Chapter 6, Problem 6F.2E

Interpretation Introduction

Interpretation:

The ratio of concentration of product 1 to product 2 whether increases or decreases with temperature that has to be determined for a reaction showing kinetic control of product and one product has more activation energy than the other.

Concept Introduction:

Arrhenius equation:

Arrhenius equation gives the temperature dependence of reaction rates.  It can be represented as,

  $\text{k}\text{=}\text{A}{\text{e}}^{\text{-}\frac{{\text{E}}_{\text{a}}}{\text{RT}}}$

Where,

$\text{k}$ is the rate constant

A is the pre-exponential factor

T is temperature

R is the Universal gas constant

${\text{E}}_{\text{a}}$ is the activation energy

Kinetic control:

In some reactions the reactants give rise to various products.  Here the reactions considered are,

  $\begin{array}{l}\text{A}\text{+}\text{B}\to {\text{P}}_{\text{1}}\\ \\ \text{A}\text{+}\text{B}\to {\text{P}}_{\text{2}}\end{array}$

Now the rate of formation of ${\text{P}}_{\text{1}}$ is, ${\text{R}}_{\text{1}}\text{=}{\text{k}}_{\text{1}}\text{[A]}\text{[B]}$

Now the rate of formation of ${\text{P}}_{\text{1}}$ is, ${\text{R}}_{\text{2}}\text{=}{\text{k}}_{\text{2}}\text{[A]}\text{[B]}$

The relative proportion in which two products have been produced at a given stage of reaction before it reaches equilibrium is given by the ration of the two rates.  This ratio is known as kinetic control over the proportions of the products.

  $\begin{array}{l}\frac{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}=\frac{{\text{R}}_{\text{1}}}{{\text{R}}_{\text{2}}}\\ \\ \frac{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}=\frac{{\text{k}}_{\text{1}}\text{[A]}\text{[B]}}{{\text{k}}_{\text{2}}\text{[A]}\text{[B]}}\\ \\ \frac{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}=\frac{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}\end{array}$

Hence the kinetic control for the reaction is,

  $\frac{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}=\frac{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}$