Thermodynamics and Chemistry

CHAPTER 2 SYSTEMS AND THEIR PROPERTIES

2.2 PHASES ANDPHYSICALSTATES OFMATTER 35

Then we equate coefficients of equal powers of1=Vmin Eqs.2.2.2and2.2.5(since both
equations must yield the same value ofpVmfor any value of1=Vm):

BDRTBp (2.2.6)

CDBpRTBCCp.RT /^2 D.RT /^2 .Bp^2 CCp/ (2.2.7)

In the last equation, we have substituted forBfrom Eq.2.2.6.
At pressures up to at least one bar, the terms beyondBppin the pressure power series
of Eq.2.2.3are negligible; thenpVmmay be approximated byRT .1CBpp/, giving, with
the help of Eq.2.2.6, the simple approximate equation of state^3

Vm

RT

p

CB (2.2.8)

(pure gas,p 1 bar)

Thecompression factor(or compressibility factor)Zof a gas is defined by

Z

def

D

pV

nRT

D

pVm

RT

(2.2.9)

(gas)

When a gas at a particular temperature and pressure satisfies the ideal gas equation, the
value ofZis 1. The virial equations rewritten usingZare

ZD 1 C

B

Vm

C

C

Vm^2

C


(2.2.10)

ZD 1 CBppCCpp^2 C


 (2.2.11)

These equations show that the second virial coefficientBis the initial slope of the curve of
a plot ofZversus1=Vmat constantT, andBpis the initial slope ofZversuspat constant
T.
The way in whichZvaries withpat different temperatures is shown for the case of
carbon dioxide in Fig.2.3(a) on the next page.
A temperature at which the initial slope is zero is called theBoyle temperature, which
for CO 2 is 710 K. BothBandBpmust be zero at the Boyle temperature. At lower temper-
aturesBandBpare negative, and at higher temperatures they are positive—see Fig.2.3(b).
This kind of temperature dependence is typical for other gases. Experimentally, and also
according to statistical mechanical theory,BandBpfor a gas can be zero only at a single
Boyle temperature.

The fact that at any temperature other than the Boyle temperatureBis nonzero is

significant since it means that in the limit aspapproaches zero at constantTand the

gas approaches ideal-gas behavior, thedifferencebetween the actual molar volumeVm

and the ideal-gas molar volumeRT=pdoes not approach zero. Instead,VmRT=p

approaches the nonzero valueB(see Eq.2.2.8). However, theratioof the actual and

ideal molar volumes,Vm=.RT=p/, approaches unity in this limit.

Virial equations of gasmixtureswill be discussed in Sec.9.3.4.

(^3) Guggenheim (Ref. [ 71 ]) calls a gas with this equation of state aslightly imperfect gas.