Chapter 5.3, Problem 41P

Suppose you have a liquid (say, water) in equilibrium with its gas phase, inside some closed container. You then pump in an inert gas (say, air), thus raising the pressure exerted on the liquid. What happens?

(a) For the liquid to remain in diffusive equilibrium with its gas phase, the chemical potentials of each must change by the same amount: $d{\mu }_l=d{\mu }_g$. Use this fact and equation 5.40 to derive a differential equation for the equilibrium vapor pressure, $P_{\upsilon }$, as a function of the total pressure P. (Treat the gases as ideal, and assume that none of the inert gas dissolves in the liquid.)

(b) Solve the differential equation to obtain $P_{\upsilon }\left(P\right)=P_{\upsilon }\left(P_{\upsilon }\right)\cdot e^{\left(P-P_{\upsilon }\right)V/N\kappa T}$ where the ratio V/N in the exponent is that of the liquid. (The quantity $P_{\upsilon }\left(P_{\upsilon }\right)$ is just the vapor pressure in the absence of the inert gas.) Thus, the presence of the inert gas leads to a slight increase in the vapor pressure: It causes more of the liquid to evaporate.

(c) Calculate the percent increase in vapor pressure when air at atmospheric pressure is added to a system of water and water vapor in equilibrium at 25°C. Argue more generally that the increase in vapor pressure due to the presence of an inert gas will be negligible except under extreme conditions.