Second Law that the entropy of the universe can only increase or remain the same, is
seen to operate.
At the heart of this view is the relationship between Eqs. (11.1) and (11.3) and the
notion that entropy is property of a body with a unique value in a thermodynamic
state. A body is defined here as simply any collection of atoms or molecules in
whatever state, solid liquid or gas. This phrasing makes the distinction between
entropy as a state function, which is essentially a mathematical notion, and entropy
as a property of a body, which is a physical idea. The connection between the two is
an area that has been neglected in the literature on entropy, but lies at the heart of the
present discussion. In relation to physics education research, the link between
mathematics and physics should be central to any physics education programme,
but I will argue here that the history of entropy shows that even professional
physicists can put mathematics first ahead of physics. In this chapter, the connection
between the law of increasing entropy, as expressed by Eq. (11.1) and the law of
conservation of energy will be examined with reference to particular examples. In
addition, the relationship between the mathematical idea of a state function and the
corresponding physical properties will also be examined and examples of students’
confusion over entropy will be presented.
11.2 Entropy and Energy Conservation
The preceding example on the free expansion illustrates the fundamental problem of
the notion of the entropy of a body. As there is no change in internal energy during
the free expansion, there is also no change in temperature. Even if it is argued that
during the expansion the state of the gas is not well defined, the quantity TS, which
has the units of energy, must be defined for the initial and final states. The change in
the Gibbs free energy is given by the difference, yet there is no change in U, no heat
flow and no work done. We can meaningfully ask about the physical meaning of the
Gibbs free energy in the light of this change.
This difficulty was built into the structure of thermodynamics by Clausius, as
summarised in the 1898 (Clausius 1898) collection of his nine Memoirs. It is evident
in all his early writings that Clausius was interested in what he referred to as,
“internal work”, which is the work associated with inter-particle forces when a gas
is either compressed or expands. In treating internal work, Clausius borrowed from
his earlier work on cyclic processes: “. . . as there is no essential difference between
interior and exterior work, we may assume with certainty that a theorem which is so
generally applicable to exterior work cannot be restricted to this alone”. In fact,
there were two theorems in Clausius’ view of thermodynamics, which Clausius
explained as the equivalence of heat and work, or Joule’s principle, and the equivalence of transformations. The latter will be unfamiliar to the modern physicist, as it
is an obscure concept not taken up by Clausius’ contemporaries and which has
subsequently disappeared altogether from the thermodynamics lexicon. Clausius
regarded two transformations as being equivalent in some way: the conversion of
120
D. Sands
seen to operate.
At the heart of this view is the relationship between Eqs. (11.1) and (11.3) and the
notion that entropy is property of a body with a unique value in a thermodynamic
state. A body is defined here as simply any collection of atoms or molecules in
whatever state, solid liquid or gas. This phrasing makes the distinction between
entropy as a state function, which is essentially a mathematical notion, and entropy
as a property of a body, which is a physical idea. The connection between the two is
an area that has been neglected in the literature on entropy, but lies at the heart of the
present discussion. In relation to physics education research, the link between
mathematics and physics should be central to any physics education programme,
but I will argue here that the history of entropy shows that even professional
physicists can put mathematics first ahead of physics. In this chapter, the connection
between the law of increasing entropy, as expressed by Eq. (11.1) and the law of
conservation of energy will be examined with reference to particular examples. In
addition, the relationship between the mathematical idea of a state function and the
corresponding physical properties will also be examined and examples of students’
confusion over entropy will be presented.
11.2 Entropy and Energy Conservation
The preceding example on the free expansion illustrates the fundamental problem of
the notion of the entropy of a body. As there is no change in internal energy during
the free expansion, there is also no change in temperature. Even if it is argued that
during the expansion the state of the gas is not well defined, the quantity TS, which
has the units of energy, must be defined for the initial and final states. The change in
the Gibbs free energy is given by the difference, yet there is no change in U, no heat
flow and no work done. We can meaningfully ask about the physical meaning of the
Gibbs free energy in the light of this change.
This difficulty was built into the structure of thermodynamics by Clausius, as
summarised in the 1898 (Clausius 1898) collection of his nine Memoirs. It is evident
in all his early writings that Clausius was interested in what he referred to as,
“internal work”, which is the work associated with inter-particle forces when a gas
is either compressed or expands. In treating internal work, Clausius borrowed from
his earlier work on cyclic processes: “. . . as there is no essential difference between
interior and exterior work, we may assume with certainty that a theorem which is so
generally applicable to exterior work cannot be restricted to this alone”. In fact,
there were two theorems in Clausius’ view of thermodynamics, which Clausius
explained as the equivalence of heat and work, or Joule’s principle, and the equivalence of transformations. The latter will be unfamiliar to the modern physicist, as it
is an obscure concept not taken up by Clausius’ contemporaries and which has
subsequently disappeared altogether from the thermodynamics lexicon. Clausius
regarded two transformations as being equivalent in some way: the conversion of
120
D. Sands