5266
J. Phys. Chem. B 2002, 106, 5266-5274
Low-Density Expansion of the Solution of Mean Spherical Approximation for Ion-Dipole Mixtures Zhi-Ping Liu, Yi-Gui Li,* and Jiu-Fang Lu Department of Chemical Engineering, Tsinghua UniVersity, Beijing, 100084, China ReceiVed: NoVember 1, 2001; In Final Form: March 4, 2002
A low-density expansion of the solution of the mean spherical approximation (MSA) for ion-dipole mixtures in the semi-restricted case (with equal size hard sphere cations and anions and different size hard sphere dipoles) is proposed. From the expansion, the explicit approximations of the three energy parameters (b0, b1, b2) and the thermodynamic properties are obtained. A reasonable explanation is offered as to why the dielectric constant of the vacuum cannot be used in the long-range term in the successful EOSs for real electrolytes solutions, although the dipolar solvent is treated as explicit molecule in the models. The energy of the solvation and the relationship between the primitive model (PM) and the nonprimitive model (NPM) are discussed in some limit cases. Comparisons of our approximation with the molecular simulation are also carried out. An EOS with rather simple expressions based on the expansion is applied to the calculations of activity coefficients, osmotic coefficients, and densities of the sodium chloride aqueous solutions.
I. Introduction Among the different approaches to obtain the properties of the ionic fluids, the mean spherical approximation (MSA) is an attractive one because the integral equations can be solved analytically. In the primitive model (PM), only the ions are modeled as molecules and the solvent is treated as a dielectric continuum. Blum1 has solved the MSA for charged hard spheres. The thermodynamic properties of the system depend on one single screening parameter Γ, which plays the similar role in the Debye-Huckel inverse length κ. The results have been applied successfully to calculate the activity and osmotic coefficients for the real electrolytes when the diameters are treated as adjustable parameters.2,3 But in the PM, the ionsolvent and the solvent-solvent interactions are neglected, so the microscopic structures of the electrolyte solution cannot be exactly described. The solvent must be incorporated into the model to establish an equation of state (EOS) for the electrolytes. The simplest nonprimitive model (NPM) is represented by the mixtures of charged hard spheres and hard spheres with embedded dipoles (ion-dipole mixtures). It has been solved in MSA in the restricted case (all the diameters are the same) by Blum4 and Adelman and Deutch.5 Hoye and Stell6,7 have used the statistical mechanics to yield the explicit form for the ionion, ion-dipole, and dipole-dipole pair distribution functions and also the potentials of mean forces in the limit of vanishing ion charge or ion concentration. Blum and Wei8 have given the solutions in the case of arbitrary diameters. The results depend on the coupled nonlinear equations. Compared with the electrolyte perturbation theory, the main advantages in the NPM of MSA is that the vacuum dielectric constant can be used for real electrolyte aqueous solutions.9,10 In this work, we consider a mixture in a semi-restricted case (with equal size hard sphere ions and different size hard sphere dipoles) because the equations are more complex in the more general case. The results of the semi-restricted MSA are introduced in section II. In section III, a low ionic density expansion is proposed to obtain the explicit expressions for the thermodynamic properties.
