During an adiabatic process, the pressure of a gas is found...

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $C_{P} / C_{V}$ for the gas is [2003]
(A) $4 / 3$
(B) 2
(C) $5 / 3$
(D) $3 / 2$

Key Concepts

Adiabatic Process

An adiabatic process is a thermodynamic transformation 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, leading to specific relationships among pressure, volume, and temperature.

Ideal Gas Law

The ideal gas law establishes a fundamental relationship between pressure, volume, and temperature of a gas, expressed as pV = nRT. It serves as the basis for connecting thermodynamic properties and is pivotal when relating state changes under different conditions.

Heat Capacity Ratio (?)

The heat capacity ratio, denoted as ? (gamma), is defined as the ratio of the specific heat at constant pressure (CP) to the specific heat at constant volume (CV). This ratio is essential in characterizing the behavior of gases during adiabatic processes, influencing the steepness of pressure and temperature changes.

Derivation of Pressure-Temperature Relations in Adiabatic Processes

By combining the ideal gas law with the adiabatic condition (pV^? = constant), one can derive specific relationships between pressure and temperature during adiabatic transformations. This derivation highlights how the exponent in the pressure-temperature relationship is linked to the heat capacity ratio, thus providing insight into the thermodynamic characteristics of the gas.

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