The diffusion and simultaneous first$-$order irreversible chemical reaction in a

single phase containing only reactant A and product $B$ results in a second$-$order

ordinary differential equation given by

$\frac{d^2{C}_A}{d{z}^2}=\frac{k}{D_{AB}}{C}_A$

where $C_A$ is the concentration of reactant $A((kg)-mo\frac{l}{m^3}),z$ is the distance vari$-$

able $($ m $),k$ is the homogeneous reaction rate constant $(s^{-1}),$ and $D_{AB}$ is the binary

diffusion coefficient $(\frac{m^2}{s}).$ A typical geometry for Equation $(6-20)$ is that of a

one$-$dimensional layer that has its surface exposed to a known concentration and

allows no diffusion across its bottom surface. Thus the initial and boundary con$-$

ditions are

$C_A=C_{A0}$ for $z=0$

$\frac{d{C}_A}{dz}=0$ for $z=L$

where $C_{A0}$ is the constant concentration at the surface $)=(0$ and there is no

transport across the bottom surface $(z=L),$ so the derivative is zero.

This differential equation has an analytical solution given by

$C_A=C_{A0}\frac{cosh[L(\sqrt[\phantom{2}]{\frac{k}{D_{AB}}})(1-\frac{z}{L})]}{cosh(L\sqrt[\phantom{2}]{\frac{k}{D_{AB}}})}$