Norman 0. Smith
Fordham University New York, New York
The Variation of Vapor Pressure with Total Pressure
I t is common knowledge that a volatile liquid, confined under the pressure of an inert gas, has a higher vapor pressure than when confined under the pressure of its own vapor. The relation between the total pressure and the vapor pressure is almost always derived bv imaeininx the liauid to be confined in a closed cylinder with-a the head of which is permeable only t o the vapor of the liquid. Only the vapor is then present on the other side of the piston head. The criterion of equilibrium, dF = 0 , is then applied and the familiar relation first derived by Poynting,'is obtained. There are two difficulties encountered in teaching this topic according t o the above approach. First, the student must visualize a piston head which is permeable to vapor but not to liquid-an unrealistic model. Secondly, the criterion of equilibrium, dF = 0 , is applied to a system which is not at constant total vressure and therefore not a t comnlete equilibrium. if he has learned that the condition dF = 0 is to be applied a t constant temperature and total pressure, he mill be confused. The following simple derivation avoids both difficulties; and the use of an inert, insoluble gas approximates better the conditions under which this phenomenon is usually experienced. Consider a closed system containing the liquid ( A ) in equilibrium with a gas phase containing the vapor of A and an inert, insoluble gas. At equilibrium = f i r , so, for a virtual displacement, d~,' = dp,,, where the primes refer to the liquid phase and the quantities without primes refer t o the gas phase. Now, since an' = f ( T , P ) and
PA =
f(T, P , XA),
X , being the mole fraction of A , and P and T the total pressure and temperature of both phases, PA'
=
+
(aa~'lbT)pdT (bpr'/bP)~dP
and d m = ( b P n l b T ) ~xdT ,
+ (ban/aP)r,(apalbxl)T. xdP +
pdXr
But ( b a ~ ' / b P= ) ~VA' and ( b a r / b P ) ~ , = x VA
Furthermore, if the gas is ideal,
(2)
since p , = X,P, where f, is the fugacity of d and p, its partial pressure at the total pressure P. Equating the right-hand sides of (1) and ( Z ) , imposing the condition of constant temperature (dT = O ) , recognizing that = R T / P , and substituting for the partial differentialcoefficientsvields
vA
+
Vl'dP = ( R T / P ) d P ( R T / X A ) ~ X= A RTd In P RTd In Xn = RTd In p~ = ( R T l p r Idpr
+
(3)
Therefore dpn/dP = Vl'prlRT
or
d ln pr/dP = VA'IRT
This relation is valid if the gas phase is ideal and if the gas is insoluble in the liquid. If, hon-ever, the gas (R) is soluble, PA'
=
f(T, P , XA')
and equation (1) will have the additional term ( b r r l b x r ' ) ~~, X A '
For a gas which obeys Henry's law it can be shown that, dpr/dP = f?~'pr/RT - p n / k X ~ '
(4)
where k = dp,/dX,', the Henry's law constant for B in A . If the gas is only slightly soluble X,' 1, VA' E V,', and me have dpa/dP = v n ' p ~ / X T- p ~ / k
which is effectively the same as equation (3), since the last term is usually negligible. Actually, there have been few experimental studies of the effect of high pressures on the vapor pressure of liquids, and the results of different investigators do not always agree. Among these may be mentioned the systems HTHIO, N2-H20 and Ndiquid hydroc a r b o n ~ . ~ -Bartlett's ~ investigations2 with water, for example, show that, although in the absence of inert eases the water content of the eauilibrium vanor at 50" is 83 mg/l, under a total pressure of 200 i t m of hydrogen it is raised to 95 mg/l. When nitrogen at 200 atm is used, the water content is 126 mg/l. At total pressures of 400 atm, hydrogen and nitrogen give water contents of 102 and 148 mg/l. The values predicted by eauation (3) are 94 and 108 me/l for total pressureH of 500 and 400 atm, respectively. - ' ~ h u seqnation ( 3 ) is valid for hydrogen a t -200 atm but nobfor hydrogen at 400 atm, and it fails for nitrogen a t both measures. The chief cause for this failure does not
-
BARTLETT, E. P., J. Am. C k . Soc., 49, 65 (1927). A. W., AND KRASE,N. W., J. Am. Chem. Soc., 56, 353 (1934). ' L~wrs,W. K., AND LUKE,C. D., Ind. Eng. C h a . , 2 5 , 725 (1933). a
So
"ADDINGTON,
( a a a l b X a ) ~T. = R T ( b In p ~ l a x ~T )=~ RTIXA .
POYNTING, J. H., Phil. Mag., 12, 32 (1881).
Volume 40, Number 6, lune 1963
/
317
appear t o lie in the use of VH.O'instead of PE,,~', for, even at 400 atm the two are nearly equal, because of the small solubility of both gases. In fact, the use of equation (4) to include the effect of the solubility of the gas in the liquid makes no significant improvement. The real cause of the failure lies in the nonideality of the gas mixtures, which does not permit equating PAto R T / P or f, to p,, as was done in the derivation of equation (3) above. One may even say that in the light of available data the latter relation, although widely quoted in textbooks, is only of theoretical interest in this sense: As soon as the effect of total pressure on vapor pressure becomes large enough to be of importance, it can no longer be relied upon to give anything but a qualitative prediction of the effect.
31 8
/
Journal of Chemical Education
Satisfactory predictions can be expected only by eliminating the assumption of ideal behavior in the gas phase, and thus by not replacing V , by R T / P or f, by p,. One then obtains, for only slightly soluble gases, dXa/dP
=
(V,' - F A ) / R T (in ~ an/bXn)p.~
the use of u-hich requires more thermodynamic data 011 gas mixtures than are generally a ~ a i l a b l e . ~Deviations from ideal behavior arise, of course, from intermolecular forces, and attempts have been made to correlate these with the experimental result^.^ For a. method of handling such data the reader is referred to R.H.,Ind. Eng. Chem., the paper of DODGE, B. F., AND NEWTON, 29, 720 (1937). See, for example, ROBIN, S., AND VODAR, B.,Discussions Faraday Soe., 15, 233 (1953).
