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A knowledge of the equilibrium conditions for an isolated system permits us to deduce similar conditions for
other situations of physical interest. For instance, much experimental work is performed under conditions of
constant temperature. Thus, it would be interesting to investigate the equilibrium of some system,
, in thermal
contact with a heat reservoir,
, that is held at the constant absolute temperature
.
The combined system,
, consisting of the system
and the heat reservoir
, is an isolated system of the type
discussed in the previous section. The entropy,
, of
therefore satisfies condition (9.1). In other
words, in any spontaneous process,
|
(9.6) |
However, this condition can also be expressed in terms of quantities that only refer to the system
. In fact,
|
(9.7) |
where
is the entropy change of
, and
that of the reservoir. But, if
absorbs heat
from the reservoir
during the process then
absorbs heat
, and suffers a
corresponding entropy change
|
(9.8) |
(because it remains in equilibrium at the constant temperature
). Furthermore, the first law of thermodynamics implies that
|
(9.9) |
where
is the change in internal energy of
, and
is the work done by
. Thus,
Equation (9.7) can be written
|
(9.10) |
or
|
(9.11) |
where use has been made of the fact that
is a constant. Here,
|
(9.12) |
reduces to the Helmholtz free energy,
, of system
, when the latter has a temperature,
, equal to that of the
heat reservoir,
. Of course, in the general case in which
is not in equilibrium with
, the former system's temperature is
not necessarily equal to
.
The fundamental condition (9.6) can be combined with Equation (9.11) to give
|
(9.13) |
(assuming that
is positive, as is normally the case). This relation implies that the maximum work that
can be done by a system in contact with a heat reservoir is
. (Incidentally, this is the reason for the name
``free energy'' given to
.) The maximum work corresponds to the equality sign in the preceding equation, and
is obtained when the process used is quasi-static (so that
is always in equilibrium with
, and
).
Equation (9.13) should be compared to the rather different relation (9.3) that pertains to an isolated system.
If the external parameters of system
are held constant then
, and Equation (9.13) yields the condition
|
(9.14) |
This equation is analogous to Equation (9.1) for an isolated system. It implies that if a system is in thermal contact with a heat reservoir then
its Helmholtz free energy tends to decrease. Thus, we arrive at the following statement:
If a system, whose external parameters are fixed, is in thermal contact with a heat reservoir then the stable equilibrium state
is such that
The preceding statement can again be phrased in more explicit statistical terms. Suppose that the external parameters of
are fixed, so that
. Furthermore, let
be described by some parameter
. The thermodynamic functions of
--namely,
and
--have the
definite values
and
, respectively, when
has a given value
. If the parameter changes to some other value,
,
then these functions change by the corresponding amounts
and
.
The entropy of the heat reservoir,
, also changes because it absorbs heat. The corresponding change in the total entropy of
is given by
Equation (9.11) (with
):
|
(9.15) |
But, in an equilibrium state, the probability,
, that the parameter lies between
and
is proportional to the number of states,
, accessible to the isolated total system,
, when the parameter lies in this range. Thus, by analogy with Equation (9.5),
we have
|
(9.16) |
However, from Equation (9.15),
|
(9.17) |
Now, because
is just some arbitrary constant, the corresponding constant terms can be absorbed into the constant of
proportionality in Equation (9.16), which then becomes
|
(9.18) |
This equation shows explicitly that the most probable state is one in which
attains a minimum value, and also allows us to determine the relative probability of fluctuations about this state.
Next: Equilibrium of Constant-Temperature Constant-Pressure
Up: Multi-Phase Systems
Previous: Equilibrium of Isolated System
Richard Fitzpatrick
2016-01-25