Question

A first-order reaction is carried out at two different temperatures of \(300\,K\) and \(310\,K\), where the rate constants are \(k_1\) and \(k_2\) respectively. The activation energies of the reaction in absence and presence of catalyst are \(E_{a1}\) and \(E_{a2}\) respectively. Which of the following options is correct?

• A. \(k_1 \gt k_2,\; E_{a1}\gt E_{a2}\)
• B. \(k_1 \lt k_2,\; E_{a1}\lt E_{a2}\)
• C. \(k_1 \lt k_2,\; E_{a1}\gt E_{a2}\)
• D. \(k_1 \lt k_2,\; E_{a1}=E_{a2}\)

Hint:

\[ \text{Higher Temperature} \Rightarrow \text{Higher } k \] \[ \text{Catalyst} \Rightarrow \text{Lower } E_a \] A catalyst changes the rate of reaction but does not change the equilibrium constant.

Answer 1

The Correct Option is C

Concept: According to the Arrhenius equation, \[ k = A e^{-E_a/RT} \] Rate constant increases with increase in temperature. A catalyst lowers the activation energy of a reaction.

Step 1: Compare \(k_1\) and \(k_2\). Given, \[ T_1=300\,K,\qquad T_2=310\,K \] Since temperature increases, \[\begin{aligned} k_2\gt k_1 \end{aligned}\] Therefore, \[\begin{aligned} k_1\lt k_2 \end{aligned}\]

Step 2: Compare activation energies. A catalyst provides an alternative pathway having lower activation energy. Hence, \[\begin{aligned} E_{a2}\lt E_{a1} \end{aligned}\] or \[\begin{aligned} E_{a1}\gt E_{a2} \end{aligned}\]

Step 3: Select the correct option. \[\begin{aligned} \boxed{ k_1\lt k_2,\qquad E_{a1}\gt E_{a2} } \end{aligned}\] Hence, option \(\mathbf{(C)}\) is correct.