Real Gases, Change of Phase and van der Waals Equation

Squeeze a real gas hard enough and it turns into a liquid, something no ideal gas can do. Andrews' isotherms show where the ideal model breaks, and the van der Waals equation fixes it with two corrections.

Builds on: Part 2 · Gas Laws.

Real Gases, Change of Phase and van der Waals Equation

Video coming soon

Andrews' isotherms

Liquefaction and the critical point

At high pressure a real gas has less volume than an ideal one: attractions pull it together.
Below T_c there is a flat region: liquid and vapour coexist and volume falls at constant pressure.
The steep left branch is the liquid, which is nearly incompressible.
At the critical point C the isotherm has a horizontal inflection. Above T_c no pressure can liquefy the gas.

P–V isotherms with a liquid–vapour dome and the critical point C. — Schematic isotherms below, at and above T_c.

van der Waals

$(P + \frac{a n ^{2}}{V ^{2}})$ $(V - nb)$ = nRT

b corrects the volume: the molecules themselves take up space, so the free volume is $V - nb$ (b ≈ volume of one mole of molecules). a corrects the pressure: attractions pull wall-bound molecules back, lowering the pressure by an amount $\propto \frac{n ^{2}}{V ^{2}}$ . At low pressure or high temperature both corrections vanish and PV = nRT returns.

Summary

Key formulas

(P + \frac{a n ^{2}}{V ^{2}}) (V - nb) = n R T

van der Waals gas equation

JEE-style question

Your turn

In the van der Waals equation (P + an²/V²)(V − nb) = nRT, what does the constant a account for?

1)Attraction between molecules

2)Finite size of molecules

3)Elastic collisions

4)Gravity on molecules

Show the answer and the traps

Attraction between molecules. Option 1.

Option 2, finite molecular size, is what b accounts for. Options 3 and 4 aren't corrections in the equation at all.

Watch out

Common mistakes

Getting the signs backwards.Pressure is added to

(P + \frac{a n ^{2}}{V ^{2}})

; volume is subtracted from

(V - nb)

"Any gas liquefies under enough pressure."Only below its critical temperature T_c.

Practice

Try these

1. What are the SI units of the van der Waals constants a and b?

a: $Pa m^{6} /mo l^{2}$ (so $\frac{a n ^{2}}{V ^{2}}$ is a pressure); b: m³/mol (so nb is a volume).

2. The critical temperature of CO₂ is 304 K. Can CO₂ at 320 K be liquefied by compressing it?

No. Above T_c no pressure can liquefy a gas; it must first be cooled below 304 K.