Pure Substances and Phase Transitions

Pure substances have distinct phases (solid, liquid, vapor) with phase boundaries in pressure–temperature (p–T) space. Phase equilibrium occurs when the intensive potentials (chemical potentials or molar Gibbs free energies) of coexisting phases are equal. Important characteristic points are the triple point (all three phases coexist) and the critical point (the distinction between liquid and vapor disappears). Phase-change processes (melting, vaporization, sublimation) involve latent heat. The Clapeyron equation relates the slope of a phase boundary to the latent heat and the specific volume change; the Clausius–Clapeyron relation is an approximate integrated form useful for vaporization when the vapor behaves approximately as an ideal gas. Metastable states (superheating, supercooling) can occur when nucleation is suppressed.

Governing FormulaGibbs phase rule: F = C - P + 2 (for a single-component C=1 → F = 3 - P). Clapeyron equation (general): dp_sat/dT = L / (T Δv), where L is latent heat per mass (J·kg⁻¹) or per mole (J·mol⁻¹) consistent with Δv. Clausius–Clapeyron (approx., for vaporization with ideal-gas vapor): ln(p2/p1) = - (L_m / R) (1/T2 - 1/T1), where L_m is molar latent heat (J·mol⁻¹) and R = 8.314 J·mol⁻¹·K⁻¹. Using ideal-gas v_g ≈ RT/p, Clapeyron simplifies to dp/dT ≈ (L_m p) / (R T^2) (molar form). Quality (mass fraction of vapor) x for a saturated liquid–vapor mixture: x = (v - v_f) / (v_g - v_f), where v is mixture specific volume, v_f is saturated liquid specific volume, v_g is saturated vapor specific volume. Phase equilibrium condition: equality of chemical potentials (μ^α = μ^β) or equality of molar Gibbs free energy between phases.

Knowledge Check

10 Questions

1.According to the Gibbs phase rule, what is the number of independent intensive degrees of freedom (F) at the triple point of a pure substance?

2.Which of the following statements is true at the critical point of a pure substance?

3.A pure substance has a molar latent heat of vaporization L_m = 30.0 kJ·mol⁻¹. At T1 = 300.0 K its saturation pressure is p1 = 100.0 kPa. Using the Clausius–Clapeyron approximation and R = 8.314 J·mol⁻¹·K⁻¹, estimate the saturation pressure p2 at T2 = 310.0 K. (Assume vapor behaves as an ideal gas and L_m is constant over this interval.)

4.For a phase boundary where the vapor behaves approximately as an ideal gas, the Clapeyron equation can be written in terms of pressure as dp/dT ≈ (L_m p)/(R T^2). For a substance with L_m = 30.0 kJ·mol⁻¹ at T = 350 K and saturation pressure p = 200 kPa, compute dp/dT (in Pa·K⁻¹). Use R = 8.314 J·mol⁻¹·K⁻¹ and assume ideal-gas vapor.

5.During an isothermal (constant temperature), isobaric (constant pressure) vaporization of a pure substance, which of the following quantities for the two-phase mixture changes continuously as vaporization proceeds?

6.A saturated liquid–vapor mixture has saturated liquid specific volume v_f = 0.001000 m^3·kg⁻¹ and saturated vapor specific volume v_g = 0.500000 m^3·kg⁻¹. If the mixture specific volume is v = 0.200000 m^3·kg⁻¹, what is the vapor quality x (mass fraction of vapor)?

7.Which phase boundary in a pure-substance p–T diagram terminates at the critical point?

8.At phase equilibrium between two phases of a pure substance (e.g., liquid and vapor) which thermodynamic condition must be satisfied?

9.Using the Clapeyron equation for melting of ice, take latent heat of fusion L = 333.5 kJ·kg⁻¹, density of liquid water ρ_liq = 999.8 kg·m⁻³ and density of ice ρ_ice = 917.0 kg·m⁻³ at T = 273.15 K. Compute dp/dT for the solid–liquid line (in kPa·K⁻¹).

10.Which statement correctly describes a superheated liquid (liquid heated above its equilibrium boiling temperature at a given pressure)?