By combining the ideal gas equation of state with (7.43),...

By combining the ideal gas equation of state with (7.43), show that during an adiabatic expansion $T V^{\gamma-1}$ is constant and $T^\gamma p^{1-\gamma}$ is constant, when $\gamma$ is independent of temperature.

Key Concepts

Ideal Gas Law

The ideal gas law, typically written as pV = nRT, is a fundamental equation in thermodynamics that relates pressure, volume, and temperature for an ideal gas. It assumes that the molecules of the gas are point particles that interact only through elastic collisions and without intermolecular forces. This relation provides the basis for analyzing how these state variables change under various thermodynamic processes, including adiabatic processes.

Adiabatic Process

An adiabatic process is one in which there is no heat exchange with the surroundings (Q = 0). In such a process, any change in the internal energy of the system results solely from work done by or on the system. For an ideal gas undergoing an adiabatic process, this constraint leads to specific relationships between pressure, volume, and temperature, which in turn produce constants or invariants when certain combinations of these variables are considered.

Adiabatic Index (?)

The adiabatic index, denoted as ?, is defined as the ratio of the specific heat at constant pressure to that at constant volume (Cp/Cv). This parameter characterizes the thermal properties of the gas during adiabatic transformations. When ? is constant (i.e., independent of temperature), it greatly simplifies the derivation of relationships among state variables and the identification of invariants like T·V^(?-1) and T^?·p^(1-?).

Adiabatic Invariants

Adiabatic invariants are combinations of thermodynamic variables that remain constant during an adiabatic process. By combining the ideal gas law with the energy conservation principles specific to adiabatic processes, one can demonstrate that certain expressions, namely T·V^(?-1) and T^?·p^(1-?), do not change during the process. This result provides a powerful tool for predicting the behavior of a gas when it expands or compresses without exchanging heat with its surroundings.

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