Although it is difficult to obtain an accurate explicit approximation, the expansion is valuable at two aspects. First, a reasonable approximation is provided for electrolyte in NPM. Second, it is helpful to understand the relationship between the PM and the NPM. Some of the problems are discussed in section IV. In section V, the approximation is compared with molecular simulations and the full MSA. In section VI, some equations of state based on the approximation are proposed. Calculations of activity coefficients and osmotic coefficients are compared with the experimental data for sodium chloride aqueous solutions. II. Results of MSA in the Semi-Restricted Case The system is a mixture of ions and dipoles. The ions are hard spheres of diameter σi, half with charge +q and half with charge -q. The diploes are hard spheres of diameter σd, with a central point dipole of magnitude µ. The ratio between the diameter of dipoles and diameter of ions is defined
p ) σd/σi
(2.1)
The pair potential outside the hard core can be expressed as
uii(12) ) q2/r
(2.2a)
uid(12) ) (qµ/r2)(rˆ ‚ sˆ)
(2.2b)
udd(12) ) (µ2/r3)[3(rˆ ‚ sˆ1)(rˆ ‚ sˆ2) - (sˆ1 ‚ sˆ2)] (2.2c) where rˆ and sˆ are the unit vectors of r12 and the dipole, respectively. In MSA the assumptions are follows
cij(12) ) - uij(12)/kT (r12 > σij)
(2.3a)
hij(12) ) - 1 (r12 e σij)
(2.3b)
The system is decided by two independent variables, which
10.1021/jp0140264 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/25/2002
MSA for Ion-Dipole Mixtures
J. Phys. Chem. B, Vol. 106, No. 20, 2002 5267
represent the intensity of the ions and the dipoles, respectively
4πq Fiσi kT
(2.4a)
4πFdµ2 )y 3kT
(2.4b)
2
d02 ) d22 )
Λ)
2
where the reduced parameters are defined
q2 kTσi3
(q*)2 )
(µ*)2 )
(2.5a)
2
µ kTσd3
(2.5b)
∫0∞ drhii(r)rσi
(2.6a)
∫0∞ drhid(r)σiσd
b2 ) 3πFdx2/15
hˆ
∫0∞ dr
β6 ) 1 - b2/6
(2.12b)
β12 ) 1 + b2/12
(2.12c)
β24 ) 1 - b2/24
(2.12d)
The excess internal energy of the system is
1 UMSA ) [p3d02b0 - 2p2d0d2b1 - 2d22b2] NkT 4πFσ 3
r
(r) 3 σd
(2.6c)
(2.7a)
a1K10 - a2[1 - K11] ) d0d2
(2.7b)
K102 + [1 - K11]2 - y12 ) d22
(2.7c)
AMSA 1 [2p3d02b0 - 2p2d0d2b1 - J′] ) NkT 12πFσ 3
(2.14a)
J′ ) p3(Q′ii)2 + p(p + 1)(Q′id)2 + (Q′dd)2 + 2(q′)2
(2.14b)
d
Q′ii ) - a1 - 2 + β6/DF
2
Q′id )
[∆ - 2β6DF] 2
2DF
[
]
- b1 ∆ β3DF + a2 ) p 2β6DF2 2 b1 K10 ) p [1 + a1Λ] 2∆ 1 - K11 )
[
[
]
β6 β122
1 Q′dd ) [2β32 - pb1a2(3Λ - 2DF)] - 2 ∆
(2.15c)
q′ ) b2β24/β122
(2.15d)
The excess chemical potentials for ion and dipole are expressed as
(
)
d2b1 µMSA i ) (q*)2 b0 kT d0p
(
(2.16a)
)
(2.8a)
(2.8b)
In MSA, the excess Gibbs free energy equals the excess internal energy. Thus, the compressibility factor can be given as
(2.9b) (2.10)
]
(2.15b)
d0b1p2 µMSA (µ*)2 d 2b2 + )kT 3 d2
(2.9a)
1 p β - abΛ ∆ 3 2 2 1
y1 )
(2.15a)
b1 [β + a1(3Λ - 2DF)] ∆ 3
where
a1 )
(2.13)
d
112
a 1 + a2 ) d0 2
(2.12a)
where
(2.6b)
Because there are some misprints in the thermodynamic properties of MSA in the literature presentations,8,11 we reproduced the expressions in this paper. The three parameters b0, b1 and b2 satisfy the following set of equations 2
β3 ) 1 + b2/3
(2.5c)
The solution of MSA8 is given in terms of the three energy parameters b0 (ion-ion), b1(ion-dipole) and b2 (dipole-dipole)
b1 ) 2πxFiFd/3
(2.11c)
The excess Helmholtz free energy is given by
F* ) Fσi3
b0 ) 2πFi
1 + b0 p + β6 2 6
b12p 1 DF ) β6(1 + b0) 2 12
(2.11a)
∆ ) b12/4 + β62
(2.11b)
ZMSA )
pMSA UMSA - AMSA ) FkT NkT
(2.16b)
(2.17)
III. Low-Density Expansion of MSA Blum and Wei8 expanded MSA in the low ionic density region to obtain an explicit solution, which is verified accurate up to 0.1 M. But the expansion was only to the second order, and there were no more discussions based on the expansion. Here, we use the similar technique and expand MSA to the third order. In the next section, more discussions and comparisons will be carried out. Because the three parameters b0, b1, and b2 depend on the two variables d0 and d2, the Taylor expansion can be made at
5268 J. Phys. Chem. B, Vol. 106, No. 20, 2002
Liu et al.
the point d0 ) 0. The set of equations (2.7) can be written as
F0(d0,d2) ) d02
(3.1a)
F1(d0,d2) ) d0d2
(3.1b)
F2(d0,d2) ) d22
(3.1c)
