Physics · Molecular physics and thermodynamics · Grade 10

Real gases

Why does the pressure in a propane cylinder stay constant until it is empty, and why can CO₂ not be liquefied above 31 °C?

Open the experiment

The situation

The pressure gauge on a kitchen propane cylinder reads almost the same whether the cylinder is full or half empty — roughly 7–8 bar — and drops only when the gas is nearly gone. In 1869 Thomas Andrews compressed carbon dioxide and showed that below 31 °C it turns into a liquid, while above that temperature it will not become liquid however hard it is compressed. The ideal gas law cannot explain this: real molecules have a size and they attract one another. These two properties make it possible to liquefy gases, to obtain liquid nitrogen and oxygen from air, and even to cool helium down to 4 K. How does a real gas differ from an ideal gas, and how can a gas be turned into a liquid?

Topic

Real gases, the van der Waals equation. Propane cylinder, the critical point of CO₂ and liquefaction of gases

Key ideas

Van der Waals equation
Critical point
Joule–Thomson effect

Formulas

(p + \frac{a}{V_{m}^{2}}) (V_{m} - b) = R T

Van der Waals equation (for 1 mol): $a / V_{m}^{2}$ — attraction between molecules (“internal pressure”), $b$ — their own volume.

V_{k} = 3 b, T_{k} = \frac{8 a}{27 R b}, p_{k} = \frac{a}{27 b^{2}}

Critical parameters: the point of inflection on the isotherm, $\partial p / \partial V = \partial^{2} p / \partial V^{2} = 0$ .

(π + \frac{3}{φ^{2}}) (3 φ - 1) = 8 τ

Reduced equation: $π = p / p_{c}$ , $φ = V / V_{c}$ , $τ = T / T_{c}$ — the constants of the substance disappear (law of corresponding states).

\frac{p V_{m}}{R T} = 1 + \frac{B (T)}{V_{m}} + \frac{C (T)}{V_{m}^{2}} + \dots, B = b - \frac{a}{R T}

Virial expansion; the Boyle temperature, where $B (T_{B}) = 0$ , is $T_{B} = a / (R b)$ for van der Waals.

μ_{J T} = {(\frac{\partial T}{\partial p})}_{H} \approx \frac{1}{c_{p}} (\frac{2 a}{R T} - b), T_{i n v} = \frac{2 a}{R b}

Joule–Thomson coefficient: at $T < T_{i n v}$ a throttled gas cools, above it — it warms up.

Theory

How molecules interact

When molecules come close, their electron shells cannot overlap (Pauli principle) — a strong repulsion appears. Farther away, mutual polarisation of the electron clouds gives a weak attraction (dispersion, orientation and induction van der Waals forces, ≈ 1/r⁶). Both are expressed by the Lennard-Jones potential $U = 4 ε [{(σ / r)}^{12} - {(σ / r)}^{6}]$ : the well depth $ε$ for argon is ≈ 0.01 eV — hundreds of times weaker than a chemical (valence) bond. The energy of a molecule itself is stepped: electronic levels (eV, visible and UV light), vibrational (≈ 0.1 eV, infrared) and rotational (≈ 0.001 eV, microwave) — molecular spectra come from these. Here only the intermolecular forces matter.

Van der Waals equation and metastable states

Two corrections are made to the ideal gas equation: the “free space” for molecules $V - b$ ( $b$ is about 4 times the volume of the molecules themselves) and the internal pressure $a / V^{2}$ , which pulls molecules near the wall inwards. At $T < T_{c}$ the isotherm is wavy. The part with $d p / d V > 0$ (red) is completely unstable. The parts adjacent to it are metastable: superheated liquid and supercooled vapour; they can exist in a pure substance, but boil or condense at a small disturbance (bubble chambers and Wilson cloud chambers are based on this). A real isotherm has a horizontal saturated vapour line; its height is found by the Maxwell construction: the line cuts equal areas from the wave.

