Page 200 - Advanced thermodynamics for engineers

186 CHAPTER 9 THERMODYNAMIC PROPERTIES OF IDEAL GASES

Gibbs energy
The Gibbs energy can be evaluated from Eqn (9.40) or Eqn (9.26)
g m ðTÞ¼ h m ðTÞ Ts m ðTÞ (9.26)
The values obtained from this approach, using Eqns (9.31)–(9.42) are given in Table 9.4 for
oxygen, nitrogen, hydrogen, water, carbon monoxide, carbon dioxide, nitric oxide and methane. The
tables have been evaluated up to 3500 K, slightly beyond the range stated in relation to Table 9.3, but
the error is not significant. This enables more combustion problems to be solved using this data. Other
commonly used tables are those of JANAF (1971).

9.3.1 TABLES OF MEAN SPECIFIC HEAT
Sometimes data are given in terms of mean specific heat rather than the actual specific heat at a
particular temperature. This approach will now be described.
Consider the change of enthalpy, h, between temperatures T 1 and T 2 . This is
T
Z 2
h 2 h 1 ¼ c p ðTÞdT ¼ hðT 2 Þ hðT 1 Þ (9.43)
T 1
Now, this can be written
c p ðT 2 T 1 Þ¼ hðT 2 Þ hðT 1 Þ; (9.44)

where c p is the mean specific heat between T 1 and T 2 . Normally c p is given at a particular temperature,
T, and it is then defined as the mean specific heat between the temperature T and a reference
temperature T ref .
hðTÞ hðT ref Þ

i:e: c p ¼ (9.45)
T
T T ref
It is also possible to write the mean specific heat as a polynomial function of temperature, in which
case
2
c p ¼ a þ bT þ cT þ . (9.46)
T
where a, b and c are tabulated coefficients.
If it is required to calculate the enthalpy difference between two temperatures T 1 and T 2 then

h 2 h 1 ¼ c p ðT 2 T ref Þ c p ðT 1 T ref Þ (9.47)
T 2 T 1
Having calculated the change in enthalpy, h 2 h 1 , then the change in internal energy, u 2 u 1 , may
be evaluated as

u 2 u 1 ¼ h 2 RT 2 ðh 1 RT 1 Þ¼ h 2 h 1 RðT 2 T 1 Þ (9.48a)
or, in molar terms by

u m;2 u m;2 ¼ h m;2 <T 2 h m;1 <T 1 ¼ h m;2 h m;1 <ðT 2 T 1 Þ (9.48b)