7478
J . Phys. Chem. 1989, 93, 7478-7482
The reason why sample 3 deviates at higher temperatures may be due to differences in the temperature dependence of S. In the analysis above we have tacitly assumed that the temperature dependence in S is equal in the three samples. Now sample 3 is close to the lamellar phase. In fact, at temperatures above 75 OC a lamellar phase is formed with a I4N quadrupolar splitting of 19.4 kHz (at around 100 "C). The splitting, and thus S, increases slowly with increasing temperature. In the hexagonal phase, the order parameter as determined from the I4N quadrupolar splitting is close to the one found in the lamellar phase. However, the temperature dependence of the order parameter in the hexagonal phase is opposite to that in the lamellar phase. Inasmuch as the order parameter in the cubic phase varies over the region and bears some resemblence to the values found in the neighboring anisotropic phases one would thus expect a slightly varying temperature dependence of S as the concentration is increased over the cubic phase region. Another possibility is that the quadrupolar interaction constant x changes. As argued above, the cubic phase is poor in water and the number of water molecules per surfactant is not even enough to fully hydrate the counterions, much less to hydrate the polar head-groups. It is thus conceivable that the counterions can come in close proximity to the charged nitrogen and thus give a intermolecular contribution to the quadrupolar coupling constant, a contribution that may very well vary with concentration and temperature. The fact that R , does not vary in the same way as R2 rules out the concentration dependence of x. To investigate the possible temperature dependence of x,we have performed a 2H relaxation study of a sample with the composition 86.3% which is made up of a C,,TACl molecule deuterated on the methylene group adjacent to the nitrogen. The quantity In (AR)for both 14N and ZHare presented in Figure 5. It is clear that the two nuclei display a very similar temperature
3.0
1
I
3.2
3.4
lOOO/T (IiK)
Figure 5. Natural logarithm of the difference between R2and R I for 2H and 14N at Bo = 2.35 T for a sample of concentration 86.3%plotted vs the inverse of the absolute temperature.
dependence. It seems unlikely that intermolecular contributions would display the same temperature dependence for both I4Nand 2H and thus we rule out this possibility. The reason why the data sets of I4N and *H give almost the same values of AR is presumably due to the fact that the product XS happens to be almost equal in the two cases. An indication that this is indeed the fact is given by the splittings in the hexagonal phase, which are 12.3 kHz for 2H and 12.1 kHz for 14Nfor samples with nearly identical compositions at 28 OC.lo In conclusion the relaxation data are compatible with the two-step model where the slow motion is the lateral surfactant diffusion over the cubic unit cell. Regisby NO. CIZTACI, 112-00-5.
Explicit Approximation for the Unrestricted Mean Spherical Approximation for Ionic Solutions C. Sanchez-Castro and L. Blum* Department of Physics, College of Natural Sciences, University of Puerto Rico, Rio Piedras, Puerto Rico 00931 (Received: January 24, 1989; In Final Form: April 27, 1989)
A new scheme for approximating the MSA solution for the general primitive model of electrolytes is proposed. This scheme is based on the successive estimation of a mean diameter, with use of first a simple weighted average of the diameters of the ions in solution and then a Newton-Raphson improvementof this first estimate. The results for a large variety of situations of different concentrations and size ratios are quite good.
I. Introduction The mean spherical approximation (MSA) is a simple analytical theory for electrolytes that is very useful in representing the thermodynamic properties of ionic solutions over a wide range of concentrations.'v2 Just as in the classical Debye-Hiickel (DH) t h e ~ r ythe , ~ properties of the solutions are represented in terms of one simple parameter: r, and the charges, diameters, and concentrations of the ions, even for complex mixtures involving many different kinds of ions. Because the MSA is, essentially, the correct way of including the size effects of the ions into the DH theory, it is more accurate than the latter. (1)Blum, L.Mol. Phys. 1975,30, 1529. (2)Blum,L. Theor. Chem. (N. Y.)1980,5, 1. (3) Debye, P.; Huckel, E. Phys. 2.1923,24, 185. (4) Blum, L.;Hoye, J. S. J . Phys. Chem. 1977,81, 1311.
