Tungsten is deposited into a small, cylindrical cavity (2.0 ?m depth, 0.5 um diameter) by the following chemical vapor deposition reaction:

WF₆(g) + 3H₂(g) + W(s) + 6HF(g)

As the tungsten metal forms, it slowly fills the cavity. The tungsten metal does not coat the sidewalls of the cavity; it only grows upward from the bottom of the cavity. The temperature is maintained at 700 K, the total system pressure is 75 Pa, and the molar fraction of WF₆ in the bulk gas space over the cavity is 0.1. The reactants within the pore are significantly dilute in inert Helium (He) gas to lower the deposition rate. Assume that the tungsten deposition is limited by molecular diffusion, and the total concentration of the gas and other properties are constant. The molecular weight of tungsten and fluoride are 184 g/mol and 19 g/mol, respectively, and the density of tungsten is 19.4 g/cm³.

a) Develop an equation using a pseudo-steady-state model to express the thickness (delta) of the tungsten metal within the cavity as a function of time (t). Use the bulk concentration of WF₆,C_A0, the diffusion coefficient of WF₆ in the pore, D_A-mix, the density of W, rho_w, and the molecular weight of W, MW.

b) Estimate the time required to completely fill the cavity assuming Knudsen diffusion for WF₆ vapor.