k/An experiment for deter-
mining the shapes of molecules.very sophisticated experiments could vindicate him definitively. Af-
ter all, the microscopic and macroscopic definitions of entropy are
equivalent, so it might seem as though there was no real advantage
to the microscopic approach. Surprisingly, very simple experiments
are capable of revealing a picture of the microscopic world, and there
is no possible macroscopic explanation for their results.
In 1819, before Boltzmann was born, Cl ́ement and Desormes did
an experiment like the one shown in figure k. The gas in the flask
is pressurized using the syringe. This heats it slightly, so it is then
allowed to cool back down to room temperature. Its pressure is
measured using the manometer. The stopper on the flask is popped
and then immediately reinserted. Its pressure is now equalized with
that in the room, and the gas’s expansion has cooled it a little,
because it did mechanical work on its way out of the flask, causing
it to lose some of its internal energyE. The expansion is carried out
quickly enough so that there is not enough time for any significant
amount of heat to flow in through the walls of the flask before the
stopper is reinserted. The gas is now allowed to come back up
to room temperature (which takes a much longer time), and as a
result regains a fractionbof its original overpressure. During this
constant-volume reheating, we havePV =nkT, so the amount of
pressure regained is a direct indication of how much the gas cooled
down when it lost an amount of energy ∆E.l/The differing shapes of a
helium atom (1), a nitrogen
molecule (2), and a difluo-
roethane molecule (3) have
surprising macroscopic effects.If the gas is monoatomic, then we know what to expect for this
relationship between energy and temperature: ∆E= (3/2)nk∆T,
where the factor of 3 came ultimately from the fact that the gas
was in a three-dimensional space, l/1. Moving in this space, each
molecule can have momentum in the x, y, and z directions. It has
three degrees of freedom. What if the gas is not monoatomic?
Air, for example, is made of diatomic molecules, l/2. There is a
subtle difference between the two cases. An individual atom of a
monoatomic gas is a perfect sphere, so it is exactly the same no
Section 5.4 Entropy as a microscopic quantity 335