A thermally insulated box is separated into two compartments...

A thermally insulated box is separated into two compartments (volumes $V_1$ and $V_2$ ! by a membrane. One of the compartments contains an ideal gas at temperature $T$; the other is empty (vacuum). The membrane is suddenly removed, and the gas fills up the two comparments and reaches equilibrium. (a) What is the final temperature of the gas? (b) Show that the gas expansion process is irreversible. FIGURE CAN'T COPY.

A thermally insulated box is separated into two compartments (volumes $V_1$ and $V_2$ ! by a membrane. One of the compartments contains an ideal gas at temperature $T$; the other is empty (vacuum). The membrane is suddenly removed, and the gas fills up the two comparments and reaches equilibrium.
(a) What is the final temperature of the gas?
(b) Show that the gas expansion process is irreversible.
FIGURE CAN'T COPY.

Key Concepts

Free Expansion

Free expansion refers to the rapid expansion of a gas into an evacuated space (a vacuum) where no work is performed by or on the gas. For an ideal gas undergoing free expansion in an adiabatic environment, the internal energy remains constant, leading to no change in temperature even though the volume increases.

Irreversible Process

An irreversible process is one that cannot return both the system and the surroundings to their original conditions once the process has occurred. The spontaneous expansion of the gas into an available vacuum is non-quasi-static and dissipative, meaning that it cannot be reversed by infinitesimally changing the external conditions, hence it is intrinsically irreversible.

Ideal Gas Behavior

An ideal gas is a theoretical model in which particles do not interact except through elastic collisions and occupy negligible volume relative to the container. In such a system, the internal energy is solely a function of temperature, which means that under certain processes, such as free expansion, the temperature may remain unchanged if no work is done and no heat is exchanged.

Adiabatic Process

An adiabatic process occurs when a system does not exchange heat with its surroundings. In the context of a thermally insulated box, the entire process is adiabatic, meaning any changes in the state variables (such as temperature or pressure) occur without the influence of external heat transfer.

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