A thermally insulated and evacuated chamber is placed in a...

A thermally insulated and evacuated chamber is placed in a room where the pressure and temperature $T_0$ are maintained constant. Gas leaks slowly into the chamber through a small hole. Show that when the pressures are equalized, the temperature of the air in the chamber is $\gamma T_0$, where $\gamma=C_p / C_v$ and you may assume the heat capacities are independent of temperature. [Hint: imagine placing a bag around the chamber, just large enough to enclose the chamber and all the gas that finally ends up inside the chamber, and calculate the work done by the rest of the atmosphere as this bag collapses.] Consider the case of argon gas $(\gamma=5 / 3)$ at $50^{\circ} \mathrm{C}$ leaking into a flask made of tin. What happens to the flask?

Key Concepts

Work Done by External Pressure

When a system undergoes compression or expansion under the influence of external pressure, work is performed on or by the gas. In an adiabatically insulated system, this work directly translates into a change in the internal energy of the gas, hence its temperature.

Specific Heats and Their Ratio (?)

The specific heats at constant volume (C_v) and constant pressure (C_p) quantify the amount of energy required to raise the temperature of a substance under specific conditions. Their ratio, ? (gamma = C_p/C_v), is a critical parameter in thermodynamics that governs the relationship between pressure, volume, and temperature during adiabatic processes.

Pressure Equalization in Open Systems

In open systems where gas gradually leaks or enters a chamber, achieving pressure equalization involves complex interactions such as the work performed by the external atmosphere on the gas. This concept is important for understanding how energy transfers and transformations occur during slow gas inflows.

Thermal Insulation

Thermal insulation refers to the condition where a system does not exchange heat with its surroundings. This property ensures that any change in the system’s energy is solely due to work done on or by the system, and it is fundamental in analyzing processes like adiabatic expansion or compression.

Adiabatic Process

An adiabatic process is one in which no heat enters or leaves the system because of perfect thermal insulation. In such processes, any work performed on the system changes its internal energy, resulting in a temperature change that can be predicted using the gas’s specific heat capacities.

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