An Equation of State for Nonaqueous Electrolyte Solutions

doi:10.4236/aces.2012.24061

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Advances in Chemical Engineering and Science, 2012, 2, 504-507

http://dx.doi.org/10.4236/aces.2012.24061 Published Online October 2012 (http://www.SciRP.org/journal/aces)

An Equation of State for Nonaqueous Electr olyte Solutions

Zheng Han

The Laboratory of Molecular and Materials Simulation, Department of Chemical Engineering, College of Chemical Engineering,

Beijing University of Chemical Technology, Beijing, China

Email: [email protected]

Received August 29, 2012; revised September 30, 2012; accepted November 11, 2012

ABSTRACT

A two parameters equation of state (EOS) for nonaqueous electrolyte solutions system has been developed. The equa-

tion is in terms of Helmholtz free en ergy and incorp orated with results of low density expansion of non-primitive mean

spherical approximation. The EOS was tested for experimental data reported in literatures of 9 nonaqueous single elec-

trolyte solutions of which the temperature was 298.15 K, and it also has a good predictive capability for nonaqueous

electrolyte solutions at different temperature in this work. The comparisons with EOSs published earlier by other re-

searchers in literatures are carried out in detail.

Keywords: EOS; Nonaqueous; Electrolyte Solutions

1. Introduction

Electrolyte solutions are encoun tered in a wide variety of

industrial processes, for example, wastewater treatment,

extraction, seawater desalinization, distillation and geo-

logical processes. It is very important for us to describe

the thermodynamic properties of such systems accurately.

Phase equilibrium in electrolyte systems is very impor-

tant to chemical industry.

Past a few decades, people made a lot of progress on

describing thermodynamic properties of electrolyte sys-

tems with some models [1-4]. But most of studies in lit-

eratures were restricted in aqueous electrolyte systems.

There is little attention on nonaqueous electrolyte sys-

tems until now. Although we can get some data of prop-

erties from literature, the data about nonaqueous electro-

lyte systems is much less than the one about aqueous

electrolyte systems. So in engineering, we need a simple

predictive model in order to describe phase behavior of

nonaqueous electrolyte systems.

EOSs of nonaqueous electrolytes have been developed

successfully since the late 1970s. Pitzer’s models [5,6]

have also extended to nonaqueous electrolyte solutions

and the adjustable parameters are needed in all of these

models. But up to now, there are still few models to rep-

resent phase equilibria properties of nonaqueous electro-

lyte solution.

In general, EOS can be derived by differentiating the

Helmholtz free energy with respect to the density. The

Helmholtz free energy is divided into several contribu-

tions, typically inclu ding solvent-so lvent, ion -solv ent and

ion-ion terms. In this work , we expanded Helmholtz free

energy as several contributions (including electrostatic

contribution and association contribution) according to

perturbation theory. On the other hand, the EOS pro-

posed is tested for 9 nonaqueous solutions of alkali metal

halides. The parameters can be obtained by fitting the

vapor pressure of solvents. In addition, we also compared

our results with the results of Mock et al., Youxiang Zuo

and Tzujen Chou.

2. Model and Theory

The systems of interest in this work are limited in non-

aqueous solu tions (methanol solvent) of alkali metal hal-

ides. Since the salts are fully dissociated, there are three

components in the solution, including cation, anion and

methanol solvent respectively. The ions are treated as

charged Lennard-Jones (LJ) spheres with additional as-

sociating sites forming h ydrogen bond s with methano l. A

methanol molecule is regarded as the LJ sphere with

embedded a point dipole and three associating sites, two

of which represent lone pair electrons and the others rep-

resent prot ons.

At temperature T and volume V, the system consists of

N particles, and the number of species i is Ni. By using

the perturbation theory [7-10], the differences of the

Helmholtz free energies (A – Ahs) between the syste m and

the corresponding hard sphere system can be expanded as

elect assochs LJ

AA A AA

NkTNkT NkTNkT

 (1)

where k is the Boltzmann constant. The superscripts hs,

LJ, elect and assoc represent the contributions from hard

Z. HAN 505

sphere, Lennard-Jones, electrostatic (including ion-ion,

ion-dipole and dipole-dipole terms) and association in-

teractions, respectively. Then we can get this equation:

elect assoc

Jhs

NkTNkT NkT

 A A

NkT NkT

  (2)

The equation of state expressed as the compressibility

factor can be derived from above equations by differenti-

ating the free energ y with respect to the density,

NkT

















 (3)

The chemical potential of species k is derived from the

following differentiation,

,Tk

kT NkT























 (4)

Note although the differentiation s of Equations (3) and

(4) can be derived analytically, for convenience, the nu-

merical ones are used for our calculations in this work

directly.

3. Results and Discussion

As all the ions are removed, the system is regressed to

pure methanol, i.e., LJ spheres with a point dipole and

three associating sites. Its Helmholtz free energy can be

expressed by equations. The dipole moment is set as 2.49

Debye to reproduce the experimental dielectric constant

of methanol, which is predicted as 32.49 for our EOS and

is very close to the experimental value 32.70 at 298.15 K.

There are still two kinds of parameters need to be

fixed. The first one is the effective average ionic diame-

ter (σi), which is assumed adjustable here for each salt.

Another one is the ion-methanol association parameter

for each ion. Th is parameter can be obtained by fitting it

as a salt-dependent parameter. Furthermore, it is found

that the association of the anion and methanol can be

ignored in this model without notable losses of accura-

cies. The anion-methanol association term was therefore

removed from our EOS either. Consequently, only two

parameters are required in our model.

