Three moles of an ideal monoatomic gas performs a cycle as...

Three moles of an ideal monoatomic gas performs a cycle as shown in the fig. The gas temperature in different states are $T_{1}=400 \mathrm{~K}, \quad T_{2}=800 \mathrm{~K}$, $T_{3}=2400 \mathrm{~K}, T_{4}=1200 \mathrm{~K} .$ What is the work done by the gas during the cycle ? (a) $10 \mathrm{~kJ}$ (b) $20 \mathrm{~kJ}$ (c) $5 \mathrm{~kJ}$ (d) $8.3 \mathrm{~kJ}$

Three moles of an ideal monoatomic gas performs a cycle as shown in the fig. The gas temperature in different states are $T_{1}=400 \mathrm{~K}, \quad T_{2}=800 \mathrm{~K}$,
$T_{3}=2400 \mathrm{~K}, T_{4}=1200 \mathrm{~K} .$ What
is the work done by the gas during the cycle ?
(a) $10 \mathrm{~kJ}$
(b) $20 \mathrm{~kJ}$
(c) $5 \mathrm{~kJ}$
(d) $8.3 \mathrm{~kJ}$

Key Concepts

Work Calculation in Thermodynamic Processes

Work in thermodynamic processes is calculated by integrating the pressure with respect to volume (W = ?P dV) along the path of the process. Since work is a path-dependent quantity, the specific details of each process within the cycle determine the total work done by the gas, which is visually represented by the area enclosed by the cycle on a P-V diagram.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, temperature, and number of moles of a gas. It is fundamental for analyzing the behavior of gases in thermodynamic processes, allowing one to transition between these variables to describe how a gas expands or compresses under various conditions.

First Law of Thermodynamics

The first law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another. In the context of a thermodynamic cycle, this principle implies that the net change in the internal energy of the system over one complete cycle is zero, so the net work done by the gas is equal to the net heat absorbed by the gas.

Thermodynamic Cycle

A thermodynamic cycle is a series of processes that eventually return a system to its original state. Despite the internal energy returning to its starting value over a complete cycle, work can be done by or on the system during the individual processes, and the net work done is represented by the area enclosed by the cycle on a pressure-volume diagram.

Internal Energy of a Monoatomic Ideal Gas

For a monoatomic ideal gas, the internal energy depends solely on temperature and is given by U = (3/2)nRT. This relationship is essential in thermodynamics because, during a cycle, even though the system’s internal energy returns to its original value (resulting in zero net change), the differences in energy during individual processes determine how much work is done or heat is exchanged.

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