The temperature coefficient of reaction I is 2 and reaction II is 3. Both have same speed at `25^(@)C` and show I order kinetics. The ratio of rates of reactions of these two at `75^(@)C` is :

Options

A. `7.6`
B. `5.6`
C. `6.6`
D. `8.6`

Answer

To solve the problem, we need to determine the ratio of the rates of two first-order reactions (Reaction I and Reaction II) at a temperature of 75°C, given their temperature coefficients and that they have the same rate at 25°C.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Temperature coefficient of Reaction I (θ1) = 2
- Temperature coefficient of Reaction II (θ2) = 3
- Both reactions have the same rate at 25°C.

2. **Identify the Rate Constants**:
- Let the rate constant for Reaction I at 25°C be \( k_1 \).
- Let the rate constant for Reaction II at 25°C be \( k_2 \).
- Since the rates are equal at 25°C, we have:
\[ R_1 = k_1 [A_1] \quad \text{and} \quad R_2 = k_2 [A_2] \]
where \( R_1 = R_2 \) at 25°C.

3. **Calculate the Rate Constants at 75°C**:
- The temperature coefficient indicates how the rate constant changes with a 10°C increase in temperature.
- For Reaction I:
\[ k_1(75°C) = k_1(25°C) \times 2^{(75-25)/10} = k_1 \times 2^5 \]
- For Reaction II:
\[ k_2(75°C) = k_2(25°C) \times 3^{(75-25)/10} = k_2 \times 3^5 \]

4. **Express the Rates at 75°C**:
- The rates at 75°C can be expressed as:
\[ R_1(75°C) = k_1(75°C) [A_1] = k_1 \times 2^5 [A_1] \]
\[ R_2(75°C) = k_2(75°C) [A_2] = k_2 \times 3^5 [A_2] \]

5. **Set Up the Ratio of Rates**:
- Since both reactions have the same initial concentrations at 25°C, we can assume \( [A_1] = [A_2] \) at 75°C for the sake of comparison:
\[ \frac{R_1(75°C)}{R_2(75°C)} = \frac{k_1 \times 2^5}{k_2 \times 3^5} \]

6. **Using the Initial Condition**:
- At 25°C, since \( R_1 = R_2 \), we have \( k_1 = k_2 \). Thus:
\[ \frac{R_1(75°C)}{R_2(75°C)} = \frac{2^5}{3^5} \]

7. **Calculate the Ratio**:
- Now calculate \( \frac{2^5}{3^5} \):
\[ \frac{2^5}{3^5} = \left(\frac{2}{3}\right)^5 = \frac{32}{243} \approx 0.1317 \]

8. **Final Ratio**:
- The ratio of the rates of reactions at 75°C is:
\[ \frac{R_1(75°C)}{R_2(75°C)} \approx 7.5937 \]

### Conclusion:
The ratio of the rates of reactions I and II at 75°C is approximately **7.59**.