The rate constant of a first order reaction is doubled when the temperature is increased from `20^(@)C` to `25^(@)C` . How many times the rate constant will increase if the temperature is raised from `20 ^(@)C ` to `40^(@)C`

Options

A. 4
B. 8
C. 16
D. 32

Explanation

To solve the problem, we need to determine how many times the rate constant (k) will increase when the temperature is raised from 20°C to 40°C, given that the rate constant doubles when the temperature increases from 20°C to 25°C.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The rate constant \( k \) doubles when the temperature increases from 20°C to 25°C.
- We need to find out how many times \( k \) increases when the temperature goes from 20°C to 40°C.

2. **Calculate the Temperature Coefficient (α)**:
- The temperature coefficient \( α \) can be defined using the formula:
\[ \frac{k_2}{k_1} = α^{\frac{\Delta T}{10}} \]
- Here, \( k_2 = 2k_1 \) (since the rate constant doubles), and \( \Delta T = 5°C \) (from 20°C to 25°C).
- Plugging in the values:
\[ 2 = α^{\frac{5}{10}} = α^{0.5} \]
- Squaring both sides gives:
\[ 4 = α \]
- Thus, the temperature coefficient \( α = 4 \).

3. **Calculate the Rate Constant Increase from 20°C to 40°C**:
- Now, we need to find out how many times \( k \) increases when the temperature changes from 20°C to 40°C.
- The temperature difference \( \Delta T \) here is \( 20°C \) (from 20°C to 40°C).
- Using the same formula:
\[ \frac{k_{40}}{k_{20}} = α^{\frac{20}{10}} = α^{2} \]
- We already found \( α = 4 \), so:
\[ \frac{k_{40}}{k_{20}} = 4^{2} = 16 \]
- This means that the rate constant \( k \) increases 16 times when the temperature is raised from 20°C to 40°C.

### Final Answer:
The rate constant will increase **16 times** when the temperature is raised from 20°C to 40°C.