When d0 is set to zero, from eqs 2.6a and 2.6b, b0 and b1 will be zero. The left sides of eqs 3.1a and 3.1b will be zero. The eq 3.1c become
F2(0,d2) )
β32
β62
β6
β12
4
) d22 ) y 4
1 + b2/3 β3 ) 1 - b2/6 β6
(
)
|
| |
0
3
3
d0 ∂ Fk 6 ∂d 3 0 For example, from eq 3.1a, we obtain
[
+ ‚‚‚ (3.5)
d0)0
|
d0)0
)
]
∂ibj ∂d0i
)0
β62 + β6β12 3β125
(3.9)
β32
By comparing the coefficients of d02 in eq 3.1a, we get
b(0) 01 )
|
∂b0 ∂d0
)d0)0
x
1 2
1-
d22β64
(3.10a)
β32
1
b(0) 01 ) -
(3.10b)
2xw
where w is the Wertheim’s dielectric constant12
λ2(1 + λ)4 16
(3.11)
Similarly, after some tedious algebraic calculations, the other (0) (0) expressions for bj2 and bj3 can also be deduced. Now, we define
d0
xw
4d0
)
(3.12)
λ(1 + λ)2
Thus, bj (j)0, 1, 2) can be written as the polynomials of κ as follows
|
∂bj bj ) bj|d0)0 + d0 ∂d0
d02 ∂2bj + 2 ∂d 2 d0)0 0
|
+ d0)0
0
+
) bj0 +
3β65
(0) d0bj1
(3.7b)
]
2β32 + 2β3β6
b(0) 21
|
d03 ∂3bj 6 ∂d 3
(3.7a)
|
|
2d22β64
d0)0
(3.6a)
(0) + ) bj0
d0
b(0) 2 j2
+ d0
2
(0) 1/2 bj1 w +
xw
b(0) 3 j3
+ d0
( )
6
) bj0 + κbj1 +
2
+ ‚‚‚
d0)0
+ ‚‚‚
d0
2b(0) j2 w
xw
2
(3.7c)
Here, we define
b(0) ji )
(3.8b)
d0)0
β3(β3 + pβ6) (0) ) b11 pβ64 d0)0
[
|
κ)
Similarly, we obtain
∂F2 ∂d0
∂d02
d0)0
Substituting eqs 3.6 to 3.5 and setting d0 ) 0, we get
∂F1 ∂d0
)0 d0)0
2 ) 8[b(0) 01 ] +
w )
∂ak ∂ak ∂b0 ∂ak ∂b1 ∂ak ∂b2 ) + + (k ) 1, 2) (3.6b) ∂d0 ∂b0 ∂d0 ∂b1 ∂d0 ∂b2 ∂d0
|
∂2F0
+
∂F0 ∂a1 ∂a2 ) 2 a1 + a2 ∂d0 ∂d0 ∂d0
∂F0 ∂d0
|
(3.8a)
β3(λ + p)
The second-order partial differential of F0 to d0 is
(3.4)
Here, we defined a function F(λ), which will be used later. The Taylor series of Fk (k ) 0, 1, 2) at d0 ) 0 are
d02 ∂2Fk + 2 ∂d 2 d0)0
)
Equation 3.10a can be simplified by use of eq 3.2
16 ) 9y F(λ) ) 9F2(0,d2) ) (λ + 2)2 λ2 (1 + λ)4
∂Fk Fk(d0,d2) ) Fk(0,d2) + d0 ∂d0
pd2β63
d0)0
∂b2 ∂d0
b(0) 21 )
(3.3)
We rewrite eq 3.2 as
|
∂b1 ) ∂d0
b(0) 11
(3.2)
Then b20 can be calculated from eq 3.2. It will turn out convenient to use the parameter
λ)
of d0 in both sides of the equations, we obtain the values of ∂b1/∂d0 and ∂b2/∂d0 at the point d0 ) 0
+
( ) d0
xw
3b(0) j3
3/2 w + 6 ‚‚‚
3
κ κ b + b + ‚‚‚ 2 j2 6 j3
(3.13)
i/2 bji ) b(0) ji w
(3.14)
where
(i ) 0, 1, 2, 3; j ) 0, 1, 2)
d0)0
Substituting eqs 3.5 to eqs 3.1 and comparing the coefficients
It is noticed that the values of λ, w, b20 and βn are all calculated
MSA for Ion-Dipole Mixtures
J. Phys. Chem. B, Vol. 106, No. 20, 2002 5269
at d0 ) 0, i.e., at the point of pure fluid of dipoles. Equation 3.4 is satisfied in the point. The coefficients bji (j ) 0, 1, 2; i ) 0, 1, 2, 3) are listed in the following
b00 ) 0 (From eq 2.6a)
(3.15a)
b01 ) -1/2
(3.15b)
b02 ) 1 - R1
(3.15c)
[
5 3R1(p(λ - 2) - 4λ) + b03 ) - 3 + 4 8λ R1 S1(λ + 2)(λ + 3)(p + λ)
]
AMSA
[
1
)
∞
-2C20 +
4πFσd3
(3.16a)
b11 ) R2
(3.16b)
b12 ) - R2(1 + p/λ)
(3.16c)
3
)]
p + (p + 3)λ + 2λ2 R1 λ 8S2(λ + p) κ p b1 ) κR2 1 - 1 + + ‚‚‚ 2 λ
(3.16d)
]
(3.16e)
6(λ - 1) (From eq 3.3) λ+2
(3.17a)
(
)
b21 ) 0
(3.17b)
b22 ) - S1R3/S2
(3.17c)
R3 6(1 + λ)(p + λ)2 b23 ) S2 λ
[
]
C20 )
U
1
)
NkT
-2d2 b20 +
∑ i)0
κ
(3.17e)
]
gi+2
i!
2p d2xwb1,i+1 2d2 b2,i+2 i+1 (i + 1)(i + 2) 2
gi+2 ) p3wb0i -
[
4πFσd3
∞
2
A2
+
NkT
+
NkT
+ ‚‚‚ (3.20a)
[
]
2 2 32 16 (λ - 1)2 + 2 + 4 3 (λ + 1) (λ + 1)2
2C2,i+2 2p2C1,i+1 fi+2 ) p C0i i+1 (i + 1)(i + 2) 3
[ ] F(λ)
∫1λ bji(λ)F′(λ) (λ)
1 27
(3.18a)
(3.20b)
(3.20c)
i/2
dλ
(3.20d)
w
The integration in eq 3.20d seems quite complicate, but the expressions of fi+2 are rather simple when i)0 and 1
f2 )
-9p3d24
,
λ(λ + p)(λ + 2)2
f3 ) -
( )
p3 d2 3 , 3 xw
(3.21a)
(3.21b)
Then in the eq 3.20a, the Helmholtz free energy is expanded as
A0 -1 ) NkT 3πFσ
3 d
[
(λ2 - 1)2 + 2 +
]
32 16 (λ + 1)4 (λ + 1)2 (3.22a)
()
d0 2 - 9yd02 A2 1 1 f ) ) ) NkT 4πFσ 3 2 d2 4πFσi3 λ(λ + p)(λ + 2)2 d - κ 2 w - 1
where Rn and Sn are the intermediate variables to simplify the expressions. They are listed in the Appendix. From eqs 2.13 and 3.13, the excess internal energy is derived i+2
)
d2
(3.17d)
κ2R3 S1 κ(1 + λ) b2 ) b20 + ‚‚‚ S2 2(p + λ)2 λ
MSA
∑ i)0 i!
