Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

Specific Heats cvand cp

Recall that the specific heats of an ideal gas depend on temperature only.

For a general pure substance, however, the specific heats depend on specific

volume or pressure as well as the temperature. Below we develop some gen-

eral relations to relate the specific heats of a substance to pressure, specific

volume, and temperature.

At low pressures gases behave as ideal gases, and their specific heats

essentially depend on temperature only. These specific heats are called zero

pressure,or ideal-gas, specific heats(denoted cv 0 and cp 0 ), and they are rel-

atively easier to determine. Thus it is desirable to have some general rela-

tions that enable us to calculate the specific heats at higher pressures (or

lower specific volumes) from a knowledge of cv 0 or cp 0 and the P-v-T

behavior of the substance. Such relations are obtained by applying the test

of exactness (Eq. 12–5) on Eqs. 12–38 and 12–40, which yields

(12–42)

and

(12–43)

The deviation of cpfrom cp 0 with increasing pressure, for example, is deter-

mined by integrating Eq. 12–43 from zero pressure to any pressure Palong

an isothermal path:

(12–44)

The integration on the right-hand side requires a knowledge of the P-v-T

behavior of the substance alone. The notation indicates that vshould be dif-

ferentiated twice with respect to Twhile Pis held constant. The resulting

expression should be integrated with respect to Pwhile Tis held constant.

Another desirable general relation involving specific heats is one that relates

the two specific heats cpand cv. The advantage of such a relation is obvious:

We will need to determine only one specific heat (usually cp) and calculate

the other one using that relation and the P-v-Tdata of the substance. We

start the development of such a relation by equating the two dsrelations

(Eqs. 12–38 and 12–40) and solving for dT:

Choosing TT(v,P) and differentiating, we get

Equating the coefficient of either dvor dPof the above two equations gives

the desired result:

cpcvTa (12–45)

0 v

0 T

b

P

a

0 P

0 T

b

v

dTa

0 T

0 v

b

P

dva

0 T

0 P

b

v

dP

dT

T 10 P> 0 T (^2) v
cpcv
dv
T 10 v> 0 T (^2) P
cpcv
dP
1 cpcp 02 TT (^) 
P
0
a
02 v
0 T^2
b
P
dP
a
0 cp
0 P
b
T
Ta
02 v
0 T^2
b
P
a
0 cv
0 v
b
T
Ta
02 P
0 T^2
b
v
664 | Thermodynamics