Page 194 - Advanced thermodynamics for engineers

180 CHAPTER 9 THERMODYNAMIC PROPERTIES OF IDEAL GASES

(e.g. v m ,u m ,h m ,g m , etc.). The molar properties are more useful for combustion calculations, and
these will be considered here.
The molar internal energy and molar enthalpy can be evaluated by integrating Eqn (9.7), giving
Z T
u m ¼ c v;m dT þ u 0;m
T 0
(9.9)
Z T
h m ¼ c p;m dT þ h 0;m
T 0
where u 0,m and h 0,m are the values of u m and h m at the datum temperature T 0 .
Now, by definition,

h m ¼ u m þ pv m ; and for an ideal gas pv m ¼<T; thus h m ¼ u m þ<T: (9.10)
Hence, at T ¼ 0, u m ¼ h m , i.e. u 0,m ¼ h 0,m . (A similar relationship exists for the specific properties,
u and h.)
To be able to evaluate the internal energy or enthalpy of an ideal gas using Eqn (9.10),it is
necessary to know the variation of specific heat with temperature. It is possible to derive such a
function from empirical data by curve-fitting techniques, if such data is available. In some regions of
the state diagram it is necessary to use quantum mechanics to evaluate the data, but this is beyond the
scope of this text. It will be assumed here that the values of c v are known in the form
2 3
c v;m ¼ a þ bT þ cT þ dT (9.11)
where a, b, c and d have been evaluated from experimental data. Since c p;m c v;m ¼< then an
expression for c p;m can be easily obtained. Hence it is possible to find the values of internal energy and
enthalpy at any temperature if the values at T 0 can be evaluated. This problem is not a major one if the
composition of the gas remains the same because the datum levels will cancel out. However, if the
composition varies during the process it is necessary to know the individual datum values. They can be
measured by calorimetric or spectrographic techniques. The ‘thermal’ part of the internal energy and
enthalpy, i.e. that which is a function of temperature, will be denoted
Z T
u m ðTÞ¼ c v;m dT
T 0
(9.12)
Z T
h m ðTÞ¼ c p;m dT
T 0
and then Eqns (9.9) become

u m ¼ u m T þ u 0;m
(9.13)

h m ¼ h m T þ h 0;m
assuming that the base temperature is absolute zero.