The EOS proposed was tested for 9 nonaqueous elec-

trolyte solutions of alkali metal halides. The parameters

were obtained by fitting the experimental data of the va-

por pressure and activity of electrolyte solutions at 1 bar

and 298.15 K. The regressed parameters and the average

absolute deviations (AADs, see definitions in Table 1) of

the vapor pressure data are listed in Table 1.

As can be seen from Table 1, our EOS gives a good

correlation of vapor pressure and activity with an average

AAD of 1.120% and 0.106%. Mean while, it is o bviou s to

see that the predicted activities are in good agreement

with the experimental d ata over from low molality ran g es

to high molality ranges. And the agreement with experi-

mental data is very good when the maximum molality up

to 4.58 mol/kg methanol. So it reveals that our EOS is

very successful in activity calculation over molality

range about 0 - 5 mol/kg methanol in general although

the AADs about vapor pressure are little higher than the

nes obtaine d by Z uo . o

Table 1. Regressed parameters for EOS in this work and the average absolute deviations (AADs) in the vapor pressure (P)

and activity (a), from this work and other models at 1 bar and 298.15 K.

EOS parameters1 AAD%2

This work

Salt σi (Å) εassoc/k (K) Zuo, P3 Mock et al., P4 Chou, P5 P a

Molarity range

(mol/kg)

LiCl 5.326 3215.46 2.33 2.90 0.42 1.825 0.387 0 - 4.580

LiBr 5.316 3137.97 1.99 3.17 0.59 1.800 0.282 0 - 4.345

NaCl 6.126 2106.30 0.17 0.19 0.01 0.942 0.023 0.041 - 0.219

NaBr 5.651 3623.06 0.36 0.22 0.08 0.839 0.053 0.042 - 0.649

NaI 5.111 2717.21 0.84 0.84 0.26 0.907 0.063 0.024 - 0.755

KBr 6.992 4471.59 0.19 0.12 0.00 0.961 0.016 0.044 - 0.134

KI 6.008 5129.05 0.24 0.25 0.06 0.889 0.066 0.022 - 0.735

RbI 6.473 5241.23 0.20 0.24 0.01 0.945 0.045 0.02 - 0.436

CsI 7.421 5787.42 0.21 0.16 0.00 0.975 0.020 0.033 - 0.130

Average 0.99 1.63 0.26 1.120 0.106

1There are two p arameters for each salt . One is the ef fectiv e average io n diamet er, σi, and the other is the cation-methanol associating parameter, εassoc. The two

parameters are all salt dependent. 2cal exp

exp

100

%NP

AAD NP f





, where NP is the number of the experimental points and f is the property of interest (P and a).

The supers cripts cal and ex p indicate th e value is fr om the calcul ation and exp eriment, respecti vely. 3The AADs% were reported for the electrolyte EOS pro-

posed by Julia n Youxiang Zuo, Dan Zhang and Walter Furst [8] . 4The AADs% were reported for the electrolyte NRTL model proposed by Mock et al. [9]. 5The

AADs% were reported for the two-parameter ACM proposed by Tzu-Jen Chou and Akihiko Tanioka [10].

Z. HAN

506

10000

15000

20000

25000

30000

35000

40000

45000

298.15K

318.15K

308.15K

vapor pressure (pa)

molality (mol/kg methano l)

(a) 024

10000

15000

20000

25000

30000

35000

40000

45000

298.15K

318.15K

308.15K

vapor pressure (pa)

molality (mol/kg methanol)

(b)

Figure 1. Predicted vapor pressure of (a) LiCl and (b) LiBr nonaqueous electrolyte solutions as a function of salt molality.

The lines are calculated from equation of state with the parameters in Table 1, which were obtained by fitting the experi-

mental data at 298.15 K. The points represent the experimental data. For average absolute deviations (ADDs), see Table 2.

Table 2. The average absolute deviations (AADs) about the vapor pressure (P) for same regressed parameters of EOS in dif-

ferent temperature, from this work at 1 bar.

EOS parameters1

Salt σi (Å) εassoc/k (K) Molarity range (mol/kg) AAD%2 for P T (K)

1.825 298.15

2.802 308.15 LiCl 5.326 3215.46 0 - 4.580

4.045 318.15

1.800 298.15

2.733 308.15 LiBr 5.316 3137.97 0 - 4.345

3.630 318.15

1There are two parameter s for each salt . One is th e effecti ve averag e ion d iameter σi, and the other is the cation-methanol associating parameter: εassoc. The two

parameters are all salt dependent. 2cal exp

exp

100 NP

NP f







AAD , where NP is the number of the experimental points and f is the property of interest (P). The

superscript cal and exp indicate the value is from the calculation and experiment, respectively.

The predictive capability of EOS in this work can be

demonstrated by extrapolating the temperature to a little

higher value. For example, Figures 1(a) and (b) show the

predictive vapor pressures by using the parameters given

in Table 1, which are correlated from experimental vapor

pressures with a temperature of 298.15 K. Strikingly,

even up to 308.15 K, our EOS can still accurately repre-

sent the non-ideality of the nonaqueous electrolyte solu-

tions and the AADs are shown in Table 2.

4. Conclusion

A fundamental two-parameter equation of state for non-

aqueous electrolyte solutions is proposed by incorpora-

tion of low density expansion of nonprimitive mean

spherical approximation and statistical associating fluid

theory. The EOS has been tested for 9 nonaqueous alkali

halide solutions at ambient condition . The parameters are

obtained by fitting the vapor pressures and activities with

the average absolute deviation (AAD, see definition in

Table 1) of 1.120% and 0.106%. With the parameters

given by 298.15 K, the EOS can also well predict the

vapor pressure data of nonaqueous electrolyte solutions

at different temperature points and over the same mo-

lality range accurately.

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