Hello. In this video, we're going to
discuss dense gases. Dense gases are gases where the density is high
enough that molecules are close enough together so that they feel the electrostatic
field from other molecules, at least the molecules that
are reasonably close to them. We're just going to
look at the case for a relatively small departure
from the ideal gas limit. We'll start by considering in the general form of
the partition function, and that will lead to an integral called the
configuration integral, that contains the interaction
potential energy terms. So if we start with the
canonical partition function, this particularly simple form, then for a dense gas, if we calculate U, there's an internal energy term, a translational energy term, and then there's an
interparticle potential energy, which is written symbolically as Phi of r1, r2, and so on, where r1, r2 are the coordinates of all the
molecules in the system. If we factor out the
internal energy, then it comes out, and so we note that
we can write U of the internal energy separately. Then we have that the sum of
e to the minus Beta U over the internal energy
can be written in terms of the so-called
internal partition function, or molecular partition
function for internal motion, which we introduced at the beginning of our
discussion of ideal gases. So if we plug that
into the summation, we get q internal to the n, where n is the number of
molecules in our system. So that the partition function becomes q internal to the n, times the sum over e to the
minus Beta U translation, plus the potential energy term. So it pays to replace this
sum with an integral, and this is okay because translational energy states
are very closely spaced. So it turns out
that one can show, and you can see
this in McQuarrie, I'll give you the
reference in a minute. But you can write that sum in terms of the integral
expression on the right, where H is the
classical Hamiltonian. The Hamiltonian is the sum of the kinetic and
potential energy. So it turns out we
can separate this integral into the product
of two integrals, one over momentum, and
the other over space. But the derivation
for all of this can be found in these references, the McQuarrie statistical
thermo textbook Chapter 10, Section 7, and the
papers by Kirkwood, which were the formative
papers back in 1933 and 1934. So the first term is just
the translational energy. The second term, we call
the Configuration Integral. So it's the integral over all position of all molecules of e to the minus Phi over kT. It's called the
configuration integral because it depends on
the configuration. In other words, where
are all the atoms and molecules located. So if we put all this together, the Canonical Partition Function becomes as shown
in this equation. So you have the product of the internal
partition function to the one over n factorial,
times this term, 2pi mkT over h squared, to the three-halves N times
the configuration integral, I'm sorry, which was
denoted as Z sub Phi. This result is identical
to an ideal gas, except for the presence of the configuration integral
in place of the volume. So Phi is zero, the integral, in fact, of Z is equal to the
volume to the nth power. So you can then write the canonical form of the
Massieu function as shown here, and you can then get all
the properties from this, if you knew Z, of course. So the problem devolves
to evaluating Z. Z is taking into
account the effect of inter particle forces that we did not need to deal
with for an ideal gas. So we require some
simplifying assumptions, and these are typically that the orientation of
molecules is unimportant. So the force between
any two molecules is a function of distance only. This is actually not true
for non-symmetric molecules. Another simplifying
assumption is that the total potential energy is the sum of the pairwise
potential energies between molecules, and that's only strictly true
for a not too dense gas, or just a relatively
small departure from ideal gas behavior. So if we do that, we can write the potential energy term, Phi, is equal to the sum over
Phi for rj interactions, and then we have to multiply that sum by one-half
because otherwise, we'd be counting each
interaction twice. Or we can write it as shown on the right
side of this equation, where we've got some restrictions on the indices that
we're summing over. Here, rij is the distance between molecule
i and molecule j. Now, we can make a simplification if we
introduce this new function f as e to the minus
Phi, all minus 1, and if you're astute, you would think that
if Phi is small, then there are some
great simplifications to be made, and that is the case. So it turns out that
we can write e to the minus Phi over kT as equal to this product of e to
the minus Phi ij, which is the potential energy between molecule i and
molecule j over kT, and that in turn can
be written in terms of this function f that
we just introduced. So what we've done is we've eliminated the
exponential in this product. If we assume that the gas
only weakly interacts, that means f is going to be
much, much less than one. This is where the
simplification comes in, and if we drop all the
higher-order terms, then e to the minus Phi
over kT is just equal to 1, plus the sum over all the fij's. Then plugging that into the integral expression for
the configuration integral, we get the equation on the top, and if we integrate
the first term, we get V to the N, and then the second term
we're going to integrate over the sum of the f_ij's. Since each f_ij
depends only on r_ij, it turns out we can
write this integral as V to the N plus one over 2N squared times the integral, from 0 to infinity of f of rdr. This is where we've
assumed that all of the f_ij are the same function. That is that the
intermolecular potential between any two
particles is the same. This obviously is not
true in the general case. If you have a
mixture, for example, the potential is
going to be different between different
types of molecules. So this is what we call the weakly
interacting configuration, and it is then
typical to write Z. This first order
approximation to Z, as V to the N times
1 minus N squared, times something called B over V, and B is a function of
T, it's got f in it. It happens to be called the
second Virial coefficient, and is equal to minus two pi, the integral from 0
to infinity f of rdr. Now, Virial means force in Latin, and we'll come back to that idea that it's force in Latin. So more generally, we can write that Z is equal to V to the N, all times one minus N squared B over V plus
higher-order terms. Of course, people have explored those higher-order
terms and so on, we will not do that. But it provides a context for the discussion of so-called
Virial equation of state, which are equations
of state which have terms in the increasing order of one over the volume to the power. So that's it for this video. Thanks for listening,
have a great day.