Historical burdens on physics
77 Names of the ideal gas law
Subject:
The equation p · V = n · R · T is introduced under different names: gas
equation, general gas equation, universal gas law, thermal equation of state
of the ideal gas, and others. Since the equation relates more than two variables, one may be interested in the relationship between only two of these
quantities, keeping the remaining variables constant. The corresponding
relations are known under particular names. The relation between p and V
is Boyle’s law, the V-T relation is called Charles’ law, the p-T proportionality
is Amontons’ law and the V-n relationship is Avogadro’s law. In the French
and German literature Boyle’ law is called Boyle-Mariotte’s law and Charles’
law is called Gay-Lussac’s law.
Deficiencies:
1. The importance of an equation can be emphasized by giving it a proper
name. Such a name also facilitates the reference to the equation. The gas
equation (let us here call it so) is important. It is valid for matter in a very
large sense, provided that the corresponding substance is sufficiently diluted and/or the temperature is high enough. The equation not only applies
to gases in the usual sense, as for instance the air around us, but also for
the solute in a diluted solution or for the compressed plasma in the central
region of the Sun. Thus, the equation is worthy of a name. It is another
question wether the attributes “general” or “universal” are appropriate, since
such a classification can hardly be topped.
2. It is a nice custom to name equations after an important scientist. However, as the gas equation shows, this can also be overdone. In our case six
researchers are honored by means of one single equation. The problem of
baptizing an equation with the name of a scientist is known from street
names. Someone may come to unexpected honors since a small alleyway
that carried his name transformed later into a main artery. On the contrary,
there are great scientist who never became the patron of an important
equation, as for instance Leibniz or Descartes. Still others are honored for
something that was relatively unimportant in their work, as for instance
Huygens for the elementary waves or Faraday for the somewhat puny
Faraday effect or the curious Faraday cup.
3. Let us come back to the gas equation. It is equivalent to various other
relations that seemingly state something rather different from the gas equation in its usual form:
(a) E(V) – E(V0) = 0 for T = const,
in words: At fixed temperature the energy of a gas is independent of the
volume.
V
(b) S(V ) – S(V0 ) = n · R · ln
, for T = const,
V0
in words: At fixed temperature the entropy depends logarithmically on the
volume.
(c) µ(p) – µ(p0 ) = R ·T · ln
p
, for T = const,
p0
in words: At fixed temperature the chemical potential depends logarithmically on the pressure. (From this equation one easily obtains the law of
mass action and the barometric formula.)
All of these three equations can be derived without any further physical input from the “gas equation”. Therefore, each of them could also be called
“gas equation”, what is not done.
4. A gas is not fully described by the gas equation or “thermal equation of
state”. The thermal equation of state is just one of several equations of
state that are needed to completely characterize a particular gas. So it does
not describe the caloric properties of an ideal gas: How does the temperature depend on the heat (entropy) content of the gas? The answer to this
question is given by the “caloric equation of state”. The effects that it describes are as striking as those described by the thermal equation of state.
Traditionally, at school it is considered less important, with the result that
many interesting processes are disregarded in the classroom: the isentropic
expansion in the steam engine and the internal combustion engine, or the
decrease of the temperature with height.
Origin:
1. In the usual treatment of the gas equation and various proportionalities
which it contains, one can recognize the various contributions of the various
epochs of its genesis. One also sees that the view from different countries
is different.
2. The thermal equation of state, that contains the variables p, V and T,
which are easy to measure, is overrated since the quantities entropy and
chemical potential, which for many processes are more important, and, by
the way are also easy to measure, never found wide acceptance.
Disposal:
1. Introduce names for equations parsimoniously. In the case of the gas
equation we propose not to give own names to the partial proportionalities.
We recommend particular caution in the award of predicates like “general”,
“universal”, “fundamental” etc.
2. If the logarithm is available, discuss the volume dependency of the entropy and the pressure dependency of the chemical potential. Treat in any
case the “caloric” properties of gases, in particular the relation between
temperature and volume at constant entropy, since it allows for an understanding of the working principle of the heat engine and the temperature
stratification of the atmosphere.
Friedrich Herrmann, Karlsruhe Institute of Technology