Page 118 - Mechanical Engineers' Handbook (Volume 4)

6 Relations among Thermodynamic Properties 107

Tv 2
c c K
P
v

v
1
v
c P T T P
The general equations relating the derived properties (u, h, s) to measurable quantities are
du cdT T
P
Pdv
v
T
dh cdT
v
v
T
v dP
P
T
v
v
c dT P c dT
ds v dv or ds P dP
T T T T
v P
These relations also suggest the following identities:
T c
s
h
s
u
T v T v T P T T P c P
The relationships between thermodynamic properties and the analyses associated with ap-
plying the laws of thermodynamics are simplified considerably in cases where the pure
substance exhibits ideal gas behavior. As shown in Fig. 3, this behavior sets in at sufficiently
high temperatures and low pressures; in this limit, the (P, v, T) surface is fitted closely by
the simple expression
Pv
R (constant)
T
where R is the ideal gas constant of the substance of interest (Table 1). The formulas for
internal energy, enthalpy, and entropy, which concluded the preceding section, assume the
following form in the ideal-gas limit:
du cdT c c (T)
v
v
v
dh cdT c c (T) c R
P
P
v
P
c R c R c c
ds v dT dv or ds P dT dP or ds v dP P dv
T v T P P v
If the coefficients c and c are constant in the temperature domain of interest, then the
v
P
changes in specific internal energy, enthalpy, and entropy relative to a reference state ( )
0
are given by the formulas
u u c (T T )
0 v 0
h h c (T T ) (where h u RT )
P
0
0
0
0
0
T v
s s c ln R ln
0
v
T 0 v 0
T P
s s c ln R ln
P
0
T 0 P 0