A closed box contains an ideal gas if temperature of gas increased which

Options

A. mean free path remain same
B. mean free path decreases
C. relaxation time decreases
D. relaxation time remain same

Answer

To solve the problem, we need to analyze the effects of increasing the temperature of an ideal gas contained in a closed box on its mean free path and relaxation time.

### Step-by-Step Solution:

1. **Understand the Mean Free Path (λ)**:
- The mean free path is defined as the average distance a gas molecule travels between successive collisions. It can be expressed mathematically as:
\[ \lambda = \frac{1}{\sqrt{2 \pi n d^2}} \]
where \( n \) is the number of molecules per unit volume and \( d \) is the diameter of the molecules.

2. **Relate Number Density to Temperature**:
- For an ideal gas, the number density \( n \) can be related to pressure \( P \), temperature \( T \), and volume \( V \) using the ideal gas law:
\[ PV = nRT \quad \Rightarrow \quad n = \frac{P}{RT} \]
- Substituting this into the mean free path equation gives:
\[ \lambda = \frac{RT}{\sqrt{2 \pi P d^2}} \]

3. **Effect of Increasing Temperature**:
- When the temperature \( T \) is increased, the average kinetic energy of the gas molecules increases, leading to higher speeds. However, the mean free path is primarily dependent on the number density \( n \) and the diameter \( d \) of the molecules.
- Since the volume of the closed box remains constant, an increase in temperature will increase the pressure \( P \) (since \( P \propto T \)), but the relationship between \( n \) and \( T \) does not change the mean free path significantly. Thus, the mean free path \( \lambda \) remains approximately the same.

4. **Understand Relaxation Time (τ)**:
- Relaxation time is defined as the average time between two successive collisions of a molecule. It is inversely related to the collision frequency.
- As temperature increases, the speed of the molecules increases, which leads to more frequent collisions. Therefore, the relaxation time decreases.

5. **Conclusion**:
- From the analysis, we conclude that:
- The mean free path remains the same.
- The relaxation time decreases.

### Final Answer:
- Mean free path remains the same.
- Relaxation time decreases.