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Application of a perturbation theory to the liquid—liquid equilibrium of normal-butyl alcohol—sea water system
The Chemical Engineering Journal, 18 (1979) 225 - 232 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
Application of a Perturbation Theory to the Liquid-Liquid Butyl Alcohol-Sea Water System
225
Equilibrium of Normal-
LUIGI MARRELLI Cattedra di Principi di Ingegneria Chimica, Facoltd di Ingegneria dell’Universitd di Roma, Rome (Italy) (Received
29 December
1978; in final form 23 March 1979)
Abstract Measurements of liquid-liquid equilibria in the system n-butyl alcohol-saline water have been carried out at 20, 30 and 40 “C. A theoretical treatment is given to calculate the salting coefficient k, by a perturbation method in the case of liquid-liquid equilibrium. An intermolecular potential is assumed to describe the polar interactions of the non-electrolytes. Calculated values of k, are compared with experimental values.
INTRODUCTION
Most of the theories proposed in the past [l - 81 to explain the salting effect are not fully satisfactory. Recently, perturbation theories have opened new possibilities in the prevision of thermodynamic properties. In particular, they have been applied successfully to evaluate the solubility of non-polar gases in aqueous solutions of electrolytes [9 - 131. Attempts were also made to calculate salting effects on polar molecules [14-171. . The general theory of the perturbation methods was first described by Zwanzig [18] and then developed by Barker et al. [ 19 - 241 The basic idea is to relate the properties of a real liquid to those of a reference fluid whose behaviour is well known. Then the total potential energy of a liquid can be divided into two contributions: a reference part and a perturbation part, coupled by one or more parameters which may take values from oto1. The hard-sphere fluid is generally chosen as reference system. In this case the perturbation or correction terms account for the
softness of the repulsive forces and for the attractive interactions. However, other reference systems can be used with different repulsive and attractive potentials. No restriction, but pairwise additivity, is required on the type of intermolecular potential. This approach allows a calculation of the macroscopic properties of real fluids by a Taylor series expansion of the coupling parameters around the reference value. Thus, chemical potential and equilibrium conditions can be related to molecular properties. A feature of perturbation theories is that the series converges rapidly even at low temperatures, so that the second-order theories are generally suitable to describe the behaviour of liquids. In this work, however, a firstorder perturbation theory is used because it is adequate in spite of its simple form. The approach of Gubbins et al. [9, 12, 131 is followed, but some adjustment is required in this case of liquid-liquid equilibrium. Furthermore, a more complex form of the intermolecular potential is assumed to account for the interactions of dipoles. The final expressions are derived in the general case of several salts and of multivalent ions. The results are compared with experimental observations.
EXPERIMENTAL
The procedure for the determination of the equilibrium conditions is the same as that used in previous works [15,16] and will not be reported here. A synthetic solution of salts according to ASTM standards [25] was used as sea water. The saline composition is reported in Table 1.
226 TABLE l-
Experiment results at 20, 30 and 40 “C are reported in Tables 2 - 4. Measurements for three weight ratios of n-butyl alcohol/ saline water were carried out at 20 “C.
Saline water composition Salt
Concentration
NaCl NazSO* MgC1,*6H,O CaClz KC1 NaHC03 KBr SrCIz
24.540 4.090 11.120 1.160 0.605 0.200 0.100 0.043 0.013
%B’h
(g/l)
THEORY
At low salt coycentrations, the solubility of a ~non-electrolyI+ in an aqueous salt solution can be expresSed by the empirical equation of Setschenovk
TABLE 2 Liquid-liquid Wt. alcohol/wt. solution
equilibrium at 20 “C salt
Organic phase
phase
Wt.% salts
Wt.% alcohol
Wt. % salts
Wt. % alcohol
80.0 81.2 81.5 83.9 84.6 85.2 85.9 86.3 86.7 88.0 88.7 89.5 90.4
-
0.021 0.046 0.060 0.074 0.091 0.094 0.102 0.120 0.141 0.159 0.167 0.176
7.8 7.2 6.6 5.8 5.3 5.2 4.6 4.5 4.3 3.3 3.0 2.6 2.1
80.0 82.2 82.8 84.2 83.3 85.4 85.0 86.2 86.9 87.0 86.5 88.8 89.0
-
0.032 0.046 0.054 0.074 0.078 0.092 0.108 0.109 0.130 0.142 0.158 0.169
80.0
-
1
l/3
Aqueous
0.044 0.063 0.096 0.102 0.112 0.135 0.132 0.142 0.160 0.184 0.194 0.201
82.0 83.1 84.9 85.2 85.6 86.8 88.1 88.2 89.1 89.9 90.5 91.4
1.2 2.3 3.3 4.4 5.4 6.4 7.5 8.5 10.4 12.2 14.0 15.8 1.0 2.0 2.9 3.9 4.8 5.8 6.7 7.6 9.5 11.3 13.0 14.8 2.4 4.2 5.6 7.0 8.2 9.4 10.5 11.6 13.8 15.6 17.5 19.3