A0
i+2
NkT
Cji(λ) )
(
()]
fi+2 d0
where
b10 ) 0 (From eq 2.6b)
b20 )
where β ) 1/kT. To complete the integration, we must rewrite the internal energy as the function of β. From eqs 2.4 and 3.15 to 3.17, we get
(3.15d)
(3.15e)
S12(λ + 1)5
(3.19)
A3
κ b0 ) - [1 - κ(1 - R1) + ‚‚‚] 2
3p p + b13 ) 3R2 1 + + 2 λ λ (λ + 2)(λ + p)
∫0β(UN)dβ
2
p3S2
[
AMSA A - Ahs ) ) NkT NkT
NkT
2
[
of internal energy
4πFσi3 (1 + p/λ)
()
d0 3 A3 - κ3 1 f ) ) NkT 4πFσ 3 3 d2 12πFσi3 d
(3.22b)
(3.22c)
The excess chemical potentials are derived from the differentiation
2
(3.18b)
The excess Helmholtz free energy is obtained by the integration
βµk )
( ( )) ∂ FA ∂Fk NkT
T,Fj*Fk
(3.23)
5270 J. Phys. Chem. B, Vol. 106, No. 20, 2002
Liu et al.
Thus MSA
βµi
βµdMSA )
[
(µ*)2 3
)
(q*)2 p3
- 2b20 +
∞
∑ i)0 ∞
()
(i/2 + 1)fi+2 d0 i!y
( )( 2
(3.24a)
d2
1 d0 i dfi + 2
∑ i)0 i! d
i
-
dy
)]
(i + 2)fi+2 2y
(3.24b)
IV. Some Explanations from the Expansion (1) Relation between the PM and NPM. A lot of equations for the real aqueous electrolyte solutions13,14 can give rather accurate results and have been successfully extended to the electrolyte mixtures15 and the mixed solvent electrolyte.16,17 In these equations, the solvent is regarded as explicit molecule and the interactions of ion-solvent and solvent-solvent must be concerned. The ion-ion interaction is usually included as a longrange (LR) term, which is expressed in the form of DebyeHuckel, Pitzer or the MSA in PM. A common characteristic of these long-range terms is that the dielectric constant of the solution must be concerned and cannot be set to unity. In a “real” NPM because the solvent is treated as molecules, the dielectric constant of the solution should not exist in the expression of the ion-ion interaction term. But all of the practices to set the dielectric constant as unity are proved to be unsuccessful, neither in the classical electrolyte theories15 nor in the perturbation theory based equations.18 Although in our earlier studies, based on the molecular simulation data, Wu et al.19 proposed a new reference system for the perturbation electrolytes EOS, and later Liu et al.20 correlated 30 real aqueous electrolyte solutions with reasonable accuracy, the dielectric constant of water was used in their work. In section III, the Helmholtz free energy of the ion-dipole system is expanded as eq 3.20a. The physical explanation of each term is: A0 is the Helmholtz free energy for pure dipoles. A2 is the Helmhlotz free energy of the ion solvation at infinite dilution. In the limit of a large ion (pf0) eq 3.22b will reduce to the classical expression given by Born. In the expression of A3 (eq 3.22c), it should be noticed that no more variable is related to λ except κ, which is expressed in eq 3.11. It means that the dipolar solvent influence A3 in the screening parameter κ by the Wertheim dielectric constant w. As we know, the PM is just based on this concept. The expression of A3 is just the first term of the PM for ionic fluids. By comparing the expansion with the practical EOSs mentioned above, it is evident that A0, A2, and A3 are in the positions of the solvent-solvent, the ionsolvent and the ion-ion terms, respectively. Actually, from eqs 3.22c and 3.20c, A3 is an integral including all the three interactions of ion-ion, ion-dipole, and dipole-dipole. It is to say that the ‘long-range’ terms in those EOSs are not contributed only by ion-ion interactions and can be regarded as the average of all the three interactions. As deduced by Hoye and Stell,6,7 the pair correlation function between the ions would exactly be (1/w)e-κruii(r) in the limit of κf0. It is the reason the dielectric constant cannot be set to unity in those EOSs and the rationality of the practical EOSs can be demonstrated. In their work,6 Hoye and Stell have also exhibited their new expressions of Helmholtz free energy and the internal energy for ion-dipole mixture in the Fourier space. In their equation (eq 50 in ref 6), the first two terms represent the energy of ions in a dielectric medium, the third is the energy change associated with moving the ions from a vacuum to the dielectric medium,
the last two terms are related to the dipole-dipole energy, which is similar to what we show in the eqs 3.22. Adelman21 established a solvent-free Ornstein-Zernike equation by introducing an effective pair potential ueff ij (r) and effective correlation function. Hoye and Stell22 found the 2 dominant term of ueff ij (r) is q /nr. n is an effective dielectric constant which goes to that of pure solvent when density of ion is set to zero. They also obtained the next-dominant contributions to the effective pair potential, which is a r-4 term screened by e-2κr and multiplied by the prefactor (1 + κr)2. This term plays an important role in the calculation of thermodynamic properties and indicates the additional influences of the dipolar solvent besides those by dielectric constant. In this paper, the higher order expansions are not obtained, so we cannot compared them with the results of Hoye and Stell. (2) Limit at µf0, σd * 0. When the dipole moment goes to zero, the molecules of solvent will be hard spheres. In the MSA, when µf0, d2f0, from eqs 3.15 and 3.16, we find b1 and b2 - b20 are the infinitesimals of d2, so we get
b1 f 0, b2 f b20
(4.1a)
From eq 3.4, λ must be unity when y is set to zero. Thus, when µf0
b20f0
(4.1b)
From the eqs 2.8 to 2.12, we obtain
∆f1, DF f
1 + b0 -2b0 , a2 f 0, a1 f 2 (1 + b )2
(4.2)
0
The eq 2.7a will be
4b02 (1 + b0)4
) d02
(4.3)
Note b0 is negative, we get
b0 ) - 1 +
-1 + x1 + 2d0 d0
From eq 2.13, the internal energy is shown as
[
(4.4)
]
- 1 + x1 + 2d0 d02 UMSA ) -1 + 3 NkT d0 4πFσ i
(4.5)
In the primitive model of MSA with equal ion diameters, the screening parameter Γ is given as
1 Γσ ) (-1 + x1 + 2κDσ) 2
(4.6)
where κD is the Debye-Huckel screening parameter. The internal energy is
[
]
2 -1 + x1 + 2κDσ UMSA (κDσ) -1 + ) NkT κDσ 4πFσ3
(4.7)
Comparing eq 4.5 with eq 4.7 we can see that the difference between them is only to use d0 instead of κDσ in the PM of MSA. So eqs 4.3 to 4.7 show that the PM is recovered but with the dielectric constant being unity when µ is set to zero in the NPM.
