padb can be formed to speed up the convergence. A possible rearrangement” for eq 14 is
PI
and
+
In applying equation 13 to the ion-dipole mixture, the decrease of D with ionic strength is due to diminishing dipole concentration as y decreases with x in eq 1 1 . For small y, the third term in eq 13 can be truncated and D is linear with y or in turn linear with x. This linear dependence of D on ionic strength is in agreement with experimental observation for low-concentration electrolyte solutions. While eq 13 was originally developed for a homogeneous dipole fluid, a more proper equation for the dielectric constant in an inhomogeneous mixture is desirable. We assume here that eq 13 is a good approximation for moderate concentration of ions. After this substitution of eq 13 an alternative to eq 9 is formed:
2
-aY= - = aNd o
api
d~~
we obtain Bpi
-
- BPP =
+
~ ~ ( n,Kq) 1
+ 8np1(l + dlKq + d2K2q2)
n,yn2ut (1
+ d$)pl(l + d4K2.i2)
+ ... (19)
where pI denotes the number density of all ions. Equation 19 now has pI appearing in the denominator instead of p. If we only consider the leading ion-ion term of eq 19, this would be close to the primitive model result. The coefficients nl, dl, d2, and pps, however, are evaluated at the total number density p and not at pr as in the primitive model. The reference chemical potential, #’, can be evaluated assuming a homogeneous hard-sphere fluid or a mixture of hard spheres. This leads to two methods of calculating the chemical potential and activity coefficient. (1) Homogeneous Hard-Sphere Reference. In the ion-dipole perturbation theory, the sizes of ions and dipoles are assumed to be the same. The first method of evaluating pps may be to use the Carnahan-Starling equation1*for a pure hard-sphere fluid: Bpps
= In ( p A 3 ) - 3
3-11 +-
(20) (1 - 7)’ where 11 = (n/6)pu3 and p is the total number density of ions and dipoles. The activity coefficient, ti,can be obtained by taking the limiting function, @pi*,as pI 0 (but p density of pure dipolar fluid) and subtracting this from Bpi: ~ ~ ( nlKq) 1 In ti = @(pl- pi*) = 87rp1(l + dlKq + d2K2q2) nlyK2q2 3-11 3-11, + In (P/Pd) + -(1 + d$)p~(1 + d4K2q2) (1 - d 3 (1 - tol3 (21) where 7, is the original packing factor of the pure solvent. Except for the hard-sphere repulsion parts, this is similar to the primitive model result14 of
- -
+
+
-
njK2uty Pqd(l
+ d4K2q2)(1+ d 9 ) (15)
This equation has three Pad& representing the ion-ion, ion-dipole, and dipole-dipole interactions. The leading term is similar to that of eq 7 for the primitive model. These Pad& have been tested to give convergent behavior. The eight constants nlrn2, and n3 and dl, ..., and ds are chosen so that, upon expansion, the terms in eq 14 will be reproduced. Equation 15 also gives the correct limits when KU 0, KU a,y 0, or y a. In general, perturbation theory of the ion-dipole system results in a twoparameter expansion. Several alternatives exist in resumming, rearranging, and truncating the expansion to achieve convergence and acceleration in convergence. Equations 9, 14, and 15 are some of these alternatives. The physical framework for these approximations and mathematical manipulation is that resulting terms can be identified to represent ion and dipole effects and limiting behavior will agree with pure ionic and pure dipolar behavior. The ultimate test of these ad hoc approximations will be comparison with experimental results.
- - - -
Chemical Potential Expressions from Perturbation Theory The chemical potential of ions can be obtained by differentiating the total free energy with respect to Ni,the number of ion species I:
(22) where pI and qI are those of a system with only ions present. In eq 21, only the first term is negative and the remaining terms represent the repulsive contribution due to a change in packing when ions are added to the dipoles. This contribution is positive since we would expect an increase in density when ions are added. In the primitive model or eq 22, the repulsion is due to direct hard-sphere interaction in an unrealistically low-density system. (2) Hard-Sphere Mixture Reference. While the perturbation theory is developed with the assumption of equal-size ions and dipoles, we recognize that almost all the aqueous electrolytes have ionic diameters that are different and different from that of water. Since the hard-sphere chemical potential in a mixture is affected by the dissimilarity in size, a correct expression for mixtures should be used instead of the one-component hard-sphere expression. Several equations are available for the chemical potential of a (18) Eyring, H.; Henderson, D.; Stover, B. J.; Eyring, E. M. StafIsfJcd Mechanics and Dynamics; J. Wiley and Sons: New York, 1982; p 268.
