heat into work, and vice versa, in a cyclic process and the conversion of “heat at one
temperature to heat at another temperature”. Mathematically, the theorem of the
equivalence of transformations is expressed by Eq. (11.3), though originally heat
was defined such that the integral is positive for an irreversible cycle. It was
sometime later that Clausius adopted the modern convention that negative heat
corresponds to a heat flowing out of a body.
In referring to exterior work, Clausius meant the work produced by a heat engine.
In an ideal reversible engine of the kind considered by Carnot, all the transformations are, to use Clausius’ terminology, compensated by equivalent transformations
and the equality in Eq. (11.3) applies. For example, the heat taken in from the hot
reservoir is converted to work, but in returning the piston to its starting state work is
converted to heat which is ejected to the cold reservoir. Both processes have the
same equivalence value, Q/T. In an irreversible cycle, at least one of the transformations is uncompensated, leading to the inequality as previously discussed. In
comparing internal work to external work, Clausius believed that there must be a
similarly uncompensated transformation and actively sought an equivalent inequality. In consequence, he derived Eq. (11.1).
The essential difficulty with Clausius’ work is that he did not base it on conservation of energy. The concept was not fully developed at the time and this can be
seen in his approach to irreversible, noncyclic processes. Equation (11.3) for cyclic
processes is fully compatible with energy conservation whereas Eq. (11.1) for
noncyclic processes is not. In a cyclic process, an irreversible stage can by offset
by some other process within the cycle in which heat is extracted to restore the
original state, but this cannot occur in a single, noncyclic process. Clausius overcame
this incompatibility in his Sixth Memoir by disregarding the work that is actually
done in a process, which of course is governed by energy conservation, and looking
instead at the work that might be done: “The law does not speak of the work which
the heat does, but of the work which it can do . . .”; “. . .similarly, in the first form of
the law, it is not of the resistances which the heat overcomes, but those of which it
can overcome that mention is made”. The emphasis is Clausius’ and by this
reasoning he introduced an inequality into the First Law of thermodynamics. In
the following pages the consequence of this inequality are explored for a classical
ideal gas subject to a change in the number of particles.
11.3 Extensivity, Entropy and Open Systems
Consider two systems at the same temperature and pressure. One contains N particles
of a classical ideal gas in a volume V and the other N + δN in a volume V + δV.
Clearly, the difference in internal energy between the two systems is directly
proportional to δN, so we have U, V and N all increasing by the same factor. Write,
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temperature to heat at another temperature”. Mathematically, the theorem of the
equivalence of transformations is expressed by Eq. (11.3), though originally heat
was defined such that the integral is positive for an irreversible cycle. It was
sometime later that Clausius adopted the modern convention that negative heat
corresponds to a heat flowing out of a body.
In referring to exterior work, Clausius meant the work produced by a heat engine.
In an ideal reversible engine of the kind considered by Carnot, all the transformations are, to use Clausius’ terminology, compensated by equivalent transformations
and the equality in Eq. (11.3) applies. For example, the heat taken in from the hot
reservoir is converted to work, but in returning the piston to its starting state work is
converted to heat which is ejected to the cold reservoir. Both processes have the
same equivalence value, Q/T. In an irreversible cycle, at least one of the transformations is uncompensated, leading to the inequality as previously discussed. In
comparing internal work to external work, Clausius believed that there must be a
similarly uncompensated transformation and actively sought an equivalent inequality. In consequence, he derived Eq. (11.1).
The essential difficulty with Clausius’ work is that he did not base it on conservation of energy. The concept was not fully developed at the time and this can be
seen in his approach to irreversible, noncyclic processes. Equation (11.3) for cyclic
processes is fully compatible with energy conservation whereas Eq. (11.1) for
noncyclic processes is not. In a cyclic process, an irreversible stage can by offset
by some other process within the cycle in which heat is extracted to restore the
original state, but this cannot occur in a single, noncyclic process. Clausius overcame
this incompatibility in his Sixth Memoir by disregarding the work that is actually
done in a process, which of course is governed by energy conservation, and looking
instead at the work that might be done: “The law does not speak of the work which
the heat does, but of the work which it can do . . .”; “. . .similarly, in the first form of
the law, it is not of the resistances which the heat overcomes, but those of which it
can overcome that mention is made”. The emphasis is Clausius’ and by this
reasoning he introduced an inequality into the First Law of thermodynamics. In
the following pages the consequence of this inequality are explored for a classical
ideal gas subject to a change in the number of particles.
11.3 Extensivity, Entropy and Open Systems
Consider two systems at the same temperature and pressure. One contains N particles
of a classical ideal gas in a volume V and the other N + δN in a volume V + δV.
Clearly, the difference in internal energy between the two systems is directly
proportional to δN, so we have U, V and N all increasing by the same factor. Write,
11 Physics Education Research and the Foundations of Physics: A Case Study from. . .
121