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We discussed gases, and in particular, real gases this week.
But we did start with the ideal gas equation of state.
And recall that that's PV equals nRT. Where P is pressure.
V is volume. n is the number of moles.
R is the universal gas constant. And T is the temperature.
It's a little bit more convenient to work with molar volume which is an extensive
property, not an intensive property. And that would be expressed as PV bar
equals RT. And indeed, the ratio of PV bar to RT
itself has a name. That's called the compressibility and
it's usually indicated by capital Z. Gases are in fact rarely ideal.
That is, they do not obey the ideal gas equation of state except at very low
pressure. The van der Waals equation of state, on
the other hand, is a better predictor of non-ideal gas behavior, and there are
other equations of state that do well, as well, and they include the Redlich-Kwong
and the Peng-Robinson equations, which we looked at briefly.
1:44
A single value, value of molar volume that satisfies the cubic equation.
And, once we know that critical volume, we can determine the van der Waals
constants, a and b from that position, the critical volume, the critical
temperature, and the critical pressure. The Law of Corresponding States says that
all gases have the same properties if compared at their corresponding states.
Where under corresponding conditions that is, where corresponding conditions means
that the conditions for that gas, relative to its own critical conditions
are the same as the conditions for a different gas relative to its critical
conditions. So that is a corresponding state or
corresponding conditions. We also saw another equation of state,
the virial equation of state. And the virial expansion that appears in
that equation of states has coefficients that are directly related to
intermolecular interactions. The second virial coefficient, B2v,
measures the deviation of the volume of a real gas compared to an ideal one under
the same temperature and pressure conditions.
So, if B2v is a positive number, that describes how much more volume the real
gas occupies than an ideal one would under the same conditions and vice versa.
When B2v is negative, that means that the gas is occupying less volume than an
ideal one at the same conditions. We looked at the sorts of attractive
interactions that could occur between two different molecules in a real gas.
And in particular, we looked at the Lennard-Jones potential where the
intermolecular u is expressed in terms of the strength of a molecular interaction
epsilon and the size of the molecules themselves sigma.
And the Lennard-Jones potential has an attractive term that drops off as r to
the 6th, that is, as it goes as r to the minus 6, and a repulsive terms that goes
as r to the minus 12. So at very short distances, the repulsion
rises very, very steeply. The Lennard-Jones parameters themselves,
epsilon and sigma, can be determined through analysis of experimental second
virial coefficient values. We also talked about the physical
underpinnings of the r to the minus 6 attractive term and the most dominant
contribution to it, namely dispersion. We talked about London's development of a
quantum mechanical explanation for dispersion.
And his equation that produces its magnitude based on atomic or molecular
ionization potentials and the polarizabilities of the individual
molecules. Finally, we looked at some simpler
potentials than the Lennard-Jones potential that allowed us to solve for
the second virial coefficient analytically in order to gain some
physical insight. One potential was the square-well
potential, where there is a region of attractive interaction followed by a hard
wall. And determined that the predicted second
virial coefficients are not bad. And compare reasonably favorably to the
full Lennard-Jones potential over reasonable ranges of temperature and
pressure, and I showed an example for nitrogen gas.
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The simplest potential, the hard-sphere potential, where there is no attractive
force, and at the point of two molecules touching, there is an infinitely
repulsive potential. is not very appropriate under most
conditions, but at very high temperatures, it does describe reasonably
accurate gas behavior, and it says that the second virial coefficient is positive
and independent of temperature. So it's that region in a plot of the
second virial coefficient versus temperature, where the, the plot has
turned over and is almost a straight line horizontally.
The square-well potential that has a free parameter lambda in it, includes the
length over which a constant potential is effective, and this emphasizes again,
that we can get good agreement with experimental data.
All right. Well, that completes the summary of the
key concepts from this week. you'll have an opportunity to explore the
concepts in more detail on the homework associated with this week.
Good luck with that. And I look forward to continuing as we
move to next week. And, to talk about what we will be
looking at next week, we're going to start with the concept of Boltzmann
probability. See you then.
Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics
