j/Heat capacities of solids
cluster around 3k T per atom.
The elemental solids plotted are
the ones originally used by Du-
long and Petit to infer empirically
that the heat capacity of solids
per atom was constant. (Modern
data.)
We would now like to generalize this theorem. Example 20,
p. 332, tells us that it doesn’t matter whether the gas is a mixture
of two different types of atoms — Betsy doesn’t give a raccoon more
love just because it’s big and fat. Equal energy sharing might seem
obvious by symmetry if all the atoms are identical, but we see that
it still holds when they are not identical. Symmetry was not a
necessary assumption.
To generalize even further, let’s look at what the necessary as-
sumptions really were in example 20. For simplicity, suppose we
have only one argon atom, named Alice, and one helium atom,
named Harry. Their total kinetic energy isE=p^2 x/ 2 m+p^2 y/ 2 m+
p^2 z/ 2 m+p′^2 x/ 2 m′+p′^2 y/ 2 m′+p′^2 z/ 2 m′, where the primes indicate
Harry. The system consisting of Alice and Harry has six degrees
of freedom (the six momenta), and the six terms in the energy all
look alike. The only difference among them is that the constant
factors attached to the squares of the momenta have different val-
ues, but we’ve just proved that those differences don’t matter. In
other words, if we have any system at all whose energy is of the form
E= (...)p^21 + (...)p^22 +..., with any number of terms, then each
term holds, on average, the same amount of energy,^12 kT.
It doesn’t even matter whether the things being squared are
momenta: if you look back over the logical steps that went into the
argument, you’ll see that none of them depended on that. In a solid,
for example, the atoms aren’t free to wander around, but they can
vibrate from side to side. If an atom moves away from its equilibrium
position atx= 0 to some other value ofx, then its electrical energy
is (1/2)κx^2 , whereκis the spring constant (written as the Greek
letter kappa to distinguish it from the Boltzmann constantk). We
can conclude that each atom in the solid, on average, has^12 kT of
energy in the electrical energy due to itsxdisplacement along the
xaxis, and equal amounts foryandz. Thus for a solid, we expect
there to be a total ofsixenergies per atom (three kinetic energies
and six interaction energies), each of which carries an average energy
1
2 kT, for a total of 3kT. In other words, a solid should have twice the
heat capacity of a monoatomic gas. This was discovered empirically
by Dulong and Petit in 1819, as shown in figure j.
Equipartition theorem: general form
For a system whose energy can be written as the sum of the
squares ofnvariables, the average value of each term in the
energy is^12 kT.An unexpected glimpse of the microcosm
These ideas about equipartition now lead us to some surprising
insights into how the microscopic world manifests itself on the hu-
man scale. You may have the feeling at this point that of course
Boltzmann was right about the literal existence of atoms, but only334 Chapter 5 Thermodynamics