The elementary rate constant for a reaction at temperature can be expressed as a…

CHEMISTRY · Multi-part · CBSE · Chemistry · Chemical Kinetics · Chemical Kinetics

Problem

The elementary rate constant

k

for a reaction at temperature

T

can be expressed as a function of the activation energy

E_{a}

and a temperature-independent pre-exponential factor

A

. For the esterification of acetic acid with methanol at

298\,\text{K}

, measurements give the following values. Uncatalyzed pathway:

E_{a}=125\,\text{kJ\,mol}^{-1}

Sulfuric-acid-catalyzed pathway:

E_{a}=55\,\text{kJ\,mol}^{-1}

Assume that

A

is the same for both pathways. Use

R=8.314\,\text{J\,mol}^{-1}\,\text{K}^{-1}

. (a) Derive an explicit expression for the ratio

\dfrac{k_{\text{cat}}}{k_{\text{uncat}}}

in terms of

E_{a,\,\text{uncat}}

E_{a,\,\text{cat}}

R

, and

T

. (b) Calculate the numerical value of

\dfrac{k_{\text{cat}}}{k_{\text{uncat}}}

T=298\,\text{K}

. Report your answer to two significant figures. (c) The uncatalyzed reaction has a half-life of

3.0\times10^{5}\,\text{s}

under standard conditions. Estimate the half-life of the catalyzed reaction at the same temperature and explain any assumption you make.