7468 The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 species in a mixture of hard spheres. A simple expression is due to Percus-Yevick theory and is obtained through the compressibility route:I9
where
is the pressure of the hard-sphere fluid
r" = (*/6)CPPr I
(25)
hi is the thermal de Broglie wavelength of i, qj"' is the partition
function due to the internal coordinates of i, and {,, is the density-averaged moments of diameter. Other equations of state for mixtures, for example, the Mansoori et aLm equation, can be used. In the limit of pi --* 0, the reference chemical potential of eq 23 becomes
Chan pole-dipole interaction does not contribute to the chemical potential of ions as shown in eq 18. The ion-dipole interaction should be negative and smaller than the ion-ion interaction in magnitude. The ion-dipole energy is shown to be small compared to the ion-ion term in the computer simulation studies." The only positive contribution comes from the hard-sphere repulsion. From the experimental data of common electrolytes, activity coefficients are much greater than unity beyond moderate concentrations. The prediction of the hard-sphere interaction term appears to be very important. In the nonprimitive model, even using only the ion-ion term, the packing effect can be more realistic and the effect of the dipole size is included in the overall packing factor. The nonprimitive model results of eqs 21 and 27 offer alternatives to the primitive result14 for the hard-sphere interaction. We present a simple application where only the leading ion-ion interaction term is considered with the hard-sphere interaction. Further improvements may be possible by considering the other terms of ion-dipole and dipoldipole interactions. In the previous the Stell-Lebowitz expansion with Pad6 (SL6P) is found to be give results very close to those of the Debye-Huckel theory.z1 To further simplify, we can use the Debye-Huckel result: K3(1
u3 N
+ KQ)
(30)
as the ion-ion interaction term. In using eqs 21 and 27, it is necessary to know the density of the mixture. This can be determined by assuming a constant-pressure mixing process when the ions are added to the dipoles. Neglecting the dipole-dipole interaction, the initial pressure is known from the Percus-Yevick equation of a one-component hard-sphere fluid:
(-$lo[
-TP =6kT Subtracting eq 26 from eq 23 and taking only the leading term of eq 19, the activity coefficient is In ti = -
+ nlKq)
8 ~ p I1( + d l q + d z ~ 2 q 2 ) 8*pdl
K3(1+ nlKq) 8Tpl(l
+
dlKUi
+ 1 n ( 2 ) +
70 (Td3
(1
+ 70 + vo2 + vO3) (1 - 70)'
(31)
where qo is the known packing factor of the pure solvent. This pressure can be equated to the equation for the mixture of hard spheres in eq 24:
+ dzKzq2)
+ (27) and the mean ionic coefficient is In Ft = 0.5 In (F+[-) (28) In effect, eq 27 is the result of a semiprimitive model in which the ions interact in a dielectric medium but the structure of the solvent is at the same time represented by neutral hard spheres. The Henry law constant, Ki, is the limiting chemical potential of species i at infinite dilution and is
Application of the Ion-Dipole Perturbation Theory In the previous work,I4 the ion-ion interaction term and the hard-sphere repulsion term were compared for the primitive model in eq 22. At moderate concentrations, the hard-sphere term increases rapidly and dominates. In a nonprimitive model, di(19) Reed, T. M.; Gubbins, K. E. Applied Statistical Mechanics; McOraw-Hill: New York, 1973; p 258. Lebowitz, J. L. Phys. Reu. 1964, 133, A 8 9 5 Lebowitz, J. L.; Rowlinson, J . S. J . Chem. Phys. 1964, 41, 133. (20) Mansoori, G. A.; Carnahan, N . F.; Starling, K. E.; Leland, T. W. J . Chem. Phys. 1971,34, 1523.
and l3and p can be solved a t a given pr. Equation 32 only accounts for the change of packing due to changes in the sizes of hard spheres. The attractive forces between the ions should also lead to a change in density. The effect of attractive interaction on pressure can also be included by differentiating eq 14 or 15 with respect to volume. Considering only the leading term as in the Debye-Huckel equation, this excess pressure is
-n P =X- L ( ! 2 2 ? ) 6kT 247rpl 1 + K U
(33)
Assuming the total pressure is the sum of the hard-sphere term and this excess term, the density can be determined by TPX+ -*P"S =-
6kT
6kT
*Po 6kT
(34)
Comparison with Experimental Data Experimental data of simple chloride solutions reported in Robinson and Stokes2zand Parsonsz3 are used to test the equation for activity coefficients. The conversion of data from the molality scale to molarity scale is performed in ref 14. In the previous comparison of primitive model theories with data,I4 the SL6P and the Debye-Huckel plus hard-sphere model are found to fit the (21) Debye, P.; Huckel, E. Phys. Z . 1923.24, 185.
