Consider a second-order reaction in a catalyst in the form of a slab. Show that the differential equation can be expressed as

\[\begin{equation*}\frac{d^{2} c_{\mathrm{A}}}{d \xi^{2}}=\phi^{2} c_{\mathrm{A}}^{2} \tag{10.91}\end{equation*}\]

How is the \(\phi\) parameter defined for this case? Use the \(p\)-substitution method and derive an implicit integral representation to the solution of this problem.

Calculate the concentration profiles and the effectiveness factor for \(\phi\) equal to 1 and 3 .