MSA for Ion-Dipole Mixtures
J. Phys. Chem. B, Vol. 106, No. 20, 2002 5271
(3) Limit at σdf0 (pf0), µ * 0. If we let the diameter of the solvent shrink to zero, then pf0. From eqs 3.16 and 3.17, we find b1 and b2 - b20 are the infinitesimals of p, i.e., b1f0, b2fb20. Then
(1 + b0)β6 - 2b0 ∆fβ62, DFf , a1f 2 (1 + b )2 0
1 + b0 β3 , K10f0, K11f 2 ) y0 2 β
Λf
(4.8)
[
-2b0
]
2
(1 + b0)2
+ a2 ) d0 2
2
(4.9a)
- a2y0 ) d0d2
(4.9b)
y02 - y12 ) d22
(4.9c)
Substituting the eqs 4.9b and 4.9c into the eq 4.9a, we get 2
2
2
d0 y1 ) d02 2 ) ) κ2 w (1 + b0) y0 4b0
4
(4.10)
Comparing eq 4.10 with eq 4.3, we find that this time we recovered the PM with the Wertheim dielectric constant. The same results were deduced in a different way by Blum et al. in 1992.11 (4) Limit at Infinite Dilution. Another limit of interest is the infinite dilution limit. We will consider the ion solvation energy, USol. It is calculated by the expression given by Garisto et al.23
U(+) Sol ) lim lim
F+ f0F- f0
(
)
U - U(0) N+
(4.11)
Because in this paper the anions and cations have the same charge except their sign, the superscript (+) can be dropped in eq 4.11. U(0) is the internal energy of pure solvent in the absence of ions. Using eqs 3.17 and 2.4, we get
USol ) g2
)p3
q2σi2
∞
∑
κigi+2
lim wσs3κf0 i)0 i!
)
()
q2 g2 wσi p3
In eq 4.12b, the first term is the direct ion-solvent interaction energy, and the second term is the change of the solvent-solvent interaction energy when an ion is added into the solvent. We write
Udd Sol )
2q2 (1 - 1/w) σi 1 + p/λ
4q2(1 + λ)(w - 1)2 S1 S2 σ (p + λ)2
(4.14)
Interestingly, at infinite dilution, the Helmholtz free energy is just the half of the internal energy, which was also obtained from a charging method by Garisto et al.23
In section III, the MSA is expanded in the low ionic density. In the eqs 3.20 to 3.22, we give the expressions of the Helmholtz free energy of the ion-dipole system. Here, two approximations are proposed to establish the EOS of the mixtures of ion-dipole. The first one is based on the Pade approximation
(
A0 A2 1 AappI ) + NkT NkT NkT 1 - A3/A2
(4.13a)
(4.13b)
w i
The results agree with those of Garisto et al.23 and Blum et al.24
)
(5.1)
In section III, we show that A3 is just the first term in the PM for ionic fluids. So in the second approximation, we substitute A3 in eq 5.1 with Acc, which is given in the PM of MSA1
Acc -1 [3κ2 + 6κ + 2 - 2x1 + 2κ(1 + 2κ)] ) NkT 12πFσ 3
(5.2a)
i
(
)
A0 A2 AappII 1 ) + NkT NkT NkT 1 - Acc/A2
(5.2b)
It is noticed that the first term of Acc is just A3 when κf0. Lo et al.25 and Eggebrecht and Ozler26 have studied the iondipole mixtures using Monte Carlo simulation. Some of the data is used to verify the approximations. The results of the full MSA are obtained from eqs 2.7 to 2.17. The set of nonlinear eqs 2.7 are solved by the simplex method. It is verified that the following initial guesses are effective in the calculations
b(0) 0 )
(4.12a)
2λ(w - 1) 4(1 + λ)(w - 1)2 S1 + (4.12b) λ+p S2 (p + λ)2
Uid Sol ) -
q2 (1 - 1/w) USol ) σi 1 + p/λ 2 id
ASol ) -
V. Comparisons with the Molecular Simulation and the Full MSA
6
We rewrite eqs 2.7 as
From eqs 3.19a and 3.21, it is easy to obtain the Helmholtz free energy of the solvation
- 2d0(1 + d0) 4 + 8d0 + 3d02
b(0) 2 )
F1/2 H
3y F 2+y H
(5.3a)
(5.3b)
(0) (0) 1/2 b(0) 1 ) - b0 (2b2 )
(5.3c)
FH ) 1-1.5xixdF*1/2
(5.3d)
where
These initial guesses are first proposed by Blum4 and later improved by Harvey.27 In Table 1, the reduced internal energies, calculated from the full MSA, the approximation I (appI) and the approximation II (appII), are compared with the simulations given by Lo et al.25 It is shown that the results of the full MSA and the appII are reasonably in agreement with those of Monte Carlo, but the appI cannot yield the right results when the mole fraction of the ion is high enough. The reduced Helmholtz free energy, internal energy and the compressibility factor are listed in Table 2. The results of the
5272 J. Phys. Chem. B, Vol. 106, No. 20, 2002
Liu et al.
