The rate of a reaction quadruples when the temperature changes from `293K` to `313K`. Calculate the energy of activation of the reaction assuming that it does not change with temperature.

Explanation

To solve the problem of calculating the energy of activation (Ea) of the reaction given that the rate quadruples when the temperature changes from 293 K to 313 K, we can follow these steps:

### Step 1: Understand the relationship between rate and temperature
We know that the rate of a reaction is related to the rate constant (k) by the Arrhenius equation. When the temperature changes, the rate constant also changes. In this case, we are given that the rate quadruples, which means:

\[ k_2 = 4 \times k_1 \]

### Step 2: Use the Arrhenius equation
The Arrhenius equation is given by:

\[ k = A e^{-\frac{E_a}{RT}} \]

Where:
- \( k \) is the rate constant,
- \( A \) is the pre-exponential factor,
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant (8.314 J/mol·K),
- \( T \) is the temperature in Kelvin.

Taking the ratio of the rate constants at two different temperatures, we can express it as:

\[ \frac{k_2}{k_1} = e^{-\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)} \]

### Step 3: Substitute known values
From the problem, we have:
- \( T_1 = 293 \, K \)
- \( T_2 = 313 \, K \)
- \( \frac{k_2}{k_1} = 4 \)

Taking the natural logarithm of both sides gives us:

\[ \ln(4) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]

### Step 4: Calculate the temperature difference
Calculate \( \frac{1}{T_2} - \frac{1}{T_1} \):

\[ \frac{1}{T_2} - \frac{1}{T_1} = \frac{1}{313} - \frac{1}{293} \]

Calculating the above expression:

\[ \frac{1}{313} \approx 0.003194 \, K^{-1} \]
\[ \frac{1}{293} \approx 0.003414 \, K^{-1} \]
\[ \frac{1}{313} - \frac{1}{293} \approx 0.003194 - 0.003414 = -0.000220 \, K^{-1} \]

### Step 5: Substitute into the equation
Now substituting back into the equation:

\[ \ln(4) = -\frac{E_a}{8.314} \left(-0.000220\right) \]

Calculating \( \ln(4) \):

\[ \ln(4) \approx 1.386 \]

Now we have:

\[ 1.386 = \frac{E_a \times 0.000220}{8.314} \]

### Step 6: Solve for \( E_a \)
Rearranging gives:

\[ E_a = \frac{1.386 \times 8.314}{0.000220} \]

Calculating this:

\[ E_a \approx \frac{11.515}{0.000220} \approx 52386.36 \, J/mol \]

### Step 7: Convert to kJ/mol
To convert Joules to kilojoules:

\[ E_a \approx 52.39 \, kJ/mol \]

### Final Answer
The energy of activation \( E_a \) is approximately **52.39 kJ/mol**.

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