A prefect gas is found to obey the relation P V^3 / 2=...

A prefect gas is found to obey the relation $P V^{3 / 2}=$ constant during an adiabatic process. if such a gas initially at a temperature $T$, is compressed to half of its initial volume, then its final temperature will be
(a) $2 T$
(b) $4 T$
(c) $(2)^{1 / 2} T$
(d) $2(2)^{1 / 2} T$

Key Concepts

Specific Heat Ratio (?)

The specific heat ratio, denoted as ? (gamma), is the ratio of the specific heat at constant pressure (Cp) to that at constant volume (Cv). This ratio is key in determining the exponent in the adiabatic equation, PV^? = constant, and it plays a significant role in relating changes in pressure, volume, and temperature during an adiabatic process.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. In thermodynamics, this means that any change in the internal energy of an ideal gas during compression or expansion is solely due to the work done on or by the gas. This principle is crucial for analyzing processes where temperature changes occur without heat input or loss.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. It provides the foundational relationship that connects these macroscopic properties, and it is used in conjunction with other thermodynamic equations to analyze and predict the behavior of gases under various process conditions.

Temperature-Volume Relationship in Adiabatic Processes

For an ideal gas undergoing an adiabatic process, the relationship TV^(?-1) = constant connects the temperature and volume of the gas. This relationship allows one to calculate the final temperature when the volume changes, given that no heat is transferred during the process, by directly relating the initial and final states of the gas.

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