For gas absorption with a semi-infinite region with reaction show that the following limiting values for the flux into the system can be derived starting from the detailed equation in the text:

\[N_{\mathrm{A}, \mathrm{s}}=C_{\mathrm{A}, \mathrm{s}} \sqrt{\frac{D_{\mathrm{A}}}{\pi t}}(1+k t) \text { for } k t \ll 1\]

The solution error is \(5 \%\) for \(k t=0.5\).

Find the total quantity of gas absorbed up to \(t_{\mathrm{E}}\) for this case.

Find the average rate of absorption.

If the average rate is divided by \(k_{\mathrm{L}} C_{\mathrm{A}, \mathrm{s}}\) we can obtain an expression for the enhancement factor. Here \(k_{\mathrm{L}}\) can be calculated using the penetration model.