82
2 Macroscopic Thermodynamics
of the hotter heat reservoir and a minimal temperature T min that lies below the
temperature T low of the cooler heat reservoir;
(b) the range of temperatures (T min < T < T low ) over which the cycle imports heat
lies entirely below the range of temperatures (T max > T > T high ) over which it
exports heat;
(c) the value of ε COP satisfies the Second Law requirement that ε Carnot
COP > ε COP ;
(d) the comparison Carnot cycle proceeds in the CCW direction,
must be met in order that a CCW cycle other than a CCW Carnot cycle be a valid
refrigeration cycle.
2.6 The Second Law and Stability
We have noted earlier that entropy is a thermodynamic state function, so that
the entropy change S that occurs when a thermodynamic system passes from
an initial thermodynamic state i to a final thermodynamic state f by any valid
thermodynamic process depends only upon the values of the entropy for the initial
and final thermodynamic states and is independent of the path taken between them.
We also have seen (see Eqs. (2.2.9)) that an integrating factor T −1 can be introduced
for the inexact differential δQ for a reversible process so that δQ/T ≡ dS. It thus
follows that for a reversible isothermal process, the ratio Q rev /T of a reversible heat
flow to the surroundings gives the entropy change, S, for that process.
Because entropy is a state function, the net entropy change for a closed cycle
in which the system is first taken from state i to state f by a reversible isothermal
process, then returned from state f to state i by another reversible isothermal process
must be zero, i.e., ((S) rev
cycle = 0. Consider now what happens when a final system
state f is obtained from the initial state i of an isolated system via a spontaneous
irreversible process, after which the system is placed in contact with its surroundings
and returned to the initial state via a reversible process. However, as this process is
a cyclic process between thermodynamic states i and f, i.e., i → f → i, we have
((S) cycle = 0. The Second Law of Thermodynamics, in the form (2.2.10b), tells us
that the overall entropy change, ((S) cycle , is given by
((S) cycle = 0 >
f
i
δQ irr
T
+
i
f
δQ rev
T
≡
δQ
T
.
This expression, when read in reverse and considered together with ((S) rev
cycle = 0,
is often referred to as the Clausius inequality.
As the first integral vanishes because δQ irr = 0 for an isolated system (as no
energy exchange can occur between an isolated system and its surroundings), while
the second integral gives S i − S f , we thus find that
((S) i → f = S f − S i > 0 ,
2 Macroscopic Thermodynamics
of the hotter heat reservoir and a minimal temperature T min that lies below the
temperature T low of the cooler heat reservoir;
(b) the range of temperatures (T min < T < T low ) over which the cycle imports heat
lies entirely below the range of temperatures (T max > T > T high ) over which it
exports heat;
(c) the value of ε COP satisfies the Second Law requirement that ε Carnot
COP > ε COP ;
(d) the comparison Carnot cycle proceeds in the CCW direction,
must be met in order that a CCW cycle other than a CCW Carnot cycle be a valid
refrigeration cycle.
2.6 The Second Law and Stability
We have noted earlier that entropy is a thermodynamic state function, so that
the entropy change S that occurs when a thermodynamic system passes from
an initial thermodynamic state i to a final thermodynamic state f by any valid
thermodynamic process depends only upon the values of the entropy for the initial
and final thermodynamic states and is independent of the path taken between them.
We also have seen (see Eqs. (2.2.9)) that an integrating factor T −1 can be introduced
for the inexact differential δQ for a reversible process so that δQ/T ≡ dS. It thus
follows that for a reversible isothermal process, the ratio Q rev /T of a reversible heat
flow to the surroundings gives the entropy change, S, for that process.
Because entropy is a state function, the net entropy change for a closed cycle
in which the system is first taken from state i to state f by a reversible isothermal
process, then returned from state f to state i by another reversible isothermal process
must be zero, i.e., ((S) rev
cycle = 0. Consider now what happens when a final system
state f is obtained from the initial state i of an isolated system via a spontaneous
irreversible process, after which the system is placed in contact with its surroundings
and returned to the initial state via a reversible process. However, as this process is
a cyclic process between thermodynamic states i and f, i.e., i → f → i, we have
((S) cycle = 0. The Second Law of Thermodynamics, in the form (2.2.10b), tells us
that the overall entropy change, ((S) cycle , is given by
((S) cycle = 0 >
f
i
δQ irr
T
+
i
f
δQ rev
T
≡
δQ
T
.
This expression, when read in reverse and considered together with ((S) rev
cycle = 0,
is often referred to as the Clausius inequality.
As the first integral vanishes because δQ irr = 0 for an isolated system (as no
energy exchange can occur between an isolated system and its surroundings), while
the second integral gives S i − S f , we thus find that
((S) i → f = S f − S i > 0 ,