ABSTRACT
When deriving the first cubic equation of state, van der Waals used the phase behavior of a pure component as the starting point. Figure 4.1 shows schematically pressure (P) versus molar volume (V) curves for a pure component at various temperatures. At temperatures far above the critical (T1 in Figure 4.1), the pressure-volume (PV) curves exhibit a hyperbolic shape suggesting that the pressure is inversely proportional to the molar volume. This behavior is known from the ideal gas law:
P RT
V = (4.1)
where R equals the gas constant and T the absolute temperature. The molar volume of a component behaving like an ideal gas also at high pressures would asymptotically approach zero for the pressure going toward infinity. As seen from Figure 4.1, this is not the case in reality. With increasing pressure, the molar volume approaches a limiting value, which van der Waals named b. Rearrangement of Equation 4.1 to
V RT
P = (4.2)
suggests that the b-parameter should enter the equation as follows:
V RT
P b= + (4.3)
which would give the following expression for P:
P RT
V b =
−
(4.4)
At temperatures below the critical (T3 in Figure 4.1), a vapor-to-liquid phase transition may take place. Consider a component at temperature T3 initially at a low pressure and in vapor form. By decreasing the volume while maintaining a constant temperature, T3, the pressure will increase and at some stage a liquid phase may start to form showing that the dew point pressure has been reached. A further lowering of the volume will take place at a constant pressure until all the vapor has been transformed into liquid. As a liquid is almost incompressible, a further reduction of the volume will be associated with a steep increase in pressure, as is also illustrated in Figure 4.1. The fact that the substance may undergo a transition from a gaseous form with the molecules far apart to a liquid form with the molecules much closer together shows the existence of attractive forces acting between the molecules. These attractive forces are not accounted for in Equation 4.4, which is therefore incapable of describing a vapor-to-liquid phase transition.
