Liquefaction of Gases

An isotherm is a curve on a pressure–volume (P–V) graph drawn at a constant temperature. For an ideal gas, every isotherm is a smooth hyperbola described by PV = nRT. For a real gas such as CO₂, the isotherms deviate significantly from this ideal behaviour, especially at lower temperatures and higher pressures.

Thomas Andrews sealed CO₂ in a glass tube and measured pressure as he reduced the volume at several fixed temperatures. At high temperatures (e.g. 48 °C), the isotherm resembles an ideal gas curve — smooth and continuously decreasing. At 31.1 °C (the critical isotherm), the curve develops a horizontal inflection point. Below 31.1 °C (e.g. at 21 °C or 13 °C), the isotherm shows a distinct flat, horizontal plateau — a region where volume decreases but pressure stays constant. This plateau corresponds to the coexistence of liquid and gas (the two-phase region), where the gas is condensing into liquid at constant pressure.

The critical temperature T_c is the temperature above which a gas cannot be liquefied by pressure alone, no matter how high that pressure is. At T_c, the density of the liquid and the gas phases become identical — the meniscus (liquid–gas boundary) disappears. For CO₂, T_c = 31.1 °C (304.2 K). For nitrogen, T_c = −147 °C (126 K), meaning nitrogen must be cooled below −147 °C before it can be liquefied. This is why early attempts to liquefy gases like O₂ and N₂ failed — scientists were working above their critical temperatures.

The critical pressure P_c is the minimum pressure required to liquefy a gas exactly at its critical temperature T_c. For CO₂, P_c = 73.8 atm. For water, P_c = 218 atm and T_c = 374 °C. At pressures above P_c and temperatures above T_c, the substance enters the supercritical fluid region, where it has characteristics of both liquid and gas — it can diffuse through solids like a gas but dissolve substances like a liquid. Supercritical CO₂ is used industrially to decaffeinate coffee and extract essential oils.

On a P–V diagram, the flat horizontal segments of sub-critical isotherms have endpoints that, when connected, trace out a dome-shaped curve called the saturation envelope or coexistence curve. Inside the dome, gas and liquid coexist in equilibrium. The left boundary of the dome is the bubble-point curve (all liquid), and the right boundary is the dew-point curve (all vapour). The very top of the dome is the critical point, defined by coordinates (V_c, P_c, T_c). For CO₂: T_c = 304.2 K, P_c = 73.8 atm, V_c = 94 cm³ mol⁻¹.

Carl von Linde exploited Andrews' findings to liquefy air in 1895. His process uses the Joule–Thomson effect: when a real gas expands adiabatically through a small nozzle (throttling), it cools if the temperature is below the inversion temperature. The cooled gas then pre-cools incoming compressed gas in a heat exchanger, creating a feedback loop that progressively cools the gas until it liquefies. Nitrogen (bp = −196 °C) and oxygen (bp = −183 °C) are separated from liquid air by fractional distillation.

• P — Pressure (Pa or atm): The measured pressure of the real gas.
• a — Van der Waals constant (attraction) (Pa·m⁶·mol⁻²): Accounts for intermolecular attractive forces; larger a means stronger attraction (e.g. CO₂: a = 0.3658 Pa·m⁶·mol⁻²).
• V_m — Molar volume (m³·mol⁻¹): Volume occupied per mole of the gas.
• b — Van der Waals constant (volume exclusion) (m³·mol⁻¹): Represents the volume excluded due to finite molecular size (e.g. CO₂: b = 4.27 × 10⁻⁵ m³·mol⁻¹).
• R — Universal gas constant (8.314 J·mol⁻¹·K⁻¹): Relates energy to temperature per mole.
• T — Absolute temperature (K): Temperature in Kelvin; must be below T_c for liquefaction to be possible.

Problem. The van der Waals constants for CO₂ are a = 3.658 L²·atm·mol⁻² and b = 0.04286 L·mol⁻¹. Using the relations derived from the van der Waals equation at the critical point, calculate the critical temperature (T_c) and critical pressure (P_c) of CO₂. (R = 0.08206 L·atm·mol⁻¹·K⁻¹)

Answer. Critical temperature T_c ≈ 308 K (≈ 35 °C); Critical pressure P_c ≈ 73.8 atm

• Real Gases and the Van der Waals Equation — The van der Waals equation quantitatively explains why real gases deviate from ideal behaviour and predicts T_c and P_c from molecular constants a and b.
• Joule–Thomson Effect — The Joule–Thomson effect describes the temperature change of a real gas during free expansion and is the physical basis of industrial liquefaction processes such as Linde's method.
• Vapour Pressure and Boiling Point — Vapour pressure is the pressure exerted by a gas in equilibrium with its liquid phase; understanding it is essential to interpreting the horizontal plateaus (phase-coexistence regions) in Andrews isotherms.
• Kinetic Molecular Theory of Gases — KMT explains why temperature governs whether intermolecular forces can overcome molecular kinetic energy to hold molecules in the liquid state — the molecular basis of the critical temperature.
• Intermolecular Forces (van der Waals Forces) — The strength of dispersion, dipole–dipole, and hydrogen-bonding forces determines the value of T_c; gases with stronger intermolecular forces have higher critical temperatures and are easier to liquefy.