Gas permeability coefficients of polymer films are typically measured by placing a film of uniform thickness in a

permeation cell, which allows one side of the film $($the so$-$called upstream side$)$ to be exposed to a desired gas pressure,

and the other side of the film $($the downstream side$)$ is connected to a rigid $($e$.$g$.,$ stainless steel$)$ container of known

volume. Initially, the volume connected to the downstream side of the film is under vacuum. The pressure difference

across the film drives permeation of gas from the upstream side of the film through the film and into the downstream

volume. The downstream pressure is monitored as a function of time, and once the measurement comes to pseudo$-$steady

state, the rate of pressure rise in the downstream volume is constant $($i$.$e$.,$ independent of time$).$ The entire system is kept

at constant temperature. Knowing the rate of pressure rise in a container of fixed volume and temperature, as well as the

cross$-$sectional area of the film, one can use the ideal gas law to calculate the gas flux through the film. Permeability, P$,$ is

defined as:

$P=\frac{(\text{F}\text{l}\text{u}\text{x})(\text{t}\text{h}\text{i}\text{c}\text{k}\text{n}\text{e}\text{s}\text{s})}{p_{\text{u}\text{p}\text{u}\text{t}\text{r}\text{e}\text{a}\text{m}}-p_{\text{d}\text{o}\text{w}\text{n}\text{s}\text{t}\text{r}\text{e}\text{a}\text{w}}}$

The experiments are typically organized so that the upstream pressure is orders of magnitude higher than the downstream

pressure, so

$p_{\text{d}\text{i}\text{r}\text{n}\text{u}\text{s}\text{t}\text{r}\text{e}\text{a}\text{w}}$ in the equation above is negligible.

A student is conducting a permeation experiment on a $23m$ thick polysulfone film with pure oxygen. The upstream

pressure is set to $10$atm of oxygen, and the downstream pressure is initially $0$atm $($absolute$).$ After a sufficient amount of

time passes to achieve steady$-$state permeation of oxygen through the film, the student notes that the pseudo$-$steady state

rate of pressure rise in the downstream volume is $5\mathrm{.}16{10}^{-4}$ torr$/$s$.$ From this steady state pressure change, calculate the

oxygen permeability of this sample in Barrer.

The downstream volume of the system is $46\mathrm{.}5c{m}^3.$ The system is maintained at $35C.$ The sample area is $0\mathrm{.}904c{m}^2.$

Recall that $1$ Barrer $={10}^{-10}\frac{c{m}^3(STP)cm}{c{m}^{2}s((cm)Hg)}.$ A useful value for the gas constant, $R,$ is $0\mathrm{.}278\frac{cmHg**c{m}^3}{c{m}_{STP}^3(K)}.$