Kinetics of Unimolecular Surface Reaction

A unimolecular surface reaction may involve a reaction between molecule A of the reactant and a vacant site S on the surface. The mechanism is represented as follows;

\[\displaystyle A+S\underset{{}}{\overset{{{{k}_{1}}}}{\longrightarrow}}AS\text{ (adsorption)}\]

\[\displaystyle \text{AS}\underset{{}}{\overset{{{{k}_{{-1}}}}}{\longrightarrow}}A+S\text{ (desorption)}\]

\[\displaystyle \text{AS}\underset{{}}{\overset{{{{k}_{2}}}}{\longrightarrow}}\text{Product (decomposition)}\]

If r is the rate of the reaction, then according to the Langmuir-Hinshelwood hypothesis, r is proportional to the fraction of the surface covered. So,

\[\displaystyle r={{k}_{2}}\theta ----(1)\]

Assuming steady state approximation for the concentration of AS, we get

\[\displaystyle r=\frac{{d\left[ {AS} \right]}}{{dt}}\]

\[\displaystyle ={{k}_{1}}\left[ A \right]\left[ S \right]-{{k}_{{-1}}}\left[ {AS} \right]-{{k}_{2}}\left[ {AS} \right]=0---(2)\]

If Cs is the total concentration of active sites on the surface, then the concentration [S] of the vacant sites on the surface is equal to the product of Cs and (1-θ), the fraction of places remaining uncovered. So,

\[\displaystyle \left[ S \right]={{C}_{s}}\left( {1-\theta } \right)----(3)\]

From equation (2),

\[\displaystyle \left[ {AS} \right]=\frac{{{{k}_{1}}\left[ A \right]\left[ S \right]}}{{{{k}_{{-1}}}+{{k}_{2}}}}---(4)\]

Also, the concentration of AS on the surface, [AS], is given by,

\[\displaystyle \left[ {AS} \right]={{C}_{s}}\theta ---(5)\]

From equations (3), (4) and (5), we get

\[\displaystyle {{C}_{s}}\theta =\frac{{{{k}_{1}}\left[ A \right]{{C}_{s}}\left( {1-\theta } \right)}}{{{{k}_{{-1}}}+{{k}_{2}}}}---(6)\]

\[\displaystyle \frac{{\left( {1-\theta } \right)}}{\theta }=\frac{{{{k}_{{-1}}}+{{k}_{2}}}}{{{{k}_{1}}\left[ A \right]}}\text{ or }\frac{1}{\theta }-1=\frac{{{{k}_{{-1}}}+{{k}_{2}}}}{{{{k}_{1}}\left[ A \right]}}\text{ }---(7)\]

\[\displaystyle \frac{1}{\theta }=\frac{{{{k}_{{-1}}}+{{k}_{2}}}}{{{{k}_{1}}\left[ A \right]}}+1=\frac{{{{k}_{1}}\left[ A \right]+{{k}_{{-1}}}+{{k}_{2}}}}{{{{k}_{1}}\left[ A \right]}}---(8)\]

\[\displaystyle \theta =\frac{{{{k}_{1}}\left[ A \right]}}{{{{k}_{1}}\left[ A \right]+{{k}_{{-1}}}+{{k}_{2}}}}----(9)\]

Substituting this value of in equation (1), we obtained

\[\displaystyle r=\frac{{{{k}_{1}}{{k}_{2}}\left[ A \right]}}{{{{k}_{1}}\left[ A \right]+{{k}_{{-1}}}+{{k}_{2}}}}----(10)\]

Equation (10) may be rewritten in the form

\[\displaystyle \frac{1}{r}=\frac{1}{{{{k}_{2}}}}+\frac{{{{k}_{{-1}}}+{{k}_{2}}}}{{{{k}_{1}}{{k}_{2}}\left[ A \right]}}---(11)\]

According to equation (11), a plot of 1/r versus 1/[A] would give a straight line with an intercept equal to 1/k₂ and slope equal to (k_-1+k₂)/k₁k₂.

For gaseous adsorbates, the concentration is conveniently expressed in partial pressure. Therefore, equations (10) and (11) can be written as follows,

\[\displaystyle r=\frac{{{{k}_{1}}{{k}_{2}}{{p}_{A}}}}{{{{k}_{1}}{{p}_{A}}+{{k}_{{-1}}}+{{k}_{2}}}}---(12)\]

\[\displaystyle \frac{1}{r}=\frac{1}{{{{k}_{2}}}}+\frac{{{{k}_{{-1}}}+{{k}_{2}}}}{{{{k}_{1}}{{k}_{2}}{{p}_{A}}}}---(13)\]

Consider the limiting cases of equation (10)

Case 1. If the rate of decomposition is enormous in comparison with the rate of adsorption and desorption, i.e. k₂>>(k₁[A]+k_-1), then equation (1) reduces to.

\[\displaystyle r={{k}_{1}}[A]----(14)\]

So, the reaction is the first order concerning A.

Case 2. If the decomposition rate is minimal compared with the rate of adsorption and desorption. i.e. k₂ << (k₁[A] + k_-1), then equation (10) becomes

\[\displaystyle r=\frac{{{{k}_{1}}{{k}_{2}}\left[ A \right]}}{{{{k}_{1}}\left[ A \right]+{{k}_{{-1}}}}}=\frac{{{{k}_{2}}K\left[ A \right]}}{{{{k}_{1}}\left[ A \right]+1}}=\frac{{{{k}_{2}}{{K}^{'}}{{p}_{A}}}}{{{{K}^{'}}{{p}_{A}}+1}}----(15)\]

Here, K or K’ = k₁/k_-1 is the adsorption equilibrium constant.

Some Unimolecular surface reactions are the decomposition of N₂O on gold, NH3 on molybdenum, and HI on platinum.