6.4.2 Temperature of Clausius’ Inequality
Substituting inequality Eq. (88B) into (84B) yields
dS ! d E S ¼
Z
A
ÀJ Q
T
dA
2
4
3
5 dt
ð89AÞ
If one defines an effective mean temperature T mean in accordance with
T mean
R
A J Q dA
R
A
J Q
T dA
ð90Þ
inequality Eq. (89A) becomes (note
R
A
ÀJ Q dA
!
dt ¼ dQ)
dS !
R
A ÀJ Q dA
Â
Ã
dt
T mean
¼
dQ
T mean
ð89BÞ
S B
ð Þ À SðAÞ !
Z B
A
dQ
T mean
!
Quasistatic
ð89CÞ
Inequality Eq. (89C) in terms of an effective mean temperature defined by Eq. (90)
supersedes the ill-defined Eq. (73).
6.4.3 Internal Reversibility as the Condition for Defining
Entropy
In the case of vanishing entropy production, d G S ¼ 0 in (84B), the system entropy
change dS is balanced solely with the system entropy exchange, d E S. That is, the
condition for equality
dS ¼
Z
A
ÀJ Q
T
dA
2
4
3
5 dt
ð91Þ
depends on the interior condition of the system alone—independent of any other
specifics concerning how the system interacts with its surroundings. We call this
interior condition of vanishing entropy production within the system by the name
internal reversibility (see Sect. 6.5 for a precise definition).
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6 Reversible Processes Versus Quasi-static Processes …