Page 195 - Advanced Thermodynamics for Engineers, Second Edition

182 CHAPTER 9 THERMODYNAMIC PROPERTIES OF IDEAL GASES

If c p;m ðT 0 Þ 0 then the expression is either infinite or indeterminate. However, before reaching
absolute zero the substance will cease to be an ideal gas and will become a solid. It can be shown by the
3
Debye T law (Atkins (1996)) and experiment that the specific heat of a solid is given by the law
3
c p ¼ aT .
Hence
3
dh m aT dT 2
¼ ¼ aT dT /0: (9.18)
lim T lim T lim
T/0 T/0 T/0
To integrate from absolute zero to T it is necessary to include the latent heats, but still it is possible
R dh m
to evaluate :
T
If s m (T) is defined as
Z T
dh m
s m ðTÞ¼ dT (9.19)
T
T 0
then entropy
p
s m ¼ s m ðTÞ <ln þ s 0;m : (9.20)
p 0
The term s 0,m is the constant of integration and this can be compared with u 0,m and h 0,m .Ifthe
composition is invariant then the value of s 0,m will cancel out when evaluating changes in s and it is
not necessary to know its value. If the composition varies then it is necessary to know, at least, the
difference between s 0 values for the substances involved. It is not possible to obtain any information
about s 0 from classical (macroscopic) thermodynamics but statistical thermodynamics shows that
‘for an isothermal process involving only phases in internal equilibrium the change in entropy
approaches zero at absolute zero’. This means that for substances that exist in crystalline or liquid
form at low temperatures it is possible to evaluate entropy changes by assuming that differences in
s 0,m are zero.
It should also be noted that the pressure term, p,in Eqn (9.20) is the partial pressure of the gas
if it is contained in a mixture, and p 0 is a datum pressure (often chosen as 1 bar or, in the past, 1 atm).

9.2.4 THE GIBBS ENERGY OF AN IDEAL GAS
The Gibbs energy will now be derived for use later when considering the equilibrium composition of
mixtures (dissociation).
By definition

g m ¼ h m Ts m (9.21)
and
(9.22)
h m ¼ h m ðTÞþ h 0;m
If the pressure ratio, p/p 0 , is denoted by the symbol p r , i.e. p r ¼ p=p 0 , then

s m ¼ s m ðTÞ < ln p r þ s 0;m (9.23)