(22) Robinson, R. A.; Stokes, R. H. Electrolytes Solurions; Academic Press: New York, 1955. ( ~ 3 Parsons, ) R. Handbook of Elecrrochemical Consranis; Butterworths: London, 1959. (24) Lange, N . A. hnge's Handbook of Chemistry; McGraw-Hill: New York, 1987.
The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 7469
Ion-Dipole Model Perturbation Theory
0.50 I
1 0.5 1
+o
+
+ ++o
b
+-
rd
%
/
....... LlCl
CSCl data
A
KCI data
0
LlCl data
0
/ /
E
....... LlCl
0
-KCI
eqn(34)
-KCI
0.49
eqn(34)
-_--
LlCl eqn(34)
eqn(34)
Y
rd Q
eqn(34)
CSCl eqn(34)
_..-.KCI 0.48
eqn(32) CSCl eqn(32)
CSCl eqn(34)
0.47
0.47 0
1
2
3
4
5
6
7
8
9
0
concentration (moles/liter)
0
LlCl data
+
CSCl data
A
KCI dal a
0
3.60
0
0
iij
0
0
3.50
_e.
...
3.40
7
8
9
1.oo
eqn(34)
3.30
-__-
CSCl eqn(34)
1
2
3
4
5
6
7
8
(35)
and
+ [ d - c(MW)]/(MW
Of
0
CSCl data
lines are results
-0.50
density data
0.5
1.0
sqrtk)
data well except the best fitted ionic diameters are unrealistic and inconsistent with the Pauling diameters. While an ion-dipole theory is expected to be a more physically reasonable theory, approximations used in the application can lead to deficiencies. A preliminary test is performed here to understand the behavior of the leading terms and to identify the major deficiencies of the model. Rediction of Der&@ chmse& The hard-sphere repulsion which rises sharply with density is an important contribution to the activity at high concentration. The determination of density changes using eqs 32 and 34 is tested by comparison with experimental density data. Using the Pauling diameters for ions and water, the packing factors and number densities of electrolyte solutions are calculated from the density data?
1000)
KCI data
CSCl
0.0
9
Figure 2. Variation of total number density in chloride solutions.
(cNA)(
A
-KCI
0.00
- 1.oo
concentration (moles/l i ter)
PI
LlCl data
of eqn(271 with
t
0
+
....... LlCl
eqn(341
-KCI
{C
6
0.50
....... LlCl
p
5
1.50
0 0
3.10
4
2.00
3.70
3.20
3
Figure 3. Prediction of packing factors by two methods.
3.80
*
2
concentration (moles/liter)
Figure 1. Variation of packing factors of chloride solutions.
N
1
H20)I(NA)(1000) (36)
1.5
2.0
2.5
3.0
(molar**O.5)
Figure 4. Comparison with activity coefficient data of chlorides. [Lines are results of cq 27 with density data.]
