General formula for the entropy of the gas

The tool that allows the four reversible transformations to be derived from a single condition is the general formula for the entropy change of an ideal gas, already derived in the chapter on entropy. Going from a state $A$ to a state $B$, the entropy change of the gas depends only on the end states, not on the path followed:

$\Delta S_\text{gas} = nC_v\,\ln\frac{T_B}{T_A} + nR\,\ln\frac{V_B}{V_A}$

The first term measures the contribution of the temperature change, the second that of the volume change. Alongside this we must add the entropy change of every reservoir with which the gas exchanges an amount of heat $Q$ while remaining at constant temperature $T_\text{term}$:

$\Delta S_\text{term} = \frac{Q}{T_\text{term}}$

Key formula

$\Delta S_\text{gas} = nC_v\,\ln\frac{T_B}{T_A} + nR\,\ln\frac{V_B}{V_A} \qquad \Delta S_\text{term} = \frac{Q}{T_\text{term}}$

The mechanism that generates the fundamental transformations is now purely algebraic. We impose that the entropy of the universe does not change,

$\Delta S_\text{tot} = \Delta S_\text{gas} + \Delta S_\text{term} = 0$

and each time we add the specific constraint of the process under consideration: constant $V$, or constant $P$, or constant $T$, or $Q=0$. Each choice closes the system and produces one of the four fundamental reversible transformations — isochoric, isobaric, isothermal and adiabatic. These are not four cases picked by hand, but the four faces of a single condition.

Connections

Topics: Thermodynamics · Entropy and the second law Concepts: Entropy · Second law of thermodynamics · Thermodynamic transformations Skills: Entropy balance Objects: Ideal gas

Related exercises: Worked exercise — reversible vs irreversible isothermal expansion · Worked exercise — free expansion is not reversible · Gas in reversible adiabatic expansion