Solutions and Phase Equilibria

Intuition [Beginner]

When you dissolve salt in water, something subtle happens to the thermodynamics. The vapor pressure above the solution drops below that of pure water. The boiling point rises. The freezing point falls. These colligative properties depend only on how many solute particles are present, not on their identity. A mole of glucose has the same effect as a mole of urea -- what matters is the count.

The reason is entropy. Dissolving a solute increases the disorder of the liquid phase relative to the vapor and solid phases. This shifts every phase equilibrium. Raoult's law captures the simplest case: in an ideal solution, each component's vapor pressure is proportional to its mole fraction. Real solutions deviate from this ideal, and those deviations carry information about intermolecular forces.

Phase diagrams are the map of where each phase (solid, liquid, gas) is stable as a function of temperature, pressure, and composition. Reading a phase diagram tells you what happens when you cool a mixture, compress a gas, or boil a solution. The lever rule lets you extract quantitative information about how much of each phase is present at any point.

Visual [Beginner]

   Phase diagram for a single substance (P vs T):

   Pressure
   |          * (critical point)
   |         / \
   |        /   \     supercritical fluid
   |       /     \
   |      / solid  \  liquid
   |     /           \
   |    /             \
   |   * (triple point)
   |  / gas            \
   | /                   \
   |/_____________________\___________
                            Temperature


   Binary phase diagram (T vs composition, constant P):

   T
   |  A . . . . . . . . . . . . . . . B
   |  |         liquid
   |  |      . . . . . . . .
   |  |    /               \
   |  |   /  liquid + solid  \
   |  |  /                    \
   |  | .   liquid + solid B    .
   |  |                           \
   |  |  * eutectic point          \
   |  |/                            \
   |  |solid A + solid B             \
   |  |_______________________________
   |         composition (A -> B)

   At the eutectic point, liquid coexists with both
   solids simultaneously -- it is the lowest melting
   composition in the system.

Worked example [Beginner]

Problem: What is the freezing point depression when 5.00 g of glucose (C6H12O6, MW = 180.16 g/mol) is dissolved in 250.0 g of water? K_f for water = 1.86 K kg/mol.

Solution:

Moles of glucose:

n = 5.00 g / 180.16 g/mol = 0.02775 mol

Molality:

m = 0.02775 mol / 0.2500 kg = 0.1110 mol/kg

Freezing point depression:

Delta T_f = K_f * m = (1.86 K kg/mol)(0.1110 mol/kg) = 0.206 K

The solution freezes at -0.206 degrees C. Notice that the identity of the solute (glucose) did not matter -- only the number of dissolved particles entered the calculation. Had the solute been NaCl, which dissociates into two ions, the van't Hoff factor i = 2 would double the effect: Delta T_f = 0.413 K.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Raoult's law. For component i in an ideal solution at temperature T:

P_i = x_i * P_i*

where P_i is the partial vapor pressure of component i above the solution, x_i is its mole fraction in the liquid, and P_i* is the vapor pressure of pure i at the same temperature.

Henry's law. For a dilute solute B dissolved in a solvent:

P_B = x_B * K_H

where K_H is the Henry's law constant. Unlike Raoult's law, K_H is not the vapor pressure of pure B -- it is an empirical constant that accounts for the different intermolecular environment a dilute solute experiences compared to a pure substance.

Activity and activity coefficient. For a real (non-ideal) solution, Raoult's law generalizes to:

P_i = a_i * P_i* = gamma_i * x_i * P_i*

where a_i is the activity of component i and gamma_i is its activity coefficient. In the limit x_i approaches 1 (Raoult's law convention), gamma_i approaches 1. For the solute at high dilution (Henry's law convention), a different reference state is used.

Colligative properties. All four colligative properties follow from the condition for chemical equilibrium between the solution and the adjacent phase:

1. Vapor pressure lowering: Delta P = x_solute * P_solvent* (Raoult's law)
2. Boiling point elevation: Delta T_b = K_b * m (non-volatile solute)
3. Freezing point depression: Delta T_f = K_f * m
4. Osmotic pressure: Pi V = n_solute R T (van't Hoff equation), or Pi = c R T

where m is molality, K_b and K_f are ebullioscopic and cryoscopic constants, and c is the molar concentration of solute.

