A monatomic gas is kept at room temperature 300 K. Calculate the average kinetic energy of gas molecule. (Use K = 1.38 x `10^(-23)` M.K.S. units)

Options

A. 0.138 eV
B. 0.062 eV
C. 0.039 eV
D. 0.013eV

Answer

To calculate the average kinetic energy of a gas molecule in a monatomic gas at room temperature (300 K), we can use the formula for the average kinetic energy (E) of a gas molecule:

\[ E = \frac{3}{2} k T \]

Where:
- \( E \) is the average kinetic energy,
- \( k \) is the Boltzmann constant, and
- \( T \) is the absolute temperature in Kelvin.

### Step-by-Step Solution:

1. **Identify the given values**:
- Temperature \( T = 300 \, \text{K} \)
- Boltzmann constant \( k = 1.38 \times 10^{-23} \, \text{J/K} \)

2. **Substitute the values into the formula**:
\[ E = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times (300 \, \text{K}) \]

3. **Calculate the product**:
- First, calculate \( k \times T \):
\[ k \times T = 1.38 \times 10^{-23} \times 300 = 4.14 \times 10^{-21} \, \text{J} \]

4. **Multiply by \( \frac{3}{2} \)**:
\[ E = \frac{3}{2} \times 4.14 \times 10^{-21} = 6.21 \times 10^{-21} \, \text{J} \]

5. **Convert Joules to electron volts**:
- Since 1 electron volt (eV) is equal to \( 1.6 \times 10^{-19} \, \text{J} \), we convert Joules to electron volts:
\[ E = \frac{6.21 \times 10^{-21} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} = 0.0388125 \, \text{eV} \]

6. **Round off the answer**:
\[ E \approx 0.039 \, \text{eV} \]

### Final Answer:
The average kinetic energy of a gas molecule at room temperature (300 K) is approximately \( 0.039 \, \text{eV} \).

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