Review

1. Checklist of Key Concepts

1. Phase‐Equilibrium Condition

   • Gibbs Equality: At equilibrium between any two phases \(\alpha\) and \(\beta\), their molar Gibbs free energies are equal:

     \[G_\alpha = G_\beta.\]

     This condition holds at a phase‐transition point (e.g. melting point, boiling point) and underpins all phase‐diagram constructions.

2. Latent Heat & Entropy Jump

   • Enthalpy Change: \(\Delta H_\text{trs}\) is nonzero

     \[\Delta H_\text{trs} = H_\beta - H_\alpha,\]

     and known as the latent heat.

   • Entropy Change: \(\Delta S_\text{trs}\) likewise jumps:

     \[\Delta S_\text{trs} = S_\beta - S_\alpha = \frac{\Delta H_\text{trs}}{T_\text{trs}}.\]

3. Chemical Potential in Open Systems

   • Definition: For component \(i\) in an open, multi‐component system,

     \[\mu_i \;=\;\Bigl(\frac{\partial G}{\partial N_i}\Bigr)_{T,P,N_{j\neq i}},\]

     appears in the fundamental relation

     \[dG = -S\,dT + V\,dP + \sum_i \mu_i\,dN_i.\]

   • Role: \(\mu_i\) is the “driving force” for matter exchange across a boundary; at phase equilibrium, the chemical potentials of each component are equal in all phases.

4. Clausius–Clapeyron Equation

   • Differential Form: Relates the slope of a phase boundary on a \(P\)–\(T\) diagram to latent heat and volume change:

     \[\frac{dP}{dT} \;=\; \frac{\Delta S_\text{trs}}{\Delta V_\text{trs}} \;=\; \frac{\Delta H_\text{trs}}{T\,\Delta V_\text{trs}}.\]

   • Sketching Phase Diagrams: By knowing \(\Delta H_\text{trs}\) and \(\Delta V_\text{trs}\), one can approximate the coexistence curve between two phases (e.g. solid–liquid, liquid–gas).

   • Integrated Form (Ideal Gas Approximation):

     \[\ln\frac{P_2}{P_1} \approx -\frac{\Delta H_\text{vap}}{R}\,\Bigl(\frac{1}{T_2}-\frac{1}{T_1}\Bigr),\]

     useful for extracting \(\Delta H_\text{vap}\) from vapor‐pressure data.

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2. Checklist of Most Important Equations

Equation	Variables	Meaning & Use Cases	Caveats/Approximations
\(G_\alpha = G_\beta\)	\(G\): molar Gibbs free energy	Fundamental phase‐equilibrium criterion. Use to locate transition points in \(P\)–\(T\) space.	Applies only at the equilibrium curve.
\(\Delta H_\text{trs} = H_\beta - H_\alpha\)	\(H\): enthalpy	Latent heat associated with phase change (fusion, vaporization). Determines heat required/​released at transition.	Measured at constant \(P\).
\(\Delta S_\text{trs} = S_\beta - S_\alpha = \dfrac{\Delta H_\text{trs}}{T_\text{trs}}\)	\(S\): entropy; \(T_\text{trs}\): transition temperature	Quantifies entropy jump at a first‐order transition. Essential for Clapeyron equation.	—
\(\dfrac{dP}{dT} = \dfrac{\Delta S_\text{trs}}{\Delta V_\text{trs}} = \dfrac{\Delta H_\text{trs}}{T\,\Delta V_\text{trs}}\)	\(\Delta V_\text{trs} = V_\beta - V_\alpha\): molar volume change	Clausius–Clapeyron: slope of coexistence curve. Use to predict how \(P\) changes with \(T\) for phase boundaries (e.g., vapor pressure curve).	Requires knowledge of \(\Delta H_\text{trs}\) and \(\Delta V_\text{trs}\).
\(\ln\frac{P_2}{P_1} \approx -\frac{\Delta H_\text{vap}}{R}\Bigl(\frac{1}{T_2}-\frac{1}{T_1}\Bigr)\)	\(R\): gas constant	Integrated Clapeyron under ideal‐gas assumption for vapor: use vapor‐pressure measurements at two \(T\) to find \(\Delta H_\text{vap}\) or predict \(P\).	Assumes vapor behaves ideally and \(\Delta H_\text{vap}\) is constant over \(T_1\)–\(T_2\).
\(\mu_i = \Bigl(\frac{\partial G}{\partial N_i}\Bigr)_{T,P,N_{j\neq i}}\)	\(\mu_i\): chemical potential of component \(i\) \(N_i\): number of moles	Defines the intensive variable controlling mass exchange. Equal in all phases at multi‐phase equilibrium (e.g., for water in coexisting liquid & vapor).	Valid for systems where composition can vary; neglects real gas corrections.
\(dG = -S\,dT + V\,dP + \sum_i \mu_i\,dN_i\)	—	Fundamental differential for open, multi‐component systems. Basis for deriving all equilibrium conditions (thermal, mechanical, chemical).	Assumes only \(PV\) work and matter exchange; neglects electric, magnetic, surface‐tension work, etc.
\(G(P,T) = G^\circ(T) + RT\ln\frac{P}{P^\circ}\)	\(P^\circ\): standard pressure	Adjusts the Gibbs free energy of an ideal gas from standard pressure to \(P\).	Ideal‐gas assumption.