This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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分子热力学统计

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This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Module 2

This module begins our acquaintance with gases, and especially the concept of an "equation of state," which expresses a mathematical relationship between the pressure, volume, temperature, and number of particles for a given gas. We will consider the ideal, van der Waals, and virial equations of state, as well as others. The use of equations of state to predict liquid-vapor diagrams for real gases will be discussed, as will the commonality of real gas behaviors when subject to corresponding state conditions. We will finish by examining how interparticle interactions in real gases, which are by definition not present in ideal gases, lead to variations in gas properties and behavior. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts.

- Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics

Chemistry

Well, we've now discussed the Lennard-Jones inter-molecular potential,

and hopefully you enjoyed the demonstration of the liquefaction of

oxygen from the air. It is inter-molecular interactions that

permit you to liquefy a gas, because the molecules just stick together.

And we got to see some of the more spectacular properties of liquid oxygen.

Today I'd like to spend some time focused on other inter molecular potentials.

Besides Deleanar Jones, because they can provide us with some information that can

be interesting as well. And let me remind you that the virtue of

having the intermolecular potentials potential function.

Is that we have a relationship between the potential.

Written on this slide in terms of u, u is a potential that depends on a in, inter

particle separation r and the second virial coefficient, in this case B2vt.

And so by plugging in a given potential u into that integral expression, we can in

fact compute by solution of the integral B2v, and understand how a real gas will

behave. So, I'd like to pause for a moment before

looking at some different potentials by actually discussing the physics behind

the various pieces of the Leonard Jones potential.

And in particular I'd like to look at the attractive term.

Which has r to the minus 6 dependents. And if we ask what, what sort of physical

interactions do in fact diminish as r to the minus 6 as things grow further apart.

The first is dipole-dipole interactions. So, if our molecule has a permanent

dipole moment, then there are two extreme possibilities for alignment.

One is that the two dipoles are opposed to one another.

So dipole here represented by a negative charge and a positive charge.

And since like charges repel, this would be a bad arrangement of these dipoles.

On the other hand, they can also be head to tail, that is the maximally attractive

arrangement of two dipoles. And, it turns out that the dipole-dipole

interactions between molecules are really quite small compared to thermal energy

for typical molecules and typical temperatures we'd work with And so, the

two dipoles are in fact tumbling with thermal energy, and, as a result we have

to average over the many different accessible orientations.

When one does that averaging, one discovers that the potential of

interaction is given here, it depends on the square of the two individual dipole

moments. Here's where you see temperature playing

a role, because it is causing these dipoles to tumble.

And then here's the r to the minus 6 dependence, and this is the permittivity

of free space. Another r to the 6 interaction is a

dipole-induced dipole interaction. So when a molecule with a permanent

electrical moment, like this one with a dipole is brought up to a molecule that

does not have a permanent moment, maybe it's an atom.

It will polarize the electron cloud of that atom and introduce an induced

dipole. And so, to emphasize that it's induction

I've put these little delta symbols here. It's sort of a small increase in positive

charge, to be near the region of negative charge in the permanent dipole.

And when one works through the electro-statics in that one find that if

you have two systems each of which, each of which does have a permanent dipole but

the drawing here only one of them does. But if they do, this is the most general

formula. Each permanent dipole can induce some

additional dipole in the other. And the net interaction then goes as,

square of the individual dipole moments times the polarizability, that's what

alpha is. So it's the ability to be polarized.

it, that's what it's a measure of. Then again permittivity of free space.

And an r to the minus 6 dependence. And then finally, a very important

interaction that a physical chemist would call dispersion.

And that is induced dipole-induced dipole interactions.

So when two particles with no permanent electrical moments are brought together,

particles with electron clouds, then because those clouds of electrons can

move in a correlated fashion, they will instantaneously arrange themselves to

have a favorable induced dipole, induced dipole interaction.

That's an electron correlation phenomenon.

And again, the equation has an r to the minus 6 dependence.

It involves the ionization potentials of the two particles.

There are polarize abilities and again the permittivity of free space.

So, it turns out that although all of these different kinds of interactions

show r to the minus 6 behavior, really the dispersion interactions dominate.

They form up the largest percentage of the total interaction energy between two

molecules. Dispersion are important, let's take one

more moment to take a look at it. It was first

Given this relatively simple formula and described by Fritz London.

It's a purely quantum mechanical effect. It has no classical analog.

So, it happens because of the correlated motions of electrons in quantum

particles. If you just bring two uncharged classical

species together in physics, they have no electrical interaction, they have no

electrical moments. Nothing.

But when the electrons are in motion about a nucleus, you can induce these

moments. And as I mentioned on the on the last

slide, these i terms appearing in the numerator are ionization energies, and

they could be given in joules, for instance.

