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Quasistatic processes
Consider the special case of an interaction of the system with its surroundings
which is carried out so slowly that remains arbitrarily close to equilibrium
at all times. Such a process is said to be quasistatic for
the system . In practice, a quasistatic process must be carried out on
a timescale which is much longer than the relaxation time of the system.
Recall
that the relaxation time is the typical timescale for the system to return
to equilibrium after being suddenly disturbed (see Sect. 3.5).
A finite quasistatic change can be built up out of many infinitesimal changes.
The infinitesimal heat
absorbed by the system when infinitesimal work
is done on its environment
and its average energy changes by is
given by

(124) 
The special symbols
and
are introduced to emphasize that the
work done and heat absorbed are infinitesimal quantities which
do not correspond to the
difference between two works or two heats. Instead, the work done and heat absorbed
depend on the interaction process itself. Thus, it makes no sense to talk
about the work in the system before and after the process, or the difference between
these.
If the external parameters of the system have the values , then
the energy of the system in a definite microstate can be written

(125) 
Hence, if the external parameters are changed by infinitesimal amounts, so that
for in the range 1 to , then
the corresponding change in the energy of the microstate is

(126) 
The work
done by the system when it remains in this particular
state is

(127) 
where

(128) 
is termed the generalized force (conjugate to the external parameter )
in the state . Note that if is a displacement then is
an ordinary force.
Consider now an ensemble of systems. Provided that the external
parameters of the system are changed quasistatically, the generalized forces
have well defined mean values which are calculable from the
distribution of systems in the ensemble characteristic of the instantaneous
macrostate.
The macroscopic work
resulting
from an infinitesimal quasistatic change of the external parameters is obtained by
calculating the decrease in the mean energy resulting from the parameter change.
Thus,

(129) 
where

(130) 
is the mean generalized force conjugate to . The mean value is
calculated from the equilibrium distribution of systems in the ensemble corresponding to the external parameter values . The macroscopic work resulting
from a finite quasistatic change of external parameters can be obtained by
integrating Eq. (129).
The most wellknown example of quasistatic work in thermodynamics is that done by
pressure when the volume changes. For simplicity, suppose that the
volume is the only external parameter of any consequence. The work done in
changing the volume from to is simply the product of the force and
the displacement (along the line of action of the force). By definition, the mean
equilibrium pressure
of a given macrostate is equal to the normal force per unit area acting
on any surface element. Thus, the normal force acting on a surface element
is
. Suppose that the surface element is subject
to a
displacement . The work done by the element is
.
The total work done by the system is obtained by summing over all of the surface
elements. Thus,

(131) 
where

(132) 
is the infinitesimal volume change due to the displacement of the surface.
It follows from (130) that

(133) 
so the mean pressure is the generalized force conjugate to the volume .
Suppose that a quasistatic process is carried out in which the volume is changed
from to . In general, the mean pressure is a function of the volume, so
. It follows that the macroscopic work done by the
system is given by

(134) 
This quantity is just the ``area under the curve'' in a plot of versus
.
Next: Exact and inexact differentials
Up: Heat and work
Previous: The microscopic interpretation of
Richard Fitzpatrick
20060202