Concise Physical Chemistry

c02 JWBS043-Rogers September 13, 2010 11:23 Printer Name: Yet to Come

24 REAL GASES: EMPIRICAL EQUATIONS

Vm (dm^3 )

0 1 2

p (bar)

73.8

FIGURE 2.4 Three isotherms of a van der Waals gas. The top isotherm is aboveTc,the

middle isotherm is at the critical temperatureTcand the bottom curve is belowTc. The critical

pressure is 73.8 bar.

The apparent paradox that there are only two degrees of freedom in the equation of

state of a pure substance which may have an infinite number of terms in an equation

of state is removed by noting that each term contains only the pressure,p, and an

adjustable parameter (not a variable) that is a function of the temperature. Hence the

only true variables in the equation arepandT.

2.4 THE CRITICAL TEMPERATURE

At lower temperatures, gas molecules occupy a smaller volume and move more

slowly than they do at higher temperatures. Attractive forces among molecules or

atoms become more important at lower temperatures. Ultimately, they become so

strong that the gas liquefies. Thus, a useful physical picture of the liquid state is that

liquefaction is thelimiting behaviorof an extremely nonideal gas and it results from

large interparticle attractive forces.

As the temperature of a real gas is lowered, its deviation from hyperbolic (Boyle’s

law) behavior becomes more pronounced until thep–Vcurve has become so distorted

that it goes through a horizontal inflection point. The temperature at which this occurs

is called thecritical temperature,Tc.

The curves in Fig. 2.4 arise from plotting the van der Waals equation at each of

three temperatures above, at, and belowTc. The locus ofp–Vpoints at the same

temperature is called anisotherm. The pressure, volume, and temperature at the

inflection of the critical isotherm define a point called thecritical point, which is

unique to each real gas. The coordinates of the critical point (thecritical constants)

are the critical pressurepc, critical volumeVc, and critical temperatureTc. Critical

constants vary widely. For example, the critical temperature of helium is 5.19 K while

that for CO 2 is 304 K.

Below the critical temperature, the system may be in the liquid or gaseous state,

or it may exist as an equilibrium between liquid and vapor. The term “vapor” means