Example: Dilute A diffuses through a stagnant liquid film onto a plane surface of solid B$.$ On this plain surface, A and $B$ react to yield liquid product $R$ which then diffuses back through the film into the main liquid stream. Develop the overall rate expression for the $\frac{L}{S}$

Solu

w$.$r$.$t$.$ A$.$

Dr$.$ Munaf Adnan $-$ Basra University for Oil and Gas

Steps in series :

$($i$)$ Diffusion of A through the liquid film to the surface.

$($ii$)$ Reaction of A with $B$ on the surface of $B.$

Mass transfer step :

The molar diffusional flux of A to the surface is given by

$-r_{Al}^{\text{'}\text{'}}=-\frac{1}{(S)}\frac{d{N}_A}{dt}=\frac{D}{x}(C_N-C_{As})=k_l(C_{Al}-C_{As})$

where

$k_l-$ mass transfer coefficient of the liquid film, $\frac{m}{s}$

$-r_{Al}^{\text{'}\text{'}}-$ molar flux $ofA,mo\frac{l}{m^2*s}$

$C_{Al}-$ concentration $ofAin$ the bulk liquid, $mo\frac{l}{m^3}$

$C_{As}-$ concentration $ofAat$ the surface $ofB,mo\frac{l}{m^3}$

Reaction step :

The reaction is first order w$.$r$.$t$.$ A$.$ The rate of reaction based on unit surface is given by

$-r_{A2}^{\text{'}\text{'}}=-\frac{1}{(S)}\frac{d{N}_A}{dt}=k^{\text{'}\text{'}}{C}_{As}$

At steady state, these steps will proceed at the same rate $($steps in series$).$ Thus,

$-r_{\text{A}\text{o}\text{v}\text{e}\text{r}\text{a}\text{l}\text{l}}^{\text{'}\text{'}}=-r_{A1}^{\text{'}\text{'}}=-r_{A2}^{\text{'}\text{'}}$

$k_l(C_{Al}-C_{As})=k^{\text{'}\text{'}}{C}_{As}$

$k_l{C}_{Al}=(k^{\text{'}\text{'}}+k_l)C_{As}$

$C_{As}=\frac{k_l{C}_{Al}}{k^{\text{'}\text{'}}+k_l}$

Substituting the value of $C_{As}$ from Equation $(3)$ into Equation $(2),$ we get

$-r_{A2}^{\text{'}\text{'}}=\frac{k_l{k}^{\text{'}\text{'}}{C}_{Al}}{k^{\text{'}\text{'}}+k_l}$

$-r_{A2}^{\text{'}\text{'}}=-r_{A1}^{\text{'}\text{'}}=-r_A^{\text{'}\text{'}}=(\frac{1}{\frac{1}{k_l}+\frac{1}{k^{\text{'}\text{'}}}})C_{Al},$mol$\frac{A}{m^2*s}$

$-r_A^{\text{'}\text{'}}=k_{\text{o}\text{v}\text{e}\text{r}\text{a}\text{l}\text{l}}{C}_{Al}$

$k_{\text{o}\text{v}\text{e}\text{r}\text{a}\text{l}\text{l}}=\frac{1}{\frac{1}{k_l}+\frac{1}{k^{\text{'}\text{'}}}}$

ssignment: Repeat the above example considering the reaction step to be second order with respect to $A-$ all else remain unchanged