The critical state and the law of corresponding states

As the temperature rises, the densities of liquid and vapour approach each other and become equal at the critical point: the meniscus disappears and the substance turns milky (critical opalescence). At $T > T_{c}$ no pressure can liquefy the gas — it is a supercritical fluid. For CO₂, $T_{c} = 31.0$ °C, $p_{c} = 73.8$ bar. In reduced variables ( $p / p_{c}$ , $V / V_{c}$ , $T / T_{c}$ ) the van der Waals equation is the same for all substances — the law of corresponding states: at the same $τ$ and $π$ the compressibility of gases $Z = p V / R T$ is almost the same (CO₂ and N₂ on the Z graph). The equation is qualitative: at the critical point it gives $Z_{c} = 3 / 8$ , whereas in reality it is ≈ 0.27–0.29.

Andrews' experiment, the propane cylinder and Le Chatelier's principle

Andrews compressed CO₂ with mercury and measured isotherms at different temperatures (from 13.1 °C to 48.1 °C). Once liquid appears, the pressure does not change on compression: vapour keeps turning into liquid. This is an example of Le Chatelier's principle: the system answers an external influence (compression) with a process that weakens it — condensation. A propane cylinder also works on this “plateau”: as long as there is liquid inside, the pressure depends only on temperature (≈ 8.4 bar at 20 °C). Liquid propane expands strongly when heated (≈ 0.3–0.4 % per degree), which is why the cylinder is filled to 80–85 % of its volume. If the liquid fills the whole volume, no vapour “cushion” remains: the almost incompressible liquid raises the pressure by several bar (≈ 4–6 bar) for every 1 °C, and the cylinder may burst. Model: the “real” gas is calculated with the Peng–Robinson equation (it gives the saturated vapour pressure of CO₂ and propane to ≈ 1 % accuracy); the density and compressibility of the liquid propane in the cylinder are taken from formulas fitted to NIST data — Peng–Robinson makes the liquid too “soft”.

Internal energy and the Joule–Thomson effect

The internal energy of a real gas also includes the potential energy of the molecules: $U = c_{V} T - a / V$ . When the gas expands and the molecules move apart, work is done against attraction and the kinetic energy decreases. In the Joule–Thomson experiment, a gas is slowly passed from high pressure to low pressure through a porous plug in a thermally insulated pipe — the enthalpy $H = U + p V$ is conserved. It is a state function: its value depends only on $T$ and $p$ , so we find $T_{2}$ from the condition $H (T_{2}, p_{2}) = H (T_{1}, p_{1})$ . Below the inversion temperature the gas cools (air and nitrogen at room temperature ≈ 0.2 K/bar), above it, it warms up (hydrogen and helium at room temperature). This process is irreversible. If the gas expands reversibly and adiabatically, turning a piston or turbine, it does external work and cools much more strongly (compare in the panel) — the maximum work is obtained in this reversible process; the Claude machine works this way.

Liquefaction of gases and liquid helium

Linde machine (1895): the compressed gas is throttled, and the cooled gas in a counter-flow heat exchanger pre-cools the next portion — the temperature falls with each cycle, and finally part of the gas turns into liquid. By liquefying air and distilling it, oxygen, nitrogen and argon are obtained; liquid nitrogen boils at −195.8 °C (77 K) at 1 atm. Hydrogen and helium must first be cooled below the inversion temperature: for helium ≈ 40 K, so it is cooled with liquid hydrogen and then throttled. Liquid helium boils at 4.2 K and becomes superfluid below 2.17 K. Model simplifications: effective Peng–Robinson parameters for hydrogen and helium, an ideal heat exchanger; a 2D molecular model — argon, cut-off radius 2.5σ. Peng–Robinson gives the inversion temperature for nitrogen and air about 20 % too high (for nitrogen it is actually ≈ 620 K).

Did you know?

The Dutch physicist Heike Kamerlingh Onnes was the first to liquefy helium (4.2 K), in Leiden in 1908, and received the Nobel Prize in Physics in 1913.
Supercritical carbon dioxide (above 31 °C and 74 bar) is used in industry as a solvent to extract caffeine from coffee beans.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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