- Email: [email protected]
Prediction of finite-concentration activity coefficients from a single g.l.c.-determined activity coefficient at infinite dilution
3. Chem. Thermodynamics 1975, I, 353-357
Prediction of finite-concentration activity coefficients from a single g.l.c.-determined activity coefficient at infinite dilution T. M. LETCHER
and G. NETHERTON
Department of Chemistry, University of Witwatersrand, Johannesburg, South Africa (Received 26 July 1974) The Guggenheim-Miller-Flory-Huggins equation and a one-parameter form of the Wilson equation can be used to fit activity coefficients for a range of hydrocarbon mixtures. In this work these two equations are shown to provide a simple and accurate method of estimating finite-concentration activity coefficients from a single g.l.c.-determined value of the infinite-dilution activity coefficient. The mixtures studied were cyclohexane + n-hexadecane, cyclohexane + n-eicosane, cyclohexane f squalane, n-hexane + n-hexadecane, and nhexane + squalane. These mixtures were chosen because, in all cases, reliable activity coefficients at finite concentrations were available in the literature.
1. Introduction Two equations which have been used to fit activity coefficients are (a) the GuggenheimMiller-Flory-Huggins equation”) and (b) the Wilson equation.@’ The first equation is a one-parameter equation (if z is allowed to tend to infinity) and the second equation, normally a two-parameter equation, can be simplified and treated as a one-parameter equation. (3) In this work, each of these equations has been used to predict finiteconcentration activity coefficients from a single g.l.c.-determined activity coefficient at infinite dilution. The success of this method in predicting activity coefficients for a variety of hydrocarbon mixtures suggests that it could be used in general for nonpolar hydrocarbon mixtures.
2. Experimental APPARATUS
The g.1.c. apparatus used in this work has been described.(4) The column specifications are given in table 1. The carrier gas used was nitrogen (supplied by African Oxygen with a quoted purity of 99 moles per cent). MATERIALS The stationary
phases, n-hexadecane,
n-eicosane,
and squalane
were supplied by
B.D.H. and were used without further purification. The quoted purities were better than 99 moles per cent. The celite was supplied by B.D.H. @O/100 mesh) and was
T. M. LETCHER AND G. NETHERTON
354
TABLE 1. Column specifications. In all casescopper tubing of 4.2 mm bore was used. I is the column length, ms the mass of solvent + celite on the column, e the percentage loading, and n3 the amount of solvent on the column solvent
l/m
n-hexadecane n-eicosane squalane
2.01 1.98 2.02
mlg 19.5218 21.8638 21.3424
e
nJmmo1
7.12 8.33 9.40
6.138 6.446 4.745
used without further treatment. The solutes, n-hexane and cyclohexane were supplied by B.D.H. It was not necessary to purify the solutes as the chromatograph column separated impurities from the major solute component. 3. Results The retention volumes were determined as described by Cruickshank, Young,(5) and by Letcher and Marsicano.‘4’ The activity coefficients 77 were calculated from the relations:
Windsor, and
In V, = ln(k”VS) f flpOJi4),
(1)
where In k” = ln(n,RT/v,r~p9-pXB1, p = w3,z
- Vy)/RT,
(2)
- WRT,
(3)
and (4) where V, is the net retention volume, V, the volume of solvent, Vi the molar volume of pure solvent, B,, the second virial coefficient of pure solute vapour, p; the vapour pressure of the solute at temperature T, n3 the amount of solvent on the column, pi the column inlet pressure, and p,, the column outlet pressure. The additional inforA4)
mation
required
=
(3/4){(Pi/P34-
to calculate the activity
1)l{(Pi/Po)3
coefficient
-
l>P
at infinite dilution
is given in
table 2 and the activity coefficients are tabulated in table 3. TABLE 2. Vapour pressures J$ and second virial coefficients PI1 for the solutes. Molar volumes V,” aregivenin table4 Solute cyclohexane
n-hexane
Ref. -B&m3
mol-1 Ref.
T/K
Pl”/kPa
298.07 303.15 317.40
12.967 16.234 29.130
8
1600 1550 1500
9
293.15 303.15
6.061 24.985
8
2100 1800
9
The column loadings were checked by determining activity coefficients of solutes which were considered reliable. The estimated uncertainty in the activity coefficient r?
