An insulated chamber is divided into two halves of volumes....

An insulated chamber is divided into two halves of volumes. The left half contains an ideal gas at temperature $T_0$ and the right half is evacuated. A small hole is opened between the two halves, allowing the gas to flow through, and the system comes to equilibrium. No heat is exchanged with the walls. Find the final temperature of the system.

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Key Concepts

Ideal Gas Properties

An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions and whose internal energy is solely a function of temperature. This concept implies that any change in internal energy, and therefore temperature, is independent of changes in volume provided no work or heat is exchanged.

Internal Energy Dependence

For an ideal gas, the internal energy is a function only of temperature. This means that even when the volume changes, as long as no external work is done or heat is exchanged, the internal energy and thus the temperature of the gas remain constant.

First Law of Thermodynamics

The first law establishes the conservation of energy principle, asserting that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In an insulated (adiabatic) system with no work done due to free expansion, the internal energy remains constant.

Free Expansion

Free expansion refers to the process where a gas expands into a vacuum without doing work on its surroundings and without heat exchange with the environment. Although the process is irreversible, for an ideal gas the temperature remains unchanged because its internal energy, which depends only on temperature, does not change.

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