The rate constants of a reaction at 500 K and 700 K are `0.02 s^(-1)` and `0.07 s^(-1)` respectively. Calculate the values of `E_a` and A.

Explanation

To solve the problem, we need to calculate the activation energy (E_a) and the pre-exponential factor (A) for the given reaction using the Arrhenius equation and the provided rate constants at two different temperatures.

### Step-by-Step Solution

1. **Identify the Given Values**:
- Rate constant at T1 (500 K), k1 = 0.02 s^(-1)
- Rate constant at T2 (700 K), k2 = 0.07 s^(-1)
- Universal gas constant, R = 8.314 J/(mol·K)

2. **Use the Arrhenius Equation**:
The Arrhenius equation relates the rate constants at two different temperatures:
\[ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \]

3. **Substitute the Known Values**:
\[ \ln\left(\frac{0.07}{0.02}\right) = \frac{E_a}{8.314} \left(\frac{1}{500} - \frac{1}{700}\right) \]

4. **Calculate the Left Side**:
\[ \frac{0.07}{0.02} = 3.5 \quad \Rightarrow \quad \ln(3.5) \approx 1.25276 \]

5. **Calculate the Right Side**:
\[ \frac{1}{500} - \frac{1}{700} = \frac{7 - 5}{3500} = \frac{2}{3500} = \frac{1}{1750} \]

6. **Substitute and Rearrange**:
\[ 1.25276 = \frac{E_a}{8.314} \cdot \frac{1}{1750} \]
Rearranging gives:
\[ E_a = 1.25276 \cdot 8.314 \cdot 1750 \]

7. **Calculate E_a**:
\[ E_a \approx 1.25276 \cdot 8.314 \cdot 1750 \approx 18186.8 \text{ J/mol} \approx 18.186 \text{ kJ/mol} \]

8. **Calculate the Pre-exponential Factor (A)**:
Using the Arrhenius equation:
\[ k = A e^{-\frac{E_a}{RT}} \]
Rearranging gives:
\[ A = \frac{k}{e^{-\frac{E_a}{RT}}} \]
Using k1 at T1:
\[ A = \frac{0.02}{e^{-\frac{18186.8}{8.314 \cdot 500}}} \]

9. **Calculate the Exponential Term**:
\[ -\frac{18186.8}{8.314 \cdot 500} \approx -4.373 \]
Thus:
\[ e^{-4.373} \approx 0.0125 \]

10. **Calculate A**:
\[ A = \frac{0.02}{0.0125} \approx 1.6 \]

### Final Results:
- Activation Energy, \( E_a \approx 18.186 \text{ kJ/mol} \)
- Pre-exponential Factor, \( A \approx 1.6 \text{ s}^{-1} \)