7.8 7.4 6.5 6.1 6.0 5.6 5.0 4.4 4.1 3.9 3.1 2.9 2.5 7.8 6.0 5.8 5.5 4.4 4.2 4.0 3.6 3.0 2.6 2.3 1.9 1.4
227 TABLE 3 Liquid-liquid equilibrium at 30 “C Organic phase
Aqueous
Wt.% salts
Wt.% alcohol
Wt.% salts
Wt.% alcohol
_
79.4
_
0.035 0.052 0.069 0.085 0.095 0.106 0.123 0.135 0.145 0.166 0.176 0.188
81.6 82.3 83.0 83.9 84.5 85.3 85.5 86.0 87.2 88.1 88.6 89.5
1.2 2.3 3.4 4.4 5.5 6.5 7.6 8.6 10.5 12.4 14.2 16.0
7.1 6.3 5.8 5.7 4.9 4.6 4.2 4.1 3.9 3.0 2.3 2.2 1.9
phase
TABLE 4
was required to be the same in the case of pure water and of the salt solution. The evaluation of k, can be performed by the perturbation theory and classical thermodynamics. For most highly non-ideal systems, over the range of concentrations in the organic phase encountered here, it can be observed experimentally that the nonelectrolyte activity in the aqueous phase is fortuitously unchanged by addition of a salt to water. Therefore the following equality of the chemical potentials holds: cl! = PS
(3)
where superscripts 0 and s indicate pure water and salt solution, respectively. By adding and subtracting kT In p1 to the right-hand side of eqn. (3) and observing that x1 = PllEPj
(4)
Liquid-liquid equilibrium at 40 “C
one obtains Organic phase
Aqueous
Wt.% salts
Wt.% alcohol
Wt.% salts
Wt. % alcohol
_
78.6 80.4 81.4 82.4 83.4 83.8 84.4 84.8 85.5 86.3 87.5 88.3 88.8
_
6.6 5.8 5.7 5.0 4.3 4.2 4.0 3.7 3.1 2.8 1.9 1.8 1.3
0.041 0.064 0.082 0.097 0.107 0.118 0.136 0.140 0.165 0.175 0.188 0.200
1.2 2.3 3.4 4.5 5.6 6.6 7.7 8.7 10.7 12.6 14.4 16.2
phase
where X: and 3tl are the mole fractions of the non-electrolyte in pure water and in the salt solution of molarity C,; k, is the salting coefficient defined by the following equation:
-1n3ti
=pi/kT-lnp,-p$kT+ln
Epj
(5)
This equation can be substituted in eqn. (2) to give the value of k,*. The chemical potential p1 can be expressed by the first-order perturbation theory of Leonard et al. [21]. The expansion is made about the value for a fluid of hard spheres with the diameters of the various components given by d,(T) = r[l
- exp{-@g(r)/kT)]
dr
(6)
0 In eqn. (6) @jj(r) is the pair potential and ojj the collision diameter of the species j. The form of @j](r) chosen for numerical calculations is the (6-12) Lennard-Jones potential. The expression for the chemical potential [12] is
dr]
Pjpk f@jk(')&(')r2 Ojk
(7) A theoretical expression, similar to eqn. (l), was derived by Long and McDevit [3] for the case of a low non-electrolyte concentration. The chemical potential of the nonelectrolyte
*By performing the differentiation with respect to C,, the term containing /.I: does not appear since this quantity does not depend on C,.
228
Since it is difficult distribution function uniform distribution assumed :
to calculate the radial &i(r), the following approximation was
&W = 1
r>
ujk
&Z(r)= 0
r<
(Jjk
Therefore
(8)
eqn. (7) becomes (9)
To calculate the integral in eqn. (9) we must assume an expression for the intermolecular potential Q1j(r) to account for the various interactions between the molecules of solute 1 and the other- chemical species in the solution. If all the electrolytes are considered to be dissociated, the species present are the molecules of the solute, those of water, and the cations and anions of the various salts*. For the solute-water interaction the following form for the intermolecular potential was assumed: @r2 = 4Er2 [(F)12 - i&l --_-__ r6
i&2
($2)6] 2
r6
The intermolecular potential due to the ion-induced dipole and to the ion-permanent dipole can be neglected if the charge distribution around the solute molecule is assumed uniform [9] . Furthermore, the following mixing rules can be assumed: Eij = (Eiej)l”
(12)
Uij = (Ui + Uj)/2
(13)
If eqns. (10 - 13) are introduced one obtains /.ll =Pt-g
32~
m F Pjelj”fj i
into eqn. (9),
-
1
3 P2 #ii& -_ 477
42 +&X2
--
Sn 9
p2 $
-
12
(14)
The quantity r_ly can be calculated by the scaled particle theory of Reiss et al. [ 26 - 291 or by the Carnahan and Starling equation of state for hard spheres [30, 311. The expression given by the scaled particle theory is
-
i&i;
3 kTr6
In eqn. (10) the non-polar part is expressed by the (6-12) Lennard-Jones potential. Account was taken of the following angleaveraged interactions: (1) permanent dipole of water (,!i2)permanent dipole of solute (iI); (2) permanent dipole of water (,ii2)induced dipole of solute (cx~); (3) permanent dipole of solute (,Zr)induced dipole of water (a2). As in the case of non-polar solutes [9] , the only type of interaction considered between ions and solute molecules is the non-polar contribution
*The interaction (l-l) can be omitted since p1 and its concentration dependence are negligibly small compared with ~2, p3, . . . . pm [lo]. For this reason p1 will also be dropped in the last summation of eqn. (5).
