Effects of doubling the temperature and pressure on the mean free path of gas molecules:

Introduction:
The mean free path of gas molecules is defined as the average distance traveled by a molecule between two successive collisions. It is an important parameter that determines the behavior of gases and is influenced by various factors, including temperature and pressure.

Effect of doubling the temperature:
When the temperature of a gas is doubled, the average kinetic energy of the gas molecules also doubles according to the kinetic theory of gases. This increase in kinetic energy results in higher molecular speeds. Consequently, the molecules collide with each other more frequently, reducing the mean free path.

Effect of doubling the pressure:
Doubling the pressure of a gas increases the number of gas molecules in a given volume. As a result, the density of the gas increases. With a higher density, the gas molecules have a higher probability of colliding with each other. Therefore, the mean free path decreases.

Combined effect:
When both the temperature and pressure of a gas are doubled, the mean free path is affected by the combined influence of these factors. The increase in temperature leads to a decrease in the mean free path, as explained earlier. However, the increase in pressure also contributes to the reduction in mean free path.

Explanation:
Initially, when only the temperature is doubled, the mean free path decreases. However, as the pressure is doubled simultaneously, the density of the gas increases, and the mean free path further decreases. Therefore, the combined effect is a reduction in the mean free path.

Conclusion:
When the temperature and pressure of a gas are both doubled, the mean free path of the gas molecules is halved. This is because the increase in temperature leads to more frequent collisions between molecules, while the increase in pressure increases the density of the gas, resulting in a higher probability of collisions.