PREDICTION
OF FINITE-CONCENTRATION
ACTIVITY
COEFFICIENTS
355
TABLE 3. The parameters a and b of the equation: loglo(V,/m3) = a - bp,J’? together with the results for loglOy? and &. Also included is the standard deviation u for the straight-line fit solute
solvent
cyclohexane cyclohexane cyclohexane hexane hexane
n-hexadecane n-eicosane squalane n-hexadecane squalane
T/K
-U
298.07 317.40 303.15 293.15 303.15
2.8272 3.0552 2.8564 2.5609 3.1232
--lO~lOY,” -&a pIGPa- 1 10% this work literature --cm3mol-’ 68.116 55.197 63.99 66.765 58.325
15 17 9 12 14
0.100 0.171 0.272 0.044 0.188
0.271 (6) o.044’6’ 0.187 m) 0.193 (‘)
139 112 131 122 103 103
was estimated to be less than 0.004. The activity coefficients in three of the five mixtures have been reported”’ and our g.1.c. results agree, within experimental error, with these literature values.
4. Discussion The Guggenheim-Miller-Flory-Huggin@ lo--’ ‘) equation relates the activity coefficient of a component of a binary mixture of molecules of different sizes to composition according to; In y1 = 141 -d2Wxl)+(l
- l/r)42 +x&,
(5)
where & is the volume fraction of component 2, x1 the mole fraction of component 1, and x the interaction parameter. We have chosen r to be the ratio of the size of molecules, (VJV,), because of its convenience.(‘3’ The Wilson equation is a semi-theoretical extension of the Flory-Huggins(“$ 12) theory for athermal solutions. It relates the activity coefficient y1 to mole fractions .x1 and x2 for a binary mixture by; ln y1 = -Wl where
+~12x2)+x2C~12{~l(~l
+n12x2>I -~21{W21xI
+x2)~1~
(6)
Al2
= (I/,(/~l)exp(-(g12-gll)/RT),
(7)
A,,
= (vl/V,)exp{-(g12-g22)/RT},
(8)
and VI and V, are the molar volumes of components 1 and 2 respectively, and g12, gII, and g22 pair-interaction energies for molecules of species 1 and 2, 1 and 1, and 2 and 2, respectively. Equation (6) can be regarded as a two-parameter equation with A,, and A,, the parameters. The pair-interaction energies have been related to energies of vaporization AHV(14) or configurational energies.(15* 16) The general expression is RT)/z, Sii z -2(AH,(9) where z is the coordination number. Substituting equations (7), (8), and (9) into equation (6) and fixing z, we obtain a one-parameter equation with g 12 the adjustable parameter.
356
T. M. LETCHER
At infinite dilution,
AND G. NETHERTON
equation (5) becomes: In yy(GMFH)
= ln(l/r)+(l
-l/r-)+x,
and equation (6) becomes: my?(W)=-lnn,,+l-A,, = ln(llr>+l-(llr)exP(-g~2lRT)+(g~2/RT). TABLE
4. Molar enthalpies of vaporization component n-hexadecane n-eicosane squalane n-hexane cyclohexane
AH, and molar volumes Y” for the compounds used in this work
T/K 293.15 298.07 317.40 303.15 293.15 303.15 298.07 303.15 317.40
AH&J
mol-1
V/cm3 mol-’
81.09 80.33 100.83 153.55 31.95 31.21 33.06 33.06 33.06
292.83 294.08 365.96 528.52 130.68 132.47 108.75 109.42 111.36
In this work equations (10) and (1 1), together with the data from table 4 are used to determine the adjustable parameters (x and g1 & which in turn are used to calculate the finite-concentration activity coefficients. These calculated results are compared with literature values and the standard deviation o, calculated from CT*= C{y(expt) -r(calc.)}‘/(n
- l),
(12)
where n is the number of results. The results are tabulated in table 5. In the case of the Wilson equation the best results are obtained when z is large. This is interesting because z has usually been taken as 10.‘3’ For z = co, equation (6) becomes Iny,(W)=In{l/(x,+A)}+A/(x,+A)-A/(x,r’--++/xz), where A = x2r exp(-g,,/RT). The non-random two-liquid equation derived by Renon”‘) from Scott’s two-fluid theory(‘s) can also be written in a one-parameter form. This was also tested but the standard deviation o was much greater and ranged between 0.002 and 0.08. Except for cyclohexane + n-eicosane the deviations given in table 4 are less than the deviations of the g.l.c.-determined activity coefficients. For cyclohexane + n-eicosane, using all the results given by Gbmez-Ibaiiez and Wang,t2’) we found that the deviation for either the Wilson or the Guggenheim equation was 0.008. However a plot of their experimental results, showed that one measurement deviated considerabIy from a smooth curve. When we ignored this value (x, = 0.40883; y = 0.7896) the deviations were 0.0043 and 0.0040 for the Wilson and Guggenheim equations, respectively (bracketed values of table 5).
