Entropy and the Second Law Reference Cheat Sheet

Key Facts

• For a reversible heat transfer, entropy change is $dS = \frac{\delta Q_{\mathrm{rev}}}{T}$, where $T$ is the absolute temperature in kelvins.
• For an isothermal reversible process at constant temperature, the entropy change is $\Delta S = \frac{Q_{\mathrm{rev}}}{T}$.
• For a phase change at temperature $T$, the entropy change is $\Delta S = \frac{L}{T}$ for heat absorbed and $\Delta S = -\frac{L}{T}$ for heat released.
• For an ideal gas changing between equilibrium states, $\Delta S = nC_V \ln\left(\frac{T_2}{T_1}\right) + nR \ln\left(\frac{V_2}{V_1}\right)$.
• The statistical definition of entropy is $S = k_B \ln \Omega$, where $\Omega$ is the number of accessible microstates.
• The second law for an isolated system states that $\Delta S_{\mathrm{univ}} \ge 0$, with equality only for a reversible process.
• The Clausius inequality is $\oint \frac{\delta Q}{T} \le 0$, with equality for a reversible cycle.
• For a heat engine, the maximum possible efficiency between reservoirs is the Carnot efficiency $\eta_C = 1 - \frac{T_C}{T_H}$.

Vocabulary

Entropy

Entropy is a thermodynamic state function that measures energy dispersal and is related to microscopic disorder by $S = k_B \ln \Omega$.

Reversible process

A reversible process is an ideal process that can be undone through infinitesimal changes while leaving no net change in the system and surroundings.

Irreversible process

An irreversible process is a real process with entropy production, so the total entropy change of the universe is positive.

Clausius inequality

The Clausius inequality, $\oint \frac{\delta Q}{T} \le 0$, states the entropy condition that every cyclic process must satisfy.

Microstate

A microstate is one specific microscopic arrangement of particles and energies that produces the observed macroscopic state.

Carnot efficiency

Carnot efficiency is the greatest possible heat engine efficiency between two reservoirs, given by $\eta_C = 1 - \frac{T_C}{T_H}$.