Gibbs phase rule. The number of degrees of freedom in a system at equilibrium:

F = C - P + 2

where C is the number of components and P is the number of phases. For a single-component system (C = 1), the triple point (P = 3) has F = 0 -- it is an invariant point. A two-phase equilibrium (e.g., liquid-vapor, P = 2) has F = 1 -- you can vary temperature or pressure, but not both independently.

Lever rule. In a two-phase region of a binary phase diagram, the relative amounts of the two phases are determined by the lever rule. If the overall composition is x_0 and the phase boundaries are at x_alpha and x_beta:

(n_alpha / n_beta) = (x_beta - x_0) / (x_0 - x_alpha)

where n_alpha and n_beta are the moles in each phase. The rule is analogous to a lever balanced on a fulcrum at x_0.

Key results [Intermediate+]

1. Ideal solution thermodynamics. Mixing two components to form an ideal solution produces:

   Delta H_mix = 0 (no enthalpy change) Delta S_mix = -R (n_A ln x_A + n_B ln x_B) > 0 Delta G_mix = RT (n_A ln x_A + n_B ln x_B) < 0

   The driving force for mixing is purely entropic. Real solutions have Delta H_mix not equal to 0, which leads to deviations from Raoult's law and, if sufficiently positive, to phase separation (partial miscibility).

2. Positive and negative deviations from Raoult's law. When A-B intermolecular forces are weaker than A-A and B-B forces, Delta H_mix > 0 and the solution shows positive deviations (vapor pressure higher than ideal). When A-B forces are stronger, Delta H_mix < 0 and negative deviations occur. Extreme positive deviations can produce a minimum-boiling azeotrope; extreme negative deviations can produce a maximum-boiling azeotrope.

3. Eutectic systems. In a binary system where the two components are immiscible in the solid state, the phase diagram shows a eutectic point -- the composition with the lowest melting temperature. At the eutectic, the liquid is in equilibrium with both pure solid phases simultaneously (a three-phase equilibrium, F = 0 at constant pressure). The eutectic composition solidifies at a single temperature like a pure substance, but over a range of compositions the mixture solidifies over a temperature interval.

4. Distillation and azeotropes. Fractional distillation separates components based on differences in volatility. For an ideal solution, repeated vaporization-condensation cycles can achieve any desired purity. An azeotrope -- a composition where the vapor and liquid have the same composition -- cannot be separated by simple distillation. Breaking an azeotrope requires adding a third component (azeotropic distillation) or changing the pressure (pressure-swing distillation).

Advanced treatment [Master]

Regular solution theory. For solutions where the entropy of mixing is ideal but the enthalpy is not, the regular solution model gives:

Delta G_mix = RT (x_A ln x_A + x_B ln x_B) + Omega x_A x_B

where Omega is an interaction parameter proportional to the difference (epsilon_AB - (epsilon_AA + epsilon_BB)/2). When Omega > 2RT, the system exhibits an upper critical solution temperature (UCST) below which two liquid phases coexist. The spinodal curve, where (d^2 G_mix / dx^2)_T = 0, marks the limit of metastability. Between the binodal and spinodal curves, phase separation occurs by nucleation and growth; inside the spinodal, it occurs by spinodal decomposition.

Gibbs-Duhem equation. The chemical potentials of components in a solution are not independent. At constant T and P:

x_A d(mu_A) + x_B d(mu_B) = 0

This constrains activity coefficients: if you know gamma_A over a range of compositions, you can calculate gamma_B (and vice versa). The Gibbs-Duhem equation provides a rigorous consistency check on vapor-liquid equilibrium data.

