CALC Two moles of an ideal gas occupy a volume V. The gas expands isothermally and reversibly to a volume 3V. (a) Is the velocity distribution changed by the isothermal expansion? Explain.

Verified step by step guidance

1. Understand the concept of isothermal processes: In an isothermal process, the temperature of the system remains constant. For an ideal gas, this implies that the kinetic energy of the gas molecules, which is related to the temperature, also remains constant.
2. Recall the relationship between temperature and molecular velocity: The average kinetic energy of gas molecules is given by \(\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the temperature. Since temperature is constant in an isothermal process, the average kinetic energy, and hence the root mean square velocity of the molecules, remains unchanged.
3. Consider the Maxwell-Boltzmann distribution: This distribution describes the probability of finding a gas molecule with a certain velocity at a given temperature. Since the temperature is constant in this scenario, the form of the Maxwell-Boltzmann distribution, which depends solely on temperature, does not change.
4. Analyze the effect of volume change on velocity: The expansion of the gas to three times its original volume at constant temperature means that while the gas molecules have more space to move, their average speed does not increase as the thermal energy (related to temperature) does not change.
5. Conclude the effect on velocity distribution: Since both the temperature and the form of the Maxwell-Boltzmann distribution remain constant, the velocity distribution of the molecules in the gas does not change during the isothermal expansion.

Key Concepts

Ideal Gas Law

The Ideal Gas Law describes the relationship between pressure, volume, temperature, and the number of moles of a gas, expressed as PV = nRT. In this context, it helps understand how the gas behaves during isothermal expansion, where temperature remains constant, affecting pressure and volume but not the number of moles.

Isothermal Process

An isothermal process occurs at a constant temperature, meaning that any heat added to the system is used to do work during expansion. For an ideal gas, this implies that as the gas expands to a larger volume, its pressure decreases, but the average kinetic energy of the gas molecules remains unchanged, which is crucial for understanding velocity distribution.

Velocity Distribution

Velocity distribution refers to how the speeds of gas molecules are spread out in a sample. In an ideal gas undergoing isothermal expansion, the temperature remains constant, which means the average kinetic energy—and thus the velocity distribution—of the gas molecules does not change, even though the volume increases.