Exact and inexact differentials

Consider the purely mathematical problem where is some general
function of two
independent variables and . Consider the change in in
going from the point , in the - plane to the neighbouring point
(, ). This is given by

(135) |

where and . Clearly, is simply the infinitesimal difference between two adjacent values of the function . This type of infinitesimal quantity is termed an

(137) |

(138) |

Of course, not every infinitesimal quantity is an exact differential. Consider
the infinitesimal object

(140) |

(141) |

(142) |

(143) |

Consider, for the moment, the solution of

(144) |

Since the right-hand side is a known function of and , the above equation defines a definite direction (

The elimination of between Eqs. (145) and (146) yields

(147) |

Inserting Eq. (148) into Eq. (139) gives

(149) |

(150) |

After this mathematical excursion, let us return to 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 . Alternatively,
we can specify the external parameters and the mean pressure, which fixes the
mean energy. Quantities such as and are infinitesimal
differences between well-defined quantities: *i.e.*,
they are exact differentials.
For example,
is just the difference between the
mean energy of the system in the final macrostate and the
initial macrostate , in the limit where these two states are nearly the same.
It follows that if the system is taken from an initial macrostate to any
final macrostate the mean energy change is given by

(151) |

Consider, now, the infinitesimal work done by the system in going from some
initial macrostate to some neighbouring final macrostate . In general,
is *not* the difference between two
numbers referring to the properties of two neighbouring macrostates. Instead,
it is merely an infinitesimal quantity characteristic of the process of going
from state to state . In other words, the work
is in general
an inexact differential. The total work done by the system in going from any
macrostate to some other macrostate can be written as

(152) |

Recall that in going from macrostate to macrostate the change
*does not*
depend on the process used whereas the work , in general, *does*.
Thus, it follows from
the first law of thermodynamics, Eq. (123),
that the heat , in general, also depends on the
process used. It follows that

(153) |

(154) |

Suppose that the system is thermally insulated, so that . In this case, the
first law of thermodynamics implies that

(155) |

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,
then
, Eq. (124) reduces
to

(156) |