0022-365418912093-7478$01.50/0
In the MSA for the primitive model of electrolytes, the Ornstein-Zernike5 equation is solved for particular boundary conditions, which are equivalent to the DH theory when the diameters of the ions are vanishingly small and which become the very successful Percus-Yevick theory6 when the charges are turned off. In spite of the complexity of the mathematical solution of the MSA for the general ionic mixture, the remarkable result is that one gets the same expressions for the thermodynamic properties as in the DH, but with the new screening parameter
r.
One of the inconveniences of the MSA is that, for the general mixture, the parameter r has to be obtained by solving an al(5)See for example: McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1973. ( 6 ) Percus, J. K.; Yevick, G . Phys. Rev. 1958,110, 1.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 21, 1989 7479
Explicit Mean Spherical Approximation for Ionic Solutions gebraic equation of high order. For this reason, several approximate but explicit expressions for r have been proposed It is the purpose of this paper to give a convenient and rather accurate expression for^ r, which is obtained by a method described in an earlier communi~ation.~ In the equal-size case,I0 this parameter is related to the Debye screening parameter K by the relation
r = [(I + 2KU)'/2-
1]/(2U)
(1.1)
where u is the diameter of the ions. The Debye parameter defined by
K
-cPid
is
Here, /3 = l/(kBT), where kBis the Boltzmann constant, Tis the temperature, e is the elementary charge, t is the dielectric constant of the solvent, pi and zi are the number density and the electrovalence of the ith ionic species, respectively. 2 r satisfies the nonlinear equationlv2 4r2 = -Cpi 4r/3e2 2
(-)
c
Furthermore, for low concentrations, an estimate for u is given by9 (To
(2.4)
-ci Pi$
(2.5a) where
i
(2.5b) dr
-)+rui
L-c(
-.( ")+
= 1 + 2A
1
i
dzir dr
rui
This equation can be solved by iteration starting with the initial guess 2r N K . Several approximation^'^ have been proposed that express 2I' in a closed form. Harvey, Copeman, and Prausnitz* studied the accuracy of two of these approximations compared with the exact MSA. However, their study was limited to twocomponent systems having opposite charges. In this paper, we compare the most accurate (at low concentrations) of the two approximations studied by Harvey et al., namely, the approximation of Copeman and Stein,' with two new approximations, A1 and A2 for r derived from a more accurate estimation of the mean diameter u. 11. Approximations for 2r In earlier work,9 we showed that solving eq 1.2 is equivalent to solving 4172 =
K'
(1
1
-+
PiZPi
1
Q Q
i -
This estimate turns out to be quite good. In the present work, we find the first correction to the mean diameter estimate using the Newton-Raphson method for eq 2.2
.u;P, zir(r) = zi - 2A P, =
=
.P,
piziu;
TU;
2AQ
(2.5~)
i
(1
+
-
--c 2A
pia; i
(1
(2.5d)
+
We will call this the A1 approximation. Certainly the exact value of u can be obtained by iterating (2.5a). This is necessary only in extreme cases. For ionic solutions, (2.5a) is quite accurate. A considerably simpler formula, which is valid when ai are not too large (excludes colloids) and pi are small, is obtained by assuming zir(r)= zi in eq 2.2. Explicitly, this approximation, which we will call the A2 approximation, yields
(-)
-cPiZ?[ i
-11
roui(i+ rouo)2
(1
+ 2KU0)'I2 Kuo )FPiZ?( (1 +
+ 1
+
(2.6)
(2.la)
+ ru)2
Once 2 r is obtained, the thermodynamic excess properties are given by the full MSA f o r m ~ l a s for ~ , ~the unequal-size case.
or equivalently
r = [(I + 2KU)1/2- 1]/(2U)
(2.lb)
with a mean ionic diameter u given by the solution off(u) = 0, where
f(.) =
1
(2.7b)
+ ru
Solving this equation explicitly for u gives (7) Copeman, T. W.; Stein, F. P. Fluid Phase Equilib. 1986,30,237; Ibid. 1987, 35, 165. ( 8 ) Harvey, A. H.; Copeman, T. W.; Prausnitz,J. M. J . Phys. Chem. 1988, 92, 6432. (9) Blum, L. J . Phys. Chem. 1988, 92, 2969. (IO) Waisman, E.; Lebowitz, J. L.J . Chem. Phys. 1970,52,4307; 1972, 56, 3086, 3093.
