$69$ A membrane test cell is designed and operates along the lines suggested in Fig. P$4\mathrm{.}69.$ A pure liquid is under its own vapor pressure in the closed container below the membrane. The upper volume is initially pumped down to $($nearly$)$ zero absolute pressure, after which vapor from the lower compartment dissolves in the membrane and diffuses across the membrane. Assume that both vapor phases are well mixed. Given the membrane area normal to the diffusion flux, the membrane thickness $,$ the volume of the upper compartment, and the vapor pres$-1$ sure of the liquid, derive an expression from which you can calculate the product of solubility and diffusivity of the vapor in the membrane, from data on the time at which the pressure in the upper compartment rises to the value $.$ Do not assume that $.$ The goal is to express in terms of parameters given. is the partition coefficient between the concentration in membrane and in vapor. i$.$e$.$ at the interfaces.