Figure 1. Comparisons of internal energies obtained by computer simulation (MC), the full MSA (MSA), the approximation I (appI) and the approximation II (appII).
Figure 2. Comparisons of residual Helmholtz free energies obtained by computer simulation (MC), the full MSA (MSA), the approximation I (appI) and the approximation II (appII).
TABLE 1: Reduced Internal Energy Obtained by Computer Simulation (MC),19 the Full MSA (MSA), the Approximation I (appI) and the Approximation II (appII) with G* ) 0.6, σi ) σs, q* ) 13.6 and µ* ) 1.775
and the Helmholtz free energies of the full MSA and appII are smaller in magnitude than those of the simulations. The values from the appII are always smaller than those from the full MSA. The internal energies obtained from the full MSA are the most accurate and those from appI have the largest deviation. But the Helmholtz free energies calculated from the appI sometimes are the most accurate. It should be pointed out that µ*2 for water is about 4, at which the results from all the three methods are nearly the same for the internal energy and the Helmholtz free energy, respectively. All the calculation results of the compressibility factors have poor agreement with the simulation data except that the value of the dipole moment is very low.
xi
MC
MSA
appI
appII
0.0625 0.222 0.5
-13.2 -39.9 -74.6
-11.822 -33.939 -73.628
-11.513 -29.461 64.104
-11.678 -32.184 -63.890
calculations are compared with those of the simulations given by Eggebrecht and Ozler.26 As shown in Table 2, Figure 1 and Figure 2, the results of the appI have large deviations when the value of the dipole moment is low. Both the internal energies
TABLE 2: Reduced Thermodynamic Properties Obtained by Computer Simulation (MC),20 the Full MSA (MSA), the Approximation I (appI), and the Approximation II (appII) with G* ) 0.6786 and σi ) σs. χ ) q*2/µ*2 - β(Ares - Ahs)/N
-βU/N
- βp/F
µ*
χ
Fi
MC
MSA
appI
appII
MC
MSA
appI
appII
MC
MSA
appI
appII
.50 .50 .50 .50 .50 1.75 1.75 1.75 1.75 1.75 2.50 2.50 2.50 2.50 2.50 .80 .80 .80 .80 .80 2.40 2.40 2.40 2.40 2.40 4.00 4.00 4.00 4.00 4.00
0.0 64.0 64.0 64.0 64.0 0.0 64.0 64.0 64.0 64.0 0.0 64.0 64.0 64.0 64.0 0.0 40.0 40.0 40.0 40.0 0.0 40.0 40.0 40.0 40.0 0.0 40.0 40.0 40.0 40.0
.0000 .0217 .0434 .0814 .1194 .0000 .0217 .0434 .0814 .1194 .0000 .0217 .0434 .0814 .1194 .0000 .0217 .0434 .0814 .1194 .0000 .0217 .0434 .0814 .1194 .0000 .0217 .0434 .0814 .1194
.14 .80 1.39 2.60 3.70 1.27 3.80 6.20 10.60 15.10 2.21 5.90 9.48 16.00 22.40 .34 .90 1.65 2.80 3.90 2.08 4.20 6.30 10.10 13.80 4.40 8.10 11.80 18.20 24.70
.102 .593 1.136 2.142 3.193 .918 3.112 5.393 9.508 13.727 1.627 4.918 8.308 14.392 20.608 .240 .768 1.330 2.354 3.411 1.526 3.444 5.404 8.904 12.467 3.339 6.729 10.168 16.275 22.468
.102 .872 3.193 -8.769 -3.335 .918 3.150 5.534 10.093 15.356 1.627 4.922 8.316 14.485 21.045 .240 .827 1.577 3.490 7.355 1.526 3.449 5.418 8.966 12.687 3.339 6.716 10.110 16.082 22.106
.102 .608 1.191 2.354 3.754 .918 3.069 5.234 9.016 12.777 1.627 4.862 8.097 13.725 19.292 .240 .764 1.313 2.306 3.342 1.526 3.422 5.318 8.615 11.876 3.339 6.699 10.047 15.861 21.602
.28 .97 1.66 2.92 4.16 2.01 4.72 7.38 12.08 16.80 3.33 7.30 11.21 18.06 24.80 .63 1.31 2.02 3.27 4.53 3.15 5.46 7.78 11.77 15.76 6.31 10.33 14.24 21.07 27.92
.189 .801 1.440 2.596 3.783 1.505 4.023 6.571 11.106 15.710 2.558 6.261 10.002 16.642 23.371 .428 1.074 1.730 2.900 4.089 2.410 4.578 6.754 10.597 14.477 4.970 8.704 12.454 19.063 25.723
.189 .323 -7.594 -97.506 -16.881 1.505 3.921 6.163 9.531 11.649 2.558 6.172 9.657 15.408 20.521 .428 1.007 1.376 .548 -9.503 2.410 4.541 6.607 10.060 13.222 4.970 8.670 12.318 18.581 24.647
.189 .732 1.186 1.700 1.591 1.505 3.988 6.433 10.635 14.728 2.558 6.224 9.852 16.124 22.284 .428 1.053 1.652 2.640 3.531 2.410 4.564 6.694 10.378 13.996 4.970 8.686 12.376 18.783 25.111
5.20 5.03 4.91 4.68 4.46 4.17 3.70 3.30 2.60 1.90 3.53 2.92 2.30 1.20 .20 4.95 4.82 4.66 4.47 4.26 3.64 3.30 2.96 2.40 1.90 2.29 1.70 1.20 .20 -0.50
5.275 5.154 5.058 4.909 4.772 4.775 4.452 4.184 3.764 3.378 4.431 4.019 3.667 3.111 2.599 5.174 5.056 4.962 4.816 4.684 4.477 4.228 4.012 3.669 3.353 3.731 3.387 3.076 2.574 2.107
5.275 5.911 16.149 94.099 18.907 4.775 4.591 4.733 5.924 9.068 4.431 4.111 4.021 4.439 5.885 5.174 5.182 5.563 8.303 22.220 4.477 4.270 4.172 4.269 4.827 3.731 3.408 3.154 2.863 2.821
