Hello. In this video, we're going to talk
about liquids. In this module, we'll
talk about liquids. Liquids are much more difficult to theoretically describe them. Ideal gases are
crystalline solids. For a particular particle or
a molecule in the liquid, interatomic forces are
constantly present, essentially making
the configuration integral functionally complex. They're two theoretical
approaches that have been used, one are cell-based methods, and they work so-so not great, and distribution function
based, and they worked better. Then molecular dynamics is a computational tool that can
be used to study liquids. I should point out at
this point that many of the empirical equations
of state that we discussed earlier
under the section on dense gases apply
to liquids. In fact, that is usually from a
practical point of view, engineering point of
view we typically, describe liquids with
empirical equations of state. So cell theories are
based on the idea that molecules in a liquid are
restricted in their motion. They are not locked in place like they would be in
a solid structure. They can move around, but they are somewhat
stricted in their motion. So early workers that develop these
cell-based techniques, suggested that they were confined to an area that was perhaps seven- several not seven, several collision diameters
across, roughly speaking. Then they assume
that the molecule behaves like a particle in a box. Perhaps not with a simple square well potential
energy function, but something more realistic. Here's a sketch
from [inaudible] , The Green Bible that
shows on the top, the assumption of just
a particle in a box or a box potential energy function, square well potential
energy function. Then what the potential energy function for an individual molecule
would look like in a crystalline solid. Then perhaps what it
would look like in a real liquid where it can move a little bit
with no potential, but then if it moves to the side, some potential is
going to kick in. It just isn't a square function. So that's cell methods, and they don't work
all that great. You can, if you are interested, look in [inaudible] they have an extensive discussion
of cell methods. Distribution function
methods are much more common and much more modern than cell theories and they're
based on evaluating the configuration integral using statistical
distributions. The key idea is that a
distribution function can describe the liquid
structure in an average way. Experimentally, it turns
out that you can determine the radial distribution function
using x-ray scattering. So the most common
function that's used is the so-called radial
distribution function. This is just the density
that a molecule would experience as a function of
distance from the molecule divided by the average density. Essentially what
you're looking at in the image to the left is the black circle
is the molecule of interest and as you go
away from that what is the density in some
differential shell around that molecule that's
rho of r and then the so-called radial
distribution function is that normalized on
the average density. So if you've got far enough away from the
molecule of interest, you would expect that things
would smear together and the radial density function would eventually asymptote out
to the average density. As you get closer, there's
going to be bumps because you get a sort of ordering around because of the
potential energy functions. You get a little bit of ordering around that target molecule. So you see this oscillatory
approach to a peak that is roughly one mean
free path away from the target molecule
and then it goes to zero at the at the
molecule itself. If you integrate the radial
distribution function over or actually
you're integrating the radially dependent
density over all rs away from the molecule
in question you get approximately the total
number of molecules that are in the system. So it turns out that one can relate the radial
distribution function to thermodynamic properties. If you assume that the
total potential energy of the N-body system is pairwise additive as
shown in the equation, then all the
thermodynamic functions can be written in terms of g of r. The number density of molecules whose centered between r and r plus are relative to a specific molecule
is 4 Pi r squared, N over V, g of r, dr. So for a spherically symmetric pairwise
potential function, it follows that time, so potential energy
is the function, is the intermolecular
potential energy between the central molecules and the other molecules so center between r and r plus dr. So it does turn out
that you can use that to calculate the
internal energy as given by the equation
on the top and you can use the same idea
to get the pressure, which is shown on
the bottom slide. So that's it for this video. We'll discuss more of this. I should say that
all of this theory can be found in the
book by McQuarrie, which is in your reference list