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Adiabatic Expansion Cooling of
Gases
Using
simple adiabatic expansion, the ratio of the constant pressure to
constant volume heat capacities of gases will be measured. This
property will be correlated to the internal motion and molecular
structure of the gases studied. Several gases will be compared and
the contributions of the the different types internal molecular
motions to the total thermal energy of the gas molecules will be
gleaned.
Theory
For
a perfect (ideal) gas, Cp = CV + R ,
where Cp and CV are the molar
heat capacities at constant pressure and volume, respectively (see
on-line lecture
notes for a derivation of this and related formulae). For an
arbitrary real gas a slightly more complicated relationship
between these heat capacities may be derived from the equation of
state. Essentially, however, the difference between heating a gas
at constant volume and constant pressure is expansion work. Thus,
the ratio Cp / CV is related to the
capacity of the system to do work upon expansion. This ratio is
usually given the symbol 
[lower case greek gamma].
Properties
of CV
The heat capacity of a molecule is clearly related to the way that
the molecule can accomidate energy at a given temperature. The
energy of the molecule is partitioned among the types of motion
the molecule can exhibit. It is important in thermodynamics to
count and categorize such molecular motion in a systematic way.
The number of degrees of freedom (DOF) for a given molecule is the
number of independent coordinates needed to specify all its
nuclear positions (what do I ignore in this statement?). A
molecule of n atoms therefore has 3n DOF. These
could be assigned to the coordinates of the individual n
atoms, or alternatively they can be classified as follows:
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Translational degrees of freedom: 3
independent coordinates specify the center of mass of the molecule.
-
Rotational degrees of freedom: All
molecules containing more than a single atom require specification of their
orientation in space. Rotation of a diatomic molecule can be described by
two rotational degrees of freedom since rotation about the internuclear axis
leaves the molecule unchanged. Non-linear molecules require three rotational
degrees of freedom.
-
Vibrational degrees of freedom: The
displacements of the atoms from their equilibrium positions can be
described by 3n-5 DOF for linear molecules and 3n-6 for
non-linear molecules. These values are determined by the fact that the
total number of DOF must be 3n. For each vibrational DOF there is
an associated normal mode of vibration of the molecule with characteristic
symmetry properties and a characteristic harmonic frequency.
From classical statistical mechanics the 'equipartition
of energy' theorem can be derived which associates an energy of RT/2
per mole with each quadratic term in the Hamiltonian or per degree of
translational or rotational freedom (again, see my lecture
notes for elaboration of this point). Here, R is the Molar Gas Constant (Boltzmann's
constant times Avogadro's number) and T is the absolute thermodynamic
temperature. Vibrational DOF have two quadratic terms: one potential energy
term and one kinetic term per vibration. Therefore an energy of RT per mole is
associated with each vibrational DOF. This is in contrast to rotational and
translational DOF's which are 'free' motions and thus have no potential energy
term.
CONTINUE with the MATH?
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