Question:

For a sample of perfect gas when its pressure is changed isothermally from $p_i$ to $p_f$, the entropy change is given by

Updated On: Apr 20, 2025

$\Delta S = nR \, ln \bigg( \frac{P_f}{p_i}\bigg)$
$\Delta S = nR \, ln \bigg( \frac{p_i}{p_f}\bigg)$
$\Delta S = nRT\, ln \bigg( \frac{p_f}{p_i}\bigg)$
$\Delta S = RT \, ln \bigg( \frac{p_i}{p_f}\bigg)$

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The Correct Option is B

Solution and Explanation

Entropy Change in Isothermal Processes

Heat transmission causes an entropy shift, which is measurable. The ratio of heat to the temperature at which transfer occurred gives the change in entropy. This process is isothermal if the heat transfer $Q$ occurs at a single temperature, and the change in entropy is represented by the equation:

$\Delta S = \frac{Q}{T}$

According to the second law of thermodynamics, the entropy of a closed system increases for irreversible processes while remaining constant for reversible ones. In a closed system, entropy can never decrease:

$\Delta S \geq 0$

Measurable initial and final states are $i$ and $f$ for an ideal gas, respectively.

Entropy Change:

The entropy change is given by:

$\Delta S = \int_{i}^{f} \frac{dQ}{T}$

If the temperature is constant, we can simplify this as:

$\Delta S = \frac{1}{T} \int_{i}^{f} dQ_{rev}$

For a Reversible Process:

The reversible heat transfer is given by:

$dQ_{rev} = -dW_{rev} = nRT \left( \frac{dV}{V} \right)$

Substituting this into the equation for entropy change:

$\Delta S = \frac{1}{T} \int_{i}^{f} nRT \frac{dV}{V}$

This simplifies to:

$\Delta S = \frac{nRT}{T} \int_{i}^{f} \left( \frac{dV}{V} \right)$

Integrating, we get the entropy change as:

$\Delta S = nR \ln \left( \frac{V_f}{V_i} \right)$ --- (1)

Using Charle’s Law:

From Charle’s law, we know:

$P_i V_i = P_f V_f$

Rearranging this equation gives:

$\frac{V_f}{V_i} = \frac{P_f}{P_i}$ --- (2)

Final Result:

Substituting equation (2) into equation (1), we get:

$\Delta S = nR \ln \left( \frac{P_f}{P_i} \right)$

Conclusion:

Therefore, for a sample of a perfect gas when its pressure is changed isothermally from $ P_i $ to $ P_f $, the entropy change is:

$\Delta S = nR \ln \left( \frac{P_f}{P_i} \right)$

The correct answer is Option B.

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Concepts Used:

Thermodynamics

Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.

Important Terms

System

A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.

There are three types of systems:

Isolated System – An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
Open System – In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.

Thermodynamic Process

A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.

There are four types of thermodynamic process that have their unique properties, and they are:

Adiabatic Process – A process in which no heat transfer takes place.
Isochoric Process – A thermodynamic process taking place at constant volume is known as the isochoric process.
Isobaric Process – A process in which no change in pressure occurs.
Isothermal Process – A process in which no change in temperature occurs.

Laws of Thermodynamics

Zeroth Law of Thermodynamics

The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.

First Law of Thermodynamics

The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.

Second Law of Thermodynamics

The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.

Third Law of Thermodynamics

Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.