where d is the density of the solution in grams per liter, c is in moles of salt per liter, N A is the Avogadro number, and pI is the The different tn number density of anions and cations in can then be determined from eq 25. Table I gives the change of p and f3 with concentration for three chloride solutions. The corresponding reduced density is high, with a minimum value of 0.9 for the pure solvent state. The packing factor and number density are plotted in Figures 1 and 2, respectively, as functions of concentration of salts. A 10% increase in packing is shown from zero to 8 M concentration. For the cesium chloride solution, the number density decreases since both chloride and cesium ions have diameters larger than that of water. For lithium chloride, the average ionic diameter is smaller than that of water and the number density increases. Although the average ion diameter of KCI is larger than the diameter of water, the number density in KCI solution increases slightly with concentration. The overall packing factors of all three solutions, however, are almost the same in the range of concentrations. Equation 34 underestimates the
7470 The Journal of Physical Chemistry, Vol. 95, No. 19, I991
Chan
TABLE 1. Packing Factors of Chloride Solutions Determined from Density Data (Solvent Diameter Is 3.0 A, MW = 18.011) c3
concn of salt c, mol L-I
packing factor
o.oO0oo 0.1 I807 0.23682 0.47637 0.7 1866 0.96368 1.21132 1.46179 1.71483 1.97057 2.22920
0.47 137 0.47273 0.47357 0.47522 0.47686 0.47849 0.48006 0.48167 0.48321 0.48475 0.48631
0.13474 0.27 1 19 0.54930 0.83442 1.I2661 1.42610 1.73305
0.47294 0.47401 0.4761 2 0.478 17 0.4801 7 0.48216 0.4841 3
0.02976 0.05975 0.08995 0.12042 0.15 I07 0.18 I99 0.24463 0.30807 0.37278 0.50506 0.64166 0.78276 0.92859 1.07936 1.23530 1.39669 I S6378 1.73687 1.91653 2.10270
0.47209 0.47237 0.47257 0.47288 0.47300 0.47327 0.47391 0.47422 0.47494 0.47598 0.47703 0.47810 0.47919 0.48029 0.48141 0.48254 0.48370 0.48488 0.48614 0.48739
2.00
1.50
1.00 ti
.llF
C -
0.50
= ( * / ~ ) I P I ( O ++ ~ g-? 4-
number density p , mass density concn of salt 102' m-3 d, g L-l c, mol L-' (A) LiCl (cation diam = 1.20 A. anion diam = 3.62 A. '3.33427 . 0.99723 2.49057 3.34519 1.00120 3.02123 3.35235 I .004 IO 3.56340 1.00990 4.1 1698 3.36664 1.01570 4.6821 2 3.38086 1.0215 5.25943 3.39503 1.0272 5.84972 3.40881 3.42286 1.0330 6.45226 1.0387 7.06844 3.43652 3.45012 1.0444 7.69868 1.0502 8.34269 3.46400
c3
(B) KCI (cation diam 2.66 3.34172 1.0046 1.0110 3.34553 3.35283 1.0239 3.35955 1.0369 3.36570 1.0500 3.37158 1.0633 3.37716 1.0768
+ I
1 t-
+
2.04761 2.36974 2.70024 3.03863 3.38557 3.74131
I
+ I1
+
+
LlCl
A
KCI
data 0
+
CsCl data
....... LlCl
-KCI
0.00
---_CsCl -0.50
0.0
, , ,
,
, , , ,
0.5
sqrt(c)
,,,
1.0
, ,
,
, , , ,
1.5
,
I ,
2.0
,
,
,
,
,
,
2.5
, ,
lines are re%i+,S of eqn(27) with
lbnny
W.bOId
by
04
3.0
(molar**O.5)
Figure 5. Prediction of activity cocfficients of chlorides. [Lines are results of eq 27 with density predicted by eq 34.1
c3
0.48609 0.48799 0.48996 0.49187 0.49379 0.49573 MW = 168.37) 0.48852 0.48995 0.49122 0.49173 0.49254 0.49384 0.495 18 0.49654 0.49789 0.49874 0.49929 0.50074 0.50257 0.50370 0.50529 0.50697 0.50873 0.5 1062 0.5 1268 0.54197
mass density d, g L-l 1.0565 1.0675 1.0792 1.0910 1.1029 1.1154 1.1274 1.1399 1.1527 1.1658 1.1791
3.38243 3.38707 3.39196 3.396 16 3.40027 3.40425
1.0905 1.1043 1.1185 1.1328 1.1474 1.1623
3.29152 3.287 19 3.28136 3.27737 3.27520 3.26835 3.26110 3.25328 3.2447 3 3.24125 3.23570 3.22618 3.21830 3.20503 3.19370 3.181 89 3.16949 3.15670 3.14368 3.13068
1.28817 1.3 1350 1.33930 1.35218 1.36610 1.39380 1.42260 1.45250 1.48350 1.49993 1.51580 1S4950 1.58575 1.62120 1.65960 1.69990 1.74220 1.78680 1.83400 1.88420
