p~
Component Vapor Pressures as a Function of lnitial ~uantityof Solution Edward Koubek and Mark L. Elert U.S. Naval Academy. Annapolis. MD 21402
Raoult's Law is usually expressed in the form PA
=&PA
(1)
where P i is the vapor pressure of pure liquid A and P A is the vapor pressure of A above a solution in which the mole fraction of A is x i . It is important to note that the quantity x i in eqn. (1)represents the equilibrium mole fraction of component A in the liquid; it is possible to develop an alternative formulation of Raoult's Law in which P Ais expressed in terms of the initial composition x i (before any evaporation occurs) and the initial quantity of solution. We use this formulation to point out that the component vapor pressures can vary quite substantially as a function of the initial quantity of solution for fixed initial composition. This effect is most pronounced for "small" solutions in which the fraction of liquid which evanorates is laree. Furthermore. we show that the algebraic rnachinerv im predirriny equilibrium v.#por prtssurts as a function of ~nitialconditiuns remains srraightforward evrn when activity coefficients are explicitly taken into account. Ideal Behavior for "Small" Solutions Consider a series of experiments in which successively smaller amounts of a liquid solution are injected into an evacuated chamber while the composition x i of the injected solution remains constant. One can then pose several naive questions about the resulting liquid and vapor compositions: Do the partial pressures PA and Pg of the two components change as the initial amount of solution is lowered, and if so, in which direction? At what point will all of the injected solution evaporate? What is the composition of the last drop of liouid to eva~orate?Is it a Dure c o m ~ o n e nor t a solution? .~ualitativkanswers to these questions are easy to deduce bv referrine to the phase diagram of an ideal solution as shown in Figure i. A decrease in the initial number of moles no is analogous to a decrease in the total pressure in such a diagram. Consider a container of fixed volume, into which is placed some quantity no of a liquid solution whose composition is given by the vertical line through point C in Figure 1. If no is large, the fraction of the solution which evaporates will he nedieiblv small. so that the comoosition of the licluid a t fixed volume) is equivalent to a decrease in total presBure (or an increase in container volume for fixed no). In this case the liquid and vapor compositions a t equilibrium might he given by points E and F, respectively, in Figure 1. Finally, if no is made small enough so that all of the liquid just evaporates a t equilibrium, the last remaining liquid will have a composition given by point G. It is apparent, therefore, that the mole fraction-and hence also the partial pressure-of the more volatile component will decrease as no is made progressively smaller. The behavior of the component vapor pressures PAand PB as a function of initial liquid amount can he investigated quantitatively. Consider an ideal solution whose initial composition (before any evaporation occurs) is given by x i , the mole fraction of A in the initial solution. Let nl and ny he the
number of moles of component i (A or B) in the liquid and vapor phases, respectively, a t equilibrium. If both vapors are ideal gases and if the solution is ideal, then
and
The liquid mole fractions x i and x e can be expressed in terms of nX and nb by using the relationships x i = n i l ( n i nb), n i = n i - n;, and similar equations for component B. Therefore, eqns. (2) and (3) may he viewed as two simultaneous equations for the two unknowns nX and nk or (using the ideal gas law) for PA and Pe. The result can most easily he expressed by defining a new quantity Po which is proportional to the initial amount no of solution
+
PO represents the total "potential" pressure in the container, i.e., the pressure which would exist if the solution evaporated completely. Then the vapor pressure of component A as a function of initial amount (Po) is given by
Figure 1. Liquidvapor phase diagram for a two-component ideal solution in which component A is more volatile than component B (P; > $). As the total pressure is decreased for a liquid of composition given by point C,the first vapor to appear has comoosition 0.As the oressure is further decreased. more liouid evaoorates
liquid composition (Q.
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Number 5
May 1982
357
Figure 2.Cnnponem vapor pressures versus initial liquid amount for s solution in which P, = 1. q = 3, and 4 = 0.6. As Ihe InRiI amount Increases, Ihe vspw pressures approach the limiting values PA = xiP, = 0.6 and fs = & = 1.2.
