Hello. In this video, we're going to explore
potential energy functions. These are required if we're
going to be able to evaluate B(T) and therefore be able to get an equation of
state for the dense gas case. So we'll look at some
simple algebraic forms here along with a more realistic form called
the Lennard Jones potential. So here are plots of three very simple potentials
and Lennard Jones. The first is rigid sphere. You might think of this as the billiard ball approximation. There is no force between particles until they
actually hit each other and they can't penetrate. So the potential energy just
looks like a Delta function at the distance when
the two balls touch. Square well has an
attractive portion and a repulsive portion. The attractive portion
is just represented as a square well hence
the name square well. So algebraically, it's
extremely simple. Weakly attracting
potential has some kind of curved section
of attraction here, and typically, it's represented as some kind of exponential. Then the more realistic potential is the so-called Lennard
Jones potential. We'll see what it
looks like in a bit. The function that is called
the Lennard Jones potential. But it looks like what we know, the real potentials
look like more or less. So recall that that
B of T is equal to minus two Pi the integral from zero to infinity of f of rdr. So we can just go
ahead and plug in the potential into f of
r. For a rigid sphere, this was the
expression we've got. We can divide it into two parts the repulsive part
and the attractive part. If you just run
the integrals out, you get the B(T) is two
Pi Sigma cubed over three where Sigma is
the collision distance. It's the distance between the centers when two particles
are touching each other. So note that the
volume occupied by a spherical molecules for
four-thirds Pi r cubed, which can be written as
four Pi or four-thirds Pi Sigma over two cubed because Sigma corresponds
really to a diameter. If you have two equal
sized particles, Sigma's the diameter
of one the particles. This is called the co-volume. So it's the volume excluded
to other molecules. Think about a bunch of pool balls or billiard balls on a table. Once they're touching, you
can't get them any close together because it can't
fit inside of each other. So B of T in the hard sphere approximation
turns out to be four times the volume of
a spherical molecule. Now, we can write p is
equal to RT over v plus RTB over v squared plus higher-order
terms if we so desire. If p is much less than v, and if we substitute the lowercase b for capital B and I'll
explain why in a moment, then we can write p is
equal to RT over v minus b. We use lowercase b just
because it's conventional. If you lookup Van
der Waals equation, you're going to see
lowercase b there. So this is the first term
and Van der Waals equation, and it represents the addition of a repulsive term to the
ideal gas relationship. Now, if we go to weakly
attracting spheres, and you can see the function, the attractive portion
of the function in this integral and
evaluate B of T, then it turns out that B of T can be
written as lowercase b, the previous lowercase
b minus a over RT. The a over RT represents
the contribution to the attractive term that comes from adding that to the weakly
attracting sphere model. So if b is much, much less than a over RT, then you get Van
der Waals equation. P equals RT over v minus
b minus a over v squared. So the first term, adding b essentially adds
in the repulsive portion of the potential energy function and the second term minus a over v squared adds in the
attractive portion. So I just have to say I love Van der Waals
equation because it very simply illustrates the physics of PVT relationship. As you try to increase
the density of the gas from the ideal gas limit, at first, it actually
becomes easier. You need less of a
pressure increase to accomplish a
certain increase in density because of the
attractive forces. But if the particles
get too close to each other then the
repulsive force dominates and a much
greater increase in pressure is needed. So that's it for this video. Thanks for listening
and have a great day.