+125,+ 18G [l-i-3 (1-_3J2
](;)'+g(5)"
(15) where
Si = i
A’
=
jgl
(16)
PjfdjY
janri;kT
),’
(17)
and qFt is the part of the molecular partition function which depends on the internal configuration of the molecule. Equation (15) was used in this work with dj given by eqn. (6). The term containing the pressure P was omitted because it is negligibly small at atmospheric pressure. The following expressions can be used for the number density of the ions and water:
229
Pi
= j&j
i i
1
ND2 P2
to calculate the derivatives in in the case of the various salts the solution, the total saline conC, is expressed by .
(19)
=(
M2
Finally, eqn. (20) present in centration
(18)
civji
C, = Cr i hi
(21)
i=l
Substituting eqns. (14), (15), (18) and (19) into eqn. (5), and differentiating with respect to C,, the theoretical value of k, can be obtained : k =
S
where C, is the concentration of an arbitrary reference salt and hi is the ratio between the concentration of electrolyte i and that of the reference salt. Each derivative of eqn. (20) becomes
+ [ dG1c,-+o I dCs1cs-+o + d&g EPj) dcs I d(&T2.3kT)
+
d@/2.3kT)
d _=-dC,
CT,+ 0
=
k,
+
k,
+
where (see eqns. (15) and (14)) ppq’ =py
-kkTlnp,
=PI-PP
P’I
Ehi
(22)
dCr
The ratios hi of the salts in the aqueous phase at equilibrium can be assumed equal to those in the initial solution (Table 1) since the concentration of salts in the organic phase is very low at equilibrium.
(20)
k,
ld
RESULTS AND DISCUSSION
Equation (20) is formally identical to the expression obtained by Masterton and Lee [lo] from the scaled particle theory. However, some important differences concern the temperature dependency of the diameters of the hard spheres and the form (10) assumed to describe the polar interaction.
The above procedure allows the calculation of k, using only the molecular parameters of the pure components and the apparent molal volumes of the salts at infinite dilution. The values used for these quantities are shown in Tables 5 and 6.
TABLE 5 Molecular parameters cr X 1O24 (cm3/mol)
6x 101* (e.s.u.)
1.44 8.6
[37] [15]
1.84 1.66
[lo] [35] [12] [12] [36]
0.21 0.30* 0.47 0.87 0.86
[lo]
[12]
3.02 [lo] 0.7** 4.29* 4.17 [lo]
u X lo8 (cm)
elk (K)
Non-electrolytes Water N-butyl alcohol
2.98 2.649
[12] [34]
96.3 1115.3
Cations Na’ Mg2+ Cap+ K’ Sr2+
1.98 1.373 2.06 2.76 2.35
3.75 4.44* 3.95* 4.06
Anions clso;;$”
[12]
*Evaluated as reported in the text. **Evaluated together with u as reported in the text.
[12] [34]
[12] [lo] [38]
[35] [35]
230 TABLE 6 Apparent molal volumes at infinite dilution [ 391
Salt
NaCl Naz SO4 MgClz CaC12 KC1 NaHC03 * KBr SrC12
cpm( cm3/mol) 20 “C
30 “C
40 “C
15.98 10.30 15.50 17.70 26.07 22.00 33.37 17.60
16.97 12.40 15.60 18.25 26.92 22.00 34.30 18.60
17.62 14.00 15.45 18.25 27.53 22.00 34.75 19.30
0.2,//
,
T-“”
-I
T=30°C
*Only the value at 25 “C was found and is reported here. k,=0.200
For most ions the u-values were obtained from the Pauling crystal radii multiplied by 1.04, as suggested by Tiepel and Gubbins [ 121. No value of u was found for HCO, and SO: -. Therefore, the former was evaluated with the expression of Couture-Laidler [ 321, whereas u for SO:- was obtained by fitting the theory to the observed 12, for some systems containing Na2S04 [3]. The parameters e/k were obtained for the ions by the Mavroyannis-Stephen equation [9, 331, which requires the polarizabilities (Y. The values of (Ywere taken from the literature [lo, 12,15,38]. For HCO, the polarizability was calculated by [9] a = 3R/4nN
(23)
where the mole refraction R was evaluated from the atomic contributions. For Mg2’, a was obtained (as u of SO:-) by fitting the theory to the experimental values of k, for systems containing the Mg2’ ion. The molecular parameters of water were taken from Tiepel and Gubbins [12] ; those of n-butyl alcohol, calculated from the second virial coefficient, are reported by Polak and Lu [34]. Table 6 shows that no value was found for cp? of MgC12 -6HsO (the reported value refers to MgCl,). Therefore, the experimental mole fractions x1 and X: in eqn. (2) were calculated from measurements accounting for the substitution of MgCl, (and of the corresponding water) to MgCl, -6HsO. In Fig. 1, log@y/xr) is plotted for the whole range of C, explored. In the same Figure, straight lines passing through the
0.4 -
0.2 -
0
T=20°C
0.5
1.0
1.5
2.0
2.5
C, , mol/l
Fig. 1. Solubility of n-butyl alcohol in the saline water. The slopes of the straight lines are calculated by the perturbation theory (k, can be considered independent of temperature).
