A first order reaction is 50% completed in 30 minutes at 300 K and in 10 minutes at 320 K, Calculate the activation energy of the reaction (R = 8.314 `"JK mol"^(-1)`.)

Explanation

To calculate the activation energy (Ea) of the reaction, we can use the information provided about the half-lives at two different temperatures and apply the Arrhenius equation. Here’s a step-by-step solution:

### Step 1: Determine the rate constants (k1 and k2)
The half-life (T_half) for a first-order reaction is given by the formula:
\[ T_{1/2} = \frac{0.693}{k} \]
From this, we can express the rate constants:
\[ k_1 = \frac{0.693}{T_{1/2, 300K}} \]
\[ k_2 = \frac{0.693}{T_{1/2, 320K}} \]
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