Page 81 - Modern physical chemistry

70 The First Law for Energy

At constant volume, we have
dqv -C [4.33]
dT - v·
Here dqv is the infinitesimal heat needed to raise the temperature infinitesimal amount
dT with the work w equal to zero.
In common parlance, Cv is called the heat capacity at constant volume of the system.
Combining (4.33) with (4.21) leads to

dqv
dEy =-dT=Cv dT [4.34]
dT
and
v =(~;l· [4.35]
C

For a change in temperature from Tl to T2 , we obtain

[4.36]

When the system is an ideal gas, a change in volume at a given temperature does not
alter the internal energy; then
( aE) =0 [4.37]
av T
and
dE=Cv dT [4.38]

whether the volume V is fixed or not.
At constant pressure, we have
dqp C [4.39]
dT = p.

Here dqp is the infinitesimal heat needed to raise the temperature by dT when only work
of expansion is done and the pressure P is kept constant. By convention, Cp is called the
heat capacity at constant pressure of the system.
Combining (4.39) with (4.26) yields

dqp
dllp =-dT=Cp dT [4.40]
dT
and
C =[aH) . [4.41 ]
p aT
p
Integrating (4.40) from Tl to T2 leads to

[4.42]

When the system is an ideal gas, condition (4.30) applies, enthalpy H does not vary
with pressure P when the temperature T is fixed, and H varies only with T. Then we have

dll=Cp dT [4.43]