Modern Engineering Thermodynamics

18.6 Equipartition of Energy 739
We can therefore conclude that the total energy of a system of N molecules with F degrees of freedom per
molecule is given by
U = F 2 NkT = F 2 m TRT
and that its specific internal energy is u = U/m T = FRT/2: From the definition of constant volume specific heat
introduced in Chapter 3, we have
c v =
∂u
= ∂
F
∂T v ∂T 2 RT = FR (18.27)
v 2
and, since c p − c v = R for an ideal gas,
c p = R + c v = 1 + F
R (18.28)
2
finally the specific heat ratio k becomes
k = c p
c v
= F + 2
F
(18.29)
For a molecule containing b atoms, there are F = 3b degrees of freedom. If b = 1, then F = 3andthethree
degrees of freedom are all translational. If b = 2, then F = 6 and there are three degrees of freedom in translation,
two in rotation and one in vibration. If b > 2, then F = 3b and there are three degrees of freedom in translation,
three in rotation and 3b − 6 in vibration. In the case of monatomic (single-atom) molecules like He, Ne,
Ar,Kr,orXe,thereareonlythreedegreesoffreedom(alltranslational).ThenEqs.(18.27),(18.28),and
(18.29) give
9
c v = 1:5R >=
c p = 2:5R Monatomic gases
>;
k = 1:67
For diatomic (two-atom) molecules, such as H 2 ,O 2 , CO, or NO, we have b = 2 and consequently F = 3(2) = 6.
Then, we have
9
c v = 3R >=
c p = 4R Diatomic gases
>;
k = 1:33
Similarly for triatomic gases, such as CO 2 ,H 2 O, NO 2 ,orSO 2 , we have b = 3 and F = 9. Then,
9
c v = 4:5R >=
c p = 5:5R Triatomic gases
>;
k = 1:22
A comparison of these values with the measured specific heats of some real gases, given in Table 18.3, reveals
that, for simple molecules (e.g., monatomic gases), the kinetic theory works quite well. For complex molecules,
however, kinetic theory predictions are much less accurate.
In summary, the thermodynamic properties of an ideal gas as predicted by Maxwell’s kinetic theory are
where
pV = mRT
u 2 − u 1 = c v ðT 2 − T 1 Þ
h 2 − h 1 = c p ðT 2 − T 1 Þ
s 2 − s 1 = c p ln ðT 2 /T 1 Þ− R ln ðp 2 /p 1 Þ = c v ln ðT 2 /T 1 Þ+ R ln ðv 2 /v 1 Þ
c p − c v = R
c v = FR/2
c p = ð1 + F/2ÞR
F = 3b
b = number of atoms in the molecule