The slope of Arrhenius Plot $\left(\text{In k v/s}\frac{1}{T}\right)$of first order reaction is −5 × 10³ K. The value of Ea of the reaction is. Choose the correct option for your answer.

[Given R=8.314 JK⁻¹mol⁻¹]

The correct answer is: 41.5 kJ mol⁻¹

Step 1: Recall the Arrhenius Equation

The Arrhenius equation relates the rate constant k of a reaction to the temperature T:

$k=A\cdot e^{-\frac{E_a}{RT}}$

where A is the pre-exponential factor, $E_a$ is the activation energy, R is the universal gas constant, and T is the absolute temperature in Kelvin.

Step 2: Take the natural logarithm of both sides

Taking ln on both sides gives the linearized form:

$lnk=lnA-\frac{E_a}{R}\cdot \frac{1}{T}$

This is in the form of a straight line y = c + mx, where:

• y = ln k
• x = 1/T
• Slope (m) = $-\frac{E_a}{R}$
• Intercept (c) = ln A

Step 3: Use the given slope to find E_a

We are told that the slope of the Arrhenius plot (ln k vs 1/T) is −5 × 10³ K. Setting this equal to the slope expression:

$-\frac{E_a}{R}=-5\times {10}^3\text{K}$

Removing the negative sign from both sides:

$\frac{E_a}{R}=5\times {10}^3\text{K}$

Step 4: Solve for E_a

$E_a=R\times 5\times {10}^3$

Substituting R = 8.314 J K⁻¹ mol⁻¹:

$E_a=8.314\times 5\times {10}^3$

$E_a=41570\text{J mol}{}^{-1}$

Step 5: Convert to kJ mol⁻¹

Dividing by 1000 to convert joules to kilojoules:

$E_a=\frac{41570}{1000}=41.57\approx 41.5\text{kJ mol}{}^{-1}$

Therefore, the activation energy E_a = 41.5 kJ mol⁻¹.