The rate constant for a reaction is `2xx10^(-2) s^(-1)` at 300 K and `8xx10^(-2) s ^(-1)` at 340 K The energy of activation of the reaction is

Options

A. `14.695` kJ `"mol"^(-1)`
B. `29.39` kJ `"mol"^(-1)`
C. `44 kJ "mol"^(-1)`
D. `22 kJ "mol"^(-1)`

Explanation

To find the energy of activation (Ea) for the reaction, we can use the Arrhenius equation in its logarithmic form:

\[ \log \left( \frac{K_2}{K_1} \right) = \frac{E_a}{2.303 R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \]

Where:
- \( K_1 = 2 \times 10^{-2} \, s^{-1} \) at \( T_1 = 300 \, K \)
- \( K_2 = 8 \times 10^{-2} \, s^{-1} \) at \( T_2 = 340 \, K \)
- \( R = 8.314 \, J \, K^{-1} \, mol^{-1} \)

### Step 1: Calculate the ratio of rate constants
First, we need to calculate \( \frac{K_2}{K_1} \):

\[ \frac{K_2}{K_1} = \frac{8 \times 10^{-2}}{2 \times 10^{-2}} = 4 \]

### Step 2: Calculate the logarithm of the ratio
Next, we take the logarithm (base 10) of the ratio:

\[ \log \left( \frac{K_2}{K_1} \right) = \log(4) \approx 0.602 \]

### Step 3: Calculate the temperature difference
Now we need to calculate the difference in the reciprocals of the temperatures:

\[ \frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{300} - \frac{1}{340} \]

Calculating each term:

\[ \frac{1}{300} \approx 0.00333 \, K^{-1} \]
\[ \frac{1}{340} \approx 0.00294 \, K^{-1} \]

Now subtract:

\[ \frac{1}{300} - \frac{1}{340} \approx 0.00333 - 0.00294 = 0.00039 \, K^{-1} \]

### Step 4: Substitute values into the equation
Now we substitute the values into the logarithmic form of the Arrhenius equation:

\[ 0.602 = \frac{E_a}{2.303 \times 8.314} \times 0.00039 \]

### Step 5: Solve for Ea
Rearranging the equation to solve for \( E_a \):

\[ E_a = 0.602 \times 2.303 \times 8.314 \times \frac{1}{0.00039} \]

Calculating the right-hand side:

1. Calculate \( 2.303 \times 8.314 \approx 19.1 \)
2. Now calculate \( E_a \):

\[ E_a \approx 0.602 \times 19.1 \times \frac{1}{0.00039} \approx 29397.66 \, J/mol \]

### Step 6: Convert to kJ/mol
Finally, convert \( E_a \) to kJ/mol:

\[ E_a \approx 29.397 \, kJ/mol \]

### Final Answer
The energy of activation of the reaction is approximately **29.397 kJ/mol**.

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