Kinetics of Photochemical hydrogen–chlorine reaction below,
\[\displaystyle {{H}_{{2(g)}}}+C{{l}_{{2(g)}}}\underset{{}}{\overset{{{{k}_{1}}}}{\longrightarrow}}2HC{{l}_{{(g)}}}\]
The mechanism for this reaction below as,
\[\displaystyle C{{l}_{2}}+h\nu \underset{{}}{\overset{{{{k}_{1}}}}{\longrightarrow}}2Cl\]
\[\displaystyle Cl+{{H}_{2}}\underset{{}}{\overset{{{{k}_{2}}}}{\longrightarrow}}HCl+H\]
\[\displaystyle H+C{{l}_{2}}\underset{{}}{\overset{{{{k}_{3}}}}{\longrightarrow}}HCl+Cl\]
\[\displaystyle Cl(at\text{ the walls)}\underset{{}}{\overset{{{{k}_{4}}}}{\longrightarrow}}\frac{1}{2}C{{l}_{2}}\]
Derive the rate law for the formation of HCl.
According to the mechanism given above,
\[\displaystyle r=\frac{{d\left[ {HCl} \right]}}{{dt}}\]
\[\displaystyle ={{k}_{2}}\left[ {Cl} \right]\left[ {{{H}_{2}}} \right]+{{k}_{3}}\left[ H \right]\left[ {C{{l}_{2}}} \right]---(1)\]
Applying steady state approximation to [Cl], we get,
\[\displaystyle \frac{{d\left[ {Cl} \right]}}{{dt}}={{k}_{1}}{{I}_{a}}-{{k}_{2}}\left[ {Cl} \right]\left[ {{{H}_{2}}} \right]+{{k}_{3}}\left[ H \right]\left[ {C{{l}_{2}}} \right]-{{k}_{4}}\left[ {Cl} \right]=0--(2)\]
Here, Ia =intensity of absorbed radiation,
Applying steady state approximation to [H], we get,
\[\displaystyle \frac{{d\left[ H \right]}}{{dt}}={{k}_{2}}\left[ {Cl} \right]\left[ {{{H}_{2}}} \right]-{{k}_{3}}\left[ H \right]\left[ {C{{l}_{2}}} \right]=0----(3)\]
Now, adding equation (2) to equation (3), we obtain,
\[\displaystyle {{k}_{1}}{{I}_{a}}-{{k}_{4}}\left[ {Cl} \right]=0\]
\[\displaystyle \left[ {Cl} \right]=\frac{{{{k}_{1}}{{I}_{a}}}}{{{{k}_{4}}}}----(4)\]
Now, substituting the value of [Cl] in equation (3), we get,
\[\displaystyle \left[ H \right]=\frac{{{{k}_{1}}{{k}_{2}}{{I}_{a}}\left[ {{{H}_{2}}} \right]}}{{{{k}_{3}}{{k}_{4}}\left[ {C{{l}_{2}}} \right]}}----(5)\]
Now, substituting equation (4) and equation (5) in equation (1), we get,
\[\displaystyle r=\frac{{{{k}_{1}}{{k}_{2}}{{I}_{a}}\left[ {{{H}_{2}}} \right]}}{{{{k}_{4}}}}+\frac{{{{k}_{1}}{{k}_{2}}{{I}_{a}}\left[ {{{H}_{2}}} \right]}}{{{{k}_{4}}}}\]
\[\displaystyle =\frac{{2{{k}_{1}}{{k}_{2}}{{I}_{a}}\left[ {{{H}_{2}}} \right]}}{{{{k}_{4}}}}\]
\[\displaystyle =2\left( {\frac{{{{k}_{1}}{{k}_{2}}}}{{{{k}_{4}}}}} \right){{I}_{a}}\left[ {{{H}_{2}}} \right]\]
\[\displaystyle =k{{I}_{a}}\left[ {{{H}_{2}}} \right]----(6)\]
Here,
\[\displaystyle k=2\left( {\frac{{{{k}_{1}}{{k}_{2}}}}{{{{k}_{4}}}}} \right)\]
The rate law given by equation (6) agrees with the observed rate law. The rate law is directly proportional to the intensity Ia of the absorbed radiation.
The quantum yield for this reaction is between 104 to 106.
