If the rate of reaction increases by 27 times , when temperature is increased by 30 K, then temperature coefficient of the reaction is

Options

A. 3
B. 2
C. 1
D. 2.5

Answer

To solve the problem of determining the temperature coefficient of a reaction given that the rate of reaction increases by 27 times when the temperature is raised by 30 K, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Temperature Coefficient**:
The temperature coefficient (μ) is defined as the factor by which the rate of a reaction increases for every 10 K rise in temperature.

2. **Identify the Change in Temperature**:
The problem states that the temperature is increased by 30 K. Therefore, we can express this change in temperature (ΔT) as:
\[ \Delta T = 30 \, \text{K} \]

3. **Calculate the Value of x**:
We need to calculate the value of x, which is defined as:
\[ x = \frac{\Delta T}{10} = \frac{30}{10} = 3 \]

4. **Set Up the Rate Equation**:
The rate of reaction increases by 27 times, which can be expressed as:
\[ \frac{\text{Rate}_{\text{new}}}{\text{Rate}_{\text{old}}} = 27 \]

5. **Use the Temperature Coefficient Formula**:
The relationship between the new and old rates can be expressed in terms of the temperature coefficient:
\[ \frac{\text{Rate}_{\text{new}}}{\text{Rate}_{\text{old}}} = \mu^x \]
Substituting the known values:
\[ 27 = \mu^3 \]

6. **Solve for the Temperature Coefficient (μ)**:
We can express 27 as a power of 3:
\[ 27 = 3^3 \]
Therefore, we have:
\[ \mu^3 = 3^3 \]
Since the bases are equal, we can conclude:
\[ \mu = 3 \]

7. **Final Answer**:
Thus, the temperature coefficient of the reaction is:
\[ \text{Temperature Coefficient} = 3 \]

### Conclusion:
The answer to the question is option A: 3.

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