The temperature coefficient of a reaction is 2 . The rate of this reaction , on raising the temperature by `25^(@)` ,shall increase by

Options

A. 4 times
B. 8 times
C. 6 times
D. `5.65` times

Answer

To solve the problem, we will use the formula for the temperature coefficient of a reaction, which relates the rate of reaction at two different temperatures. The formula is:

\[ \frac{R_{T+\Delta T}}{R_T} = \alpha^{\frac{\Delta T}{10}} \]

Where:
- \( R_{T+\Delta T} \) is the rate of reaction at the new temperature.
- \( R_T \) is the rate of reaction at the initial temperature.
- \( \alpha \) is the temperature coefficient.
- \( \Delta T \) is the change in temperature.

### Step-by-step solution:

1. **Identify the given values:**
- Temperature coefficient \( \alpha = 2 \)
- Change in temperature \( \Delta T = 25^\circ C \)

2. **Substitute the values into the formula:**
\[ \frac{R_{T+25}}{R_T} = 2^{\frac{25}{10}} \]

3. **Calculate \( \frac{25}{10} \):**
\[ \frac{25}{10} = 2.5 \]

4. **Calculate \( 2^{2.5} \):**
\[ 2^{2.5} = 2^{2} \times 2^{0.5} = 4 \times \sqrt{2} \approx 4 \times 1.414 \approx 5.656 \]

5. **Substituting back into the equation:**
\[ \frac{R_{T+25}}{R_T} \approx 5.656 \]

6. **Conclusion:**
The rate of the reaction increases by a factor of approximately 5.656 when the temperature is raised by \( 25^\circ C \).

### Final Answer:
The rate of the reaction shall increase by approximately 5.656 times.

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