An ideal gas in a closed container is slowly heated. As its temperature increases, which of the following statements are true?
(A) the mean free path of the molecules decreases.
(B)the mean collision time between the molecules decreases.
(C ) the mean free path remains unchanged.
(D) the mean collision time remains unchanged.

Options

A. (B) and (C )
B. (A) and (C )
C. (C ) and (D )
D. (A) and (D)

Explanation

To solve the question, we need to analyze the behavior of an ideal gas as it is heated in a closed container. We will evaluate each of the statements (A, B, C, D) regarding the mean free path and mean collision time of gas molecules.

### Step-by-Step Solution:

1. **Understanding Mean Free Path**:
- The mean free path (λ) is defined as the average distance a 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 density of molecules (number of molecules per unit volume) and \(d\) is the diameter of the molecules.

2. **Effect of Temperature on Mean Free Path**:
- When the temperature of the gas increases, the number density \(n\) and the diameter \(d\) of the molecules do not change significantly in a closed container. Therefore, the mean free path (λ) remains constant as temperature increases.
- **Conclusion**: Statement (C) is true: "the mean free path remains unchanged."

3. **Understanding Mean Collision Time**:
- The mean collision time (τ) is the average time interval between successive collisions of a molecule. It is inversely related to the collision frequency, which increases with the speed of the molecules.
- As the temperature increases, the kinetic energy of the gas molecules increases, leading to an increase in their velocity. Consequently, the rate at which collisions occur increases.

4. **Effect of Temperature on Mean Collision Time**:
- Since the mean collision time is related to how frequently molecules collide, and since the speed of the molecules has increased due to the rise in temperature, the mean collision time decreases.
- **Conclusion**: Statement (B) is true: "the mean collision time between the molecules decreases."

5. **Evaluating the Remaining Statements**:
- Statement (A): "the mean free path of the molecules decreases." This is false as we established that the mean free path remains unchanged.
- Statement (D): "the mean collision time remains unchanged." This is also false as we established that the mean collision time decreases.

### Final Conclusion:
- The true statements are (C) and (B). Therefore, the correct options are:
- (B) the mean collision time between the molecules decreases.
- (C) the mean free path remains unchanged.