A monoatomic ideal gas, intially at temperature 11 is...

A monoatomic ideal gas, intially at temperature $1_{1}$ is enclosed in a cylinders fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $\mathrm{T}_{2}$ by releasing the piston suddenly If $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ the lengths of the gas column before and after expansion respectively, then then $\left(\mathrm{T}_{1} / \mathrm{T}_{2}\right)$ is given by (A) $\left\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\}^{(2 / 3)}$ (B) $\left\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\}^{(2 / 3)}$ (C) $\left\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\}$ (D) $\left\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\}$

A monoatomic ideal gas, intially at temperature $1_{1}$ is enclosed in a cylinders fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $\mathrm{T}_{2}$ by releasing the piston suddenly If $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ the lengths of the gas column before and after expansion respectively, then then $\left(\mathrm{T}_{1} / \mathrm{T}_{2}\right)$ is given by
(A) $\left\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\}^{(2 / 3)}$
(B) $\left\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\}^{(2 / 3)}$
(C) $\left\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\}$
(D) $\left\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\}$

Key Concepts

Irreversible versus Reversible Processes

While reversible adiabatic processes follow well-defined equations, sudden expansions are typically irreversible. However, the adiabatic condition (no heat transfer) still holds, and understanding the differences between reversible and irreversible processes is important for correctly applying thermodynamic relations and interpreting their outcomes.

Specific Heat Ratio (?)

The specific heat ratio (?), defined as the ratio of the molar specific heat at constant pressure to that at constant volume, is crucial in adiabatic processes. For a monoatomic ideal gas, ? has a particular constant value (typically 5/3) that determines the relationship between pressure, volume, and temperature during expansion or compression.

Temperature-Volume Relationship in Adiabatic Expansion

For an adiabatic process involving an ideal gas, there exists a relationship between temperature and volume given by the formula T * V^(?-1) = constant. This relation is essential when quantifying the change in temperature resulting from a change in volume, especially in setups where geometric parameters can define volume changes.

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 system’s internal energy results solely from work done on or by the system, leading to specific relations between state variables like pressure, volume, and temperature.

Ideal Gas Behavior

Ideal gases are modeled using the ideal gas law, which relates pressure, volume, and temperature. In such systems, the internal energy depends only on temperature, making them a useful simplification for analyzing thermodynamic processes, including those that are adiabatic.

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