Pure hydrogen gas is stored at \(400^{\circ} \mathrm{C}\) and 9 atm in a steel tank with a wall thickness of \(1 \mathrm{~mm}\). At \(400^{\circ} \mathrm{C}\) the diffusion coefficient of atomic hydrogen in steel, \(D_{a b} \sim 1 \times 10^{-8}\) \(\mathrm{m}^{2} / \mathrm{s}\). The solubility of atomic hydrogen in steel is about 3 parts per million (by moles) under a hydrogen gas pressure of \(1 \mathrm{~atm}\). Calculate the flux of atomic hydrogen through the tank wall. The outer surface of the tank is exposed to a hydrogen atmosphere at \(1 \mathrm{~atm}\). The solubility of hydrogen in metals is described by Sievert's Law:

\[x_{h}=K_{P}\left(P_{H 2}\right)^{1 / 2}\]

where \(K_{P}\) is the equilibrium constant, \(P_{H 2}\) is the partial pressure of hydrogen in equilibrium with the metal, and \(x_{h}\) is the mole fraction of atomic hydrogen. Please answer the following questions:

a. Assuming a plane wall geometry and a dilute solution of hydrogen in steel, what is the differential equation that must be solved for the hydrogen mole fraction profile?

b. What are the boundary conditions for this problem? No numbers needed here.

c. What is the solution to the differential equation?

d. What is the expression for the mass flux?

e. What is the value for \(K_{P}\) ?

f. Using the value for \(K_{P}\) and Sievert's Law for the mole fractions of atomic hydrogen, what is the mass flux of atomic hydrogen through the tank?