Diatomic molecules

1
Diatomic molecules
The observation that the molar heat capacity of
nitrogen and oxygen is about 5R/2 enabled
Clausius to speculate (in about 1880) that these
gases must be diatomic. The additional R of heat
capacity was believed to result from two
rotational degrees of freedom obtained by
treating the molecule as a rigid dumbbell.
The fact that the rotational degree of freedom
about the molecule axis does not contribute to
CV, and the temperature dependence of CV observed
for molecular hydrogen (H2), could not be
explained at the time. This turned out to be a
limitation of the classical description of the
rotational and vibrational dynamics of molecules,
rather than a failure of the equipartition
theorem.
2
CV for solids
The equipartition theorem is also useful for
understanding the heat capacity of solids. Dulong
and Petit pointed out that the molar heat
capacity for most solids is very nearly equal to
3R. The six contributions of ½R come from the
three kinetic energy terms (½mv2, etc.) and the
three potential energy terms (½kx2, etc.).
R 2 kcal/kmol.K
At high temperatures, all solids obey the Dulong
and Petit law. For temperatures below come
critical value (called the Debye temperature,
which is a characteristic of the material), the
value of CV drops significantly. Again, this
turns out to be a limitation of the classical
theory. Also puzzling is the fact that free
electrons in metals do not appear to contribute
to CV.
3
Equipartition of energy

• Here, vi represents one component of velocity.
• On average, one expects each component to
  contribute equally to the kinetic energy, since
  the distribution function must be isotropic.

4
Gas pressure and the ideal gas law

• Kinetic theory provides a natural interpretation
  of the absolute temperature of a dilute gas.
  Namely, the temperature is proportional to the
  mean kinetic energy (e ) of the gas molecules.
• The mean kinetic energy is independent of
  pressure, volume, and the molecular species, i.e.
  it is the same for all molecules.