V. Fried
Some Comments 0n Thermodynamics of Ideal Solutiolls
Brooklyn College of the City University of New York Brooklyn, 11210
A s has been demonstrated by Williamson ( I ) , Raoult's law does not define exactly an ideal solution; it only follows from the exact definition after making some assumptions. In the cited paper deviations from Raoult's law of an ideal liquid solution in equilibrium with an ideal vapor mixture are discussed very carefully. As will be shown further these deviations are much greater if an ideal liquid solution is in equilibrium with a non-ideal vapor mixture. A solution is said to be ideal if the chemical potential of each component p,(T,P,N,) is a linear function of the logarithm of its mole fraction N, according to the relation
+
#,(T,P,N,) = aSo(T,P) RT ln N. =
+
.c
+ RT In N.
W , ~ ( T , ~ , ~ ) fr,,%P
(1)
p?(T,P) is the chemical potential of the pure component i at the temperature and pressure of the system and riO(T,p,O) is the chemical potential of the component i at the temperature of the system and its saturated vapor pressure at this temperature. Let us now consider a system composed of an ideal liquid solution and a non-ideal vapor phase. The chemical potential of a component i in the liquid solution is given by eqn. (I), and the chemical potential of the same component in the non-ideal vapor mixture, p,,(T,P,y,), at low and moderite pressures can be expressed by the formula (2, ,?)
where x, and y, denote the mole fractions in the liquid and vapor phase respectively; P is the total pressure of the system, p p the vapor pressure of the pure component i at the temperature of the system, Pi0themolar volume of component i in the liquid phase and B,, is the second virial coefficient of component i. At equilibrium the chemical potentials in both phases are equal [pir(T,P,zi) = ri,(T,P,yi); rito(T,pP) = pi,o(T,pio)l
and for a two-component system at constant temperature the following equation is derived [(P- plD)(B.*- Bn) - 2P(A)ynz P = z,pP exp RT
~
1-
( P - pl0)(V? - B d 2 exp[ ~ 2 ~ RT
+
- 2P(A)yx2 ]
(3)
For a binary solution consisting of components 1 and 2, A is given by the relation A = Bn - 'lpB1,- 1/2B2p
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Journol of Chemical Fdumfion
(4)
where BIZis known as the interaction or cross-coefficient. The second virial coefficient of a saturated vapor is always negative, so that (VSo- B,,) is positive and the deviation from Raoult's law is greater if the equilibrium vapor phase is an ideal mixture of imperfect gases (A = 0). I n the case of a non-ideal vapor phase the A term has to be considered. Experimental results show, that the value of A can be either positive or negative. In most cases the greater the differencebetween the virial coefficients of the two pure components the greater A is, nevertheless, it is much smaller than either Bn or BZ2. Under some conditions-far from the critical-the value of A can he negligibly small; nevertheless, in some cases it can contribute appreciably to the deviation from Raoult's law. A can be calculated from eqn. (4) if the values of the cross-coefficient are known. Only a few experimental values of Bn are available in the literature, therefore it is commonly calculated from the following approximate formulas:
IBn = (BnBn)'/.
Ba = '/s(B,,%
+ B%*'/I)~
(6)
(7)
Williamson (1) calculated the deviation from Raoult's law of an ideal solution in equilibrium with an ideal vapor phase for two typical cases. For comparison the same cases are used in this work, except that the vapor phase is an ideal mixture of imperfect gases and a non-ideal vapor mixture, respectively. The results for both cases are summarized in the table. For further illustration different values of A (positive and nega tive) are considered. As can be seen from these two cases deviations of an ideal solution from Raoult's law may be significant. I n many published papers on vapor-liquid equilibrium the non-ideality of the vapor-phase is ignored. This assumption can result in great errors in the calculated thermodynamic properties of solutions. I n our two cases A has the following values: calculated from eqn. (5) A = 0;from eqn. (6) A = 3; and from eqn. (7) A = 2. The deviations due to these very small values of A can be completely ignored in case 1 and represent only a very small contribution to the nonideal behavior of the vapor-phase in case 2. However, the method proposed by Prausnitz (4) gives a much higher value for A resulting in a larger deviation as evident from the table. Only experimental values of B , ~ can give more reliable values for A and a better estimate of the deviation due to the imperfection of the vapor phase.