Flory-Huggins theory. For polymer solutions, the entropy of mixing is much smaller than for small-molecule solutions because polymer chains have far fewer independent translational degrees of freedom. The Flory-Huggins model replaces mole fractions with volume fractions:

Delta G_mix / RT = (phi_1 / N_1) ln phi_1 + phi_2 ln phi_2 + chi phi_1 phi_2

where N_1 is the degree of polymerization (much greater than 1), phi_1 and phi_2 are volume fractions, and chi is the Flory interaction parameter. The chi parameter determines solvent quality: chi less than 0.5 is a good solvent, chi = 0.5 is theta conditions, and chi greater than 0.5 is a poor solvent.

Phase diagrams at variable pressure. The Clapeyron equation relates the slope of a phase boundary to the enthalpy and volume changes of the transition:

dP/dT = Delta H_trans / (T Delta V_trans)

For liquid-vapor equilibria, where the vapor is approximated as an ideal gas and V_vapor much greater than V_liquid, this simplifies to the Clausius-Clapeyron equation:

d(ln P)/dT = Delta H_vap / (R T^2)

Integration gives ln(P2/P1) = -(Delta H_vap/R)(1/T2 - 1/T1), assuming Delta H_vap is temperature-independent. Near the critical point, this assumption fails and the full Clapeyron equation must be used.

Solid-liquid equilibria in multicomponent systems. Beyond binary systems, ternary and higher phase diagrams use Gibbs triangles (equilateral triangles where each vertex is a pure component). Isothermal cross-sections show two-phase tie lines and three-phase triangles. Extractive distillation, liquid-liquid extraction, and crystallization processes are designed using these diagrams.

Connections [Master]

Solutions and phase equilibria connect to nearly every branch of chemistry and chemical engineering:

• Thermodynamics: The formalism of chemical potential, activity, and fugacity developed for phase equilibria is the same machinery used to treat chemical reaction equilibria. The condition mu_i(liquid) = mu_i(vapor) at equilibrium is structurally identical to the condition for chemical equilibrium in a reacting system.

• Physical chemistry of interfaces: Phase equilibria at curved interfaces give rise to surface tension and the Kelvin equation, which describes how vapor pressure depends on droplet size. The Gibbs adsorption isotherm relates surface excess concentrations to the surface tension.

• Materials science: Binary and ternary phase diagrams govern the design of alloys, ceramics, and semiconductors. Eutectic compositions are exploited for low-melting solders. Solid-state phase transitions (polymorphism) are critical in pharmaceuticals.

• Biochemistry: Protein solubility, membrane phase behavior, and the hydrophobic effect are all solution thermodynamics problems. The denaturation of proteins can be treated as a cooperative phase transition. Osmotic pressure drives water transport across cell membranes.

• Environmental chemistry: Henry's law governs the partitioning of gases between the atmosphere and natural waters. The solubility of CO2 in seawater and its temperature dependence are central to ocean acidification models.

• Chemical engineering: Distillation column design, liquid-liquid extraction, crystallization, and supercritical fluid extraction all rely on phase equilibrium calculations. Process simulators solve Gibbs energy minimization problems over thousands of components.

Bibliography [Master]

• Prausnitz, J. M., Lichtenthaler, R. N., and de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed. Prentice Hall, 1999. The authoritative reference for models of real solutions and phase equilibrium calculations.

• Atkins, P. W. and de Paula, J. Physical Chemistry, 11th ed. Oxford University Press, 2018. Chapters 5--6 cover solutions and phase diagrams at the intermediate level with extensive worked examples.

• Smith, J. M., Van Ness, H. C., and Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 7th ed. McGraw-Hill, 2005. Chapters 10--15 provide a thorough engineering treatment of vapor-liquid equilibrium and solution thermodynamics.

• Rowlinson, J. S. and Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed. Butterworths, 1982. A physically motivated treatment of liquid-phase thermodynamics with particular attention to critical phenomena.

• Koningsveld, R., Stockmayer, W. H., and Nies, E. Polymer Phase Diagrams. Oxford University Press, 2001. Definitive treatment of Flory-Huggins theory and its extensions to polymer solutions and blends.