Consider two reactions having same Arrhenius factor A, but different energy of activation.
(i) `A rarr B , Ea_(1)=20kJ`
(ii) `Crarr D, Ea_(2)=30kJ`
Both are at temperature `25^(@)C` It temperature in both reaction is increased slightly in such a way that change in temperature in both case is same than choose the correct options.

Options

A. The second reaction is faster
B. The second reaction is more sensitive towards temperature variation
C. If temperature increases, rate of first reaction increase more sharply
D. All the above are correct

Answer

To solve the problem, we need to analyze the two reactions given their activation energies and the effect of temperature on their rates.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Reaction 1: \( A \rightarrow B \) with \( E_{a1} = 20 \, \text{kJ/mol} \)
- Reaction 2: \( C \rightarrow D \) with \( E_{a2} = 30 \, \text{kJ/mol} \)
- Both reactions are at a temperature of \( 25^\circ C \).

2. **Understand the Arrhenius Equation:**
The rate constant \( k \) for each reaction can be described by the Arrhenius equation:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( A \) = Arrhenius factor (same for both reactions)
- \( E_a \) = activation energy
- \( R \) = universal gas constant (approximately \( 8.314 \, \text{J/mol K} \))
- \( T \) = temperature in Kelvin

3. **Convert Temperature to Kelvin:**
The temperature in Kelvin is:
\[ T = 25^\circ C + 273.15 = 298.15 \, K \]

4. **Analyze the Effect of Temperature Increase:**
Let’s denote the increase in temperature for both reactions as \( \Delta T \). The new temperature for both reactions will be \( T + \Delta T \).

5. **Evaluate the Rate Constants:**
The rate constants for both reactions at the new temperature can be expressed as:
\[ k_1' = A e^{-\frac{E_{a1}}{R(T + \Delta T)}} \]
\[ k_2' = A e^{-\frac{E_{a2}}{R(T + \Delta T)}} \]
Since \( E_{a1} < E_{a2} \), we can infer that \( k_1' > k_2' \) at the original temperature.

6. **Sensitivity to Temperature Changes:**
The reaction with the lower activation energy (Reaction 1) will generally be more sensitive to temperature changes. This means that a small increase in temperature will lead to a relatively larger increase in the rate constant \( k_1' \) compared to \( k_2' \).

7. **Conclusion:**
- The first reaction (with \( E_{a1} = 20 \, \text{kJ/mol} \)) will have a higher rate constant than the second reaction (with \( E_{a2} = 30 \, \text{kJ/mol} \)) after the temperature increase.
- The second reaction is more sensitive to temperature variations, as the increase in temperature will cause a more significant change in the rate of the slower reaction.

### Final Options Analysis:
Based on the analysis, the correct statements are:
- The first reaction is faster than the second reaction.
- The second reaction is more sensitive to temperature variations.