Modern Thermodynamics
- Chapter 1
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 =
V bN
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
in which the constant "a" is a
measure of the intermolecular forces. The expression for pressure that van der Waals proposed is:
p =
V bN
or as it is usually written:
(p + a
) (V Nb) = NRT
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
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