PREDICTION
OF FINITE-CONCENTRATION
ACTIVITY
COEFFICIENTS
357
TABLE 5. The standard deviations TV obtained by comparing the calculated finite-concentration activity coefficients with experimentallydetermined activity coefficientsobtainedfrom the literature
(references in column5)
solute
solvent
cyclohexane
n-hexadecane 298.07
TIK
y”(g.1.c.)
ref.
Wilson equation z u
0.79s
19
10 103 106 10 lo3 106 10 103 106 10 103 106 10 lo3 lo6
Guggenheim (z= co) r7
-__.
cyclohexane n-eicosane
317.40 0.675
cyclohexane squalane
303.15
20
0.535 (6) 1
n-hexane
n-hexadecane
293.15 0.903 (6) 21
n-hexane
squalane
303.15
0.648
1
0.0692 0.0021 0.0020 0.0486 (0.045) 0.0081 (0.0041) 0.0080 (0.0043) 0.0310 0.0028 0.0025 0.0685 0.0020 0.0021 0.0396 0.0032 0.0028
-.. ~
0.0023 0.0080 (0.0040) 0.0030 0.0020 0.0040
The outstanding predictive powers of these two equations for the hydrocarbon mixtures considered here, indicate that for a limited type of binary mixture, accurate predictions of activity coefficients at finite concentrations are possiblefrom one, easily determined, value at infinite dilution. The authors wish to thank the C.S.I.R.
(South Africa) for a running expense grant.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 1s. 16. 17. 18. 19. 20. 21. 25
Ashworth, A. J.; Everett, D. H. Trans. Faraday Sot. 1960, 56, 1609. Wilson, G. M. J. Amer. Chem. Sot. 1964,86,127. Hussey, C. L.; Parcher, J. F. Anal. Chem. 1973, 45, 926. Letcher, T. M.; Marsicano, F. J. Chem. T?zermodynamics 1974, 6, 501. Cruickshank, A. J. B.; Windsor, M. L.; Young, C. L. Proc. Roy. Sot. A 1%6, 295, 271. (a) Conder, J. R.; Purnell, J. H. Trans. Faraday Sot. 1969, 65, 839. (b) Private communication, Moody, R. W. University of Bristol. WitorovB, 0.; Nov&k, J.; JanBk, J. J. Chromatogr. 1972, 65, 241. Selected Values of Properties of Hydrocarbon and Related Compoun& American Petroleum Institute, Research Project 44, Thermodynamics Research Centre. Texas A & M University, College Station, Texas. Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases. Clarendon Press: Oxford. 1%9. Guggenheim, E. A. Mixtures. Oxford University Press: London. 1952. Huggins, M. L. J. Chem. Phys. 1941,9, 440. Flory, P. J. J. Chem. Phys. 1942, 10, 51. Everett, D. H.; Munn, R. J. Trans. Faraday Sot. 1964,60, 1951. Tassios, D. A. I. Ch. E. J. 1971, 17, 1367. Wong, K. F. ; Eckert, C. A. Ind. Eng. Chem., Fundam. 1971,10,20. Schreiker, L. B.; Eckert, C. A. Znd. Eng. Chem., Process Des. Develop. 1971, 10, 572. Renon, H.; Prausnitz, J. M. A.Z. Ch. E. J. 1968, 14, 135. Scott, R. L. J. Chem. Phys. 1956,2S, 193. Gbmez-Ibhfiez, J. D.; Shien, J. J. C. J. Phys. Chem. 1965, 69, 1660. G6mez-Ibfiez, J. D.; Wang, F. T. J. Chem. Thermodynamics 1971, 3, 811. McGlashan, M. L.; Williamson, A. G. Trans. Faraday Sot. 1961, 57, 588.