(2.7~) P = EPi i
-(
-)
~ i c x AEi P,u~ 7ru;a2pn keT = kBT - 4A ra, + 12A
(2.7d)
7480 The Journal of Physical Chemistry, Vol. 93, No. 21, 1989
AEi -.=kBT ai =
a2ziNi 4~
a2
+ rail
2r(1
Ni = ai
41r@e2 =-
a2
t
The approximation for 2 r proposed by Copeman and Stein7 (CS) is given by 4r2 = a
2 ~ p i ~ ; i
(2.8)
where
Sanchez-Castro and Blum TABLE I: Values of u for Various Molar Concentrations (c) for a 1-1 AQUWUSElectrolyte with c = 78.5 and T = 25 O C " c ar MSA A1 A2 Sin cs 1.9676 4.20 0.800 3.767 41 3.767 39 3.754 81 3.78000 2.901 18 1.9676 4.20 0.600 3.298 80 3.298 23 3.25 1 43 3.36000 2.497 19 1.9676 4.20 0.400 2.771 32 2.766 38 2.670 97 2.94000 2.003 34 1.9676 4.20 0.100 1.78833 1.72581 1.55480 2.3 1000 1.21661 1.moo 4.20 0.800 3.768 92 3.768 90 3.758 77 3.78000 3.17941 1.0000 4.20 0.600 3.307 13 3.306 74 3.269 47 3.36000 2.76043 1.oooo 4.20 0.400 2.797 79 2.794 68 2.71931 2.94000 2.264 44 0.4250 4.20 0.800 3.77087 3.770 86 3.763 43 3.78000 3.43374 0.4250 4.20 0.600 3.31753 3.31732 3.290 25 3.36000 3.00077 0.4250 4.20 0.400 2.82903 2.82746 2.773 19 2.94000 2.51044 0.1000 4.20 0.800 3.77404 3.77403 3.76991 3.78000 3.661 53 0.1000 4.20 0.600 3.333 25 3.33320 3.31836 3.36000 3.221 83 0.1000 4.20 0.400 2.872 77 2.872 39 2.842 93 2.94000 2.752 19 1 .oooo 3.62 0.525 2.691 38 2.69064 2.656 74 2.760 25 2.31246 0.4000 3.62 0.525 2.707 54 2.707 18 2.683 55 2.760 25 2.50007 0.1Ooo 3.62 0.525 2.727 83 2.127 74 2.7 1469 2.76025 2.652 18 1.ow0 3.78 0.275 2.205 60 2.19821 2.12222 2.409 75 1.81023 0.4000 3.78 0.275 2.258 20 2.254 97 2.202 26 2.409 75 2.029 12 0.1000 3.78 0.275 2.31965 2.31892 2.28996 2.409 75 2.227 57 3.0000 3.60 0.833 3.29095 3.290 93 3.284 11 3.29940 2.49057 4.0000 3.60 0.833 3.290 52 3.29051 3.28309 3.29940 2.367 31 5.0000 3.60 0.833 3.290 19 3.290 18 3.282 30 3.29940 2.265 50 "Here, a- is the anion diameter in angstroms and r = a+/a-. Throughout these tables, MC, MSA, Al, A2, Sig, and CS stand for Monte Carlo, mean spherical approximation, A1 approximation, A2 approximation, Sig approximation, and Copeman-Stein approximation, respectively.
