Entropy of an ideal gas

For an ideal gas that passes from a state A to a state B, the change in entropy is calculated with a formula that accounts for both the change in temperature and the change in volume.

Key formula — Entropy of an ideal gas

$\ev{\Delta S_\text{gas} = n\,C_v\,\ln\frac{T_B}{T_A} + n\,R\,\ln\frac{V_B}{V_A}}$

The two terms have distinct, independent physical meanings. The first, $nC_v\ln(T_B/T_A)$, measures the contribution of heating: heating the gas increases thermal agitation, and hence the number of ways the energy can be distributed among the molecules. The second, $nR\ln(V_B/V_A)$, measures the contribution of expansion: giving the molecules more available volume increases the number of possible positions, and hence $\Omega$.

The most important aspect is that $\Delta S_\text{gas}$ depends only on states A and B, not on the path followed to get from one to the other: $S$ is a state function. Whether the gas expands slowly, abruptly or through a thousand intermediate steps, as long as it starts from $(T_A, V_A)$ and ends at $(T_B, V_B)$, the change in entropy is the same.

Two special cases clarify the formula:

• In an isothermal transformation ($T_B = T_A$) the first term vanishes and only $\Delta S_\text{gas} = nR\ln(V_B/V_A)$ remains: expanding ($V_B > V_A$) increases entropy.
• In an isochoric transformation ($V_B = V_A$) the second term vanishes and $\Delta S_\text{gas} = nC_v\ln(T_B/T_A)$ remains.

If the problem gives pressure instead of volume, simply recall the gas law $PV = nRT$ to rewrite the ratios; and if the transformation is at constant pressure it is convenient to use $C_p = C_v + R$ in place of $C_v$, replacing the ratio of volumes with that of temperatures.

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

Related exercises: Isothermal expansion: ΔS of the gas · Ranking ΔS in three expansions · Entropy in free expansion