Page 164 - Advanced thermodynamics for engineers

150 CHAPTER 7 GENERAL THERMODYNAMIC RELATIONSHIPS

The third Tds relationship is derived by assuming that
s ¼ sðp; vÞ: (7.42)

vs vs
ds ¼ dp þ dv: (7.43)
vp vv
v p
Now, by definition,

vs vs vv
c p ¼ T ¼ T ;
vT vv vT
p p p
giving

vs 1 vT
¼ c p :
vv T vv
p p
A similar expression can be derived for c v , and this enables the equation
vT vT

Tds ¼ c v dp þ c p dv (7.44)
vp v vv p
to be obtained. This is the third Tds relationship.

7.3.1 VARIATION OF INTERNAL ENERGY AND ENTHALPY
It is now possible to investigate the variation of internal energy and enthalpy with independent
properties for gases obeying various gas laws.
The internal energy of a substance can be expressed as

u ¼ uðT; vÞ: (7.45)
Hence, the change in internal energy is
vu vu

du ¼ dT þ dv (7.46)
vT v vv T
Now

vu

c v ¼
vT
v
and from the Second Law,
du ¼ Tds pdv: (7.47)
Substituting for Tds in Eqn (7.47) from first Tds relation (Eqn (7.38)) gives,

vp
du ¼ c v dT þ T dv pdv (7.48)
vT
v