Prediction of finite-concentration activity coefficients from a single g.l.c.-determined activity coefficient at infinite dilution T. M. LETCHER
and G. NETHERTON
Department of Chemistry, University of Witwatersrand, Johannesburg, South Africa (Received 26 July 1974) The Guggenheim-Miller-Flory-Huggins equation and a one-parameter form of the Wilson equation can be used to fit activity coefficients for a range of hydrocarbon mixtures. In this work these two equations are shown to provide a simple and accurate method of estimating finite-concentration activity coefficients from a single g.l.c.-determined value of the infinite-dilution activity coefficient. The mixtures studied were cyclohexane + n-hexadecane, cyclohexane + n-eicosane, cyclohexane f squalane, n-hexane + n-hexadecane, and nhexane + squalane. These mixtures were chosen because, in all cases, reliable activity coefficients at finite concentrations were available in the literature.
1. Introduction Two equations which have been used to fit activity coefficients are (a) the GuggenheimMiller-Flory-Huggins equation”) and (b) the Wilson equation.@’ The first equation is a one-parameter equation (if z is allowed to tend to infinity) and the second equation, normally a two-parameter equation, can be simplified and treated as a one-parameter equation. (3) In this work, each of these equations has been used to predict finiteconcentration activity coefficients from a single g.l.c.-determined activity coefficient at infinite dilution. The success of this method in predicting activity coefficients for a variety of hydrocarbon mixtures suggests that it could be used in general for nonpolar hydrocarbon mixtures.
2. Experimental APPARATUS
The g.1.c. apparatus used in this work has been described.(4) The column specifications are given in table 1. The carrier gas used was nitrogen (supplied by African Oxygen with a quoted purity of 99 moles per cent). MATERIALS The stationary
phases, n-hexadecane,
n-eicosane,
and squalane
were supplied by
B.D.H. and were used without further purification. The quoted purities were better than 99 moles per cent. The celite was supplied by B.D.H. @O/100 mesh) and was
T. M. LETCHER AND G. NETHERTON
354
TABLE 1. Column specifications. In all casescopper tubing of 4.2 mm bore was used. I is the column length, ms the mass of solvent + celite on the column, e the percentage loading, and n3 the amount of solvent on the column solvent
l/m
n-hexadecane n-eicosane squalane
2.01 1.98 2.02
mlg 19.5218 21.8638 21.3424
e
nJmmo1
7.12 8.33 9.40
6.138 6.446 4.745
used without further treatment. The solutes, n-hexane and cyclohexane were supplied by B.D.H. It was not necessary to purify the solutes as the chromatograph column separated impurities from the major solute component. 3. Results The retention volumes were determined as described by Cruickshank, Young,(5) and by Letcher and Marsicano.‘4’ The activity coefficients 77 were calculated from the relations:
Windsor, and
In V, = ln(k”VS) f flpOJi4),
(1)
where In k” = ln(n,RT/v,r~p9-pXB1, p = w3,z
- Vy)/RT,
(2)
- WRT,
(3)
and (4) where V, is the net retention volume, V, the volume of solvent, Vi the molar volume of pure solvent, B,, the second virial coefficient of pure solute vapour, p; the vapour pressure of the solute at temperature T, n3 the amount of solvent on the column, pi the column inlet pressure, and p,, the column outlet pressure. The additional inforA4)
mation
required
=
(3/4){(Pi/P34-
to calculate the activity
1)l{(Pi/Po)3
coefficient
-
l>P
at infinite dilution
is given in
table 2 and the activity coefficients are tabulated in table 3. TABLE 2. Vapour pressures J$ and second virial coefficients PI1 for the solutes. Molar volumes V,” aregivenin table4 Solute cyclohexane
n-hexane
Ref. -B&m3
mol-1 Ref.
T/K
Pl”/kPa
298.07 303.15 317.40
12.967 16.234 29.130
8
1600 1550 1500
9
293.15 303.15
6.061 24.985
8
2100 1800
9
The column loadings were checked by determining activity coefficients of solutes which were considered reliable. The estimated uncertainty in the activity coefficient r?