origin are shown with slopes equal to values of k, calculated by the perturbation theory. The agreement with the experimental measurements is good at all temperatures. At the lowest temperatures, eqn. (1) holds in the whole range of saline concentrations, whereas at 40 “C the validity is limited at high dilution. However, it must be pointed out that the result cannot be considered to be as satisfactory as it appears. The calculated k, depends on the molecular parameters (in particular u ), of the chemical species in solution. Their correct choice is still a crucial problem. Different values of the molecular parameters are usually obtained, for the same potential, from different properties. These differences become smaller if the description of intermolecular potential approaches that of the actual potential [40] . The commonly used types of intermolecular potential give a rough description of the dispersion forces and overlook the non-pa&vise additivity terms. Further, angular averaging of the polar part cancels the contributions due to the asym-
231
metry of the electrostatic field. The values of E/k and (I for nonelectrolytes are usually obtained from gas properties. Since simple potentials are used for this purpose, results are not suitable to describe the behaviour of the liquid phase [15 - 171. A considerable improvement may be obtained [41 - 481 if the potential parameters are calculated by fitting the perturbation theory to the observed liquid-phase properties. A similar improvement may be obtained for the ions if the crystallographic radii are substituted by the values calculated by fitting experimental k,.
density of the pure solvent radial distribution function Planck’s constant
D2 g
h hi k k,
Ci/Cr
1121 M2
N NI r R T V Xl
Boltzmann’s constant Setschenow’s constant mass of molecule 1 molecular weight of the solvent Avogadro’s number number of molecules of component separation distance of a pair of molecules mole fraction absolute temperature volume mole fraction of component 1
1
CONCLUSIONS
The perturbation theory used can be considered adequate to describe the salting effect of the observed system. The agreement of results with experimental data shows that it is possible to handle polar compounds if a suitable intermolecular potential is chosen. However, the criticality of the choice of the molecular parameters suggests calculation of the required values of u and e/k for nonelectrolytes by fitting the perturbation theory to some observed quantity of pure components in liquid phase. This follows from oversimplification in describing the complex molecular interactions by a simple potential. For engineering applications, however, the described procedure can be useful, considering the high number of chemical species present in solution and the difficulties of describing the salting effects in phase equilibria. ACKNOWLEDGMENT
The author is very grateful to Professor A. R. Giona and to Dr. P. N. Muscetta for their help in the development of this work, and to Professor C. Mustacchi for his useful suggestions.
NOMENCLATURE Ci
C, C, di
molarity of salt i concentration of reference salt saline concentration (molarity) hard-sphere diameter of species j
Greek symbols (Y polarizability E depth of the potential minimum chemical potential of component 1 c11 dipole moment ii number of js in the formula of salt i vji number density of species j (particles/ Pi volume) a collision diameter of potential @ @ intermolecular potential apparent molal volume of salt i in Cpi solution Superscripts refers to pure solvent 0 00 at infinite dilution hard-sphere value hs perturbation term P r residual part in summations, the number S solution
of salts in
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232 and K. E. Gubbins, J. Phys. Chem., 73 (1969) 498. W. L. Masterton and T. P. Lee, J. Phys. Chem., 74 (1970) 1776. W. L. Masterton, D. Bolocofsky and T. P. Lee, J. Phys. Chem., 75 (1971) 2809. E. W. Tiepel and K. E. Gubbins, J. Phys. Chem., 76 (1972) 3044. E. W. Tiepel and K. E. Gubbins, Ind. Eng. Chem. Fundam., 12 (1973) 18. F. L. Wilcox and E. E. Schrier, J. Phys. Chem., 75 (1971) 3757. R. De Santis, L. Marrelli and P. N. Muscetta, Chem. Eng. J., 11 (1976) 207. R. De Santis, L. Marrelli and P. N. Muscetta, J. Chem. Eng. Data, 21 (1976) 324. R. De Santis, L. Marrelli and P. N. Muscetta, Proc. Congr. on Advances in Separation Science, University of Trieste, Trieste, Italy, 1978, p. 70. R. W. Zwanzig, J. Chem. Phys., 22 (1954) 1420. J. A. Barker and D. Henderson, J. Chem. Phys., 47 (1967) 2856. J. A. Barker and D. Henderson, J. Chem. Phys., 47 (1967) 4714. P. J. Leonard, D. Henderson and J. A. Barker, Trans. Faraday Sot., 66 (1970) 2439. H. C. Andersen and D. Chandler, J. Chem. Phys., 53 (1970) 547. H. C. Andersen and D. Chandler, J. Chem. Phys., 57 (1972) 1918. J. D. Weeks, D. Chandler and H. C. Andersen, J. Chem. Phys., 54 (1971) 5237. ASTM, Book ofstandards, Philadelphia, 7 (1955) 1513. H. Reiss, H. L. Frisch and J. L. Lebowitz, J. Chem. Phys., 31 (1959) 369. H. Reiss, H. L. Frisch, E. Helfand and J. L. Lebowitz, J. Chem. Phys., 32 (1960) 119.
