Two moles of an ideal gas occupy a volume VV. The gas expands isothermally and reversibly to a volume 3V3V. Is the velocity distribution changed by the isothermal expansion? Explain.

Ch 20: The Second Law of Thermodynamics

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 20: The Second Law of Thermodynamics

Problem 28a

Chapter 20, Problem 28a

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

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Understand the concept of isothermal expansion: In an isothermal process, the temperature of the system remains constant. For an ideal gas, this means that the internal energy, which depends only on temperature, does not change.

Recall the Maxwell-Boltzmann distribution: The velocity distribution of gas molecules in an ideal gas is described by the Maxwell-Boltzmann distribution, which depends on the temperature of the gas.

Since the process is isothermal, the temperature of the gas remains constant throughout the expansion. Therefore, the parameters that determine the Maxwell-Boltzmann distribution do not change.

Conclude that because the temperature remains constant, the velocity distribution of the gas molecules does not change during the isothermal expansion.

Summarize: In an isothermal expansion of an ideal gas, the velocity distribution remains unchanged because it is solely dependent on the temperature, which is constant in this process.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. It describes the relationship between these variables for an ideal gas, assuming no interactions between gas molecules and that the volume of the molecules themselves is negligible.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. For an ideal gas undergoing an isothermal expansion, the internal energy remains unchanged, and any work done by the gas is compensated by heat absorbed from the surroundings, maintaining a constant temperature throughout the process.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among particles in a gas. It is dependent on the temperature and mass of the particles. In an isothermal process, since the temperature remains constant, the velocity distribution of the gas particles does not change, meaning the shape of the distribution curve remains the same.

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