PREDICTION
OF FINITE-CONCENTRATION
ACTIVITY
COEFFICIENTS
355
TABLE 3. The parameters a and b of the equation: loglo(V,/m3) = a - bp,J’? together with the results for loglOy? and &. Also included is the standard deviation u for the straight-line fit solute
solvent
cyclohexane cyclohexane cyclohexane hexane hexane
n-hexadecane n-eicosane squalane n-hexadecane squalane
T/K
-U
298.07 317.40 303.15 293.15 303.15
2.8272 3.0552 2.8564 2.5609 3.1232
--lO~lOY,” -&a pIGPa- 1 10% this work literature --cm3mol-’ 68.116 55.197 63.99 66.765 58.325
15 17 9 12 14
0.100 0.171 0.272 0.044 0.188
0.271 (6) o.044’6’ 0.187 m) 0.193 (‘)
139 112 131 122 103 103
was estimated to be less than 0.004. The activity coefficients in three of the five mixtures have been reported”’ and our g.1.c. results agree, within experimental error, with these literature values.
4. Discussion The Guggenheim-Miller-Flory-Huggin@ lo--’ ‘) equation relates the activity coefficient of a component of a binary mixture of molecules of different sizes to composition according to; In y1 = 141 -d2Wxl)+(l
- l/r)42 +x&,
(5)
where & is the volume fraction of component 2, x1 the mole fraction of component 1, and x the interaction parameter. We have chosen r to be the ratio of the size of molecules, (VJV,), because of its convenience.(‘3’ The Wilson equation is a semi-theoretical extension of the Flory-Huggins(“$ 12) theory for athermal solutions. It relates the activity coefficient y1 to mole fractions .x1 and x2 for a binary mixture by; ln y1 = -Wl where
+~12x2)+x2C~12{~l(~l
+n12x2>I -~21{W21xI
+x2)~1~
(6)
Al2
= (I/,(/~l)exp(-(g12-gll)/RT),
(7)
A,,
= (vl/V,)exp{-(g12-g22)/RT},
(8)
and VI and V, are the molar volumes of components 1 and 2 respectively, and g12, gII, and g22 pair-interaction energies for molecules of species 1 and 2, 1 and 1, and 2 and 2, respectively. Equation (6) can be regarded as a two-parameter equation with A,, and A,, the parameters. The pair-interaction energies have been related to energies of vaporization AHV(14) or configurational energies.(15* 16) The general expression is RT)/z, Sii z -2(AH,(9) where z is the coordination number. Substituting equations (7), (8), and (9) into equation (6) and fixing z, we obtain a one-parameter equation with g 12 the adjustable parameter.
356
T. M. LETCHER
At infinite dilution,
AND G. NETHERTON
equation (5) becomes: In yy(GMFH)
= ln(l/r)+(l
-l/r-)+x,
and equation (6) becomes: my?(W)=-lnn,,+l-A,, = ln(llr>+l-(llr)exP(-g~2lRT)+(g~2/RT). TABLE
4. Molar enthalpies of vaporization component n-hexadecane n-eicosane squalane n-hexane cyclohexane
AH, and molar volumes Y” for the compounds used in this work
T/K 293.15 298.07 317.40 303.15 293.15 303.15 298.07 303.15 317.40
AH&J
mol-1
V/cm3 mol-’
81.09 80.33 100.83 153.55 31.95 31.21 33.06 33.06 33.06
292.83 294.08 365.96 528.52 130.68 132.47 108.75 109.42 111.36
In this work equations (10) and (1 1), together with the data from table 4 are used to determine the adjustable parameters (x and g1 & which in turn are used to calculate the finite-concentration activity coefficients. These calculated results are compared with literature values and the standard deviation o, calculated from CT*= C{y(expt) -r(calc.)}‘/(n
- l),
(12)
where n is the number of results. The results are tabulated in table 5. In the case of the Wilson equation the best results are obtained when z is large. This is interesting because z has usually been taken as 10.‘3’ For z = co, equation (6) becomes Iny,(W)=In{l/(x,+A)}+A/(x,+A)-A/(x,r’--++/xz), where A = x2r exp(-g,,/RT). The non-random two-liquid equation derived by Renon”‘) from Scott’s two-fluid theory(‘s) can also be written in a one-parameter form. This was also tested but the standard deviation o was much greater and ranged between 0.002 and 0.08. Except for cyclohexane + n-eicosane the deviations given in table 4 are less than the deviations of the g.l.c.-determined activity coefficients. For cyclohexane + n-eicosane, using all the results given by Gbmez-Ibaiiez and Wang,t2’) we found that the deviation for either the Wilson or the Guggenheim equation was 0.008. However a plot of their experimental results, showed that one measurement deviated considerabIy from a smooth curve. When we ignored this value (x, = 0.40883; y = 0.7896) the deviations were 0.0043 and 0.0040 for the Wilson and Guggenheim equations, respectively (bracketed values of table 5).
