THM-04

van der Waals equation

(P + a/Vm²)(Vm − b) = R T.

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Equations of statevan der Waals 1873

Governing equation

\left(P+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

where

V_m: Molar volume (L/mol)
T: Temperature (K)
a: Attraction a (L²·bar/mol²)
b: Covolume b (L/mol)
P: Pressure (bar)

Lecture brief

Historical brief

Carnot (1824), Clausius entropy, the first law and later van der Waals and Gibbs potentials turned heat into a state science. The sheets compute work, efficiency and vapour pressure. This sheet (THM-04 — van der Waals equation) is the form associated with van der Waals 1873. Working symbols: $V_m$ , $T$ , $a$ , $b$ $\rightarrow$ $P$ . van der Waals corrected the ideal gas for attraction (a/Vm²) and finite molecular volume b, the first cubic equation of state.

Purpose

Purpose: compute

P

from

V_m

T

a

b

in Thermodynamics via

\left(P+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

(P + a/Vm²)(Vm − b) = R T. Use it when a real thermodynamics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V_m = 2.000\,\mathrm{L/mol}

T = 300.000\,\mathrm{K}

a = 1.370\,\mathrm{L^{2}·bar/mol^{2}}

b = 0.039\,\mathrm{L/mol}

, the governing relation

\left(P+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

yields

P = 12.375\,\mathrm{bar}

. One mole; a in L² bar / mol², b in L/mol, P in bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Molar volume V_m2 L/molTemperature T300 KAttraction a a1.37 L²·bar/mol²Covolume b b0.0387 L/mol

Outputs

Pressure P12.375 bar

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Narration of this film

One mole; a in L² bar / mol², b in L/mol, P in bar.

van der Waals corrected the ideal gas for attraction (a/Vm²) and finite molecular volume b, the first cubic equation of state.

Reading speed

Watch on YouTube