s,*
= Cpizi".: i
The thermodynamics in this approximation8 is calculated from (2.9a)
(2.9b)
TABLE 11: -EeX/INkaTlfor the Electrolyte Described in Table I c ar MC" MSA AI A2 Sia 1.9676 4.20 0.800 0.690 f 0.002 0.679 0.679 0.680 0.679 1.9676 4.20 0.600 0.734 f 0.003 0.731 0.731 0.732 0.728 1.9676 4.20 0.400 0.789 f 0.002 0.800 0.800 0.805 0.792 1.9676 4.20 0.100 0.893 f 0.002 0.970 0.975 0.990 0.933 1.OoOo 4.20 0.800 0.581 f 0.002 0.573 0.573 0.573 0.572 1.moo 4.20 0.600 0.613 f 0.002 0.609 0.609 0.611 0.608 1 .oooo 4.20 0.400 0.657 f 0.002 0.657 0.658 0.660 0.652 0.4250 4.20 0.800 0.455 f 0.002 0.447 0.447 0.447 0.447 0.4250 4.20 0.600 0.475 f 0.001 0.470 0.470 0.470 0.469 0.4250 4.20 0.400 0.507 f 0.002 0.498 0.498 0.499 0.496 0.1000 4.20 0.800 0.283 f 0.001 0.272 0.272 0.272 0.272 0.1000 4.20 0.600 0.295 f 0.001 0.281 0.281 0.281 0.281 0.1Ooo 4.20 0.400 0.307 f 0.001 0.291 0.291 0.291 0.290 1.0000 3.62 0.525 0.677 f 0.004 0.664 0.664 0.666 0.662 0.4000 3.62 0.525 0.511 f 0.003 0.493 0.493 0.494 0.492 0.1000 3.62 0.525 0.316 f 0.002 0.293 0.293 0.293 0.293 1.oooo 3.78 0.275 0.725 f 0.003 0.721 0.721 0.724 0.712 0.4000 3.78 0.275 0.546 f 0.002 0.523 0.523 0.524 0.519 0.1000 3.78 0.275 0.332 f 0.002 0.303 0.303 0.304 0.302 3.0000 3.60 0.833 0.805 0.805 0.806 0.805 0.859 0.859 0.859 0.858 4.0000 3.60 0.833 0.900 0.900 0.901 0.900 5.0000 3.60 0.833
CS 0.708 0.760 0.831 0.996 0.587 0.624 0.671 0.453 0.475 0.502 0.273 0.281 0.291 0.676 0.496 0.293 0.731 0.524 0.303 0.841 0.904 0.955
From ref 1 1.
where
dB,
B,zt
api
2
-=-
(
1
aa;S2,2-lf2
S2,o
1 + aS2,2If2
--
111. Discussion of the Results We have studied the approximations A1 and A2 over a large variety of cases and compared the mean diameters and thermodynamics to the results of Harvey, Copeman, and Stein8 for the
systems that they have studied. We also show the simulations of Abramo et a1.I' and Rogde12 where these simulations are available. But, our aim is to provide a fast and easy way to compute the thermodynamics in the MSA, even in the case of complicated mixtures. We restrict ourselves to the ionic solution regime, although we suspect that the approximations such as A1 will be satisfactory even for the very dense ionic mixtures such as the molten salts. We should remark also that the approximations are completely explicit and that no iteration is required to find the mean diameter 0,or the screening parameter rl. However, the thermodynamic properties are always computed with the expressions for the general unrestricted case. The errors introduced by the use of the equal-size thermodynamic expressionsI0 are large in most cases. We have compared the Copeman-Stein equations to the A 1, A2, and Sig approximations (Tables I-XIV). This last one is (1 1) Abramo, M. C.; Caccamo, C.; Malescio, G.; Pizzimenti, G.; Rogde, S. A. J. Chem. Phys. 1984,80, 4396. (12) Roede. S. A. Chem. Phvs. Lett. 1983. 103. 133. (l3j Mans&, G. A,; Carnaian, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523.
