Page 121 - Mechanical Engineers' Handbook (Volume 4)

110 Thermodynamics Fundamentals

PV nMRT
where the product MR is the universal gas constant
J
R MR 8.314
mol K
is
The equivalent molar mass of a mixture of ideal gases with individual molar masses M i
M
1
i
n nM i
where n n . The molar mass of air, as a mixture of nitrogen, oxygen, and traces of
i
other gases, is 28.966 g/mol (or 28.966 kg/kmol). A more useful model of the air gas
mixture relies on only nitrogen and oxygen as constituents, in the proportion 3.76 moles of
nitrogen to every mole of oxygen; this simple model is used frequently in the field of
combustion. 1
At the opposite end of the spectrum is the incompressible substance model. At suffi-
ciently high pressures and low temperatures in Fig. 3, solids and liquids behave so that their
density or specific volume is practically constant. In this limit the (P, v, T) surface is ade-
quately represented by the equation
v v (constant)
The formulas for calculating changes in internal energy, enthalpy, and entropy become (see
the end of the section on relations among thermodynamic properties)
du cdT
dh cdT v dP
c
ds dT
T
where c is the sole specific heat of the incompressible substance,
c c c P
v
The specific heat c is a function of temperature only. In a sufficiently narrow temperature
range where c can be regarded as constant, the finite changes in internal energy, enthalpy,
and entropy relative to a reference state denoted by ( ) are
0
u u c (T T )
0
0
h h c (T T ) v (P P ) (where h u P v)
0
0
0
0
0
0
T
s s c ln
0
T 0
v
The incompressible substance model rests on two empirical constants, c and .
As shown in Fig. 3, the domains in which the pure substance behaves either as an ideal
gas or as an incompressible substance intersect over regions where the substance exists as a
mixture of two phases, liquid and vapor, solid and liquid, or solid and vapor. The two-phase
regions themselves intersect along the triple point line labeled TP-TP on the middle sketch
of Fig. 3. In engineering cycle calculations, the projections of the (P, v, T) surface on the
P-v plane or, through the relations reviewed earlier, on the T-s plane are useful. The termi-
nology associated with two-phase equilibrium states is defined on the P-v diagram of Fig.
4a, where we imagine the isothermal compression of a unit mass of substance (a closed