If the piston is moved so as to reduce the volume of gas by half keeping the temperature of gas constant, we know from the gas law that the pressure will be doubled. On microscopic level the increase in pressure on the piston is because

A. momentum change per collision is doubled while the frequency of collision remains constant
B. momentum change per collision remains constant while the frequency of collision is doubled
C. momentum change per collision and the frquency of collision both are increased
D. none of these two physical quantities are changed. It is due to some other reason.

Options

A. momentum change per collision is doubled while the frequency of collision remains constant
B. momentum change per collision remains constant while the frequency of collision is doubled
C. momentum change per collision and the frquency of collision both are increased
D. none of these two physical quantities are changed. It is due to some other reason.

Explanation

To solve the problem, we need to analyze the situation where the volume of a gas is reduced by half while keeping the temperature constant. According to Boyle's Law, if the volume decreases and the temperature remains constant, the pressure of the gas will increase. We need to understand the microscopic reasons behind this increase in pressure.

### Step-by-Step Solution:

1. **Understanding Boyle's Law**:
- Boyle's Law states that for a given mass of gas at constant temperature, the pressure (P) of the gas is inversely proportional to its volume (V). Mathematically, this is represented as \( P \propto \frac{1}{V} \).
- If the volume is reduced by half (i.e., \( V \rightarrow \frac{V}{2} \)), the pressure will double (i.e., \( P \rightarrow 2P \)).

**Hint**: Recall that Boyle's Law relates pressure and volume at constant temperature.

2. **Microscopic Interpretation**:
- At the microscopic level, pressure is caused by gas molecules colliding with the walls of the container (or piston).
- The pressure exerted by the gas is related to two factors: the frequency of collisions with the walls and the momentum change during each collision.

3. **Effect of Volume Reduction**:
- When the volume is halved, the distance between the gas molecules and the walls of the container decreases. This means that the molecules will collide with the walls more frequently.
- The frequency of collisions increases because the same number of molecules are now confined to a smaller space, leading to more frequent impacts on the walls.

**Hint**: Think about how reducing the space available to gas molecules affects their movement and interactions with the walls.

4. **Momentum Change per Collision**:
- The momentum change per collision depends on the velocity of the gas molecules. Since the temperature is constant, the average kinetic energy of the gas molecules remains constant, which implies that their velocities remain constant.
- Therefore, the momentum change per collision does not change.

**Hint**: Remember that temperature is related to the average kinetic energy of the molecules.

5. **Conclusion**:
- Since the momentum change per collision remains constant and the frequency of collisions increases (doubles), we can conclude that the increase in pressure is due to the doubling of the frequency of collisions while the momentum change per collision remains constant.

**Final Answer**: The correct option is **B**: momentum change per collision remains constant while the frequency of collision is doubled.