Well, the problem with polyatomic
molecules is they're actually pretty complicated. And as a result, we can only do
relatively simple treatments of them, at least at the level that we're
going to discuss in this course. So, a polyatomic molecule is a body with
some number of atoms, we say three or more typically because we can treat
diatomic molecules with more detail. So it will be a multi-body problem,
will have say n atoms. So a multi-body problem with 3n elements
will have 3n degrees of freedom. Translation will take up,
I'm sorry, 3n degrees of freedom. Translation will take up three
of those degrees of freedom. And that would be the center of mass, in other words can move independently
in three coordinate directions. Depending on whether the molecule
is a linear molecule or a non-linear molecule, it can have
either two or three modes of rotation. And then vibration make up the rest. So there can be either 3n minus 5,
or 3n minus 6 modes of vibration. So here's a couple of examples. CO2 is a polyatomic molecule, three atoms, its linear that is the three
atoms are on the same axis. And therefore, there are only two
independent degrees of rotation. Remember, a rotation around the axis,
there's no moment of inertia and therefore, there's no energy
associated with that. So we say that a linear
molecule has two degrees of rotational motion that store energy. Methane, CH4 on the other hand has one, two, three, four, five atoms,
and it is non-linear so it can rotate around three
independent axis, coordinate axis. So, what's left is vibrational mode. So CO2, linear triatomic has, there are four possible
independent modes of vibration. And these are the ones so called symmetric
stretch, actually symmetric stretch, and bending into independent planes. Water, which also has three atoms but
however is non-linear, has three modes of vibration,
and they are shown in the sketch. The first one is a sort of breathing mode
where all three atoms move apart from each other and come back together, and so on. You can have fun trying to mimic those modes with your fingers. But it's actually easier to
do with the computer code. Okay. So, if we assume that each vibrational and
rotational mode modes are, each of the vibrational and
rotational modes are independent. Then we can separate the partition
functions for each mode. And if we assume that the harmonic
oscillator and rigid rotator models apply. Then we can write analytic expressions for
both the vibrational and the rotational partition numbers, and
they are given here on this slide. And basically, each are just product of the of the various modes. So it's a pretty simple calculation. There's a very important
concept that can be useful in estimating specific heat of,
in polyatomic molecules or at least, the maximum specific
heat in polyatomic molecules. Each mode of energy contributes up
to one-half kT to the total energy. And therefore,
K over 2 to the specific heat, translation provides three of those modes. Rotation, one each for
each axis of rotation. And vibration, one each for
the kinetic and potential energy involved in vibration, up to 3n minus 3,
minus the number of rotational modes. And the modes must be so called fully
excited for the full K over 2 to kick in. So at normal temperatures, room
temperature and above, translational and rotational energy
are so-called fully excited. But vibration is not necessarily the case, and so I show this diagram for a diatomic molecule just because it's
easier to plot it out. We know for diatomic molecule
that at room temperature, starting around room temperature,
the specific heat is 2.5k. And that's because you have
translation and rotation for diatomic molecule, two modes of
rotation for diatomic molecule. And then if you throw in vibration,
it can vary quite a bit. Can add another K to the specific heat,
and so take you up to seven halves eventually. Well, fully excited means you're
at high enough temperatures so that you're at the asymptotic
limit of the vibrational energy. So, this is like I say,
it's an interesting way to evaluate the full potential for specific heat in a Polyatomic molecule. So that's that's it for
this for this video. Thanks, and have a great day.