Review

1. Checklist of Key Concepts¶

Section 4.1: Entropy¶

1. Exothermicity Does Not Guarantee Spontaneity

   • $\Delta H < 0$ (exothermic) often favors spontaneity but does not determine it. Counter-example: mixing of two ideal gases is spontaneous even though $\Delta H = 0$.

   • A complete account of directionality requires entropy.

2. Definition of Entropy

   • Entropy is a state function defined by

   $$dS = \frac{\delta q_{\mathrm{rev}}}{T}.$$

   (1)

   • The finite change between states $A$ and $B$ is $\Delta S = \int_A^B \delta q_{\mathrm{rev}}/T$, evaluated along any reversible path connecting those states.

3. Exactness of $\delta q_{\mathrm{rev}}/T$

   • $\delta q_{\mathrm{rev}}$ is inexact, but dividing by $T$ produces an exact differential. Verification for an ideal gas: the cross-derivative test on $C_V/T$ and $Nk_{\mathrm{B}}/V$ gives matching mixed partials (both zero).

4. Entropy of Irreversible Processes

   • Because $S$ is a state function, $\Delta S$ depends only on the endpoints. To compute $\Delta S$ for an irreversible process, find any convenient reversible path between the same initial and final states and integrate $\delta q_{\mathrm{rev}}/T$ along that path.

5. Fundamental Thermodynamic Relation

   • Combining $\delta q_{\mathrm{rev}} = T\,dS$ and $\delta w_{\mathrm{rev}} = -P\,dV$ in the First Law:

   $$dU = T\,dS - P\,dV.$$

   (2)

   • The natural variables of $U$ are $S$ and $V$. This relation replaces two inexact differentials ($\delta q$, $\delta w$) with three exact ones ($dU$, $dS$, $dV$).

Section 4.2: Carnot Cycle¶

1. Structure of the Carnot Cycle

   • A fully reversible cycle operating between a hot reservoir ($T_{\mathrm{hot}}$) and a cold reservoir ($T_{\mathrm{cold}}$), consisting of four steps:

     • $A \to B$: reversible isothermal expansion at $T_{\mathrm{hot}}$ (system absorbs heat $q_{AB} > 0$).

     • $B \to C$: reversible adiabatic expansion (system cools, $q = 0$).

     • $C \to D$: reversible isothermal compression at $T_{\mathrm{cold}}$ (system releases heat $q_{CD} < 0$).

     • $D \to A$: reversible adiabatic compression (system warms, $q = 0$).

2. Entropy Bookkeeping Around the Cycle

   • Adiabatic steps are isentropic: $\Delta S_{BC} = \Delta S_{DA} = 0$.

   • Isothermal steps carry entropy: $\Delta S_{AB} = q_{AB}/T_{\mathrm{hot}}$, $\Delta S_{CD} = q_{CD}/T_{\mathrm{cold}}$.

   • Over one complete cycle, $\Delta S_{\mathrm{cycle}} = 0$ (state function returns to initial value), so $\Delta S_{AB} = -\Delta S_{CD}$.

3. Carnot Efficiency

   $$\eta_{\mathrm{Carnot}} = 1 - \frac{T_{\mathrm{cold}}}{T_{\mathrm{hot}}}.$$

   (3)

   • This is the maximum possible efficiency for any engine operating between two temperatures. It depends only on the reservoir temperatures.

4. Direction of Spontaneous Heat Flow

   • Two subsystems at different temperatures inside an insulated boundary: $dS = dU_A(1/T_A - 1/T_B)$.

   • For $dS \ge 0$ (second law), heat must flow from hot to cold ($dU_A < 0$ when $T_A > T_B$).

   • Equilibrium ($dS = 0$) is reached when $T_A = T_B$.

Section 4.3: Microscopic View of Entropy¶

1. Entropy Increases Until Equilibrium (Isolated Systems)

   • The second law for an isolated system: $dS \ge 0$, with equality at equilibrium.