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28 J. L. Lebowitz, E. Helfand and E. Praestgaard, J. Chem. Phys., 43 (1965) 774. 29 H. Reiss, Adu. Chem. Phys., 9 (1965) 1. 30 N. F. Carnahan and K. E. Starling, J. Chem. Phys., 51 (1969) 635. 31 T. Boublik, J. Chem. Phys., 53 (1970) 471. 32 A. M. Couture and K. J. Laidler, Can. J. Chem., 34 (1956) 1209. 33 C. Mavroyannis and M. J. Stephen, Mol. Phys., 5 (1962) 629. 34 J. Polak and B. C. Y. Lu, Can. J. Chem. Eng., 50 (1972) 553. 35 R. C. Weast (ed.), Handbook of Chemistry and Physics, 50th edn., The Chemical Rubber Co., Cleveland, Ohio, 1970. 36 G. V. Samsonov (ed.), Handbook of the Physicochemical Properties of the Elements, IF1 Plenum, New York, 1968. 37 E. A. Moelwyn-Hughes, Physical Chemistry, Pergamon Press, Oxford, 1961. 38 E. D’Ans-Lax, Taschenbuch fiir Chemiker und Physiker, Springer-Verlag, Berlin, 1970. 39 R. A. Horne (ed.), Water and Aqueous Solutions, Wiley - Interscience, New York, 1972. 40 E. B. Smith, Physica, 73 (1974) 211. 41 R. A. Pierotti,J. Phys. Chem., 67 (1963) 1840. 42 R. A. Pierotti, J. Phys. Chem., 69 (1965) 281. 43 E. Wilhelm and R. Battino, J. Chem. Thermodyn., 3 (1971) 379. 44 R. Battino, F. D. Evans, W. F. Danforth and E. Wilhelm, J. Chem. Thermodyn., 3 (1971) 743. 45 E. Wilhelm and R. Battino, J. Chem. Thermodyn., 3 (1971) 761. 46 E. Wilhelm and R. Battino, J. Chem. Phys., 55 (1971) 4012. 47 S. Goldman, J. Phys. Chem., 80 (1976) 1697. 48 Y. T. Chang, M. Y. Schrier and E. E. Schrier, J. Phys. Chem., 78 (1974) 165.
Application of a Perturbation Theory to the Liquid-Liquid Butyl Alcohol-Sea Water System
225
Equilibrium of Normal-
LUIGI MARRELLI Cattedra di Principi di Ingegneria Chimica, Facoltd di Ingegneria dell’Universitd di Roma, Rome (Italy) (Received
29 December
1978; in final form 23 March 1979)
Abstract Measurements of liquid-liquid equilibria in the system n-butyl alcohol-saline water have been carried out at 20, 30 and 40 “C. A theoretical treatment is given to calculate the salting coefficient k, by a perturbation method in the case of liquid-liquid equilibrium. An intermolecular potential is assumed to describe the polar interactions of the non-electrolytes. Calculated values of k, are compared with experimental values.
INTRODUCTION
Most of the theories proposed in the past [l - 81 to explain the salting effect are not fully satisfactory. Recently, perturbation theories have opened new possibilities in the prevision of thermodynamic properties. In particular, they have been applied successfully to evaluate the solubility of non-polar gases in aqueous solutions of electrolytes [9 - 131. Attempts were also made to calculate salting effects on polar molecules [14-171. . The general theory of the perturbation methods was first described by Zwanzig [18] and then developed by Barker et al. [ 19 - 241 The basic idea is to relate the properties of a real liquid to those of a reference fluid whose behaviour is well known. Then the total potential energy of a liquid can be divided into two contributions: a reference part and a perturbation part, coupled by one or more parameters which may take values from oto1. The hard-sphere fluid is generally chosen as reference system. In this case the perturbation or correction terms account for the
softness of the repulsive forces and for the attractive interactions. However, other reference systems can be used with different repulsive and attractive potentials. No restriction, but pairwise additivity, is required on the type of intermolecular potential. This approach allows a calculation of the macroscopic properties of real fluids by a Taylor series expansion of the coupling parameters around the reference value. Thus, chemical potential and equilibrium conditions can be related to molecular properties. A feature of perturbation theories is that the series converges rapidly even at low temperatures, so that the second-order theories are generally suitable to describe the behaviour of liquids. In this work, however, a firstorder perturbation theory is used because it is adequate in spite of its simple form. The approach of Gubbins et al. [9, 12, 131 is followed, but some adjustment is required in this case of liquid-liquid equilibrium. Furthermore, a more complex form of the intermolecular potential is assumed to account for the interactions of dipoles. The final expressions are derived in the general case of several salts and of multivalent ions. The results are compared with experimental observations.
EXPERIMENTAL
The procedure for the determination of the equilibrium conditions is the same as that used in previous works [15,16] and will not be reported here. A synthetic solution of salts according to ASTM standards [25] was used as sea water. The saline composition is reported in Table 1.
226 TABLE l-
Experiment results at 20, 30 and 40 “C are reported in Tables 2 - 4. Measurements for three weight ratios of n-butyl alcohol/ saline water were carried out at 20 “C.
