PROBLEM BACKGROUND A hydrogen$-$powered fuel cell is being

developed. At one point in the fuel cell the hydrogen will be flowing

over a nickel plate. But you are concerned about the rate at which

hydrogen may diffuse through this plate, so you want to predict the

steady state hydrogen flux across this plate under various conditions.

PROBLEM STATEMENT A test cell is set up that is analogous the

illustration taken from HW#$7$ that was used to measure the solvent

diffusion flux across a glove sample. For this exam problem, the

'hazardous liquid' section in blue is the hydrogen gas feed and the

'glove sample' is the nickel plate. A vacuum is maintained on 'fresh

gas" section in white; in other words, $P=0$atm.

A difference in the hydrogen$-$metal problem from the solvent$-$glove problem is that many diatomic gases dissociate into

atoms in the surface of metals and dissolve into the metals as individual atoms. In such systems, Sievert's Law accounts

for this effect by relating the partial pressure in the gas phase to the molar concentration of atoms in metal phase with the

following function:

$c_A^{\text{'}}=H_s\sqrt[\phantom{2}]{?}{p}_A$

where: $c_A^{\text{'}}=$ concentration of gas phase atoms in the metal