5.275 5.239 5.367 6.015 7.524 4.775 4.443 4.163 3.743 3.412 4.431 4.000 3.607 2.963 2.370 5.174 5.073 5.023 5.029 5.173 4.477 4.220 3.985 3.599 3.242 3.731 3.376 3.033 2.440 1.852
2
MSA for Ion-Dipole Mixtures
J. Phys. Chem. B, Vol. 106, No. 20, 2002 5273 TABLE 3: Parameters for Water (regarded as dipolar fluid) Regressed from Experimental Data σs (nm)
µ (Debye)
0.262
2.72
TABLE 4: Comparisons of the Deviations in the Calculation of the Activity Coefficients, the Osmotic Coefficients and the Densities for the Sodium Chloride Aqueous Solutions deviations (%)
Figure 3. Comparisons of residual chemical potentials for ion calculated from the full MSA (MSA), the approximation I (appI) and the approximation II (appII).
EOS
σi (nm)
γ(
φ
F
MSA appI appII
0.345 0.230 0.220
18.1 5.97 6.83
6.22 6.91 7.29
4.16 7.24 8.74
The superscript IG and hs refer to the ideal gas and hard sphere, respectively. Aion-dipole can be calculated from the full MSA (eqs 2.7), from the appI (eq 5.1) and from the appII (eqs 5.2). In this work, our concerns are limited in the mean activity coefficients, the osmotic coefficients and the densities of the fluids under the condition of ordinary temperature (298.15 K) and ordinary pressure (1 atm). The results of the calculations are compared with the experimental data of sodium chloride aqueous solutions with different concentrations. The activity coefficient based on mole fraction of the single ion and on infinite dilute as reference state is
lnfi ) β[µires(xi) - µires(xif0)] + ln[F(xi)/F(xif0)]
(6.2)
In this work, the anions and cations are regarded as the same except the sign of their charges, so the mean activity coefficients of the ions f( ) fi. The experimental data for the activity coefficients are based on the molarity (m), which can be calculated
γ( )
Figure 4. Comparisons of residual chemical potential for dipolar solvent calculated from the full MSA (MSA), the approximation I (appI) and the approximation II (appII).
The chemical potential is the crucial property in the calculation of the phase equilibrium. There are no simulation data for the ion-dipole mixtures, so the comparisons can be only carried out between the approximations and the full MSA. The chemical potentials of the ion and the dipole are plotted in Figure 3 and Figure 4, respectively. It is evident that the appI deviate far from the full MSA and appII when the value of the dipole moment is low. With the increase of the concentration of the ion, the two approximations bring more and more deviations. VI. Equations of State for Real Aqueous Electrolyte Solution The EOS can be proposed based on the expressions of the Helmholtz free energy. For the ion-dipole mixture, the Helmholtz free energy can be written as
A AIG Ahs Aion-dipole ) + + NkT NkT NkT NkT
(6.1)
f( 1 + VmM
(6.3)
where V is the number of ions dissociated from one electrolyte molecule (2 for sodium chloride), and M is the mole mass of the solvent (0.018 02 kg/mol for water). The osmotic coefficient is obtained from the activity coefficients of water
φ ) ln ad/ln xd ) 1 + ln fd/ln xd
(6.4a)
ln fd ) β[µdres(xd) - µdres(xdf1)] + ln[F(xd)/F(xdf1)] (6.4b) The density of the system is calculated from the EOS with the given pressure and the temperature. The solvent water is regarded as pure dipolar fluid and its parameter must be regressed from the experimental vaporliquid equilibrium (VLE) data. The results of the regressions are listed in Table 3. In this work, because the anion and the cation are treated as equal sizes, we use the mean diameter of the ion as an adjustable parameter. In the calculations, we find that the deviations of the osmotic coefficients are not the minimum at the point where the deviations of the activity coefficients are the minimum, but the differences is quite small. In this work, the diameter is regressed from the experimental data of the activity coefficients. In Table 4, the regressed diameter of the ion and the deviations for the activity coefficients, the osmotic coefficients and the densities are listed for sodium chloride aqueous solutions. The range of concentration is from the infinite dilution
5274 J. Phys. Chem. B, Vol. 106, No. 20, 2002
Liu et al.
