🎓 Lesson 10
D5
Constructing P–T and P–x Phase Diagrams
A P–T (pressure–temperature) diagram shows how a substance changes between solid, liquid, and gas as pressure and temperature change; a P–x (pressure–composition) diagram shows how mixtures separate into different phases at fixed temperature.
🎯 Learning Objectives
- ✓ Explain the physical meaning of phase boundaries, critical points, and triple points on P–T diagrams
- ✓ Calculate dew-point and bubble-point pressures for binary mixtures using Raoult’s law and real-fluid equations of state
- ✓ Analyze non-ideal behavior in explosive decomposition product mixtures (e.g., CO₂–H₂O–N₂) using activity coefficients and P–x diagrams
- ✓ Design safe operating envelopes for high-pressure blasting gas containment by interpreting P–T stability limits
- ✓ Apply critical point data to assess thermal runaway risks during detonation-induced phase transitions
📖 Why This Matters
In mining and blasting engineering, understanding phase behavior is essential for predicting detonation product expansion, designing blast chamber venting systems, managing post-blast gas hazards (e.g., CO, NOₓ, steam), and modeling hydrothermal alteration near blast zones. Misinterpreting phase envelopes can lead to underestimating vapor pressures of trapped water in fractured rock — causing unexpected overpressurization — or misjudging condensation risks in exhaust ventilation ducts. Phase diagrams are the foundation for safe, predictive thermodynamic design.
📘 Core Principles
Phase diagrams emerge from the Gibbs phase rule (F = C − P + 2), which dictates degrees of freedom in multiphase equilibria. For a pure substance (C = 1), the P–T diagram has three single-phase regions (solid, liquid, vapor), separated by two-phase curves (melting, vaporization, sublimation) intersecting at the triple point. The critical point marks the end of the liquid–vapor curve, beyond which distinct phases vanish. For mixtures (C ≥ 2), P–x diagrams at fixed T reveal vapor–liquid equilibrium (VLE) behavior: ideal systems follow Raoult’s law; non-ideal ones require activity coefficients (e.g., NRTL, UNIQUAC) or cubic equations of state (Peng–Robinson). In blasting, decomposition gases (e.g., from ANFO: N₂, CO₂, H₂O, CO) form complex mixtures whose P–x envelopes govern condensation, corrosion, and toxicity exposure pathways.
📐 Bubble-Point Pressure Calculation (Ideal Mixture)
For an ideal binary mixture at known temperature and liquid composition, the bubble-point pressure is the total pressure at which the first vapor forms. It is calculated directly from Raoult’s law and serves as a baseline before introducing corrections for non-ideality.
💡 Worked Example
Problem: Given: Liquid mixture of CO₂ (x₁ = 0.45) and H₂O (x₂ = 0.55) at T = 373 K (100°C). Vapor pressures: P₁ˢᵃᵗ = 36.5 bar, P₂ˢᵃᵗ = 1.013 bar. Assume ideal behavior.
1.
Step 1: Identify mole fractions and saturation pressures for each component.
2.
Step 2: Apply Raoult’s law: P_bubble = x₁·P₁ˢᵃᵗ + x₂·P₂ˢᵃᵗ = (0.45)(36.5) + (0.55)(1.013).
3.
Step 3: Compute: 16.425 + 0.557 = 16.982 bar → round to 17.0 bar.
Answer:
The bubble-point pressure is 17.0 bar, which falls within the typical range of 15–25 bar for hot, wet explosive gas mixtures near vent outlets.
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), post-blast monitoring revealed condensation of H₂O–CO₂–N₂ mixtures inside confined haulage tunnels after wet-rock blasts. Engineers constructed a P–x diagram at 80°C using Peng–Robinson EOS to identify that at 12 bar total pressure and x_H₂O = 0.18, the mixture crossed the dew-point curve — triggering fog formation and reducing visibility. Revised venting protocols increased flow velocity to prevent residence time above dew point, eliminating safety incidents linked to reduced visibility.
🔧 Interactive Calculator
🔧 Open Thermodynamics and Equations of State Calculator📋 Case Connection
📋 Supercritical CO₂ Extraction of Caffeine
Low caffeine yield and inconsistent selectivity due to inaccurate P–T–x phase diagrams