where AP* = P; - PZ The vapor pressure of component B is then ohtained from the relation
A plot of the hehavior of PAand PB as a function of PO is shown in Figure 2, for the case when P i = 1,P; = 3, and x i = 0.60. Pressure units are arbitrary as long as PO is expressed in and PiB.Note that the vapor pressure of the same units as PA the less volatile component (A) does increase as PO decreases, as expected from the qualitative arguments presented above. and When Po is large enough that x i = xi, then PA= (0.60) PA PB = (0.40) PiBas in the "usual" form of Raoult's Law. Equation(5) is valid only as long as some liquid remains a t equilibrium. When Po is decreased sufficiently, a point will he reached a t which all of the solution just evaporates; for smaller values of PO, the partial pressures are given simply P, = nPRTIV = zPPO
(7)
The maximum value of PO a t which total evaporation occurs (the "dew point") is readily apparent in Figure 2, since the slopes of the vapor pressure curves exhibit a discontinuity a t this point. The value of POat the dew point may be found by locating the point at which the curves of eqn. (5) and eqn. (7) for PA cross. Setting the right-hand sides of the two equations equal to each other and squaring,
Figure 3. Partial pressure of H20zasafunction of initial amount of solution for a 30% (by weight) aqueous hydrogen peroxide sdution at 60%. Dashed curve is far an ideal solution: solid curve was vlculated using the activity coefficients of Scatchard,et al. See reference in footnote 3.
curve in Figure 1).The equation is well known and appears occasionally in physical chemistry texts.' It is usually derived (in a more straightforward fashion) without reference to the problem under discussion here; that is, the hehavior of partial pressures as a function of the initial amount of solution. Non-Ideal Solutions In order to obtain a more accurate calculation of vapor preawres abwe R snlution in which suhstantinl e\,apmation of the initial liquid owuri. 11 is neceiiary to go beyund the ideal solutim approximation. Huoult's T.aw should he replncerl by P, = ",;*)P: (9) where the activity coefficient yj (i = A, B) is a function of x 1. When the coefficients y~ and y~ are known, it is possible to modify the ideal-solution results presented above to incorporate the non-ideal hehavior of the actual system. This can be accomplished using a recursive numerical technique, as follows. Starting from eqn. (9) rather than Raoult's Law, we note that the derivation of eqn. (5) proceeds as before except that P i is to be replaced everywhere by ( ~ A Pwhere A ) ,y~ is the activity coefficient appropriate to the equilibrium value of xi. Using y~ = 1 (the ideal-solution limit) as a first approximation, one calculates PAby eqn. (5); then the first approximation to x i is the ideal-solution result x: = PaIPl
2
-
PO PO AP* I AP* )
2
pa + 4 x i 7AP ]
112
The factor in brackets can he eliminated by substitution from eqn. (5). After some simplification, one obtains:
The values of y~ and y~ corresponding to this composition can then he ohtained and used as a second approximation in eqn. (5),with Pi replaced by ~ A P andA AP* replaced by y& - yAPi.The resulting value of PAproduces a refined estimate of x i , and the corresponding values of y~ and ye are used in another iteration. The process is continued until x i and PA no longer change The aleorithm is easily. oro" sienificantlv. " . grammed, particularly when the activity coefficients are available as exolicit functions of comoo~ition.~ Figure 3 shows the real and ideal vapor pressures of Hz02 in a 30% (by weight) aqueous solution of hydrogen peroxide (xRzoz= 0.18) a t 60°C as a function of Po.The solid curve was calculated using the algorithm outlined ahove, with the activity coefficients given hy Scatchard, et al.Wote that the inclusion of activity coefficients has shifted the position of the dew point as a consequence of the reduction in partial pressure of H202.
-
at the dew point. This is the equation for the dew point line in the usual pressure versus composition phase diagram (lower
'
See, for example, Berry, R. S., Rice, S. A,. and Ross. J., "Physical Chemistry." John Wiley & Sons. Inc.. New York. 1980, Chap. 25. Standard analyt~calrepresentations of activity coefficients as functions of composition are discussed by King. M. 6.. "Phase Equilibrium in Mixtures." Pergamon Press. Oxford. 1969, Chap. 6 . Scatchard. G.. Kavanagh. G. M., and Ticknor. L. B., J. Amer. Chem. Soc., 74, 3715 (1952). 358
Journal of Chemical Education
Conclusion Raoult's Law gives the vapor pressure of a solution corn. ponent in terms of the equilibrium mole fraction of that component. For "large" solutions in which the amount of evaporation is negligible, the approximation x i w ; ; x i is valid and allows the vapor pressure to he expressed in terms of initial conditions. When substantial evaporation occurs, the
vapor pressures may still he expressed in terms of initial conditions, but they exhihit marked deviations from the limiting "large solution" result. Consideration of these effects leads to an alternative derivation of the dew-point equation. Finally, non-ideal effects can be accommodated in this ~mall-solutionregime by use of a simple computational algorithm.
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Number 5
May 1982
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