Real gases — van der Waals equation, critical constants, liquefaction

Question

Why do real gases deviate from ideal gas behaviour? Write the van der Waals equation and explain the significance of constants ‘a’ and ‘b’. What are critical temperature, critical pressure, and critical volume? Under what conditions can a gas be liquefied?

(JEE Main + CBSE Board — derivation + reasoning)

Solution — Step by Step

The ideal gas equation $PV = nRT$ assumes:

Gas molecules have zero volume (point particles)
There are no intermolecular forces between molecules

Both assumptions fail in reality:

At high pressure: molecules are close together, their finite volume matters
At low temperature: molecules move slowly, intermolecular attractions become significant

The compressibility factor $Z = \frac{PV}{nRT}$ measures deviation. For ideal gas, $Z = 1$ . For real gases, $Z < 1$ at moderate pressures (attractions dominate) and $Z > 1$ at very high pressures (volume effect dominates).

For $n$ moles of a real gas:

\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT

Correction	Term	What It Corrects
Pressure correction	$\frac{an^2}{V^2}$	Intermolecular attractions reduce the pressure exerted on walls — ‘a’ accounts for this
Volume correction	$nb$	Molecules have finite volume — ‘b’ is the excluded volume per mole

For 1 mole: $\left(P + \frac{a}{V^2}\right)(V - b) = RT$

Higher ‘a’ → stronger intermolecular forces → easier to liquefy

Higher ‘b’ → larger molecular size

Every gas has a critical temperature ( $T_c$ ) above which it CANNOT be liquefied, no matter how much pressure is applied.

Constant	Symbol	van der Waals Expression
Critical temperature	$T_c$	$\frac{8a}{27Rb}$
Critical pressure	$P_c$	$\frac{a}{27b^2}$
Critical volume	$V_c$	$3b$

At the critical point, the liquid and gas phases become indistinguishable — the meniscus disappears.

A gas can be liquefied ONLY below its critical temperature. Two methods:

Cool below $T_c$ and compress (increase pressure)
Joule-Thomson effect: allow compressed gas to expand through a porous plug — it cools (for most gases below their inversion temperature)

Gases like CO₂ ( $T_c = 31.1°C$ ) are easy to liquefy at room temperature. Gases like H₂ ( $T_c = -240°C$ ) and He ( $T_c = -268°C$ ) require extreme cooling first.

graph TD
    A["Real Gas Behavior"] --> B["Deviates from PV = nRT"]
    B --> C["High P: Volume effect"]
    B --> D["Low T: Attraction effect"]
    E["Van der Waals Equation"] --> F["a: intermolecular attraction"]
    E --> G["b: molecular volume"]
    H["Liquefaction"] --> I{"T below Tc?"}
    I -->|Yes| J["Apply pressure → liquid"]
    I -->|No| K["Cannot liquefy at any pressure"]
    style A fill:#fbbf24,stroke:#000,stroke-width:2px
    style K fill:#fca5a5,stroke:#000
    style J fill:#86efac,stroke:#000

Why This Works

The van der Waals equation is the simplest modification of the ideal gas law that accounts for real gas behaviour. The ‘a’ term adds back the pressure that intermolecular attractions “steal” from the walls. The ‘b’ term subtracts the volume that molecules themselves occupy from the total volume.

The critical temperature is physically meaningful — above it, the kinetic energy of molecules is too high for attractions to hold them in a liquid phase. No amount of squeezing can overcome this thermal energy.

Common Mistake

Students often confuse the conditions for ideal behaviour with those for maximum deviation. Real gases behave MOST ideally at high temperature and low pressure (molecules far apart, moving fast). They deviate MOST at low temperature and high pressure (molecules close, slow). JEE asks: “Under what conditions does a real gas approach ideal behaviour?” — High T, low P.

For JEE, the van der Waals constant ‘a’ is directly related to ease of liquefaction — higher ‘a’ means stronger intermolecular forces and easier liquefaction. So among CO₂, NH₃, and H₂, CO₂ has the highest ‘a’ and is easiest to liquefy (at room temperature). H₂ has the lowest ‘a’ and needs extreme cooling. This ranking question appears frequently.

Question

Solution — Step by Step

Why This Works

Common Mistake

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