packing factors and densities, which are shown in Figures 1 and 2 for comparison. The packing factor predicted by eq 34 increases with the size of the ions whereas the data show the opposite trend. This discrepancy of predicting packing factor and number density may be due to neglecting ion-dipole or dipole-dipole interactions or the failure of the ual diameter assumption. To compare the relative effects of PH and P", the packing factors determined by eqs 32 and 34 are compared in Figure 3. The results from the two methods differ by less than 2% and show the same dependence on the size of ions. This small difference, however, does affect the final prediction of the activity coefficients. The size dissimilarity seems to contribute to most of the increase in packing. Activity Coefficient. Once the packing factor is known as a function of concentration of the electrolyte, the number density of individual ions and dipoles can be determined. The various {,, are known, and eq 27 can be used to calculate the activity coefficient for the anion and cation. Figure 4 shows the prediction of the activity coefficient using eq 27 and density data. Figure 5 shows the In tt calculated with using eq 27 by the density determined by eq 34, which is compared with experimental data. The sizes of ions used are the Pauling diameters in Table I. In Figure 4, the error in the prediction of In [* for LiCl is about 50% at high concentrations. The aidependence of In & predicted by eq 27 shows a trend opposite to that of the data. The disagreement of the theory's prediction and data of KCI and CsCl is severe. On the other hand, by use of the densities predicted by eq 34, the theory underestimates the activity coefficients, as shown in Figure
9
data
+ + +
-
+
packing number density p, factor m-' MW = 42.40) 0.48787 3.47781 3.50495 0.49089 0.49396 3.53250 0.49703 3.56015 0.50009 3.58788 0.50319 3.6 1602 0.50637 3.64489 0.50954 3.67384 0.51279 3.70349 0.51613 3.73385 0.51949 3.76458
A, anion diam = 3.62 A, MW = 74.56)
(C) CsCl (cation diam = 3.38 A. anion diam = 3.62 A, . 3.33722 2.29525 1.0020 3.33709 2.49641 1.00593 3.33632 2.7045 3 1.00970 2.81085 1.01374 3.33635 2.9 209 2 1.O 1740 3.33503 3.33471 3. I457 1 1.02140 3.37970 1.02969 3.33472 3.62327 1.03740 3.33242 3.87682 1.04609 3.33284 4.00884 I .06297 3.33074 4.141 28 3.32844 1.08036 4.41741 1.09828 3.32591 4.7091 2 1.11676 3.32317 5.00697 1.3582 3.32018 1.1 5549 5.32271 3.31694 1.17580 5.65388 3.31343 1.I9679 6.00152 3.30967 1.21849 6.36740 3.30563 6.75346 1.241 IO 3.30173 7.16213 1.26440 3.29725
[
1, - 1.oo
PdUd']
J . Phys. Chem. 1991, 95, 7471-7477
5. Again, the dependence on ui shows a trend opposite to that shown by the data. By use of different methods of obtaining the number density of the mixture, the theory applied as in eq 27 either overestimates or underestimates the experimental activity coefficients. Similar to the primitive model, a careful choice of best fitted diameters of ions or density variation might result in a better fit of the experimental data. This, however, will result in unrealistic diameters since the theory now gives the wrong dependence on q. The ion-dipole theory as applied does not give results better than the primitive model theory. The poor agreement of the theory with experimental data can be accounted for by a number of factors: (a) model Hamiltonian; (b) approximations used in theory; (c) errors in predicting density changes; (d) errors in the calculation of the chemical potential of the hard-sphere reference. While the ion-dipole Hamiltonian is simplistic and neglects electric multipole moments, short-range forces, shape, and quantum mechanical effects, it should be more realistic than the primitive model. The major errors, therefore, must lie in other areas. The leading terms in ion-dipole theory should degenerate to the primitive model and give similar results at low concentration. 'The major deficiency probably lies in the equal-size diameter assumption and the failure to predict density changes. A computer simulation study of unequal-diameter Lennard-Jones mixtures by Shing and G u b b i n ~shows ~ ~ that the Henry law constant has a
7471
minimum a t u , , / ~= 1 in a binary mixture. Of the many theories for hard-sphere mixtures tested in ref 25, only a few give the same result. Assuming this minimum activity at equal diameter of LJ spheres applies to an ion4ipole mixture, then the theory may show correct ~i dependence. Improvements of the ion-dipole theory can be achieved by better theories of the reference mixture fluid.
