Page 58 - Modern physical chemistry

3.8 Allowing for Molecular Interactions 47

Since this is much larger than the average energy available for excitation, 1239 J, the
vibrational degrees of freedom are essentially unexcited and

Evib = 0 J.
Similarly, not enough energy is available on the average to excite any electron.
Consequently, the energy associated with the thermal agitation is

E = Eo: + Erot + Evib + Eel = 3nRT = 7437 J.

3.8 Allowing for Molecular Interactions

A gas situated so that either the volume occupied by the molecules is appreciable or
the attraction between them must be considered is no longer ideal. Thus, equation (3.24)
cannot account for the behavior at high pressures or at low temperatures. It cannot
explain condensation. So let us introduce corrections for these omissions.
Consider 1 mole of gas, for which the ideal gas equation is
PV=RT. [3.50]
Let the volume which is not available for free movement be b. The V in (3.50) should
now be replaced by V-b. When the gas is dilute enough, so that most encounters within are
binary, parameter b is about four times the total volume of the molecules. See example 3.8.
Because of attraction between the molecules, each molecule about to strike a wall is
pulled inward by a force proportional to d, the density of the gas doing the pulling. Fur-
thermore, the number of molecules striking the wall per unit area per unit time is pro-
portional to d. As a result, the pressure on the wall is lowered below the ideal value by
an amount proportional to d 2 • But d equals MIV, the molar mass divided by the volume
of 1 mole. So the pressure lowering is set equal to a/V2; and we substitute P + a/V2 for
the idealP.
The ideal gas equation is thus transformed to

[3.51 ]

the van der Waals equation. Parameters a and b chosen to fit the behavior of various
gases are listed in table 3.1. But in each case, compromises have been made. For each
substance, a and b vary considerably, especially as the condensation region is approached.
In the next section, we will see that equation (3.51) is cubic in volume V. So it has
three roots. Above a critical temperature To only one of these is real. But below Tc , all
three roots are real. The highest root applies to the gas phase; the lowest one, to the
liquid phase in equilibrium with it. Below the critical temperature, a phase of small molar
volume VI' a liquid phase, can exist in equilibrium with a phase of larger molar volume
V 3 , the gaseous phase. Above the critical temperature, no such equilibrium can exist;
there we need postulate only one phase, the gaseous. The critical temperature is the
highest temperature at which a distinct liquid phase can exist.
In studying a given substance, one might seal a sufficient amount of the substance in
a glass tube so that both liquid and gas phases are present. Then one would heat it until
the meniscus between the two phases disappears and their densities become equal. The
temperature at which this happens is the experimental critical temperature To, the cor-
responding pressure is the critical pressure Pc, the corresponding molar volume is the
critical volume Vc-