Page 169 - Advanced thermodynamics for engineers

7.4 RELATIONSHIPS BETWEEN SPECIFIC HEAT CAPACITIES 155

0
i.e. c v ¼ f (T). Hence, from Eqn (7.67), c p ¼ c v þ R ¼ f (T) þ R ¼ f (T)if R is a function of T alone.
Hence, the difference of specific heat capacities for an ideal gas is the gas constant, R.
Also

vp 1 vv
¼ and ¼ bv:
vv kv vT
T p
Thus
1 2 Tb v
2
c p c v ¼ T ðbvÞ ¼ ; (7.68)
kv k
where
b ¼ coefficient of expansion (isobaric expansivity) and
k ¼ isothermal compressibility.

Expressions for the difference between the specific heat capacities, c p c v , have been derived
above. It is also interesting to examine the ratio of specific heat capacities, k ¼ c p /c v .

vs vs
The definitions of c p and c v are c v ¼ T and c p ¼ T , and thus the ratio of specific heat
vT vT
v p
capacities is
c p ðvs=vTÞ p
¼ : (7.69)
c v ðvs=vTÞ
v
Now, from the mathematical relationship (Eqn (7.9)) for the differentials,

vs vT vp vs vT vv
¼ 1 ¼ (7.70)
vT vp vs vT vv vs
p s T v s T
giving

vT vv vp vs
c p
¼ (7.71)
c v vv vs vT vp
s T s T
From the Maxwell relationships

vs vp
¼ ; (7.19c)
vv vT
T v
and

vs vv
¼ (7.19d)
vp vT
T P
giving

c p vT vT vp vv
¼
c v vv vp vT vT
s v s p
(7.72)

vp vT vv
¼
vv vp vT
s v p