Question

The activation energies of two reactions are ${{\text{E}}_{\text{1}}}$ and ${{\text{E}}_{\text{2}}}$ with ${{\text{E}}_{\text{1}}} > {{\text{E}}_{\text{2}}}$. If the temperature of the reacting system is increased from ${{\text{T}}_{\text{1}}}$ (rate constants are ${{\text{K}}_{\text{1}}}$ and ${{\text{K}}_{\text{2}}}$) to ${{\text{T}}_{\text{2}}}$ (rate constants are ${\text{K}}_1^1$ and ${\text{K}}_2^1$) predict which of the following alternative is correct?

• (A) $\dfrac{{{\text{K}}_1^1}}{{{{\text{K}}_{\text{1}}}}} = \dfrac{{{\text{K}}_2^1}}{{{{\text{K}}_2}}}$

• (B) $\dfrac{{{{\text{K}}_{\text{1}}}}}{{{\text{K}}_1^1}} > \dfrac{{{{\text{K}}_2}}}{{{\text{K}}_2^1}}$

• (C) $\dfrac{{{\text{K}}_1^1}}{{{{\text{K}}_{\text{1}}}}} < \dfrac{{{\text{K}}_2^1}}{{{{\text{K}}_2}}}$

• (D) $\dfrac{{{{\text{K}}_{\text{1}}}}}{{{\text{K}}_1^1}} < \dfrac{{{{\text{K}}_2}}}{{{\text{K}}_2^1}}$

Answer 1

The minimum amount of energy that the reacting species must possess to undergo a specific reaction is known as the activation energy. The relation between the temperature, rate constant and activation energy is given by the Arrhenius equation.

Complete step-by-step solution: We know that the minimum amount of energy that the reacting species must possess to undergo a specific reaction is known as the activation energy. The relation between the temperature, rate constant and activation energy is given by the Arrhenius equation.
The Arrhenius equation is as follows:
$\log \dfrac{{{k_2}}}{{{k_1}}} = \dfrac{{{E_a}}}{{2.303 \times R}}\left( {\dfrac{1}{{{T_1}}} - \dfrac{1}{{{T_2}}}} \right)$
Where ${k_2}{\text{ and }}{k_1}$ are the constants for the reaction,
${E_a}$ is the energy of activation,
$R$ is the universal gas constant,
${T_1}{\text{ and }}{T_2}$ are the temperatures.
From the Arrhenius equation, we can see that higher the activation energy of the reaction greater is the temperature dependence of the rate constant of the reaction.
We are given that the activation energies of two reactions are ${{\text{E}}_{\text{1}}}$ and ${{\text{E}}_{\text{2}}}$ with ${{\text{E}}_{\text{1}}} > {{\text{E}}_{\text{2}}}$. If the temperature of the reacting system is increased from ${{\text{T}}_{\text{1}}}$ (rate constants are ${{\text{K}}_{\text{1}}}$ and ${{\text{K}}_{\text{2}}}$) to ${{\text{T}}_{\text{2}}}$ (rate constants are ${\text{K}}_1^1$ and ${\text{K}}_2^1$). Thus,
$\dfrac{{{{\text{K}}_{\text{1}}}}}{{{\text{K}}_1^1}} > \dfrac{{{{\text{K}}_2}}}{{{\text{K}}_2^1}}$

Thus, the correct option is (B) $\dfrac{{{{\text{K}}_{\text{1}}}}}{{{\text{K}}_1^1}} > \dfrac{{{{\text{K}}_2}}}{{{\text{K}}_2^1}}$.

Note: The unit of rate constant for first order reaction is ${\text{se}}{{\text{c}}^{ - 1}}$. The units do not contain concentration terms. Thus, we can say that the rate constant of a first order reaction is independent of the concentration of the reactant. The rate of any chemical reaction is inversely proportional to the activation energy. Higher the activation energy, slower is the chemical reaction.