Explicit Mean Spherical Approximation for Ionic Solutions TABLE III: -$Ox for the Electrolyte Described in Table I" c ur MC MSA AI A2 1.9676 4.20 0.800 0.142 f 0.14 0.146 0.146 0.146 1.9676 4.20 0.600 0.157f0.010 0.164 0.164 0.166 1.9676 4.20 0.400 0.184 f 0.007 0.191 0.192 0.196 1.9676 4.20 0.100 0.215 f 0.006 0.267 0.272 0.287 1.0000 4.20 0.800 0.136f0.007 0.134 0.134 0.134 1.0000 4.20 0.600 0.146f0.005 0.148 0.148 0.149 1.0000 4.20 0.400 0.166f0.003 0.168 0.168 0.171 0.4250 4.20 0.800 0.113f0.003 0.114 0.114 0.114 0.4250 4.20 0.600 0.121 f 0.002 0.124 0.124 0.124 0.4250 4.20 0.400 0.132f0.003 0.136 0.136 0.137 0.1000 4.20 0.800 0.079 f 0.001 0.078 0.078 0.078 0.1OOO 4.20 0.600 0.084 f 0.001 0.082 0.082 0.082 0.1000 4.20 0.400 0.088 f 0.001 0.087 0.087 0.087 1.oooO 3.62 0.525 0.166 f 0.004 0.169 0.169 0.170 0.4000 3.62 0.525 0.136f0.002 0.135 0.135 0.136 0.1000 3.62 0.525 0.091 f 0.001 0.087 0.087 0.087 1.0000 3.78 0.275 0.183f0.003 0.194 0.194 0.198 0.4000 3.78 0.275 0.143f0.002 0.149 0.149 0.151 0.1000 3.78 0.275 0.094 f 0.001 0.092 0.092 0.093 3.0000 3.60 0.833 0.169 0.169 0.170 4.0000 3.60 0.833 0.173 0.173 0.174 5.0000 3.60 0.833 0.176 0.176 0.177
Sin 0.146 0.162 0.184 0.233 0.133 0.146 0.163 0.114 0.123 0.134 0.078 0.082 0.086 0.166 0.134 0.087 0.185 0.145 0.091 0.169 0.173 0.176 I
CS 0.141 0.158 0.181 0.247 0.132 0.145 0.163 0.114 0.122 0.133 0.078 0.081 0.086 0.166 0.134 0.087 0.187 0.146 0.091 0.162 0.163 0.162
"The reported MC data" included a hard sphere contribution; gCawas obtained by subtracting [PVl/[NkB7'lfor the hard sphere mixture with use of the equation of Mansoori et al.I3
TABLE I V -AeX/INkRT1for the Electrolyte Described in Table I C r A1 A2 UMSA Sig cs 1.9676 4.20 0.800 0.534 0.534 0.534 0.534 0.531 1.9676 4.20 0.600 0.570 0.570 0.570 0.570 0.564 1.9676 4.20 0.400 0.617 0.6 17 0.617 0.617 0.605 1.9676 4.20 0.100 0.729 0.729 0.728 0.726 0.700 1.oooo 4.20 0.800 0.439 0.439 0.439 0.439 0.438 1.oooo 4.20 0.600 0.464 0.464 0.464 0.464 0.461 1.oooo 4.20 0.400 0.496 0.496 0.496 0.496 0.489 0.4250 4.20 0.800 0.333 0.333 0.333 0.333 0.333 0.4250 4.20 0.600 0.348 0.348 0.348 0.348 0.346 0.4250 4.20 0.400 0.366 0.366 0.366 0.366 0.362 0.1000 4.20 0.800 0.195 0.195 0.195 0.195 0.194 0.1000 4.20 0.600 0.200 0.200 0.200 0.200 0.199 0.1000 4.20 0.400 0.206 0.206 0.206 0.206 0.204 1 .oooo 3.62 0.525 0.499 0.499 0.499 0.499 0.495 0.4000 3.62 0.525 0.360 0.360 0.360 0.360 0.358 0.1000 3.62 0.525 0.207 0.207 0.207 0.207 0.206 1 .oooo 3.78 0.275 0.535 0.535 0.535 0.535 0.526 0.4000 3.78 0.275 0.378 0.378 0.378 0.378 0.373 0.1000 3.78 0.275 0.213 0.213 0.213 0.213 0.21 1 3.0000 3.60 0.833 0.637 0.637 0.637 0.637 0.632 4.0000 3.60 0.833 0.686 0.686 0.686 0.686 0.679 5.0000 3.60 0.833 0.725 0.725 0.725 0.725 0.715 TABLE V -fi+"/[kBT] for the Electrolyte Described in Table I C ur MSA A1 A2 Sig CS 1.9676 4.20 0.800 0.686 0.686 0.687 0.686 0.702 1.9676 4.20 0.600 0.755 0.755 0.757 0.752 0.789 1.9676 4.20 0.400 0.859 0.859 0.866 0.849 0.903 1.9676 4.20 0.100 1.146 1.154 1.175 1.090 1.184 1.oooo 4.20 0.800 0.578 0.578 0.579 0.578 0.590 1.oooo 4.20 0.600 0.628 0.628 0.629 0.626 0.651 1 .oooo 4.20 0.400 0.700 0.700 0.703 0.693 0.728 0.4250 4.20 0.800 0.45 1 0.451 0.45 1 0.45 1 0.458 0.4250 4.20 0.600 0.482 0.482 0.483 0.481 0.495 0.4250 4.20 0.400 0.524 0.524 0.525 0.521 0.539 0.1000 4.20 0.800 0.274 0.274 0.274 0.274 0.276 0.1000 4.20 0.600 0.286 0.286 0.286 0.285 0.290 0.1000 4.20 0.400 0.300 0.300 0.301 0.300 0.305 1.oooo 3.62 0.525 0.694 0.694 0.695 0.691 0.715 0.4000 3.62 0.525 0.510 0.510 0.51 1 0.509 0.521 0.1000 3.62 0.525 0.300 0.300 0.300 0.299 0.303 1.oooo 3.78 0.275 0.788 0.788 0.793 0.776 0.811 0.4000 3.78 0.275 0.559 0.559 0.561 0.554 0.571 0.1000 3.78 0.275 0.3 16 0.316 0.3 17 0.315 0.319 3.0000 3.60 0.833 0.814 0.814 0.8 15 0.814 0.826 4.0000 3.60 0.833 0.868 0.868 0.869 0.686 0.880 5.0000 3.60 0.833 0.910 0.910 0.911 0.910 0.923
The Journal of Physical Chemistry, Vol. 93, No. 21, 1989 7481 TABLE VI: -p2x/[kJ'] for the Electrolyte Described in Table I r C MSA A1 A2 Sig Ucs 1.9676 4.20 0.800 0.674 0.674 0.674 0.674 0.641 1.9676 4.20 0.600 0.713 0.713 0.714 0.711 0.653 1.9676 4.20 0.400 0.758 0.759 0.761 0.754 0.669 1.9676 4.20 0.100 0.846 0.848 0.854 0.830 0.71 1 1.0000 4.20 0.800 0.568 0.568 0.568 0.568 0.550 1.oooo 4.20 0.600 0.596 0.596 0.596 0.594 0.561 1.oooo 4.20 0.400 0.627 0.628 0.629 0.624 0.575 0.4250 4.20 0.800 0.444 0.444 0.444 0.443 0.435 0.4250 4.20 0.600 0.461 0.461 0.461 0.460 0.443 0.4250 4.20 0.400 0.480 0.480 0.48 1 0.478 0.452 0.1000 4.20 0.800 0.27 1 0.27 1 0.271 0.27 1 0.268 0.1000 4.20 0.600 0.277 0.277 0.277 0.277 0.27 1 0.1000 4.20 0.400 0.284 0.284 0.284 0.284 0.275 1.oooo 3.62 0.525 0.641 0.641 0.642 0.639 0.608 0.4000 3.62 0.525 0.479 0.479 0.480 0.478 0.462 0.1000 3.62 0.525 0.288 0.288 0.288 0.288 0.283 1.oooo 3.78 0.275 0.671 0.671 0.674 0.665 0.6 16 0.4000 3.78 0.275 0.496 0.496 0.497 0.493 0.468 0.1000 3.78 0.275 0.294 0.294 0.294 0.293 0.285 3.0000 3.60 0.833 0.797 0.797 0.798 0.797 0.763 4.0000 3.60 0.833 0.850 0.850 0.851 0.850 0.804 5.0000 3.60 0.833 0.892 0.892 0.892 0.89 1 0.832
-
TABLE VII: c r 2.0 0.8 2.0 0.6 2.0 0.4 1.0 0.8 1.0 0.6 1.0 0.4 0.4 0.8 0.4 0.6 0.4 0.4 0.1 0.8 0.1 0.6 0.1 0.4 OHere,
e
Values of MSA 3.75174 3.23426 2.621 36 3.75487 3.24928 2.66273 3.75921 3.26963 2.71742 3.76560 3.29869 2.79265
u
for a 2-2 A1 3.75162 3.23149 2.60009 3.75478 3.24724 2.64783 3.75915 3.26842 2.70904 3.76557 3.29826 2.78990
Aqueous Electrolytd A2 Sig 3.74634 3.78000 3.21195 3.36000 2.56073 2.94000 3.75051 3.78000 3.231 55 3.36000 2.61615 2.94000 3.75602 3.78000 3.25699 3.36000 2.68602 2.94000 3.76373 3.78000 3.291 56 3.36000 2.77652 2.94000
cs 2.18933 1.89491 1SO0 83 2.569 27 2.236 84 1.807 10 3.006 42 2.628 79 2.16766 3.452 8 1 3.032 37 2.558 70
= 78.3, T = 25 "C, and u- = 4.2
TABLE VIII: -Eex/[NkBT]for the Electrolyte Described in Table VI1 c r MC" MSA AI A2 Sig CS 2.0 0.8 3.744 f 0.011 3.641 3.641 3.642 3.636 3.949 2.0 0.6 4.089 f 0.012 3.992 3.993 3.997 3.966 4.328 2.0 0.4 4.588 f 0.018 4.506 4.512 4.524 4.415 4.897 1.0 0.8 3.359 f 0.010 3.188 3.188 3.189 3.185 3.385 1.0 0.6 3.718 f 0.012 3.461 3.461 3.464 3.441 3.667 1.0 0.4 4.235 f 0.017 3.842 3.846 3.854 3.777 4.071 0.4 0.8 2.923 f 0.012 2.592 2.592 2.593 2.590 2.686 0.4 0.6 3.280 f 0.013 2.775 2.776 2.777 2.763 2.869 0.4 0.4 3.831 f 0.018 3.017 3.018 3.023 2.979 3.114 0.1 0.8 2.331 f 0.012 1.760 1.760 1.760 1.759 1.781 0.1 0.6 2.723 f 0.014 1.846 1.846 1.847 1.842 1.866 0.1 0.4 3.322 f 0.020 1.951 1.951 1.953 1.938 1.969 a
From ref 12.
