Van der Waals Equation

Equivalent forms

P = \frac{RT}{V_m - b} - \frac{a}{V_m^2}

T_c = \frac{8a}{27Rb},\ P_c = \frac{a}{27b^2},\ V_c = 3b

Two physically-motivated constants (a, b) capture the entire gas-liquid transition and predict a universal critical point — the birth of corresponding-states theory.

Symbol	Name	SI unit	Dimension	Range
$P$	Pressureoutput Gas pressure	Pa	M L⁻¹ T⁻²	0 – 10000000
$V$	Volume Gas volume	m³	L³	0.0001 – 0.1
$T$	Temperature Absolute temperature	K	Θ	100 – 1000
$a$	Attraction parameter Strength of intermolecular attraction	Pa·m⁶/mol²	M L⁵ T⁻² N⁻²	0 – 1
$b$	Excluded volume Volume excluded per mole by molecular size	m³/mol	L³ N⁻¹	0.000001 – 0.0001

Unit systems

SI:: a in $Pa\cdot m^{6}/mol^{2}$ , b in $m^{3}/mol$
natural:: reduced variables P_r, V_r, T_r
CGS:: a in $erg\cdot cm^{3}/mol^{2}$

Where it holds

Good qualitatively for real gases near and below the critical point; quantitatively imperfect (predicts wrong critical compressibility

Z_c = 3/8\,\mathrm{vs}

measured

\sim 0.29)

.

Discovery

Johannes Diderik van der Waals · 1873

Van der Waals' 1873 doctoral thesis 'On the Continuity of the Gaseous and Liquid States' added molecular size and attraction to the ideal gas; it won him the 1910 Nobel Prize.

Try this

How do you turn an ideal gas into one that can become a liquid?

The ideal gas law has no liquid phase — ever. Add two tiny corrections for molecular size and attraction, and out pops a critical point, boiling, and condensation.

Research status: stable

Common misconceptions

The S-loop is not physical equilibrium — the real isotherm is the flat tie-line from the Maxwell equal-area construction; the loop's negative-slope region is mechanically unstable.

Derivation

Start from

PV = nRT

.

Replace V by the free volume

(V - nb)

since molecules of finite size exclude volume b each.

Reduce the measured pressure by an internal-pressure term a

n^{2}/V^{2}

representing mutual attraction (pulling molecules inward).

The result is

(P + an^{2}/V^{2})(V - nb) = nRT

.

Setting

(\partial P/\partial V)_T = (\partial ^{2}P/\partial V^{2})_T = 0

yields the critical point

T_c = 8

a/27Rb.

Limiting cases

a = 0

,

b = 0

⟶ Recovers the ideal gas law

PV = nRT

T > T_c⟶ Single smooth phase; no liquid-gas transition possible

T < T_c⟶ S-shaped loop signals liquid-gas coexistence (fixed by Maxwell construction)

Ideal Gas Law→Clausius-Clapeyron Equation→Joule-Thomson Effect→Gibbs Free Energy→