Deviation from Raoult'r Law for Two Typical Cases
B,,
BII
A
Pl"
P?
P"
Case 1 350°K, p,O = 500mm Hg, p10 = 700mm Hg, z, = 0.5, T0 , = 100ml moleCL, rXo = lOOrnl mole-', Bn Bn = -1700ml mole-' Rsoult's Law 0 0 0 250 350 Ideal vapor phase 0 0 0 250.12 349.84 Ideal gas mixture 251 .80 347.10 0 of imperfect gases - 1500 - 1700 251.77 347.07 - 1500 +lo Non-ideal vttpor - 1700 347.13 251.87 - 10 - 1500 -1700 phase 251.35 346.74 +I00 - 1500 - 1700 252.30 347.45 - 1700 -100 -1500 250.62 346.20 +250 -1500 -1700 253.02 348.00 -250 - 1700 -1500 345.31 +500 249.45 - 1700 -1500 348.87 254.23 -500 - 1500 -1700 343.60 247.11 -1700 +I000 - 1500 350.82 256.69 -1000 - 1500 -1700
T
=
Case 2 35OSK, p,O = 500mm Hg, p< = 2000mm Hg, z, = 0.5, T0 , = 200ml mole Bm = -1700ml mole-' Rmult's law Ideal vapor phase Ided gas mixture imperfect gases Nonideal vapor phase
T
=
', Vz0 = lOOml mole-',
Bll
=
-1500ml
mole-',
600 599.66 598.90 598.84 599.00 598.09 599.75 596.82 601.02 594.76 603.10 590.71 607.51
=
-1500 ml mole-',
-~-
T h val!le- in (:as? 2 and ?ome uf the V ~ I U P ill S (:aw I were d ~ t ~ h t by r d811 iterarive mrtht,d. The values of P for the tirrl approxirnntlon of the pre,sure in the expwtcr,unl rrrm of win. A , were obrmved f 1 1 m I t n o d ' s law. TIIPpret.nre thus obtained sa, used i l l rlte ~ i in n w nweswry. M I the ~ W S S I I TBTPVipleIS. I ~ r i w u w l w \ . I n wmeiwrsa third upprtminulriw>WII* r.q,u!w~~ial rr1.m tn, ~ C I I I S the llg.
Conversely, a system which is observed to obey Raoult's law is generally not an ideal solution as defined by eqn. (1). The solution benzene-ethylene dichloride is a typical example ( 5 , 6 ) . This system obeys Raoult's law exactly, nevertheless, its excess free energy G", which is defined as the excess of the increase of the Gibbs function on mixing A Gmi, over that of an ideal solution AGmix* G=
= AG,~, - AG,~,*
(8)
is not zero. It is clear that it is not correct to discuss the nonideal solutions in terms of deviations from Raoult's law. It is more exact to discuss the non-ideal solutions in terms of deviations from eqn. (I), which are the excess free energies exactly. Zero excess free energy defines an ideal solution. For more adequate discus-
sion the reader is referred to the literature (2, 3, 7-9). An excellent discussion of the deviation in terms of the excess free energy is presented by RlcGlashan (9). Literature Cited
A. G., J. CHEM. EDUC., 43,211 (1966). (1) WILLIAMSON, J. S., "Liquid and Liquid Mixtures," Butter(2) ROWLINS~N, worths Sci. Publ., London, 1959, p. 119. Thermodynamics of Nan-Elec(3) VANNESS,H. D., "Cls~~ical trolyte Solutions," Pergamon Press, New York, 1964. J. M., A. I. Ch. E. Joumd, 5.3 (1959). (4) PRAUSNITZ, (5) VON ZAWIDZKI,J., 2.physik. Chem., 35,129 (1900). J., AND JOST,W., 2. p h y ~ i k . Chem. (6) SIEO,L., CRUTZEN, (Leipzig), 198,263 (1951). (7) EVERETT, D. H., Disc Farday SOC.,No. 15, 126 (1953). E., PICK,J., FRIED, V., AND VILIM, O., "Vapor-Liquid (8) HALA, Equilibrium." Pergamon Press, London, 1958. M. L., J. CHEM. Enuc., 40,516 (1963). (9) MCGLASKAN,
Volume 45, Number 1 1, November 1968
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