Saline water composition Salt
Concentration
NaCl NazSO* MgC1,*6H,O CaClz KC1 NaHC03 KBr SrCIz
24.540 4.090 11.120 1.160 0.605 0.200 0.100 0.043 0.013
%B’h
(g/l)
THEORY
At low salt coycentrations, the solubility of a ~non-electrolyI+ in an aqueous salt solution can be expresSed by the empirical equation of Setschenovk
TABLE 2 Liquid-liquid Wt. alcohol/wt. solution
equilibrium at 20 “C salt
Organic phase
phase
Wt.% salts
Wt.% alcohol
Wt. % salts
Wt. % alcohol
80.0 81.2 81.5 83.9 84.6 85.2 85.9 86.3 86.7 88.0 88.7 89.5 90.4
-
0.021 0.046 0.060 0.074 0.091 0.094 0.102 0.120 0.141 0.159 0.167 0.176
7.8 7.2 6.6 5.8 5.3 5.2 4.6 4.5 4.3 3.3 3.0 2.6 2.1
80.0 82.2 82.8 84.2 83.3 85.4 85.0 86.2 86.9 87.0 86.5 88.8 89.0
-
0.032 0.046 0.054 0.074 0.078 0.092 0.108 0.109 0.130 0.142 0.158 0.169
80.0
-
1
l/3
Aqueous
0.044 0.063 0.096 0.102 0.112 0.135 0.132 0.142 0.160 0.184 0.194 0.201
82.0 83.1 84.9 85.2 85.6 86.8 88.1 88.2 89.1 89.9 90.5 91.4
1.2 2.3 3.3 4.4 5.4 6.4 7.5 8.5 10.4 12.2 14.0 15.8 1.0 2.0 2.9 3.9 4.8 5.8 6.7 7.6 9.5 11.3 13.0 14.8 2.4 4.2 5.6 7.0 8.2 9.4 10.5 11.6 13.8 15.6 17.5 19.3
7.8 7.4 6.5 6.1 6.0 5.6 5.0 4.4 4.1 3.9 3.1 2.9 2.5 7.8 6.0 5.8 5.5 4.4 4.2 4.0 3.6 3.0 2.6 2.3 1.9 1.4
227 TABLE 3 Liquid-liquid equilibrium at 30 “C Organic phase
Aqueous
Wt.% salts
Wt.% alcohol
Wt.% salts
Wt.% alcohol
_
79.4
_
0.035 0.052 0.069 0.085 0.095 0.106 0.123 0.135 0.145 0.166 0.176 0.188
81.6 82.3 83.0 83.9 84.5 85.3 85.5 86.0 87.2 88.1 88.6 89.5
1.2 2.3 3.4 4.4 5.5 6.5 7.6 8.6 10.5 12.4 14.2 16.0
7.1 6.3 5.8 5.7 4.9 4.6 4.2 4.1 3.9 3.0 2.3 2.2 1.9
phase
TABLE 4
was required to be the same in the case of pure water and of the salt solution. The evaluation of k, can be performed by the perturbation theory and classical thermodynamics. For most highly non-ideal systems, over the range of concentrations in the organic phase encountered here, it can be observed experimentally that the nonelectrolyte activity in the aqueous phase is fortuitously unchanged by addition of a salt to water. Therefore the following equality of the chemical potentials holds: cl! = PS
(3)
where superscripts 0 and s indicate pure water and salt solution, respectively. By adding and subtracting kT In p1 to the right-hand side of eqn. (3) and observing that x1 = PllEPj
(4)
Liquid-liquid equilibrium at 40 “C
one obtains Organic phase
Aqueous
Wt.% salts
Wt.% alcohol
Wt.% salts
Wt. % alcohol
_
78.6 80.4 81.4 82.4 83.4 83.8 84.4 84.8 85.5 86.3 87.5 88.3 88.8
_
6.6 5.8 5.7 5.0 4.3 4.2 4.0 3.7 3.1 2.8 1.9 1.8 1.3
0.041 0.064 0.082 0.097 0.107 0.118 0.136 0.140 0.165 0.175 0.188 0.200
1.2 2.3 3.4 4.5 5.6 6.6 7.7 8.7 10.7 12.6 14.4 16.2
phase
where X: and 3tl are the mole fractions of the non-electrolyte in pure water and in the salt solution of molarity C,; k, is the salting coefficient defined by the following equation:
-1n3ti
=pi/kT-lnp,-p$kT+ln
Epj
(5)
This equation can be substituted in eqn. (2) to give the value of k,*. The chemical potential p1 can be expressed by the first-order perturbation theory of Leonard et al. [21]. The expansion is made about the value for a fluid of hard spheres with the diameters of the various components given by d,(T) = r[l
- exp{-@g(r)/kT)]
dr
(6)
0 In eqn. (6) @jj(r) is the pair potential and ojj the collision diameter of the species j. The form of @j](r) chosen for numerical calculations is the (6-12) Lennard-Jones potential. The expression for the chemical potential [12] is
dr]
Pjpk f@jk(')&(')r2 Ojk
(7) A theoretical expression, similar to eqn. (l), was derived by Long and McDevit [3] for the case of a low non-electrolyte concentration. The chemical potential of the nonelectrolyte
*By performing the differentiation with respect to C,, the term containing /.I: does not appear since this quantity does not depend on C,.