   • Entropy rises during spontaneous processes and reaches its maximum at equilibrium.

2. The Clausius Inequality

   $$dS \ge \frac{\delta q}{T}.$$

   (4)

   • Equality for reversible processes; strict inequality for irreversible processes.

   • Irreversibility produces additional entropy beyond the entropy carried by heat flow.

   • For an isolated system ($\delta q = 0$): reduces to $dS \ge 0$.

3. Boltzmann’s Formula (Microcanonical Ensemble)

   $$S = k_{\mathrm{B}} \ln \Omega,$$

   (5)

   where $\Omega$ is the number of accessible microstates at fixed $U$, $V$, $N$. The logarithm ensures entropy is extensive (additive for independent subsystems).

4. Gibbs Entropy (General Probability Distribution)

   $$S = -k_{\mathrm{B}} \sum_i p_i \ln p_i.$$

   (6)

   • Reduces to Boltzmann’s formula when all $\Omega$ microstates are equally probable ($p_i = 1/\Omega$).

   • Applies to the canonical ensemble ($p_i = e^{-\beta E_i}/Q$) and any other ensemble.

5. Canonical Entropy Identity

   • Evaluating the Gibbs entropy with canonical probabilities yields

   $$S = \frac{U}{T} + k_{\mathrm{B}} \ln Q.$$

   (7)

   • Directly ties the macroscopic state function $S$ to the partition function $Q$ from Chapter 2.

6. Helmholtz Free Energy

   • Rearranging the canonical entropy identity:

   $$A \equiv U - TS = -k_{\mathrm{B}}T\ln Q.$$

   (8)

   • Natural variables of $A$ are $T$ and $V$.

   • All canonical thermodynamic quantities ($S$, $U$, $P$, $C_V$) can be derived from $A$ by differentiation.

   • Defining $A = U - TS$ is the same Legendre-transform strategy used to define $H = U + PV$ in Section 3.3: replace a hard-to-control variable ($S$) with an easy-to-control one ($T$).

7. Microscopic Consistency

   • The Gibbs entropy depends only on probabilities $\{p_i\}$. Entropy changes when probabilities change (heat), not when energy levels shift (work).

   • This is consistent with the microscopic First Law from Section 3.2: $\delta q = \sum_i E_i\,dp_i$ and $\delta w = \sum_i p_i\,dE_i$.

2. Checklist of Most Important Equations¶

Below is a unified list of the major equations from Sections 4.1–4.3.

A. Entropy Definition

• Applicability: any system. The integral must be evaluated along a reversible path, but $\Delta S$ itself depends only on the endpoints.

B. Fundamental Thermodynamic Relation

• Applicability: simple closed system with $PV$-only work. Natural variables of $U$ are $S$ and $V$.

C. Entropy Change for an Ideal Gas (from $V$ and $T$)

• For a monatomic ideal gas with constant $C_V = \tfrac{3}{2}Nk_{\mathrm{B}}$: $\Delta S = \tfrac{3}{2}Nk_{\mathrm{B}}\ln(T_2/T_1) + Nk_{\mathrm{B}}\ln(V_2/V_1)$.

D. Carnot Efficiency

• Applicability: maximum efficiency for any heat engine operating between two reservoir temperatures.

E. Clausius Inequality

• Equality for reversible processes; strict inequality for irreversible processes.

• For isolated systems ($\delta q = 0$): $dS \ge 0$.

F. Boltzmann Entropy

• Applicability: isolated systems (microcanonical ensemble) where all $\Omega$ accessible microstates are equally probable.

G. Gibbs Entropy

• Applicability: any probability distribution over microstates. Reduces to Boltzmann’s formula in the microcanonical limit.

H. Canonical Entropy Identity

• Applicability: canonical ensemble (fixed $N$, $V$, $T$). Connects macroscopic entropy to the partition function.

I. Helmholtz Free Energy

• Applicability: canonical ensemble. Natural variables are $T$ and $V$. All other canonical thermodynamic quantities can be obtained by differentiation.