Question

For a first order reaction, A$\to $P, the temperature (T) dependent rate constant (k) was found to follow the equation $\log k=-(2000)\dfrac{1}{T}+6.0$ the pre exponential factor A and the activation energy ${{E}_{a}}$, respectively, are:
(A) $1\times {{10}^{4}}{{s}^{-1}}$ and 9.2kJ/mol
(B) $1\times {{10}^{6}}{{s}^{-1}}$and 16.6 kJ/mol
(C) $6.0{{s}^{-1}}$ and 16.6 kJ/mol
(D) $1\times {{10}^{6}}{{s}^{-1}}$and 38.3 kJ/mol

Answer 1

Hint: The pre-exponential factor or A factor is the pre-exponential constant in the Arrhenius equation, an empirical relationship between temperature and rate coefficient. It is usually designated by A when determined from experiment, while Z is usually left for collision frequency.

Complete answer:
The Arrhenius equation gives the dependence of the rate constant of a chemical reaction on the absolute temperature as:
\[k=A{{e}^{-\dfrac{E_a}{RT}}}\]
where
k is the rate constant (frequency of collisions resulting in a reaction),
T is the absolute temperature (in kelvins),
A is the pre-exponential factor, a constant for each chemical reaction,
$E_a$ is the activation energy for the reaction (in the same units as RT),
R is the universal gas constant.
A first-order reaction is a reaction that proceeds at a rate that depends linearly on only one reactant concentration.
Taking the natural logarithm of Arrhenius equation yields:
\[\ln k=\ln A-\dfrac{E_a}{RT}\]
Putting the values as given in the question we get,
\[\begin{align}
  & {{\log }_{e}}k={{\log }_{e}}A-\dfrac{E_a}{RT} \\
 & \Rightarrow {{\log }_{10}}k={{\log }_{10}}A-\dfrac{E_a}{2.303RT} \\
 &\Rightarrow {{\log }_{10}}A=6 \\
 &\Rightarrow A={{10}^{6}} \\
\end{align}\]
Also, activation energy will be,
\[\begin{align}
 & \Rightarrow \dfrac{E_a}{R\times 2.303}=2000 \\
 & \Rightarrow E_a=38.3\times {{10}^{3}}J/mol \\
 & \Rightarrow E_a=38.3kJ/mol \\
\end{align}\]

Therefore, the correct answer is the D option.

Note:
Activation energy is the energy that must be provided to compounds to result in a chemical reaction. The activation energy ($E_a$) of a reaction is measured in joules per mole (J/mol), kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol).