Thermal leture 3 part 2: degrees of freedom


So hopefully when
you were listing down all the types of energy
that you could think of, you thought about kinetic
energy and potential energy. For particles, we’ve
got different types of kinetic energy. So there’s the translational
kinetic energy, which is the normal
1/2 mv squared one associated with movement. And hopefully, you
also remembered about the rotational kinetic
energy, 1/2 i omega squared, associated with the
rotation of the body. Now potential energy
is not actually too important for
particles, because one of the assumptions
of the ideal gas law was that the only forces
that we needed to consider occurred in the collisions
between the particles. So we don’t need to worry
about the gravitational force. That’s because the mass of
the particles is really small, so that gravitational
potential energy is tiny. And we also don’t have
to worry about things like electrostatic fields and
the potential energy associated with that when we are
considering a gas, which obeys those assumptions of
the kinetic theory of gases. There is another type of energy
associated with particles. When you’ve got a molecule with
one or more atoms connected with a bond, we
can actually have vibrational kinetic energy,
where the bonds vibrate. And we have energy
associated with that. So what we’re going to do now is
consider a few different types of particles. Let’s start with the
simplest– a monotonic atom. So this is something
like a helium atom, which is just one atom by itself. Now, this helium atom can store
translational kinetic energy. It can move backwards
and forwards– let’s say that’s the x direction. It can move up and down– let’s call that the y direction. And it can also move
in and out like this, so we can say that’s
in the z direction. So there’s actually
three dimensions in which it can store this
translational kinetic energy. So each of those dimensions
is associated with what we call a degree of freedom. So degree of freedom is a way
in which a molecule or an atom can store energy. So this single monotonic atom
has three translational degrees of freedom. Now this atom can also
rotate, but that’s actually got a negligible amount of
kinetic energy associated with it. The reason for that is
that atoms are very small, and the mass of the atom is
concentrated in the nucleus. The nucleus of the atom
is about the size of 10 to the minus 15
metres, so it’s tiny. The moment of
inertia for a sphere, such as this, if we model
our atom as a sphere, is given by 2/5 mr squared. So you can see that the
rotational kinetic energy, which is 1/2 i
omega squared, it’s going to be absolutely tiny
because of the very tiny mass and the very, very tiny radius. And so any rotational
kinetic energy is negligible, and we don’t have any
degrees of freedom associated with the rotational kinetic
energy for a monotonic particle or atom. Also, there’s no vibrations here
because we don’t have any bonds for it to vibrate against– vibrate about. So this monotonic particle just
has three degrees of freedom, and they’re associated with
that translational movement, so it can move in the
x, y, and z direction. Let’s now consider
a diatomic particle. So H2 is a common example,
you also got N2, O2. So this is two atoms
connected by one bond. Now in this case, it can also
move in the three directions. So it can move backwards
and forwards, up and down, in and out. So it has got three
translational degrees of freedom. Now this particle
can also rotate. When it rotates about–
let’s call this the z-axis– that has got energy
associated with it. It can also move about the
axis this way, so like this. And that has also got
energy associated with it. Now if it rotates
about this axis, like this, there’s
actually very little rotational kinetic
energy stored in that way for the same reasons as before. The mass is concentrated in
the nucleus with a tiny radius, and so we’ve got very little
rotational kinetic energy stored in that way. So this diatomic molecule
has two rotational degrees of freedom. Now the diatomic molecules
can also vibrate. So here’s another model
with a spring in between, so you can see they can
spring in and out like this. Now when we start
it springing, we can set the initial
kinetic energy and the initial potential
energy for that oscillation. You’ll be learning more about
that in the oscillations topic. But because we have
those two choices when starting it to vibrate,
that’s got two degrees of freedom associated with it. Now an important thing
about degrees of freedom is that it does actually
depend on the temperature. At low temperatures,
below 100 kelvins, the particles can
move, so they all have the translational
kinetic energy. But when they collide
with other particles, they don’t actually
have enough energy to start those are the
particles rotating. So low temperatures,
below about 100 Kelvin, these diatomic
molecules only have the translational
kinetic energy. They also do not
have enough energy to start other
particles vibrating. So they’ve just got the
three translational degrees of freedom at low temperatures. In the medium range, between
100 kelvins and 1,000 kelvins, they do have enough energy to
start other particles rotating when they collide. But they do not
have enough energy to start other
particles vibrating. So between 100 kelvins
and 1,000 kelvins, diatomic molecules have
five degrees of freedom– three translational
and two rotational. Above 1,000 kelvins,
they have enough energy to start things vibrating. And so at that point, they’ve
got seven degrees of freedom– three translational,
two rotational, and two vibrational. Now we can have more complex
molecules, such as this methane molecule here. At low temperatures,
this methane molecule can also move in
three dimensions. And so it also has three
translational degrees of freedom associated with it. This one, it can actually rotate
about three different axes, and so it also has three
rotational degrees of freedom. At high temperatures,
it’s also going to have vibrational
degrees of freedom. So for more complex
molecules, below 100 kelvins, they also only
have three degrees of freedom associated
with translation. As the temperature
increases, depending on the geometry of
the molecule, they can have additional
rotational degrees of freedom. And then when we get above
1,000 kelvins as well, they can start
vibrating, and so we’ll have additional degrees
of freedom in that way. But you won’t be
expected to calculate how many vibrational degrees of
freedom complex molecules have, because that gets difficult. But here is one for you to try. What I’ve got here is
a model of a crystal. Now I want you to
consider this particle in the middle of the crystal. So it’s surrounded
by lots of partners, and we can model the crystal
as having bonds between it, and bonds can have vibrations. So they’ve got springs here
representing those vibrations. So what I want you to do
is think about this crystal and try and work out how
many degrees of freedom associated with translation,
rotation, and vibration you think this crystal would have.

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