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

Question

Why do real gases deviate from ideal behaviour? Write the van der Waals equation and explain the physical significance of constants $a$ and $b$ . How are critical temperature, pressure, and volume related to these constants?

Ideal vs Real Gas Behaviour

flowchart TD
    A["Ideal Gas Assumptions"] --> B["No intermolecular forces"]
    A --> C["Molecules have zero volume"]
    D["Real Gas Reality"] --> E["Intermolecular forces exist — affects pressure"]
    D --> F["Molecules have finite volume — affects volume"]
    E --> G["Correction: add 'a' term to pressure"]
    F --> H["Correction: subtract 'b' from volume"]
    G --> I["van der Waals Equation"]
    H --> I
    I --> J["(P + an²/V²)(V - nb) = nRT"]

Solution — Step by Step

The ideal gas equation $PV = nRT$ assumes two things that are NOT true for real gases:

Gas molecules have no intermolecular attractive forces — FALSE. Real molecules attract each other (van der Waals forces, dipole-dipole, etc.)
Gas molecules occupy zero volume — FALSE. Real molecules have a finite size.

These deviations become significant at high pressure (molecules close together, volume matters) and low temperature (molecules slow, attractions dominate). At low P and high T, gases behave nearly ideally.

For $n$ moles of a real gas:

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

Constant $a$ — measures the strength of intermolecular attraction. Higher $a$ means stronger attraction. The term $an^2/V^2$ corrects the measured pressure upward because intermolecular forces pull molecules inward, reducing the actual pressure exerted on walls.

Constant $b$ — measures the excluded volume (effective volume occupied by molecules themselves). The term $nb$ corrects the volume downward because not all of $V$ is available for movement — some is occupied by the molecules.

Gas	$a$ (atm L $^2$ mol $^{-2}$ )	$b$ (L mol $^{-1}$ )
He	0.034	0.024
$\text{H}_2$	0.244	0.027
$\text{CO}_2$	3.59	0.043
$\text{NH}_3$	4.17	0.037

Notice: $\text{NH}_3$ has a high $a$ value because of strong hydrogen bonding.

The critical point is the temperature and pressure above which a gas cannot be liquefied regardless of pressure applied. The critical constants are:

T_c = \frac{8a}{27Rb}

P_c = \frac{a}{27b^2}

V_c = 3nb = 3b \text{ (for 1 mol)}

These are derived by applying the condition that at the critical point, the P-V isotherm has a point of inflection:

\left(\frac{\partial P}{\partial V}\right)_T = 0 \quad \text{and} \quad \left(\frac{\partial^2 P}{\partial V^2}\right)_T = 0

Also, the compressibility factor at the critical point:

Z_c = \frac{P_cV_c}{RT_c} = \frac{3}{8} = 0.375

For real gases, $Z_c$ is usually less than 0.375 (around 0.2-0.3), showing that van der Waals is an approximation.

A gas can only be liquefied below its critical temperature ( $T_c$ ). Above $T_c$ , no amount of pressure will work.

This is why gases like $\text{O}_2$ ( $T_c$ = 154 K) and $\text{N}_2$ ( $T_c$ = 126 K) need extreme cooling before compression can liquefy them, while $\text{CO}_2$ ( $T_c$ = 304 K) can be liquefied at room temperature under high pressure.

Why This Works

The van der Waals equation is a better model than the ideal gas equation because it accounts for the two main reasons real gases deviate: molecular size and intermolecular forces. While still an approximation, it captures the qualitative behaviour of real gases — including the liquid-gas phase transition — which the ideal gas equation completely fails to predict.

JEE Main commonly asks: “What do $a$ and $b$ represent?” or “Arrange gases in order of ease of liquefaction.” Higher $a$ value means stronger intermolecular forces and higher $T_c$ — easier to liquefy. So $\text{NH}_3$ (high $a$ ) is easier to liquefy than $\text{H}_2$ (low $a$ ).

Alternative Approach — Compressibility Factor

The compressibility factor $Z = \frac{PV}{nRT}$ is another way to understand real gas behaviour:

$Z = 1$ — ideal gas
$Z < 1$ — intermolecular attractions dominate (gas is more compressible than ideal)
$Z > 1$ — molecular volume dominates (gas is less compressible than ideal)

At moderate pressures, most gases show $Z < 1$ . At very high pressures, all gases show $Z > 1$ (repulsive forces and molecular volume dominate).

Common Mistake

Students mix up the corrections. The pressure correction is added to the measured pressure: $P + an^2/V^2$ (because real pressure on walls is less than ideal due to inward pull of molecules). The volume correction is subtracted from the container volume: $V - nb$ (because some volume is occupied by molecules). Writing $(P - an^2/V^2)$ or $(V + nb)$ reverses the physics completely.