Ranking ΔS in three expansions

Problem

Three processes on different systems: A) reversible isothermal expansion of $1$ mole of ideal gas, doubling the volume; B) free expansion into a vacuum of $1$ mole of ideal gas, doubling the volume; C) free expansion into a vacuum of $2$ moles, doubling the volume. Rank the entropy change of the gas from smallest to largest.

Solution

Entropy is a state function: $\Delta S$ depends only on the initial and final states, not on the path. For an ideal gas at constant $T$, $\Delta S=nR\ln\dfrac{V_B}{V_A}$. A and B have the same initial and final states (same $n=1$, same doubling, same $T$), so: $\Delta S_A=\Delta S_B=nR\ln 2=1\cdot 8{,}314\cdot\ln 2\approx 5{,}76\;\text{J/K}$ C instead has $2$ moles: $\Delta S_C=2R\ln 2\approx 11{,}5\;\text{J/K}$ Even though B and C are irreversible processes, the calculation of the gas’s $\Delta S$ uses the states, not the path. $\ev{\Delta S_A=\Delta S_B < \Delta S_C}$

Collegamenti

Argomenti: Entropy and the second law Concetti: Entropy · Thermodynamic transformations Competenze: Ranking reasoning Oggetti: Ideal gas