Page 166 - Advanced Thermodynamics for Engineers, Second Edition

7.4 RELATIONSHIPS BETWEEN SPECIFIC HEAT CAPACITIES 153

and, if s is a continuous function of temperature and volume

vs vs
ds ¼ dT þ dv: (7.58)
vT vv
v T
If Eqn (7.58) is differentiated with respect to temperature, T, with p maintained constant, then

vs vs vs vv
¼ þ : (7.59)
vT vT vv vT
p v T p

vs vs
The definitions of the specific heat capacities are c v ¼ T and c p ¼ T .
vT vT p
Hence v

c p c v vs vv
¼ : (7.60)
T vv T vT p
From Maxwell relations,

vs vp
¼ (7.19c)
vv vT
T v
Thus

c p c v vp vv
¼ : (7.61)
T vT vT
v p
The mathematical relationship in Eqn (7.9)

vp vT vv
¼ 1 (7.62)
vT vv vp
v p T
can be rearranged to give

vp vp vv
¼ (7.63)
vT vv vT
v T p
Thus
2

vp vv
c p c v ¼ T (7.64)
vv vT
T p
Examination of Eqn (7.64) can be used to define which specific heat capacity is the larger. First, it
2

vv
should be noted that T and are both positive, and hence the sign of c p c v is controlled by the
vT
p

vp
sign of . Now, for all known substances, (vp/vv) T is negative. If it were not negative then the
vv
T
substance would be completely unstable because a positive value means that as the pressure is
increased the volume increases, and vice versa. Hence, if the pressure on such a substance were
decreased its volume would decrease until it ceased to exist.