228
Since it is difficult distribution function uniform distribution assumed :
to calculate the radial &i(r), the following approximation was
&W = 1
r>
ujk
&Z(r)= 0
r<
(Jjk
Therefore
(8)
eqn. (7) becomes (9)
To calculate the integral in eqn. (9) we must assume an expression for the intermolecular potential Q1j(r) to account for the various interactions between the molecules of solute 1 and the other- chemical species in the solution. If all the electrolytes are considered to be dissociated, the species present are the molecules of the solute, those of water, and the cations and anions of the various salts*. For the solute-water interaction the following form for the intermolecular potential was assumed: @r2 = 4Er2 [(F)12 - i&l --_-__ r6
i&2
($2)6] 2
r6
The intermolecular potential due to the ion-induced dipole and to the ion-permanent dipole can be neglected if the charge distribution around the solute molecule is assumed uniform [9] . Furthermore, the following mixing rules can be assumed: Eij = (Eiej)l”
(12)
Uij = (Ui + Uj)/2
(13)
If eqns. (10 - 13) are introduced one obtains /.ll =Pt-g
32~
m F Pjelj”fj i
into eqn. (9),
-
1
3 P2 #ii& -_ 477
42 +&X2
--
Sn 9
p2 $
-
12
(14)
The quantity r_ly can be calculated by the scaled particle theory of Reiss et al. [ 26 - 291 or by the Carnahan and Starling equation of state for hard spheres [30, 311. The expression given by the scaled particle theory is
-
i&i;
3 kTr6
In eqn. (10) the non-polar part is expressed by the (6-12) Lennard-Jones potential. Account was taken of the following angleaveraged interactions: (1) permanent dipole of water (,!i2)permanent dipole of solute (iI); (2) permanent dipole of water (,ii2)induced dipole of solute (cx~); (3) permanent dipole of solute (,Zr)induced dipole of water (a2). As in the case of non-polar solutes [9] , the only type of interaction considered between ions and solute molecules is the non-polar contribution
*The interaction (l-l) can be omitted since p1 and its concentration dependence are negligibly small compared with ~2, p3, . . . . pm [lo]. For this reason p1 will also be dropped in the last summation of eqn. (5).
+125,+ 18G [l-i-3 (1-_3J2
](;)'+g(5)"
(15) where
Si = i
A’
=
jgl
(16)
PjfdjY
janri;kT
),’
(17)
and qFt is the part of the molecular partition function which depends on the internal configuration of the molecule. Equation (15) was used in this work with dj given by eqn. (6). The term containing the pressure P was omitted because it is negligibly small at atmospheric pressure. The following expressions can be used for the number density of the ions and water:
229
Pi
= j&j
i i
1
ND2 P2
to calculate the derivatives in in the case of the various salts the solution, the total saline conC, is expressed by .
(19)
=(
M2
Finally, eqn. (20) present in centration
(18)
civji
C, = Cr i hi
(21)
i=l
Substituting eqns. (14), (15), (18) and (19) into eqn. (5), and differentiating with respect to C,, the theoretical value of k, can be obtained : k =
S
where C, is the concentration of an arbitrary reference salt and hi is the ratio between the concentration of electrolyte i and that of the reference salt. Each derivative of eqn. (20) becomes
+ [ dG1c,-+o I dCs1cs-+o + d&g EPj) dcs I d(&T2.3kT)
+
d@/2.3kT)
d _=-dC,
CT,+ 0
=
k,
+
k,
+
where (see eqns. (15) and (14)) ppq’ =py
-kkTlnp,
=PI-PP
P’I
Ehi
(22)
dCr
The ratios hi of the salts in the aqueous phase at equilibrium can be assumed equal to those in the initial solution (Table 1) since the concentration of salts in the organic phase is very low at equilibrium.
(20)
k,
ld
RESULTS AND DISCUSSION
Equation (20) is formally identical to the expression obtained by Masterton and Lee [lo] from the scaled particle theory. However, some important differences concern the temperature dependency of the diameters of the hard spheres and the form (10) assumed to describe the polar interaction.
The above procedure allows the calculation of k, using only the molecular parameters of the pure components and the apparent molal volumes of the salts at infinite dilution. The values used for these quantities are shown in Tables 5 and 6.
TABLE 5 Molecular parameters cr X 1O24 (cm3/mol)
6x 101* (e.s.u.)
1.44 8.6
[37] [15]
1.84 1.66
[lo] [35] [12] [12] [36]
0.21 0.30* 0.47 0.87 0.86
[lo]
[12]
3.02 [lo] 0.7** 4.29* 4.17 [lo]
u X lo8 (cm)
elk (K)
Non-electrolytes Water N-butyl alcohol
2.98 2.649
[12] [34]
96.3 1115.3
Cations Na’ Mg2+ Cap+ K’ Sr2+
1.98 1.373 2.06 2.76 2.35
3.75 4.44* 3.95* 4.06
Anions clso;;$”
[12]
*Evaluated as reported in the text. **Evaluated together with u as reported in the text.
