An ideal gas is originally confined to a volume V1 in an...

An ideal gas is originally confined to a volume $V_1$ in an insulated container of volume $V_1+V_2$. The remainder of the container is evacuated. The partition is then removed and the gas expands to fill the entire container. If the initial temperature of the gas was $T$, what is the final temperature? Justify your answer.

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Key Concepts

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. In this problem, the insulated container ensures that the process is adiabatic, meaning that the total internal energy of the gas remains constant during expansion. Since there is no heat transfer, any change in temperature must come solely from work done by or on the gas, which in this case is minimal or none.

Free Expansion

Free expansion refers to the process in which a gas expands into a vacuum with no external pressure doing work on the gas. In the scenario described, the gas expands into an evacuated part of the container after the partition is removed. No work is performed on the surroundings during a free expansion, which is why there is no temperature change in an ideal gas undergoing this process.

Internal Energy of an Ideal Gas

For an ideal gas, the internal energy is a function solely of its temperature, independent of volume or pressure changes. Since the internal energy stays constant during the free, adiabatic expansion (with no heat exchange or work done), the temperature of the gas remains unchanged. Thus, the final temperature is the same as the initial temperature T.

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