Hello. In this video, we're going to discuss property relations
for dense gases. In the canonical
representation, you'll recall, we can write the
differential form of S of bracket 1 over T, close bracket S
minus Ud of 1 over T plus P over T dV
minus Mu over T dN. So P over T is the partial of S dV at constant 1 over T dN. Since S is equal to kIn of the partition
function on this next slide, we can write that derivative
for pressure as well. We've brought the T over
to the right-hand side. That will become P equals kT partial ln Q over V at
constant 1 over T and N. Now the only term in Q that contains the volume is the
configuration interval. So the only term that remains to take the derivative of
this partial ln z with V at constant 1 over
T and N. Now ln of Z is in the form of In of 1 plus x where x
is a small parameter. So we can expand ln of 1 plus x as x minus x squared over 2, etc. So if we use that fact, we can write P as the following: NkT over
V plus N squared kT times the second virial
coefficient B over V squared. I've got plus dot, dot, dot meaning that you could carry out higher-order terms
if you so desire. If you normalize the volume
on the number of molecules, you get a specific
volume, essentially. So you can see that here we have a PVT relationship which is P as a function of terms
of 1 over v to the N, where N is an increasing
integer power. You can convert this to molar units and so on and the
units that you're used to. So this is where the concept of the so-called
virial equation of state comes from an expansion in one over V to the N is known
as a virial expansion, hence virial equation of state, and virial again means force. So what you're doing
is expanding in higher-order terms related to the force between molecules. This type of
relationship often forms the basis for empirical
equations of state. They might not look
exactly like this. But if you want to put an
equations of state together, it might be a good first
choice to assume that this is what the functional
form is going to look like. How many terms you need
is going to of course depend on the
character of the data that you're trying to fit. Now we can also get
the internal energy. Again, this is the expression for the fundamental relationship differential
form of the fundamental, or this is the first equation is the form of the
fundamental relation. If we take the derivative, then U is the partial S with
or minus the partial S with U with one over T. You can
go ahead and evaluate that. Therefore, the energy
becomes equal to the expression given here
where the first term is the ideal gas expression
and the second term takes into account
configuration effects. So you can evaluate
this second term. It turns out to be equal to
the integral over Phi minus Phi over kT over all of the position of all
of the molecules divided by the
configuration integral. So and out of this eventually we will
get a B(T), I guess. It remains to evaluate B(T), we'll do that in the next video. So thanks for listening
and have a good day.