A mothball slowly sublimes to naphthalene gas in a flow of air:

Naphthalene (s) --> Naphthalene (g)

Assume the radius of the sphere is R, which decreases very slowly due to the loss of naphthalene on the surface (i.e. it takes a lot of gas generated and diffusing away from the surface to cause a small change in R). The flow of air around the particle causes a uniform stagnation boundary layer of thickness R/2, at the outer periphery of which the composition of gas is pure air. Assume that the concentration of naphthalene gas is C_As in the first gas layer at the mothball surface this concentration is controlled by equilibrium. The system is maintained at constant temperature and pressure. In this problem we are interested in the diffusion of naphthalene inside the boundary layer. The diffusivity of naphthalene in air is D_AB, and the total concentration of gas outside the sphere is C_tot.

a. Write a component mole balance using shells at positions r and r+dr, using the total molar flux of NA of naphthalene. Assume that transport occurs by dilute diffusion, without convection inside the boundary layer, and derive the equation that relates NA with the concentration gradient of naphthalene.

b. Given naphthalene is dilute (CA <<< Ctot), whats the differential equation for the mass transfer of naphthalene in the boundary layer in terms of CA? Solve the differential equation with appropriate boundary conditions in order to obtain an analytical expression for CA versus r within the boundary layer.

c. Starting from the mass balance of the mothball, invoke the pseudo-steady-state hypothesis and derive the equation that describes the change of R with t. How long will it take to fully sublime the mothball? ( is the density of the mothball and MA is the molar mass of naphthalene. R0 is the initial sphere radius).