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 problem, we need to analyze the behavior of an ideal gas in a closed container as its temperature increases. We will evaluate each statement provided in the question.

### Step-by-Step Solution:

1. **Understanding Mean Free Path (λ)**:
- The mean free path (λ) is the average distance traveled by a molecule between successive collisions. The formula for mean free path is given by:
\[ \lambda = \frac{1}{\sqrt{2} n \pi d^2} \]
- Here, \( n \) is the number of molecules per unit volume, and \( d \) is the diameter of the molecules.

2. **Effect of Temperature on Mean Free Path**:
- As the temperature of the gas increases, the kinetic energy of the gas molecules increases, leading to an increase in their velocities.
- However, the parameters \( n \) (number of molecules per unit volume), \( d \) (diameter of the molecules), and the constants \( \sqrt{2} \) and \( \pi \) do not change in a closed container.
- Therefore, the mean free path \( \lambda \) remains unchanged.

**Conclusion**: Statement (C) "the mean free path remains unchanged" is **True**.

3. **Understanding Mean Collision Time (τ)**:
- The mean collision time (τ) is the average time between collisions of molecules.
- As the temperature increases, the molecules move faster due to increased kinetic energy. This means they will collide more frequently.
- Since the distance between collisions remains the same (as λ is unchanged), but the speed increases, the mean collision time must decrease.

**Conclusion**: Statement (B) "the mean collision time between the molecules decreases" is **True**.

4. **Evaluating Other Statements**:
- Statement (A) "the mean free path of the molecules decreases" is **False** because we established that it remains unchanged.
- Statement (D) "the mean collision time remains unchanged" is **False** as we concluded that it decreases with increasing temperature.

### Final Answers:
- The true statements are:
- (B) the mean collision time between the molecules decreases.
- (C) the mean free path remains unchanged.

### Summary of Correct Statements:
- **True Statements**: (B) and (C)
- **False Statements**: (A) and (D)