Starting with the same initial Conditions, an ideal gas...

Starting with the same initial Conditions, an ideal gas expands from Volume $\mathrm{V}_{1}$ to $\mathrm{V}_{2}$ in three different ways. The Work done by the gas is $\mathrm{W}_{1}$ if the process is purely isothermal, $\mathrm{W}_{2}$ if purely isobaric and $\mathrm{W}_{3}$ if purely adiabatic Then (A) $\mathrm{W}_{2}>\mathrm{W}_{1}>\mathrm{W}_{3}$ (B) $\mathrm{W}_{2}>\mathrm{W}_{3}>\mathrm{W}_{1}$ (C) $\mathrm{W}_{1}>\mathrm{W}_{2}>\mathrm{W}_{3}$ (D) $\mathrm{W}_{1}>\mathrm{W}_{3}>\mathrm{W}_{2}$

Starting with the same initial Conditions, an ideal gas expands from Volume $\mathrm{V}_{1}$ to $\mathrm{V}_{2}$ in three different ways. The Work done by the gas is $\mathrm{W}_{1}$ if the process is purely isothermal, $\mathrm{W}_{2}$ if purely isobaric and $\mathrm{W}_{3}$ if purely adiabatic Then
(A) $\mathrm{W}_{2}>\mathrm{W}_{1}>\mathrm{W}_{3}$
(B) $\mathrm{W}_{2}>\mathrm{W}_{3}>\mathrm{W}_{1}$
(C) $\mathrm{W}_{1}>\mathrm{W}_{2}>\mathrm{W}_{3}$
(D) $\mathrm{W}_{1}>\mathrm{W}_{3}>\mathrm{W}_{2}$

Key Concepts

Work Done in Thermodynamic Processes

Work done by a gas in expansion is generally calculated as the integral of pressure with respect to volume (W = ?PdV). The value of this work depends on how the pressure changes during the process, which in turn depends on the specific process type (isothermal, isobaric, or adiabatic). Different paths result in different amounts of work done even when starting and ending at the same states.

Adiabatic Process

An adiabatic process is characterized by no heat exchange with the surroundings. For an ideal gas, the expansion leads to a drop in internal energy which results in lower work output. The work is determined by the change in internal energy and involves the specific heat ratio, making it less than that achieved in processes involving heat input, such as isothermal or isobaric processes.

Isobaric Process

An isobaric process is one where the pressure remains constant during the expansion. In this scenario, work is given by W = P(V2 ? V1). Because the pressure is maintained at its initial value and the volume increases, the temperature must change according to the ideal gas law, leading to a different energy exchange compared to an isothermal process.

Isothermal Process

An isothermal process occurs at constant temperature. For an ideal gas undergoing isothermal expansion, the work done is calculated by integrating pressure with respect to volume, yielding W = nRT ln(V2/V1). The constant temperature ensures that the internal energy remains unchanged, and the work done is entirely balanced by the heat exchanged.

Ideal Gas Law

The ideal gas law (PV = nRT) underpins the behavior of gases in thermodynamic processes, linking pressure (P), volume (V), and temperature (T). It is essential for determining state changes during these processes and for calculating work done when the state variables change.

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