PREDICTION
OF FINITE-CONCENTRATION
ACTIVITY
COEFFICIENTS
357
TABLE 5. The standard deviations TV obtained by comparing the calculated finite-concentration activity coefficients with experimentallydetermined activity coefficientsobtainedfrom the literature
(references in column5)
solute
solvent
cyclohexane
n-hexadecane 298.07
TIK
y”(g.1.c.)
ref.
Wilson equation z u
0.79s
19
10 103 106 10 lo3 106 10 103 106 10 103 106 10 lo3 lo6
Guggenheim (z= co) r7
-__.
cyclohexane n-eicosane
317.40 0.675
cyclohexane squalane
303.15
20
0.535 (6) 1
n-hexane
n-hexadecane
293.15 0.903 (6) 21
n-hexane
squalane
303.15
0.648
1
0.0692 0.0021 0.0020 0.0486 (0.045) 0.0081 (0.0041) 0.0080 (0.0043) 0.0310 0.0028 0.0025 0.0685 0.0020 0.0021 0.0396 0.0032 0.0028
-.. ~
0.0023 0.0080 (0.0040) 0.0030 0.0020 0.0040
The outstanding predictive powers of these two equations for the hydrocarbon mixtures considered here, indicate that for a limited type of binary mixture, accurate predictions of activity coefficients at finite concentrations are possiblefrom one, easily determined, value at infinite dilution. The authors wish to thank the C.S.I.R.
(South Africa) for a running expense grant.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 1s. 16. 17. 18. 19. 20. 21. 25
Ashworth, A. J.; Everett, D. H. Trans. Faraday Sot. 1960, 56, 1609. Wilson, G. M. J. Amer. Chem. Sot. 1964,86,127. Hussey, C. L.; Parcher, J. F. Anal. Chem. 1973, 45, 926. Letcher, T. M.; Marsicano, F. J. Chem. T?zermodynamics 1974, 6, 501. Cruickshank, A. J. B.; Windsor, M. L.; Young, C. L. Proc. Roy. Sot. A 1%6, 295, 271. (a) Conder, J. R.; Purnell, J. H. Trans. Faraday Sot. 1969, 65, 839. (b) Private communication, Moody, R. W. University of Bristol. WitorovB, 0.; Nov&k, J.; JanBk, J. J. Chromatogr. 1972, 65, 241. Selected Values of Properties of Hydrocarbon and Related Compoun& American Petroleum Institute, Research Project 44, Thermodynamics Research Centre. Texas A & M University, College Station, Texas. Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases. Clarendon Press: Oxford. 1%9. Guggenheim, E. A. Mixtures. Oxford University Press: London. 1952. Huggins, M. L. J. Chem. Phys. 1941,9, 440. Flory, P. J. J. Chem. Phys. 1942, 10, 51. Everett, D. H.; Munn, R. J. Trans. Faraday Sot. 1964,60, 1951. Tassios, D. A. I. Ch. E. J. 1971, 17, 1367. Wong, K. F. ; Eckert, C. A. Ind. Eng. Chem., Fundam. 1971,10,20. Schreiker, L. B.; Eckert, C. A. Znd. Eng. Chem., Process Des. Develop. 1971, 10, 572. Renon, H.; Prausnitz, J. M. A.Z. Ch. E. J. 1968, 14, 135. Scott, R. L. J. Chem. Phys. 1956,2S, 193. Gbmez-Ibhfiez, J. D.; Shien, J. J. C. J. Phys. Chem. 1965, 69, 1660. G6mez-Ibfiez, J. D.; Wang, F. T. J. Chem. Thermodynamics 1971, 3, 811. McGlashan, M. L.; Williamson, A. G. Trans. Faraday Sot. 1961, 57, 588.