§ C-26 Phase Diagrams P–T, Triple Point, Critical Point & Supercritical Fluid (Grade 10–11 · Phase Equilibria)
Clausius–Clapeyron equation, dry ice CO₂ sublimation, triple point, and supercritical state. Gibbs’ phase rule establishes nature’s strict constitution: how many phases may coexist in equilibrium at any given temperature and pressure.
Theory Summary & Scientific Mechanism
A single-component P–T phase diagram maps the stability fields of solid, liquid, and gas separated by coexistence curves governed by the Clapeyron equation (dP)/(dT) = (Δ Hₜr)/(T Δ Vₜr). Because ice is less dense than liquid water (Δ V_fus < 0), water’s melting curve has an anomalous negative slope (dP/dT < 0).
By Gibbs’ Phase Rule (f = 3 - Φ), solid, liquid, and vapor coexist exclusively at the invariant triple point (Tₜ, Pₜ). Because Pₜ(CO₂) = 5.11 atm > 1 atm, solid CO₂ ("dry ice") sublimes directly into gas at atmospheric pressure.
The liquid–vapor boundary terminates at the critical point (T_c, P_c); beyond this point the meniscus vanishes and the substance forms a supercritical fluid combining liquid-like solvating density with gas-like diffusivity.
Everyday Analogy
Why do ice skates glide on a thin water film while dry ice (CO₂) smokes without melting into a puddle? Because water expands on freezing (dP/dT < 0), and CO₂’s triple point (5.11 atm) lies above atmospheric pressure!
Interactive Question & Derivation
What is unique about the triple point (Tₜ, Pₜ) and critical point (T_c, P_c) on a single-component P–T phase diagram, and why does the fusion curve of water have a negative slope dP/dT < 0?
- A: At the triple point all three phases (solid, liquid, gas) coexist in equilibrium (f = 0); above (T_c, P_c) the liquid–gas boundary vanishes into a supercritical fluid; for water Vₗiquid < Vᵢce, so by the Clapeyron equation dP/dT = Δ H_fus/(TΔ V) < 0
- B: Above the critical temperature T_c, a gas can always be liquefied simply by applying slightly more pressure
- C: Carbon dioxide CO₂ melts into a liquid at 1 atm atmospheric pressure because its triple point lies below 0.1 atm
Study Hint: At the triple point all 3 phases coexist, above the critical point a supercritical fluid forms, and for water Δ V_fus < 0 ⇒ dP/dT < 0.
Proof & Explanation: By Gibbs’ Phase Rule (f = C - P + 2 = 3 - P), three phases coexist only at a single invariant triple point (f = 0). For CO₂, Pₜ = 5.11 atm > 1 atm, so at atmospheric pressure solid dry ice sublimes directly into gas at -78.5 ^°C. Above the critical point (T > T_c, P > P_c), the density of gas equals that of liquid, the meniscus disappears, and the substance forms a supercritical fluid. For water, ice is less dense than liquid water (Δ V_fus < 0), giving a negative Clapeyron melting slope dP/dT < 0.
Key Equations & 3D Simulation Parameters
(dP)/(dT) = (Δ Hₜr)/(T Δ Vₜr), ln (P₂)/(P₁) = -(Δ Hᵥap)/(R)≤ft((1)/(T₂) - (1)/(T₁) ), f = C - Φ + 2
- Substance (1=H₂O water, 2=CO₂ dry ice / supercritical): 1 … 2 (default: 2)
- Autoclave temperature T: 170 … 420 K (default: 260 K)
- External chamber pressure P: 1 … 90 atm (default: 15 atm)
- Thermal heat exchange power Q: 0.5 … 5 kJ (default: 2 kJ)