[12] [34]
[12] [lo] [38]
[35] [35]
230 TABLE 6 Apparent molal volumes at infinite dilution [ 391
Salt
NaCl Naz SO4 MgClz CaC12 KC1 NaHC03 * KBr SrC12
cpm( cm3/mol) 20 “C
30 “C
40 “C
15.98 10.30 15.50 17.70 26.07 22.00 33.37 17.60
16.97 12.40 15.60 18.25 26.92 22.00 34.30 18.60
17.62 14.00 15.45 18.25 27.53 22.00 34.75 19.30
0.2,//
,
T-“”
-I
T=30°C
*Only the value at 25 “C was found and is reported here. k,=0.200
For most ions the u-values were obtained from the Pauling crystal radii multiplied by 1.04, as suggested by Tiepel and Gubbins [ 121. No value of u was found for HCO, and SO: -. Therefore, the former was evaluated with the expression of Couture-Laidler [ 321, whereas u for SO:- was obtained by fitting the theory to the observed 12, for some systems containing Na2S04 [3]. The parameters e/k were obtained for the ions by the Mavroyannis-Stephen equation [9, 331, which requires the polarizabilities (Y. The values of (Ywere taken from the literature [lo, 12,15,38]. For HCO, the polarizability was calculated by [9] a = 3R/4nN
(23)
where the mole refraction R was evaluated from the atomic contributions. For Mg2’, a was obtained (as u of SO:-) by fitting the theory to the experimental values of k, for systems containing the Mg2’ ion. The molecular parameters of water were taken from Tiepel and Gubbins [12] ; those of n-butyl alcohol, calculated from the second virial coefficient, are reported by Polak and Lu [34]. Table 6 shows that no value was found for cp? of MgC12 -6HsO (the reported value refers to MgCl,). Therefore, the experimental mole fractions x1 and X: in eqn. (2) were calculated from measurements accounting for the substitution of MgCl, (and of the corresponding water) to MgCl, -6HsO. In Fig. 1, log@y/xr) is plotted for the whole range of C, explored. In the same Figure, straight lines passing through the
0.4 -
0.2 -
0
T=20°C
0.5
1.0
1.5
2.0
2.5
C, , mol/l
Fig. 1. Solubility of n-butyl alcohol in the saline water. The slopes of the straight lines are calculated by the perturbation theory (k, can be considered independent of temperature).
origin are shown with slopes equal to values of k, calculated by the perturbation theory. The agreement with the experimental measurements is good at all temperatures. At the lowest temperatures, eqn. (1) holds in the whole range of saline concentrations, whereas at 40 “C the validity is limited at high dilution. However, it must be pointed out that the result cannot be considered to be as satisfactory as it appears. The calculated k, depends on the molecular parameters (in particular u ), of the chemical species in solution. Their correct choice is still a crucial problem. Different values of the molecular parameters are usually obtained, for the same potential, from different properties. These differences become smaller if the description of intermolecular potential approaches that of the actual potential [40] . The commonly used types of intermolecular potential give a rough description of the dispersion forces and overlook the non-pa&vise additivity terms. Further, angular averaging of the polar part cancels the contributions due to the asym-
231
metry of the electrostatic field. The values of E/k and (I for nonelectrolytes are usually obtained from gas properties. Since simple potentials are used for this purpose, results are not suitable to describe the behaviour of the liquid phase [15 - 171. A considerable improvement may be obtained [41 - 481 if the potential parameters are calculated by fitting the perturbation theory to the observed liquid-phase properties. A similar improvement may be obtained for the ions if the crystallographic radii are substituted by the values calculated by fitting experimental k,.
density of the pure solvent radial distribution function Planck’s constant
D2 g
h hi k k,
Ci/Cr
1121 M2
N NI r R T V Xl
Boltzmann’s constant Setschenow’s constant mass of molecule 1 molecular weight of the solvent Avogadro’s number number of molecules of component separation distance of a pair of molecules mole fraction absolute temperature volume mole fraction of component 1
1
CONCLUSIONS
The perturbation theory used can be considered adequate to describe the salting effect of the observed system. The agreement of results with experimental data shows that it is possible to handle polar compounds if a suitable intermolecular potential is chosen. However, the criticality of the choice of the molecular parameters suggests calculation of the required values of u and e/k for nonelectrolytes by fitting the perturbation theory to some observed quantity of pure components in liquid phase. This follows from oversimplification in describing the complex molecular interactions by a simple potential. For engineering applications, however, the described procedure can be useful, considering the high number of chemical species present in solution and the difficulties of describing the salting effects in phase equilibria. ACKNOWLEDGMENT
The author is very grateful to Professor A. R. Giona and to Dr. P. N. Muscetta for their help in the development of this work, and to Professor C. Mustacchi for his useful suggestions.
NOMENCLATURE Ci
C, C, di
molarity of salt i concentration of reference salt saline concentration (molarity) hard-sphere diameter of species j
Greek symbols (Y polarizability E depth of the potential minimum chemical potential of component 1 c11 dipole moment ii number of js in the formula of salt i vji number density of species j (particles/ Pi volume) a collision diameter of potential @ @ intermolecular potential apparent molal volume of salt i in Cpi solution Superscripts refers to pure solvent 0 00 at infinite dilution hard-sphere value hs perturbation term P r residual part in summations, the number S solution
of salts in
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