TABLE I X -hex for the Electrolyte Described in Table MI" r C MC MSA A1 A2 Sig cs 2.0 0.8 0.67 f 0.03 0.636 0.636 0.637 0.63 1 0.519 2.0 0.6 0.71 f 0.03 0.737 0.738 0.742 0.712 0.6 12 2.0 0.4 0.77 f 0.02 0.901 0.908 0.920 0.816 0.756 1.o 0.8 0.63 f 0.02 0.619 0.619 0.620 0.6 16 0.569 1.o 0.6 0.66 f 0.02 0.706 0.706 0.709 0.686 0.654 1 .o 0.4 0.67 f 0.07 0.838 0.842 0.850 0.775 0.780 0.4 0.8 0.46 f 0.05 0.571 0.571 0.571 0.568 0.557 0.4 0.6 0.54 f 0.04 0.635 0.636 0.637 0.623 0.622 0.4 0.4 0.55 f 0.08 0.727 0.729 0.733 0.690 0.711 0.1 0.8 0.43 f 0.03 0.451 0.45 1 0.45 1 0.450 0.450 0.1 0.6 0.46 f 0.05 0.486 0.486 0.487 0.48 1 0.484 0.1 0.4 0.50 f 0.08 0.531 0.531 0.532 0.518 0.527 "The hard sphere contribution of the MC data was treated like in Table 111.
7482 The Journal of Physical Chemistry, Vol. 93, No. 21, I989 TABLE X -A =/[Nk,J] for the Electrolyte Described in Table VI1 c r MSA A1 A2 Sin CS 2.0 2.0 2.0 1.0 1.0 1.0 0.4 0.4 0.4 0.1 0.1 0.1
0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4
3.006 3.261 3.621 2.570 2.760 3.017 2.022 2.143 2.298 1.309 1.362 1.424
3.006 3.261 3.621 2.570 2.760 3.017 2.022 2.143 2.298 1.309 1.362 1.424
3.006 3.261 3.621 2.570 2.760 3.017 2.022 2.143 2.298 1.309 1.362 1.424
3.006 3.261 3.617 2.570 2.760 3.015 2.022 2.143 2.297 1.309 1.362 1.424
2.917 3.167 3.510 2.536 2.724 2.972 2.014 2.133 2.283 1.308 1.360 1.420
TABLE XI: - p + c x / [ k ~ T for ] the Electrolyte Described in Table VI1 c r MSA AI A2 Sia CS 2.0 2.0 2.0 1.0 1.0 1.0 0.4 0.4 0.4 0.1 0.1 0.1
0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4
3.789 4.366 5.244 3.305 3.749 4.392 2.672 2.965 3.364 1.798 1.933 2.100
3.789 4.366 5.253 3.305 3.749 4.397 2.672 2.966 3.366 1.798 1.933 2.101
3.790 4.372 5.270 3.306 3.753 4.408 2.673 2.967 3.371 1.798 1.933 2.102
3.784 4.332 5.115 3.301 3.724 4.303 2.670 2.951 3.315 1.797 1.928 2.085
3.683 4.329 5.232 3.271 3.754 4.416 2.674 2.985 3.394 1.804 1.944 2.1 14
TABLE XII: -l-a/lk.T1 for the Electrolyte Described in Table VI1 r Sig cs c MSA AI A2 2.0 2.0 2.0 1.o 1.o 1.o 0.4 0.4 0.4 0.1 0.1 0.1
0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4
3.495 3.631 3.801 3.073 3.183 3.318 2.5 14 2.592 2.686 1.723 1.763 1.809
3.495 3.631 3.805 3.073 3.183 3.321 2.514 2.592 2.687 1.723 1.763 1.809
3.496 3.634 3.81 1 3.074 3.185 3.326 2.514 2.593 2.690 1.723 1.763 1.810
3.491 3.612 3.751 3.070 3.167 3.277 2.5 12 2.582 2.658 1.722 1.759 1.798
3.188 3.229 3.300 2.939 3.002 3.088 2.470 2.525 2.595 1.713 1.744 1.78 1
