(i) Explain carefully why, when gas leaks slowly out of a...

(i) Explain carefully why, when gas leaks slowly out of a chamber, the expansion of the gas remaining in the chamber may be expected to be adiabatic (that is, quasistatic and without heat exchange). [Hint: choose carefully the physical system you wish to consider.] (ii) A gas with $\gamma=5 / 3$ leaks out of a chamber. If the initial pressure is $32 p_0$ and the final pressure is $p_0$, show that the temperature falls by a factor 4 , and that $1 / 8$ of the particles remain in the chamber.

(i) Explain carefully why, when gas leaks slowly out of a chamber, the expansion of the gas remaining in the chamber may be expected to be adiabatic (that is, quasistatic and without heat exchange). [Hint: choose carefully the physical system you wish to consider.]
(ii) A gas with $\gamma=5 / 3$ leaks out of a chamber. If the initial pressure is $32 p_0$ and the final pressure is $p_0$, show that the temperature falls by a factor 4 , and that $1 / 8$ of the particles remain in the chamber.

Key Concepts

Adiabatic Relations for Ideal Gases

For an ideal gas undergoing an adiabatic process, specific mathematical relationships exist between pressure, volume, and temperature. One key relation is pV^? = constant, where ? is the ratio of specific heats. Another important relation derived from this is TV^(?-1) = constant. These relations allow us to predict how a change in one variable (such as pressure) will affect the others (like temperature and volume) when no heat is exchanged, as exemplified by the leakage problem where changes in pressure lead to a factor reduction in temperature and particle number.

Adiabatic Process

An adiabatic process is one in which there is no heat exchange between the system and its surroundings. In thermodynamics, this means that the change in the internal energy of the system is solely due to work done on or by the system. When a gas expands or compresses adiabatically, its pressure, volume, and temperature change according to specific relations derived from energy conservation. This concept is crucial for understanding processes where insulation or rapid changes prevent significant heat flow, as is assumed in the slow gas leak scenario described in the problem.

Quasi-static Process

A quasi-static process is an idealized thermodynamic change that occurs so slowly that the system remains in near-equilibrium at all times. Such a process allows the state variables (pressure, volume, temperature) to be well-defined throughout the entire change. In the context of a slow gas leak, the gas expands gradually, ensuring that the expansion can be considered both adiabatic and quasi-static, because there is insufficient time for significant non-equilibrium effects or heat exchange to develop.

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