c ¼
dq
dT
:
If the process takes place at constant volume, then the heat capacity at constant
volume is
c V ¼
dq
dT
V
ð1:6Þ
and if the process takes place at constant pressure, the heat capacity is
c P ¼
dq
dT
P
:
ð1:7Þ
If the specific internal energy e is expressed in terms of υ and T, then
de ¼
∂e
∂υ
T
dυ þ
∂e
∂T
υ
dT
ð1:8Þ
and the mathematical statement of the first law can be written as
dq ¼
∂e
∂T
υ
dT þ p þ
∂e
∂υ
T
!
dυ:
For a constant volume process, we have
dq
dT
¼
∂e
∂T
υ
,
hence,
c V ¼
∂e
∂T
υ
ð1:9Þ
Defining the specific enthalpy,
h ¼ e þ pυ,
ð1:10Þ
then
dh ¼ de þ pdυ þ υdp
¼ dq þ υdp,
hence,
1.3 Some Elements of Thermodynamics
5