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. Glove Sample (Barrier Membrane) Hazardous Liquid PERMEATION TEST CELL Make-up Fresh Gas To analysis for how much chemical has permeated 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'₁ = H₂√PA where: c'A = concentration of gas phase atoms in the metal P₁ = gas phase partial pressure of diatomic gas Hs = Sievert's Law constant = 7.9 x 10-6 mol/cm³-atm1/2 at 85°C Note that this relationship is similar to Henry's law, but with a different exponent on partial pressure. Like the Henry's law constant, the Sievert's law constant Hs depends upon the gas, metal, and temperature, but is independent of pressure. a) Write down an equation previously derived in class and in the WRF textbook that can be used to predict steady state molar flux across a plate in terms of the concentration of atoms dissolved in the metal sides of the plates. Briefly note why you chose this equation. b) Rewrite this equation in terms of the partial pressure of the gas in the reservoirs on each side of the nickel plate such that flux can be calculated in terms of gas phase partial pressure instead of solid-phase molar concentration. This is desired since you can choose the gas phase partial pressures for the experiments, not the solid phase concentrations. (hint: this is analogous to using a Henry's law constant to get a flux equation in terms of permeability PA-HADAB, as used, for example, on in the In Class Problem on soft contact lenses). c) The feed is pure hydrogen at 8.0 atm and 85°C, and a vacuum is maintained on the other side. The nickel plate is 0.2 cm thick and the diffusion coefficient of the hydrogen atoms in the metal is 8 x 109 cm²/s. Calculate the molar flux of hydrogen gas across the plate. d) (was not on the original exam) Instead of using the equation in part b) to solve for the flux, it is also possible to use the equation in a) directly by solving for molar concentration directly using Sievert's law. Use this method to show that this is true. e) (was not on the original exam) Make a qualitatively reasonable sketch of the concentration profile from the bulk hydrogen gas, through the nickel plate and into the vacuum side of the device.