Thermodynamics and Chemistry

CHAPTER 2 SYSTEMS AND THEIR PROPERTIES

2.2 PHASES ANDPHYSICALSTATES OFMATTER 34

state may be expressed as a relation amongT,p, and the molar volumeVmDV=n. The
equation of state for a pure ideal gas may be writtenpDRT=Vm.
TheRedlich–Kwong equationis a two-parameter equation of state frequently used to
describe, to good accuracy, the behavior of a pure gas at a pressure where the ideal gas
equation fails:

pD

RT

Vmb

a

Vm.VmCb/T1=2

(2.2.1)

In this equation,aandbare constants that are independent of temperature and depend on
the substance.
The next section describes features ofvirialequations, an important class of equations
of state for real (nonideal) gases.

2.2.5 Virial equations of state for pure gases

In later chapters of this book there will be occasion to apply thermodynamic derivations to
virial equations of state of a pure gas or gas mixture. These formulas accurately describe
the gas at low and moderate pressures using empirically determined, temperature-dependent
parameters. The equations may be derived from statistical mechanics, so they have a theo-
retical as well as empirical foundation. There are two forms of virial equations for a pure
gas: one a series in powers of1=Vm:

pVmDRT



1 C

B

Vm

C

C

Vm^2

C



(2.2.2)

and the other a series in powers ofp:

pVmDRT

1 CBppCCpp^2 C

(2.2.3)

The parametersB,C,: : :are called thesecond,third,: : :virial coefficients, and the pa-
rametersBp,Cp,: : :are a set of pressure virial coefficients. Their values depend on the
substance and are functions of temperature. (Thefirstvirial coefficient in both power se-
ries is 1 , becausepVmmust approachRT as1=Vmorpapproach zero at constantT.)
Coefficients beyond the third virial coefficient are small and rarely evaluated.
The values of the virial coefficients for a gas at a given temperature can be determined
from the dependence ofponVmat this temperature. The value of the second virial coef-
ficientBdepends on pairwise interactions between the atoms or molecules of the gas, and
in some cases can be calculated to good accuracy from statistical mechanics theory and a
realistic intermolecular potential function.
To find the relation between the virial coefficients of Eq.2.2.2and the parametersBp,
Cp,: : :in Eq.2.2.3, we solve Eq.2.2.2forpin terms ofVm

pDRT



1

Vm

C

B

Vm^2

C



(2.2.4)

and substitute in the right side of Eq.2.2.3:

pVmDRT

"

1 CBpRT



1

Vm

C

B

Vm^2

C



CCp.RT /^2



1

Vm

C

B

Vm^2

C

 2

C

(2.2.5)