1549380323-Statistical Mechanics Theory and Molecular Simulation

176 Canonical ensemble

model, we cannot expect it to be applicable over a wide range ofP,V, andTvalues.
Nevertheless, if we plot the isotherms of the van der Waals equation, something quite
interesting emerges (see Fig. 4.8). For temperatures larger thana certain tempera-
tureTc, the isotherms resemble those of an ideal gas. AtTc, however, we see that the
isotherm is flat in a small region. That is, at this point, the “flatness”of the isotherm
is characterized by the conditions

∂P

∂V

= 0,

∂^2 P

∂V^2

= 0. (4.7.36)

The first and second conditions imply that the slope of the isotherm and its curva-
ture, respectively, vanish at the point of “flatness.” For temperatures belowTc, the

V

P

T > Tc

T > Tc

T = Tc

T < Tc

Critical point

Volume discontinuity

0

Fig. 4.8Isotherms of the van der Waals equation of state for four different temperatures.

isotherms take on an unphysical character: They all possess a region in whichPand
V simultaneously increase. As already noted, considering the many approximations
made, regions of unphysical behavior should come as no surprise. Aphysically realis-
tic isotherm forT < Tcshould have the unphysical region replaced by the thin solid
line in Fig. 4.8. From the placement of this thin line, we see that the isotherm exhibits
a discontinuous change in the volume for a very small change in pressure, signifying a
gas–liquid phase transition. The isotherm atT=Tcis a kind of “boundary” between
isotherms along whichVis continuous (T > Tc) and those that exhibit discontinuous
volume changes (T < Tc). For this reason, theT=Tcisotherm is called thecritical
isotherm. The point at which the isotherm is flat is known as thecritical point. On a
phase diagram, this would be the point at which the gas–liquid coexistence curve ter-
minates. The conditions in eqn. (4.7.36) define the temperature, volume, and pressure