Modern Thermodynamics

- Chapter 1

21

limiting volume of one mole of the gas, as p → ∞. The constant b is sometimes called the "excluded

volume". The effect of intermolecular forces, van der Waals noted, is to decrease the pressure as

illustrated in Fig. 1.4. The above "volume-corrected" equation is further modified to:

p =

NRT

V − bN

−δp

Next, van der Waals related the factor δp to the number density (N/V) using kinetic theory of gases that

showed how molecular collisions with container walls cause pressure. Pressure depends on the number of

molecules that collide with the walls per unit area, per unit time; it is therefore proportional to the number

density (N/V) (as can be seen from the ideal gas equation). In addition, each molecule that is close to a

container wall and moving towards it experiences the retarding attractive forces of molecules behind it

(see Fig. 1.4); this force would also be proportional to number density (N/V); hence δp should be

proportional to two factors of (N/V) so that one may write: δp = a

(N/V)2

in which the constant "a" is a

measure of the intermolecular forces. The expression for pressure that van der Waals proposed is:

p =

NRT

V − bN

− a

N2

V2

or as it is usually written:

(p + a

N2

V2

) (V − Nb) = NRT

(1.5.1)

This turns out to be an equation of state for both the liquid and the gas phase. van der Waals'

insight revealed that the two phases, which were considered distinct, can, in fact, be described by a single

equation. Let us see how.

For a given T, a p-V curve, called the p-V isotherm, can be plotted. Such isotherms for the van

der Waals equation (1.5.1) are shown in Fig 1.5. They show an important feature: critical temperature

Tc studied by Thomas Andrews. If the temperature T is greater than Tc the p-V curve is always single

valued, much like the ideal gas isotherm, indicating that there is no transition to the liquid state. But for

lower temperatures, TTC, the isotherm has a maximum and a minimum. There are two extrema because

van der Waals equation cubic in V. This region represents a state in which the liquid and the gas phases

coexist in thermal equilibrium. On the p-V curve shown in Fig. 5, at the point A, the gas begins to

condense into a liquid; the conversion of gas to liquid continues until the point C at which all the gas has

been converted to liquid. Between A and C, the actual state of the gas does not follow the path AA'BB'C,

along the p-V