Conclusions The perturbation theory for equal-diameter ion-dipole mixtures developed by Henderson et al. is applied for the first time to simple electrolytes. Analyses from the resulting expressions of chemical potential show that the packing of the hard spheres in the mixture plays a significant role. Increase in activity of ions at high concentrations is due to an increase in the overall packing factor of the mixture of hard spheres. A simple application of the theory, however, does not predict satisfactorily the experimental data of some simple chloride solutions. The most probable deficiency lies in the treatment of size effect in the ion-dipole mixture. Future work should be directed toward this problem besides treating long-range interactions. Computer simulation as well as theoretical work of unequal-diameter ions and dipoles should be performed to further investigate the effect of size. (25) Shing, K. S.; Gubbins, K. E. Mol. Phys. 1983,49, 1121.
Potentials of Mean Force in Charged Systems: Application to Superoxide Dismutase C. E. Woodward* and Bo R. Svensson Department of Physical Chemistry 2, Chemical Center, Lund, 27-221 00 Sweden (Received: January 22, 1991) A recently developed modification of Widom's test particle method, for calculating single ion chemical potentials, is used with Monte Carlo simulations to determine the potential of mean force between a superoxide ion and the enzyme superoxide dismutase in electrolyte solutions. A simple spherical model is used to model the enzyme, and the consequences of a low dielectric interior for the sphere are investigated. We find that traditional applications of the linearized Poisson-Boltzmann (Debye-Hackel) approximation qualitatively fail in the presence of a dielectric discontinuity. An analytic correction to the Debye-Huckel theory is derived and gives good agreement with simulations.
1. Introduction
The past 5 years or so has seen outstanding progress in the experimental techniques used to study proteins and other large biomolecules.' These breakthroughs have already had a profound affect on areas as diverse as phamaceuticals, forensic science, catalytic chemistry, and the detergent industry, to name just a few examples. Despite this success in the experimental arena, progress in the theoretical understanding and modeling of biomolecules has been rather modest. The most ambitious attempts to theoretically model biomolecules have involved detailed descriptions of the constituent atoms and their mutual interactions coupled with computer simulations? Modeling of a biomolecule in situ also requires representation of the surrounding electrolyte solution. Unfortunately, these simulations have proved to be very time-consuming, and their further development, and usefulness, is intrinsically linked to the development of fast and large computers. The formidable computational demands of detailed models have made simpler descriptions of biomolecules more attractive. Such a model was already introduced some 30 years ago by Tanford and Kirkwood3 in their classic study of protein acid constants. They described the protein as a spherical region of low dielectric constant, immersed in a higher dielectric medium (water), con-
* To whom correspondence should be addressed. Permanent address: Department of Chemistry, University College ADFA, Campbell ACT, Australia 2600. 0022-3654/91/2095-7471$02.50/0
taining salt. The linearized Poisson-Boltzmann, also known as the Debye-Huckel, approximation (DHA) was solved analytically and was used to calculate the electrostatic binding free energies of protons in the protein. The importance of electrostatic interactions in many biomolecular functions has also been emphasized in later studies. Recently, attempts have been made to improve the Tanford-Kirkwood model by including more of the structural details in the description of the biom~lecule.~These have required more complicated numerical techniques to solve the DHA; however, the essential features of the model remain the same. Simple electrostatic models have also been used in computer simulation^.^ The fact that these models are successful in some applications is no doubt linked to the long-ranged, slowly varying nature of the Coulomb potential which is rather insensitive to details of the biomolecule structure. Recently, a simple electrostatic model was used to study the rate constant for the reaction between the superoxide ion 0;with the enzyme superoxide dismutase (SOD).- Experimental work ( 1 ) Linse, S.;Brodin, P.; Johansson, C.; Thulin, E.; Grundstdm, T.; Forstn, S.Nature 1988, 335, 651.
(2) Ahlstrom, P.; Teleman, 0.; J h s o n , B. J . Am. Chem. Soc. 1988,110, 4198. (3) Tanford, C.; Kirkwood, J. G. J . Am. Chrm. Soc. 1957, 79, 4198. (4) Klapper, I.; Hagstrom, R.; Fine, R.; Sharp, K.; Honig, B. Protclns 1986, I , 47. ( 5 ) Svensson. B.; Jonsson, B.; Woodward, C. Biophys. Chrm. 1990, 38, 179. (6) Sharp, K.; Fine, R.; Honig, B. Science 1987, 236, 1460.
Q 1991 American Chemical Society
