An ideal gas at initial pressure p1 undergoes an adiabatic...

An ideal gas at initial pressure $p_1$ undergoes an adiabatic expansion from volume $V_1$ to volume $V_2$. Assuming $\gamma$ is constant, find the final pressure and show that the work done is
$$
W=\frac{p_1 V_1}{\gamma-1}\left(\left(\frac{V_1}{V_2}\right)^{\gamma-1}-1\right) .
$$

Key Concepts

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. This means that any change in the internal energy of the system is solely due to work done by or on the system, rather than by heat transfer. Adiabatic processes are commonly analyzed in thermodynamics to study systems undergoing rapid expansions or compressions where heat exchange is negligible.

Ideal Gas Law

The ideal gas law relates the pressure, volume, and temperature of an ideal gas and provides a baseline model for understanding gas behavior under various conditions. It is often used as a starting point for deriving more specialized relations in thermodynamic processes, including adiabatic transformations, where temperature variations can be linked with changes in pressure and volume.

Poisson's Equation for Adiabatic Processes

For an ideal gas undergoing an adiabatic process, Poisson's equation offers a specific relationship between pressure and volume, stating that the product of pressure and volume raised to the power of the specific heat ratio (?) is constant. This relation, expressed as pV^? = constant, is crucial for determining the final state variables after an adiabatic change and serves as the basis for further derivations such as calculating work done.

Work Done in Thermodynamic Processes

The work done by or on a system during a volume change is calculated by integrating the pressure with respect to volume, typically expressed as W = ?PdV. For adiabatic processes, the pressure can be expressed as a function of volume using Poisson's equation, and the integration then leads to expressions that account for the specific heat ratio (?), linking the macroscopic work to the underlying microscopic changes in the gas.

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