obtained computing r with the mean diameter estimate given by eq 2.4. In the tables, we show the values of u obtained from the solution of the MSA equations and those of the four approximations. Except for the concentrations above to 2M and for the largest diameter ratios (10 to I), the AI approximation is indistinguishable from the exact MSA result for aqueous 1-1 salts. The simpler A2 approximation is poorer in the estimation of the mean diameter, but when the relevant thermodynamic quantities are compared, the A1 and A2 approximations are almost equivalent. Even in the very simple minded Sig approximation performs reasonably well, and in many cases, it is as accurate or better than the CS approximation. We have also tried the A1 and A2 approximations for a model of concentrated NaCl solutions. There,
Sanchez-Castro and Blum TABLE XIII:
and -Aa/[NkBT]for a Ternary Mixture' AI A2 Sia cs
u, -Qx,
CT
cs
MSA
1.0 1.0 2.0
1.0 2.0 1.0
4.28567 4.04508 4.45445
4.283 13 4.041 73 4.45220
1.0 1.0 2.0
1.0 2.0 1.0
0.2202 0.2086 0.2360
0.2203 0.2088 0.236 1
1.0 1.0 2.0
1.0 2.0 1.0
0.9538 0.9224 1.1536
0.9538 0.9224 1.1536
U
4.27033 4.03739 4.42625
-VX
0.2209 0.2090 0.2374
4.42500 4.20000 4.58571
2.64596 2.431 91 2.28274
0.2139 0.2014 0.2293
0.1902 0.1768 0.1614
0.9537 0.9223 1.1535
0.9323 0.8993 1.0984
-Aex/[Nk~r] 0.9538 0.9224 1.1536
'The system studied consists of an aqueous mixture of CaC12 with molar concentration c, and NaCl with molar concentration cl. We used uca = 5.4 A, uNa= 3.0 A, ucI = 3.6 A, L = 78.381, and T = 25
OC. TABLE XIV Electrolvte' c
and -AeX/[NkBT] for a 2-1 Aqueous
u,
MSA
A2
Sig
cs
4.69301 4.667 84 4.65645
4.80000 4.80000 4.80000
2.98981 2.054 65 1.38688
0.2522 0.2589 0.2564
0.2472 0.251 7 0.2483
0.2190 0.0972 0.3776
1.0650 1.3447 1.4748
1.0420 1.2397 1.2079
A1 U
1.0 3.0 5.0
4.731 76 4.720 53 4.71559
4.731 20 4.719 73 4.71468
1.0 3.0 5.0
0.2503 0.2559 0.2530
0.2504 0.2560 0.2530
1.0 3.0 5.0
1.065 1 1.3448 1.4749
1.065 1 1.3448 1.4749
+X
-Aex/ [NkB r ]
'Here, u+ = 5.4
A,
u- = 3.6
1.065 1 1.3447 1.4749
A, t = 78.381, and T = 25 "C.
AI yields the best result, while A2 tends to underestimate u, and Sig, which is not concentration dependent, overestimates slightly the correct value. CS yields in general poorer estimates of the thermodynamics. For the 2-2 salts, the differences are accentuated but the trends are the same. We have studied also ternary mixtures with the results shown in the tables. IV. Conclusions Approximation A1 is the best over the entire range of parameters that were studied. However, the considerably simpler A2 approximation is sufficiently accurate to calculate most of the thermodynamic properties and is within the errors of the MSA itself. For salts with not too large size differences, the simple Sig approximation can be considered satisfactory, and certainly as good as the more elaborate CS approximation.
Acknowledgment. This work was supported by NSF Grant CHE-85-03503.
