The rate of a reaction doubles when its temperature changes form `300 K` to `310 K`. Activation energy of such a reaction will be:
`(R = 8.314 JK^(-1) mol^(-1) and log 2 = 0.301)`

Options

A. `48.6 kJ mol^(-1)`
B. (b) `58.5 kJ mol^(-1)`
C. `60.5 kJ mol^(-1)`
D. `53.6 kJ mol^(-1)`

Explanation

To find the activation energy (Ea) of the reaction, we can use the Arrhenius equation and the information given in the problem. The rate of the reaction doubles when the temperature changes from 300 K to 310 K.

### Step-by-Step Solution:

1. **Understanding the Rate Change**:
The rate of the reaction doubles, which can be expressed as:
\[ \frac{K_2}{K_1} = 2 \]
where \( K_1 \) is the rate constant at 300 K and \( K_2 \) is the rate constant at 310 K.

2. **Using the Arrhenius Equation**:
The Arrhenius equation relates the rate constants to the activation energy:
\[ K = A e^{-\frac{E_a}{RT}} \]
Taking the natural logarithm of both sides gives:
\[ \ln K = \ln A - \frac{E_a}{RT} \]
For two different temperatures, we can write:
\[ \ln K_2 - \ln K_1 = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]

3. **Substituting Known Values**:
We can express this in terms of logarithms:
\[ \ln \frac{K_2}{K_1} = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
Since \( \frac{K_2}{K_1} = 2 \), we have:
\[ \ln 2 = -\frac{E_a}{R} \left( \frac{1}{310} - \frac{1}{300} \right) \]

4. **Calculating the Temperature Difference**:
Calculate \( \frac{1}{310} - \frac{1}{300} \):
\[ \frac{1}{310} - \frac{1}{300} = \frac{300 - 310}{310 \times 300} = \frac{-10}{93000} = -\frac{1}{9300} \]

5. **Substituting Values into the Equation**:
Now substituting into the equation:
\[ \ln 2 = -\frac{E_a}{8.314} \left( -\frac{1}{9300} \right) \]
Rearranging gives:
\[ E_a = \ln 2 \times 8.314 \times 9300 \]

6. **Calculating \( E_a \)**:
Using \( \ln 2 \approx 0.693 \):
\[ E_a = 0.693 \times 8.314 \times 9300 \]
Calculating this:
\[ E_a \approx 0.693 \times 77400 \approx 53700 \text{ J/mol} = 53.7 \text{ kJ/mol} \]

### Final Answer:
The activation energy \( E_a \) of the reaction is approximately **53.7 kJ/mol**.