Ranking ΔS in expansions

Problem

Some moles of an ideal gas undergo isothermal expansions from $V_1$ to $V_2$: (A) $V_2=2V_1$, $T=300\ \text{K}$; (B) $V_2=5V_1$, $T=300\ \text{K}$; (C) $V_2=2V_1$, $T=600\ \text{K}$; (D) $V_2=10V_1$, $T=300\ \text{K}$. Rank $\Delta S$ from smallest to largest (per mole).

Solution

For an isothermal expansion of an ideal gas $\Delta S=nR\ln\dfrac{V_2}{V_1}$: it depends only on the ratio of volumes, not on the temperature. Per mole ($n=1$): $\Delta S_A=R\ln 2\approx 5{,}76\;\text{J/K}$ $\Delta S_C=R\ln 2\approx 5{,}76\;\text{J/K}\quad(\text{equal to } A:\ T\text{ does not matter})$ $\Delta S_B=R\ln 5\approx 13{,}4\;\text{J/K}$ $\Delta S_D=R\ln 10\approx 19{,}1\;\text{J/K}$ The volume ratio increases in the order $2,\,2,\,5,\,10$, so: $\ev{A = C < B < D}$

Links

Topics: Entropy and the second law Concepts: Entropy · Thermodynamic processes Skills: Ranking reasoning Objects: Ideal gas