A long cylindrical pellet of A dissolves and diffuses into stagnant and non$-$diffusing gas B$.$Assuming diffusion is only dominant in r direction, derive expressions for flux of A and time for complete dissolution.

Answer is below show the steps please.

$\frac{?}{\text{b}\text{a}\text{r}}(N_A)=2L\frac{D_{AB}}{RT}ln(\frac{P-P_{A2}}{P-P_{A1}})(ln(\frac{r_2}{r_1}){)}^{-1}$

where $\frac{?}{\text{b}\text{a}\text{r}}(N_A)=N_A*A$

$t_f=-\frac{3}{8}{r}_1^2\frac{{}_{A}RT}{m_A{D}_{AB}}ln(\frac{r_2}{r_1}(ln(\frac{P-P_{A2}}{P-P_{A1}}){)}^{-1})$