Page 162 - Advanced Thermodynamics for Engineers, Second Edition

7.3 Tds RELATIONSHIPS 149

was used. The most accurate way of performing such calculations is, in fact, to use the enthalpy or
internal energy values because these implicitly integrate the specific heat capacity over the range
involved. The following analyses will demonstrate how these values of internal energy and enthalpy
are obtained.
The two-property rule can be used to define entropy as the following functional relationship.

s ¼ sðv; TÞ (7.36)
Hence

vs vs
ds ¼ dT þ dv: (7.37)
vT vv
v T
Now, by definition

vs
c v ¼ T
vT
v

vp
and from the Maxwell relationships, Eqn (7.19c), vs ¼
vv vT
T v
Hence

vp
Tds ¼ c v dT þ T dv (7.38)
vT
v
This is called the first Tds relationship
The second Tds relationship can be derived in the following way. If it is assumed that entropy, s,isa
function of temperature and pressure alone, then
s ¼ sðT; pÞ: (7.39)
and
vs vs

ds ¼ dT þ dp: (7.40)
vT vp
p T
By definition,

vs
c p ¼ T
vT
p
and, from Eqn (7.19d)
vs vv

¼ :
vp vT
T p
Thus, the second Tds relationship is

vv
Tds ¼ c p dT T dp: (7.41)
vT
p