Review

1. Checklist of Key Concepts¶

Section 6.1: Phase Diagrams¶

1. One-Component $P$–$T$ Phase Diagram

   • Single-phase regions (solid, liquid, gas) are separated by coexistence curves on which two phases are simultaneously in equilibrium.

   • Triple point: the unique $(T_{tp}, P_{tp})$ at which solid, liquid, and gas coexist.

   • Critical point: the point at which the liquid–vapor coexistence curve terminates; beyond it, liquid and gas become indistinguishable (a supercritical fluid).

   • The solid–liquid line does not terminate in a critical point because solid and liquid differ qualitatively (long-range translational order), while liquid and gas differ only quantitatively.

2. Open Systems and the Chemical Potential

   • A closed system cannot describe two coexisting phases by itself; each phase must be treated as an open system whose amount of matter can change.

   • The Gibbs differential generalizes to

   $$dG = -S\,dT + V\,dP + \sum_i \mu_i\,dn_i,$$

   (1)

   with the chemical potential defined by

   $$\mu_i \equiv \left(\frac{\partial G}{\partial n_i}\right)_{T,P,\,n_{j\neq i}}.$$

   (2)

   • $\mu_i$ is intensive — the “price per mole” of adding substance to a phase — which makes it the natural variable for comparing phases of different size.

3. Pure Substance: $\mu = \bar g$

   • For a one-component system the Euler relation gives $G = \mu n$, so the chemical potential equals the molar Gibbs free energy:

   $$\mu = \frac{G}{n} \equiv \bar g.$$

   (3)

   • This resolves the apparent paradox in Section 5.1 that $G$ alone appeared to vanish for a one-component system at fixed $N$.

4. Phase-Equilibrium Condition

   • Transferring $dn_\alpha$ moles from phase $\alpha$ to phase $\beta$ at fixed $T$ and $P$ gives $dG = (\mu_\alpha - \mu_\beta)\,dn_\alpha$. Requiring $dG = 0$ for arbitrary transfers yields

   $$\mu_\alpha(T,P) = \mu_\beta(T,P).$$

   (4)

   • Physically: matter flows from higher $\mu$ to lower $\mu$ and stops when they are equal. Because this is a single scalar constraint on two variables, it picks out a curve in the $(T,P)$ plane — the coexistence curve.

5. Graphical View: $\bar g$ vs. $T$ at Fixed $P$

   • From $dG = -S\,dT + V\,dP$: $(\partial \bar g/\partial T)_P = -\bar s$.

   • Every phase has negative slope; the larger $\bar s$, the steeper (more negative) the slope.

   • Since $\bar s_g > \bar s_l > \bar s_s$, the gas line is steepest, then liquid, then solid. Each intersection marks a first-order transition at which the stable (lowest-$\bar g$) phase changes.

6. Graphical View: $\bar g$ vs. $P$ at Fixed $T$

   • From $dG = -S\,dT + V\,dP$: $(\partial \bar g/\partial P)_T = \bar v$.

   • Every phase has positive slope; the larger $\bar v$, the steeper the slope.

   • Since $\bar v_g \gg \bar v_l \gtrsim \bar v_s$, compressing the system favors condensed phases over gas.

   • Water anomaly: $\bar v_s > \bar v_l$, so the ice curve rises faster than the liquid curve and increasing $P$ near $T_{fus}$ can melt ice.

7. Latent Heat and Entropy Jump at a First-Order Transition

   • A first-order transition is one at which the first derivatives of $G$ ($S$ and $V$) are discontinuous across the coexistence curve.

   • At coexistence, $\mu_\alpha = \mu_\beta$ gives $\Delta \bar g_{tr} = 0$; combined with $\Delta \bar g = \Delta \bar h - T\,\Delta \bar s$:

   $$\Delta \bar h_{tr} = T_{tr}\,\Delta \bar s_{tr}.$$

   (5)

   • One calorimetric measurement ($\Delta \bar h_{tr}$) therefore supplies both quantities.

8. Trouton’s Rule and Water as an Exception

   • For many ordinary liquids at the normal boiling point,

   $$\Delta \bar s_{vap} \approx 85.9\ \mathrm{J\,mol^{-1}\,K^{-1}} \approx 10.3\,R.$$

   (6)

   • Approximate constancy reflects that the entropy cost of vaporization is dominated by the phase change itself (volume increase, loss of short-range order) rather than the liquid’s chemical identity.

   • Hydrogen-bonded liquids (water, lower alcohols) exceed the Trouton value because the liquid is more ordered than “average,” so vaporization releases more entropy. For water, $\Delta \bar s_{vap} \approx 109\ \mathrm{J\,mol^{-1}\,K^{-1}}$, roughly 25% above Trouton.

Section 6.2: Clapeyron and Clausius–Clapeyron¶

1. Derivation of the Clapeyron Equation

   • Displacing along a coexistence curve preserves $\mu_\alpha = \mu_\beta$, so $d\mu_\alpha = d\mu_\beta$.

   • Using the molar Gibbs differential $d\mu = -\bar s\,dT + \bar v\,dP$ for each phase and collecting terms gives the Clapeyron equation:

   $$\frac{dP}{dT} = \frac{\Delta\bar s_{tr}}{\Delta\bar v_{tr}} = \frac{\Delta\bar h_{tr}}{T\,\Delta\bar v_{tr}}.$$

   (7)

   • Exact for any first-order transition between pure phases.

2. Sign and Magnitude of $dP/dT$

   • The sign is set by $\Delta\bar v$, since $\Delta\bar h$ and $T$ are positive at any “normal” transition. Most substances have $\bar v_l > \bar v_s$ (positive melting slope) and $\bar v_g \gg \bar v_l$ (positive boiling slope).

