APPLICATION OF THE VIRIAL EQUATION

The equation of state pv=RT(3.12)
The Constant-Volnme Process
The equations which apply to a mechanically reversible constant-volume process
were developed in Sec. 2.10. No simplification results for an ideal gas. Thus for
one mole:
The Constant-Pressnre (Isobaric) Process
The equations which apply to a mechanically reversible, constant-pressure nont10w
process were developed in Sec. 2.10. For one mole,
The Adiabatic Process
PI
Q= W=RTlnP2
(3.19)
An adiabatic process is one for which there is no heat transfer between the system
and its surroundings; that is, dQ = O. Therefore, application of the first law to
one mole of an ideal gas in mechanically reversible nonflow processes gives
The Pnlytropic Process
This is the general case for which no specific conditions other than mechanical
reversibility are imposed. Thus only the general equations applying to an ideal
gas in a nonflow process apply. For one mole, these are:
2. An internal energy that is a function of temperature only, and as a result of
Eq. (2.20) a heat capacity Cy which is also a function of temperature only.
The ideal gas is a model fluid that is useful because it is described by simple
equations frequently applicable as good approximations for actual gases. In
engineering calculations, gases at pressures up to a few bars may often be
considered ideal. The remainder of this section is therefore devoted to the
development of thermodynamic relationships for ideal gases.
The two forms of the virial expansion given by Eqs. (3.10) and (3.11) are infinite
series. For engineering purposes their use is practical only where convergence is
very rapid, that is, where no more than two or three terms are required to yield
reasonably close approximations to the values of the series. This is realized for
gases and vapors at low to moderate pressures.
Figure 3.9 shows a compressibility·factor graph for methane. Values of the
compressibility factor Z (as calculated from PVT data for methane by the defining
equation Z = PV / RT) are plotted against pressure for various constant tem·
peratures. The resulting isotherms show graphically what the virial expansion in
P is intended to represent analytically. All isotherms originate at the value Z ~ I
for P = O. In addition the isotherms are nearly straight lines at low pressures.
Thus the tangent to an isotherm at P ~ 0 is a good approximation of the isotherm
for a finite pressure range. Differentiation of Eq. (3.10) for a given temperature
gives