Up: Heat and Work
Previous: Quasi-Static Processes
In our investigation of heat and work, we have come across
various infinitesimal objects, such as
It is instructive to examine these infinitesimals more closely.
Exact and Inexact Differentials
Consider the purely mathematical problem where
is some general
function of two
. Consider the change in
going from the point
plane to the neighboring point
). This is given by
which can also be written
simply the infinitesimal difference between two adjacent values of the function
This type of infinitesimal quantity is termed an exact differential to
distinguish it from another type to be discussed presently.
If we move in the
plane from an initial point
to a final point
then the corresponding change in
Note that because the difference on the left-hand side depends only on the initial
and final points, the integral on the right-hand side can only depend on
these points as well.
In other words, the value of the
integral is independent of the path taken in going from the initial to the final
point. This is the distinguishing feature of an exact differential.
Consider an integral taken around a closed circuit in the
plane. In this
case, the initial and final points correspond to the same point, so the difference
is clearly zero. It follows that the integral of an exact differential
over a closed circuit is always zero:
Of course, not every infinitesimal quantity is an exact differential. Consider
the infinitesimal object
general functions of
. It is easy to test whether or not
an infinitesimal quantity is an exact differential. Consider the expression
It is clear that because
(as is assumed to be the case), then
cannot be an exact differential, and
is instead termed an inexact differential.
The special symbol
is used to denote an inexact differential.
Consider the integral of
over some path in the
plane. In general, it is not true
is independent of the path taken between the initial and final points.
This is the distinguishing feature of an inexact differential. In particular,
the integral of an inexact differential around a closed circuit is not necessarily
Consider, for the moment, the solution of
which reduces to the ordinary differential equation
Because the right-hand side is a known function of
, the previous
a definite direction (i.e., gradient) at each point in the
solution simply consists of drawing a system of curves in the
plane such that, at any point, the tangent to the curve is as specified
by Equation (4.32). This defines a set of curves that can be written
is a labeling parameter.
It follows that, on a particular curve,
The elimination of
between Equations (4.32) and (4.33) yields
is function of
. The previous equation could equally
well be written
Inserting Equation (4.35) into Equation (4.26) gives
Thus, dividing the inexact differential
yields the exact
. A factor
that possesses this property is
termed an integrating factor. Because the previous analysis is quite general,
it is clear that an inexact differential involving two independent variables always
admits of an integrating factor. Note, however,
this is not generally the case for inexact
differentials involving more than two variables.
After this mathematical excursion, let us return to a physical situation of interest.
The macrostate of a macroscopic system can be specified by the values of the
external parameters (e.g., the volume), and the mean energy,
. This, in
turn, fixes other parameters, such as the mean pressure,
we can specify the external parameters and the mean pressure, which fixes the
mean energy. Quantities such as
differences between well-defined quantities: that is,
they are exact differentials.
is just the difference between the
mean energy of the system in the final macrostate
, in the limit where these two states are nearly the same.
It follows that if the system is taken from an initial macrostate
then the mean energy change is given by
However, because the mean energy is just a function of the macrostate under consideration,
depend only on the initial and final
Thus, the integral
depends only on the initial and final
states, and not on the particular process used to get between them.
Consider, now, the infinitesimal work done by the system in going from some
to some neighbouring final macrostate
. In general,
is not the difference between two
numbers referring to the properties of two neighboring macrostates. Instead,
it is merely an infinitesimal quantity characteristic of the process of going
. In other words, the work
is in general
an inexact differential. The total work done by the system in going from any
to some other macrostate
can be written as
where the integral represents the sum of the infinitesimal amounts of work
performed at each stage of the process. In general, the value of
the integral does
depend on the particular process used in going from macrostate
Recall that, in going from macrostate
, the change in energy,
depend on the process used, whereas the work done,
, in general, does.
Thus, it follows from
the first law of thermodynamics, Equation (4.10),
that the heat absorbed,
, in general, also depends on the
process used. It follows that
is an inexact differential. However, by analogy with the mathematical example
discussed previously, there must exist some integrating factor,
converts the inexact differential
into an exact differential.
It will be interesting to discover the physical quantities that correspond to
. (See Section 5.5.)
Suppose that the system is thermally insulated, so that
. In this case, the
first law of thermodynamics implies that
Thus, in this special case, the work done depends only on the energy
difference between the initial and final states, and is independent of the process.
In fact, when Clausius first formulated the first law of thermodynamics, in 1850,
If a thermally isolated system is brought from some initial to some final state
then the work done by the system is independent of the process used.
If the external parameters of the system are kept fixed, so that no work is done,
, Equation (4.11) reduces
becomes an exact differential. The amount of heat,
in going from one macrostate to another
depends only on the mean energy difference between them, and
is independent of the process used to effect the change. In this situation,
heat is a conserved quantity, and acts very much like the invisible indestructible
fluid of Lavoisier's calorific theory.
Up: Heat and Work
Previous: Quasi-Static Processes