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 find the temperature coefficient of the reaction given that the rate of reaction increases by 27 times when the temperature is increased by 30 K, we can follow these steps:

### Step 1: Understand the Temperature Coefficient
The temperature coefficient (often denoted as \( Q_{10} \)) is defined as the factor by which the rate of a reaction increases when the temperature is raised by 10 K.

### Step 2: Set Up the Relationship
We know that if the rate of reaction increases by a factor of \( 27 \) when the temperature is increased by \( 30 \) K, we can express this relationship mathematically as:
\[ \frac{R_2}{R_1} = Q_{10}^{\frac{\Delta T}{10}} \]
where \( R_2 \) is the rate at the higher temperature, \( R_1 \) is the rate at the lower temperature, and \( \Delta T \) is the change in temperature.

### Step 3: Substitute Known Values
From the problem, we have:
\[ \frac{R_2}{R_1} = 27 \quad \text{and} \quad \Delta T = 30 \, K \]
Substituting these values into the equation gives:
\[ 27 = Q_{10}^{\frac{30}{10}} = Q_{10}^{3} \]

### Step 4: Solve for \( Q_{10} \)
To find \( Q_{10} \), we need to take the cube root of both sides:
\[ Q_{10} = 27^{\frac{1}{3}} \]
Calculating this gives:
\[ Q_{10} = 3 \]

### Step 5: Conclusion
Thus, the temperature coefficient of the reaction is \( 3 \).

### Final Answer
The temperature coefficient of the reaction is \( 3 \).

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