6.6 Internal Reversibility and the C p À C V Question
in Sect. 2.3
In the caloric theory, the doctrine of latent and sensible heats of Sect. 2.3 often
gives the impression that heat, Q, is treated as a function, e.g., Q(V, T)
dQ ¼ C
V
ð Þ
T
V; T
ð
ÞdV þ C
T
ð Þ
V V; T
ð
ÞdT
ð12Þ
or, as a function of p and T,
dQ ¼ C
p
ð Þ
T p; T
ð
Þdp þ C
T
ð Þ
p
p; T
ð
ÞdT
ð13Þ
That impression is incorrect. The meaning of dQ in Eqs. (12) and (13), in fact,
should be interpreted strictly as TdS in accordance with Eq. (92) and correspondingly, Eqs. (12) and (13) are valid only if they are applied to processes that meet the
internal reversibility condition. That is, Eqs. (12) and (12) are understood exclusively as
TdS ¼ T
@S
@V
T
dV þ T
@S
@T
V
dT ¼ T
@S
@V
T
dV þ C V dT
ð94Þ
TdS ¼ T
@S
@p
T
dp þ T
@S
@T
p
dT ¼ T
@S
@p
T
dp þ C p dT
ð95Þ
The latent heat with respect to volume and the latent heat with respect to
pressure are correspondingly
C
V
ð Þ
T
V; T
ð
Þ¼ T
@S
@V
T
ð94AÞ
C
p
ð Þ
T p; T
ð
Þ ¼ T
@S
@p
T
ð95AÞ
Similarly, Eq. (15) used by Poisson in his handling of adiabatic heating in Sect. 2.4
dQ ¼
dQ
@p
V
dp þ
dQ
@V
p
dV
6.6 Internal Reversibility …
151