to 6 M (near saturation). The deviations of the results calculated from the three methods are not as low as those from the practical EOSs. Although the full MSA is more agreeable with the molecule simulations than the two approximations, the results from the approximations are better than those from the full MSA. It is probably because the ion-dipole model is too simple to describe the solvent water. The experimental dielectric constant of water is 78.3. According to eqs 3.5 and 3.10, we find the Weitherm dielectric constant will equal to the experimental data when the value of the dipole moment is 2.22 D. But the value regressed from the VLE data of water is as high as 2.72 D, with which the dielectric constant will be 144. Hoye and Stell28,29 derived a new expression for the dielectric constant of a dipolar fluid beyond that of Wertheim’s. Perhaps it will improve our calculation results. The EOSs based on the expansions of MSA in this work are not accurate enough to lead to the practical use because MSA is one of the approximations in solving the O-Z equation and the description of solvent as dipolar fluid is too simple. The dispersion interaction and the hydrogen bond association are not considered, the equal sizes of cation and anion are not reasonable, and there are more additional higher order terms in eq 3.20a neglected. But at least a framework to construct a practical EOS and a method to obtain an explicit and approximate expressions for b0, b1, and b2 are proposed. It is expected in future to obtain a rather simple and more accurate EOS to set A0, A2, and A3 (see eqs 3.22) as the first term of the solvent-solvent, the ion-solvent and the ion-ion contributions. Acknowledgment. The authors are grateful to the fundamental research fund of Tsinghua University (No. JC199038) and the National Natural Science Foundations of China (No. 29736170 and No. 20176020) for financial support. Appendix The intermediate variables Rn and Sn are listed as follows
R1 )
9d22p3(λ + 1)4 16λ(λ + 2)3(λ + p)3
(A.1)
R2 ) R3 )
9d2p(λ + 1)2 4(λ + 2)2(λ + p) 81d22p3(λ + 1)9
64(λ + 2)4(λ + p)2
(A.2)
(A.3)
S1 ) pλ + 2(λ + 1)(2λ + p)
(A.4)
S2 ) 24 + λ(3 + λ(λ + 2))(3 + λ2(λ + 2)2)
(A.5)
References and Notes (1) Blum, L. Mol. Phys. 1975, 30, 1529. (2) Triolo, R.; Grigera, J. R.; Blum, L. J. Phys. Chem. 1976, 80, 1858. (3) Lu, J.-F.; Yu Y.-X.; Li, Y.-G. Fluid Phase Equilibira 1993, 85, 81. (4) Blum, L. J. Chem. Phys. 1974, 61, 2129. (5) Adelman, S. A.; Deutch, J. M. J. Chem. Phys. 1974, 60, 3935. (6) Hoye, J. S.; Stell, G. J. Chem. Phys. 1978, 68, 4145. (7) Hoye, J. S.; Stell, G. J. Chem. Phys. 1979, 71, 1985. (8) Blum, L.; Wei, D. Q. J. Chem. Phys. 1987, 87, 555. (9) Li, C.-X.; Li, Y.-G., Lu, J.-F., Yang, L.-Y. Fluid Phase Equilibira 1996, 124, 99. (10) Liu, W.-B.; Li, Y.-G.; Lu, J.-F. Fluid Phase Equilibira 1999, 162, 131. (11) Blum, L.; Vericat, F.; Fawcett, W. R. J. Chem. Phys. 1992, 96, 3039. (12) Wertheim, M. S. J. Chem. Phys. 1971, 55, 4291. (13) Harvey, A. H.; Prausnitz, J. M. AIChE J. 1989, 35, 635. (14) Chen, C. C.; Evans, L. B. AIChE J. 1986, 32, 444. (15) Clegg, S. L.; Pitzer, K. S. J. Phys. Chem. 1992, 96, 3513. (16) Li, Y.-G.; Mather, A. E. Ind. Eng. Chem. Res. 1994, 33, 2006. (17) Zuo, J. Y.; Zhang, D.; Furst, W. Fluid Phase Equilibira 2000, 175, 285. (18) Jin, G.; Donohue, M. D. Ind. Eng. Chem. Res. 1988, 27, 1073. (19) Wu, J.-Z., Lu, J.-F., Li, Y.-G. Fluid Phase Equilibira 1994, 101, 121. (20) Liu, W.-B., Li, Y.-G., Lu, J.-F. Fluid Phase Equilibria 1999, 158160, 595. (21) Adelman, S. A. J. Chem. Phys. 1976, 64, 724. (22) Hoye, J. S.; Stell, G. J. Chem. Phys. 1995, 102, 2841. (23) Garisto, F.; Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1983, 79, 6294. (24) Blum, L.; Fawcett, W. R. J. Phys. Chem. 1992, 96, 408. (25) Lo, W.-Y.; Chan, K.-Y. Henderson, D. Mol. Phys. 1993, 80, 1021. (26) Eggebrecht, J.; Ozler, P. J. Chem. Phys. 1992, 98, 1552. (27) Harvey, A. H. J. Chem. Phys. 1991, 95, 479. (28) Hoye, J. S.; Stell, G. J. Chem. Phys. 1974, 61, 562. (29) Hoye, J. S.; Stell, G. J. Chem. Phys. 1975, 64, 1952.
