A diatomic gas initially at 18^∘ C is Compressed adiabatically to one eighth of its original vol

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A diatomic gas initially at 18^∘ C is Compressed...

A diatomic gas initially at $18^{\circ} \mathrm{C}$ is Compressed adiabatically to one eighth of its original volume. The temperature after Compression will be
(A) $10^{\circ} \mathrm{C}$
(B) $668 \mathrm{~K}$
(C) $887^{\circ} \mathrm{C}$
(D) $144^{\circ} \mathrm{C}$

Key Concepts

Ideal Gas Behavior

The ideal gas law, which relates pressure, volume, and temperature (PV = nRT), underpins the analysis of processes involving gases. While the law itself applies to a broader range of phenomena, in the context of adiabatic processes it helps in describing how state variables change in response to work done on the system. Understanding ideal gas behavior is fundamental to deriving relations like T × V^(?-1) = constant.

Specific Heat Ratio (?)

The specific heat ratio, denoted as ? (gamma), is the ratio of the specific heat at constant pressure to the specific heat at constant volume (C_P/C_V). For diatomic gases, such as the one in the problem, ? usually approximates to 7/5 (or 1.4) under normal conditions. This ratio plays a critical role in determining the behavior of the gas during adiabatic processes, particularly affecting the temperature change during compression or expansion.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is essential in thermodynamic calculations, as the Kelvin scale is the standard unit in physics for temperature. Conversions ensure that calculations are performed using absolute temperatures, which is necessary for the correct application of formulas such as the adiabatic relation. This concept is crucial to ensure accuracy in determining the final state of the gas after compression.

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 gas is solely due to the work done on or by the gas. In such processes, understanding how temperature changes with compression or expansion is crucial because the entire energy change comes from work interactions, not from heat transfer.

Temperature-Volume Relationship

In an adiabatic process for an ideal gas, the relationship between temperature and volume is given by the equation T × V^(?-1) = constant. This relationship shows that when the volume decreases (compression), the temperature increases, assuming ? (the specific heat ratio) remains constant. This formula is fundamental for calculating the final temperature after a given change in volume.

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