$f_i^v = f_i^{l2}$

Liquid–Liquid:

$f_i^{l1} = f_i^{l2}$

Thus all three equal:

$\boxed{f_i^v = f_i^{l1} = f_i^{l2}}$

Expanded form:

$y_i \phi_i P = x_{i1} \gamma_{i1} f_i^{ref} = x_{i2} \gamma_{i2} f_i^{ref}$

Use VLLE when:

Very non-ideal mixtures at low T
Alcohol–hydrocarbon systems
Refrigerant mixtures
Cryogenic systems with immiscibility

5. Solid–Liquid Equilibrium (SLE)

General condition:

$f_i^s = f_i^l$

Solid phase (pure):

$f_i^s = f_i^{s,pure}$

Liquid phase:

$f_i^l = x_i \gamma_i f_i^{l,ref}$

Thus:

$\boxed{x_i \gamma_i = \frac{f_i^{s,pure}}{f_i^{l,ref}}}$

For melting/freezing point depression:

$\ln(x_i \gamma_i) = -\frac{\Delta H_{fus}}{R}\left(\frac{1}{T} - \frac{1}{T_m}\right)$

Use SLE for:

Freezing-point depression
Eutectic diagrams
Crystallization
Solubility of solids in liquids

6. Summary Table: When to Use Each Model

Problem Type	Appropriate Model	Conditions
Ideal VLE	Raoult’s Law	Similar molecules, low P
Non-ideal VLE	Modified Raoult ($\gamma$ models)	Strong deviations, azeotropes
Gas in liquid (dilute)	Henry’s Law	Low $x_i$ for solute, sparingly soluble
High pressure VLE	$\phi–\phi$ EOS method	Compressibility high
LLE	$\gamma$ models only (activity-based)	Immiscible liquids
VLLE	$\gamma–\phi$ for vapor & liquids	Multiphase non-ideal systems
SLE	Solid–liquid fugacity equality	Solubility, freezing point

1. Fundamental Condition for Reaction Equilibrium

For any reacting system at constant T, P, the equilibrium condition is:

$\boxed{\Delta G_{rxn} = 0}$

Or equivalently:

$\boxed{\left(\frac{\partial G}{\partial \xi}\right)_{T,P} = 0}$

Where $\xi$ = extent of reaction.

The more general statement uses chemical potentials:

$\boxed{\sum_i \nu_i \mu_i = 0}$

Where $\nu_i$ = stoichiometric coefficient (products +, reactants −).

2. Equilibrium Constant (K) Definitions

The standard definition:

$\boxed{K = \exp{\left(-\frac{\Delta G^\circ}{RT}\right)}}$

Important: $(\Delta G^\circ)$ is at standard state (usually 1 bar).

2.1 Gas-Phase Reactions

For reaction:

$\sum_i \nu_i A_i \rightleftharpoons 0$

General equilibrium expression using fugacity:

$\boxed{K = \prod_i \left(\frac{f_i}{f_i^\circ}\right)^{\nu_i}}$

If ideal gas ($f_i = y_i P$):

$K = \prod_i (y_i P)^{\nu_i}$

Often separated into:

$K = K_p = \prod_i P_i^{\nu_i}$

2.2 Liquid-Phase Reactions

Ideal liquid:

$f_i = x_i f_i^\circ$

Then:

$K = \prod_i x_i^{\nu_i}$

Non-ideal liquids require activity coefficients:

$K = \prod_i (x_i \gamma_i)^{\nu_i}$

3. Reaction Quotient (Q)

For any composition: