free energy is an order of magnitude less than the accuracy of G”. As evident from Table IV, the results obtained from eq 14 are in all cases closer to the measured values than those obtained from the Wilson equation. Since we used for the determination of the temperature dependence of the parameters the regular theory approach, it must be understood that the derived equations can be applied only to mixtures of molecules not greatly differing in shape and to nonassociated mixtures in which the only negative contribution to the excess Gibbs free energy is due to the size effect. The term G* in eq 3 should always be positive and only positive excess enthalpies can be predicted. literature Cited
Baud, E., Bull. SOC.Chim. Fr. 17, 329 (1915). 9, 305 (1970). Bruin, S., ISD.ENG.CHPM.,FUNDAM. Flory, P. J., J . Chem. Phys. 10, 51 (1942). Fried, V., Franceschetti, D. R., Schneier, G. B., J . Chem. Data 13, 415 (1968). Funk, E. W., Prausnitz, J. M., Ind. Eng. Chem. 62(9), 8 (1970).
EG.
Hildebrand, J. H., Scott, R. L., “The Solubility of Nonelectrolytes,” Dover Publications, Inc., New York, N. Y., 1964. Huggins, M. , Ann. N , Y . Acad. Sci. 43, 1 (1942). Kohler, F., Chem. Tech. (Leipzig) 18(5), 272 (1966). Korvezee, A. E., private communication, 1969. Kuhn, W., Massini, P., Helv. Chim. Acta 33, 737 (1950). Liebermann, E., Kohler, F., Xonatsh. Chem. 99, 2514 (1968). Lundberg, G. W., J . Chem. Eng. Data 9, 193 (1964). Mathieson, A. R., Thynne, J. C. J., J . Chem. SOC.,London 3708 (1956). Miksch, G., Liebermann, E., Kohler, F., Momtsh. Chem. 100, 1574 (1969). \----,. Orye; V., Ph.D. dissertation, University of California, Berkeley, Calif.. 1965. Scatchard, ’G., Ticknor, L. B., Goates, J. R., McCartney, E. R., J . Amer. Chem. SOC.74, 3721 (1952). Scatchard. G.. Wood. S. E.. Mochel. J. M.. J . Amer. Chem. SOC.62,’ 712 (1940): Watson, A. E. P., McClure, I. il., Bennett, J. E., Benson, G. C.. J . Phus. Chem. 69. 2753 (1965). Wilson,’G. M.,“J. Amer. Chem. Soc. 86,’ 127 (1964).
k.
RECEIVED for review May 26, 1971 ACCEPTEDFebruary 24, 1972
The Temperature Dependence of the Excess Gibbs Free Energy of Binary Nonassociated Mixtures Ernst Liebermann’ and Vojtech Fried* Department of Chemistry, Brooklyn College of the City University of New York, Brooklyn, N . Y . 11210
Excess Gibbs free energy values have been predicted at different temperatures from single temperature data. The method has been applied to 25 systems and excellent agreement has been found between observed and calculated results within a wide temperature interval.
I n the previous paper (Liebermann and Fried, 1972) a procedure is discussed which permits the calculation of the excess Gibbs free energy from isothermal enthalpies of mixing and vice versa. The method is, however, unable to give any information on the variation of those excess properties with temperature. The theory has been, therefore, modified in such a way that it makes possible the extrapolation of the excess Gibbs free energy from a single temperature to any other temperature of concern. Theory
In the aforementioned paper the excess Gibbs free energy has been represented by a two-parameter equation. When applied to the temperature T”, this equation takes the form
[ZI
In
(21
+ ZZVZ/VJ + x2 In + xlV1/V~)l ($2
(1)
In the same paper the temperature dependence of both parameters has been derived as 1
Postdoctoral fellow from the University of Vienna, Austria.
354 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
G”” is the excess molar Gibbs free energy and Al2”, AzI”, AI2,Azl are parameters characteristic of the system a t the temperature T” and T , respectively. The molar volume ratio Vl/V2 is assumed to be temperature-independent. To evaluate the excess Gibbs free energy a t a temperature T” from its known value a t a reference temperature T’, an equation relating the parameters to the temperature must be found. Such a relationship can be obtained from eq 2a and 2b. Rearranging and subtracting of eq 2a and 2b gives
Consequently
Aiz/Azi = C where C is a temperature-independent constant.
(4)
Another relation between the parameters is obtained by rearranging and adding of eq 2a and 2b.
b In
A12
bT
+--b InbTAZI-
System
b In (AIZAZI) - 4 In (AIZAZI) bT In ( A d d - 2 This is a differential equation of the form
where y tion is
=
Table 1. Measured and Extrapolated Values of the Excess Gibbs Free Energy G E ” at x = 0.5 Extrapolation GE” in cal mole-’
In (AI2A2,)and x
=
T . The solution of this equa-
or
/ A ~ ~ ~ ’ A ~ ~ ~ ~ \
The indices ’ and ” refer to the values of the parameters a t the temperature T‘ and T”, respectively. It follows from eq 4 that
An’’ - AB’ A2I1’
Multiplying of eq 9 by
(9)
2421’
~421“/A12’
results in
Consequently Alz” =
kAiz’
(1W
Rzi” = kAzi’
(1lb) On the contrary to C, k is temperature-dependent. Upon inserting eq l l a and l l b into eq 1, the following relation is obtained for the excess Gibbs free energy a t the temperature
Pentane- benzene Neopentane- benzene Cyclopentane- benzene Hexane- benzene 2-Meth ylpentanebenzene 2,2-Dimethylbutanebenzene 2,3-Dimethylbutanebenzene Cyclohexane- benzene Methylcyclopentanebenzene Heptane- benzene 3-Methylhexane- benzene 2,CDimethylpentanebenzene 2,2,3-Trimethylbutanebenzene Methylcyclohexanebenzene Octane- benzene 2,2,4Trimethylpentanebeneene Hexane- toluene 3-Meth y lpentanetoluene Cyclohexane-toluene Methylcyclopentanetoluene Heptane- toluene Methylcyclohexanetoluene 2,2,4-Trimethylpentanetoluene 1,2-Dichloroethanecyclohexane Pyridinetetrachloroethylene
TI‘ SE =
--
range, OC
Meard
Extrapd
25 75 50 25 2 5 + 75 25 + 75
90 134 61 75
85 126 60 76
25 --t 75
97
95
25 + 75
108
99
25 + 75 25 + 75
83 65
87 67
25 + 75 25 + 75 75 25
-
67 63 70
67 68 73
25 + 75
105
98
2 5 4 75
79
73
25 + 75 25 + 75
59 68
62 67
25 + 75 75 25
-
73 73
76 67
25 + 75 25 + 75
75 60
69 62
25 + 75 25 + 75
56 46
55 39
25 + 75
37
41
25 + 75
70
68
20 + 32
176
177
60 + 80
122
120
-[=I
dGE P
+
+
+
[aIn (xl z~VZ/VI) xz In (ZZ ZIVI/VZ)I (12) I n this equation GE” is presented in terms of the parameters of GE’ and a factor k. I n order to find G”” from GE’ the value of k must be known. For this purpose we introduce eq l l a and l l b into eq 8 and obtain
+
k2
In k 2 In (~412’&’) 1n (Al2tAZ1’)
[
1’
=
(13) (F)4
After some algebraic manipulations eq 13 reduces t o the form 2 In k - k(:;,)’
In
(.412’d2
