One mole of an ideal gas expands isothermally. (a) If the...

One mole of an ideal gas expands isothermally. (a) If the gas doubles its volume, show that the work of expansion is $W=R T \ln$ 2. (b) Because the internal energy $E_{\text {int }}$ of an ideal gas depends solely on its temperature, no change in $E_{\text {int }}$ occurs during the expansion. It follows from the first law that the heat input to the gas during the expansion is equal to the energy output by work. Why does this conversion not violate the second law?

Key Concepts

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. This equation simplifies the analysis of gas behavior by assuming that the molecules do not interact and that the gas particles take up no volume. It forms the basis for understanding how pressure and volume change during various thermodynamic processes, making it a fundamental concept in thermodynamics.

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant. For an ideal gas undergoing an isothermal change, any expansion or compression happens at a fixed temperature, which means that the internal energy (dependent solely on temperature for an ideal gas) does not change. This process is characterized by the work done being directly linked to the heat exchanged with the surroundings, as dictated by the first law of thermodynamics.

Work in Thermodynamic Processes

The work done by or on a system in thermodynamic processes is determined by the integral of pressure with respect to volume change. In an isothermal process for an ideal gas, the pressure can be expressed using the ideal gas law, and the work integral becomes logarithmic in form. This calculation shows how the geometrical expansion of the gas translates into mechanical work.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation, asserting that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In the context of an isothermal process for an ideal gas, where the internal energy remains constant due to the unchanging temperature, the heat supplied to the system is entirely converted into the work done during expansion.

Second Law of Thermodynamics

The second law of thermodynamics governs the direction of spontaneous processes, stating that the total entropy of an isolated system can never decrease over time. In reversible processes, like an ideal isothermal expansion conducted slowly enough, any conversion of heat into work comes with a compensating entropy change that ensures the overall entropy of the universe does not decrease. Thus, even though heat is fully converted into work, the process does not violate the second law because it is carried out reversibly with balanced entropy considerations.

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