   • The magnitude is set by $\Delta\bar v$: solid–liquid transitions have very small $\Delta\bar v$ and therefore nearly vertical coexistence curves; liquid–vapor transitions have very large $\Delta\bar v$ and shallow coexistence curves.

   • This is why melting points are nearly pressure-independent at ordinary pressures while boiling points shift strongly.

3. Water: Negative-Slope Ice–Water Line

   • Because $\bar v_s > \bar v_l$ for water, $\Delta\bar v_{fus} < 0$ and therefore $dP/dT < 0$ along the ice–water line. Numerically, $dP/dT \approx -135\ \mathrm{bar\,K^{-1}}$, equivalently $dT/dP \approx -7.4\ \mathrm{mK\,bar^{-1}}$.

   • The effect is far too small to explain ice skating by “pressure melting”; surface friction and a pre-existing liquid layer dominate.

4. Liquid–Vapor Approximation

   • Two approximations collapse the Clapeyron equation into a far more useful form for the liquid–vapor line:

     • $\bar v_g \gg \bar v_l$, so $\Delta\bar v_{vap} \approx \bar v_g$ (for water at 1 atm the ratio exceeds 10³).

     • The vapor is treated as an ideal gas, $\bar v_g = RT/P_{sat}$.

5. Clausius–Clapeyron Equation (Differential Form)

   $$\frac{d\ln P_{sat}}{dT} = \frac{\Delta\bar h_{vap}}{RT^2}.$$

   (8)

   • The $T^{-2}$ dependence (rather than $T^{-1}$) traces to the two factors of $T$ introduced by the approximations: one from the explicit $T$ in Clapeyron, one from the $RT$ in $\bar v_g$.

6. Clausius–Clapeyron Equation (Integrated Form, Constant $\Delta\bar h_{vap}$)

   $$\ln\!\left(\frac{P_2}{P_1}\right) = -\frac{\Delta\bar h_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right).$$

   (9)

   • Two-point form: two $(T,P)$ points give $\Delta\bar h_{vap}$ directly.

   • Linearized form: a plot of $\ln P_{sat}$ vs. $1/T$ should be a straight line with slope $-\Delta\bar h_{vap}/R$, if the constant-$\Delta\bar h_{vap}$ approximation holds.

7. Diagnosing Breakdown and Recovering $\Delta\bar h_{vap}(T)$

   • A $\ln P$-vs-$1/T$ plot can look straight even when $\Delta\bar h_{vap}$ is varying. A residual plot is the reliable diagnostic: systematic curvature (not random scatter) indicates breakdown of the constant-$\Delta\bar h_{vap}$ assumption.

   • A local-slope estimate from finite differences gives

   $$\Delta\bar h_{vap}(T) = R\,T^2\,\frac{d\ln P_{sat}}{dT}.$$

   (10)

   • $\Delta\bar h_{vap}(T)$ decreases with rising $T$ and vanishes at the critical point, where the liquid and vapor phases merge.

8. Molecular Reading of $\Delta\bar h_{vap}$

   • For water, $\Delta\bar h_{vap} \approx 40\ \mathrm{kJ\,mol^{-1}}$ is consistent with breaking roughly two hydrogen bonds per molecule (each worth $\sim 20\ \mathrm{kJ\,mol^{-1}}$) on going from liquid to vapor — the macroscopic shadow of the intermolecular interactions behind the partition functions of Chapter 2.

2. Checklist of Most Important Equations¶

Below is a unified list of the major equations from Sections 6.1–6.2.

A. Open-System Gibbs Differential

• Applicability: any multicomponent system whose composition may change. Reduces to the closed-system form when all $dn_i = 0$.

B. Chemical Potential (Definition)

• Applicability: intensive quantity conjugate to the amount of component $i$.

C. Pure-Substance Identity

• Applicability: one-component systems. Chemical potential and molar Gibbs free energy are the same object.

D. Phase-Equilibrium Condition

• Applicability: two phases of a pure substance in equilibrium at fixed $T$ and $P$. A single scalar constraint on two variables defines the coexistence curve in the $(T,P)$ plane.

E. Slopes of $\bar g$ Along Natural Axes

• Applicability: pure substance, single phase. Used graphically to identify the stable phase (lowest $\bar g$) and to read off entropy and molar volume from slopes.

F. Latent Heat / Entropy Jump at a First-Order Transition

• Applicability: any first-order transition $\alpha \to \beta$ at coexistence $(T_{tr}, P_{tr})$. Follows directly from $\Delta \bar g_{tr} = 0$ on the coexistence curve.

G. Trouton’s Rule

• Applicability: empirical benchmark at the normal boiling point for ordinary (non-hydrogen-bonded, non-associated) liquids. Substantial upward deviations indicate unusual order in the liquid.

H. Clapeyron Equation

• Applicability: exact for any first-order transition between pure phases. The sign of $dP/dT$ is set by $\Delta\bar v$; the magnitude is dominated by $|\Delta\bar v|$ (small for solid–liquid, large for liquid–vapor).

I. Clausius–Clapeyron Equation (Differential Form)

• Applicability: liquid–vapor coexistence under two approximations — $\bar v_g \gg \bar v_l$ and an ideal vapor.

J. Clausius–Clapeyron Equation (Integrated Form)

• Applicability: liquid–vapor coexistence with $\Delta\bar h_{vap}$ approximately constant over the interval $[T_1, T_2]$. A plot of $\ln P_{sat}$ vs. $1/T$ is a straight line with slope $-\Delta\bar h_{vap}/R$.

K. Local-Slope Estimate of $\Delta\bar h_{vap}(T)$

• Applicability: recovers the temperature-dependent enthalpy of vaporization from vapor-pressure data via finite differences. $\Delta\bar h_{vap}(T) \to 0$ as $T \to T_c$.