A gas that can be described by the ideal gas model is...

A gas that can be described by the ideal gas model is contained in a cylinder of volume $V$. The temperature of the gas is $T$. The mass of the gas is $m,$ and the molar mass is $M .$ Write an expression for the total thermal energy of the gas. Now, imagine that the exact same gas has been placed in a container of volume $2 V .$ What happens to its pressure? What happens to its temperature? What happens to its thermal energy?

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Key Concepts

Ideal Gas Law

The ideal gas law, expressed as PV = nRT (or alternatively, using the mass m and molar mass M as P V = (m/M) R T), is a fundamental relation connecting the pressure, volume, temperature, and quantity of an ideal gas. It is used to understand how changes in one state variable (such as volume) affect the others (such as pressure and temperature) when the amount of gas remains constant.

Internal Energy of an Ideal Gas

The total thermal energy (or internal energy) of an ideal gas is a function solely of its temperature and the degrees of freedom of its molecules. For example, in a monatomic gas, the thermal energy is given by U = (3/2) nRT. This concept is crucial because it implies that if the temperature remains constant during a process, so does the internal energy, regardless of changes in volume or pressure.

Free Expansion (Joule Expansion)

Free expansion refers to the process in which a gas expands into a larger volume without exchanging heat with its surroundings and without doing external work. For an ideal gas undergoing free expansion, the temperature remains constant because the internal energy depends only on temperature. As a result, even though the pressure drops (due to the increased volume), the thermal energy stays unchanged, as dictated by the internal energy equation.

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