What is the activation energy for a reaction if its rate doubles when the temperature is raised from `20^(@)C` to `35^(@)C`? `(R = 8.314 J "mol K"^(-))`

Options

A. `269 kJ "mol"^(-1)`
B. `34.7 "kJ mol"^(-1)`
C. `15.1 "kJ mol"^(-1)`
D. `342 "kJ mol"^(-1)`

Explanation

To find the activation energy (Ea) for the reaction where the rate doubles when the temperature is raised from 20°C to 35°C, we can use the Arrhenius equation and the relationship between the rate constants at two different temperatures.

### Step-by-step Solution:

1. **Convert Temperatures to Kelvin**:
- The first temperature (T1) is 20°C, which in Kelvin is:
\[ T1 = 20 + 273 = 293 \, K \]
- The second temperature (T2) is 35°C, which in Kelvin is:
\[ T2 = 35 + 273 = 308 \, K \]

2. **Use the Rate Constant Ratio**:
- Given that the rate doubles, we can express this as:
\[ \frac{k_2}{k_1} = 2 \]
- Taking the logarithm of both sides gives:
\[ \log\left(\frac{k_2}{k_1}\right) = \log(2) \]

3. **Apply the Arrhenius Equation**:
- The Arrhenius equation in terms of two temperatures is given by:
\[ \log\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \]
- Substituting the values we have:
\[ \log(2) = \frac{E_a}{2.303 \times 8.314} \left(\frac{1}{293} - \frac{1}{308}\right) \]

4. **Calculate the Difference in Reciprocal Temperatures**:
- Calculate \( \frac{1}{T_1} - \frac{1}{T_2} \):
\[ \frac{1}{293} - \frac{1}{308} = \frac{308 - 293}{293 \times 308} = \frac{15}{90284} \approx 0.000166 \]

5. **Substitute Values and Solve for Ea**:
- Now substituting back into the equation:
\[ \log(2) = 0.301 \]
\[ 0.301 = \frac{E_a}{2.303 \times 8.314} \times 0.000166 \]
- Rearranging to solve for \( E_a \):
\[ E_a = \frac{0.301 \times 2.303 \times 8.314}{0.000166} \]

6. **Calculate the Activation Energy**:
- Performing the calculations:
\[ E_a \approx \frac{0.301 \times 19.19}{0.000166} \approx \frac{5.77}{0.000166} \approx 34700 \, J/mol \]
- Converting to kilojoules:
\[ E_a \approx 34.7 \, kJ/mol \]

### Final Answer:
The activation energy for the reaction is approximately **34.7 kJ/mol**.