Polarizability, which is the, the propensity to allow and induce dipole to

be induced in your electron cloud about a nucleus, that is given in units of

coulometer squared per volt. So, a voltage would be for instance,

field that could induce a dipole. And here's the permittivity of the vacuum

or free space. And I'll just mention again, dispersion

usually the dominant contribution to the r to the minus 6 attractive interaction

that appears as the second term in the Leonard-Jones potential.

Well, let's think about somewhat simpler potentials with the motivation being that

when we plugged the Lennard-Jones potential into the relevant integral in

order to solve for the second virial coefficient we ended up with an integral

that was impossible to solve analytically.

But maybe we can gain a little bit of intuitive insight by using somewhat

simpler forms for the potential where we really can solve that integral.

And so two potentials I want to look at, briefly.

One is the hard-sphere potential, or the billiard-ball potential, you might call

it. So in the hard-sphere potential, for a

separation r greater than sigma, and sigma can be thought of as the diameter

of a sphere, beyond that. Values sigma.

There is no interaction, it's zero. So two things approach one another.

They don't feel each other at all. And then at r equal to sigma, and for all

values below it, the potential becomes infinite.

That is if you, if you think of sigma as being the diameter.

Then if I have two particles. Think of billiard balls that have a

diameter of sigma. I will be able to bring them together

until their two centers are separated by sigma, and at that point, since their

radius is half of sigma, add together two halves of sigma, you'll get a sigma.

At that point, they kiss. And they're billiard balls, they're very

hard, they don't like each other. So, they cannot go any further towards

one another, in a real system, and then they bounce off one another.

But in any case, the potential becomes infinite.

So no interaction, no interaction, no interaction, full stop, infinite

potential. So that's a very easy one to write down.

an alternative is to still have the square wall here, the repulsive wall at

sigma. So still hard-sphere contact.

But, over some interval, as the one sphere departs from the other, there will

be an attractive interaction. And it's a constant, so it's called a

square-well potential, because there is a well below zero in the potential, but it

has a, a flat bottom. And it goes for a certain distance, and

then it ends. And so if we describe that

mathematically, we'd say for r less than sigma, infinite potential between sigma,

and let's use some multiple of sigma. So lambda's just a parameter.

How far out do you feel the attraction? It is minus epsilon.

So minus, meaning it's attractive. An then beyond that multiple of sigma,

it's zero again. So let's see how those potentials behave

when we plug them into the the integral expression for the second virial

coefficient. And let's start with the hard sphere

model. So that has the simplest mathematical

formula. And let's pause for a moment to think,

when might this be a good potential? To describe the interaction between gas

molecules. And so, you would expect it to perhaps be

relatively good at very high temperatures.

At very high temperatures, the molecules are moving with a lot of speed, and so

they don't necessarily need to feel an attraction to be drawn close to another

molecule. Instead, they just keep going till they

slam into one, and then the bounce off one another.

And they behave kind of like billiard balls, if billiard balls were moving with

a lot of kinetic energy. So if the temperature is very high

relative to, epsilon over kb. So that's a, a measure of temperature.

An attractive force divided by Boltzmann's constant.

Then we can pretty much ignore the attractive force.

And only worry about the repulsive part. So if I now take the expression for the

second virial coefficient, and I simply plug in for u here, these values, I see

that really I need to do two integrals. I need to do one integral from 0 to

sigma. And I'll plug in the potential, and it is

infinite, so I get e to the minus infinity.

So that's just 0. And then here's minus one, so I keep

minus 1 r to the 2nd dr and then a second term will go from sigma to infinity.

So, I take e to the minus u but u is now equal to 0 and so e to the 0 is 1 minus

1. This entire integral drops out because I

am just integrating over 0. So, all I am left with is the integral

from 0 to sigma of r squared dr. And of course that is r cubed over 3.

And we evaluate that at it's limits and you end up after multiplying by the

constants as 2 pi, sigma cubed, Avogadro's number divided by 3.

And if you work that out, that's, that's 4 times the volume of Avogadro's number

of spheres, having a diameter of sigma. And so that's kind of a measure of

occluded volume, can be thought of. And that is, what we expect the second

virial coefficient to be a positive number, at high temperature.

Because you can't access the whole volume in an ideal gas good because there is a

finite size to the actual gas molecules. Notice that it's independent of

temperature. Alright, so even though we have here that

B2v is a function of temperature, this says it's just a constant, but if you

remember your your plot of the second virial coefficient as a function of

temperature. You'll recall that it goes up from low

temperature, goes through the boil temperature, and then flattens out at

very high temperatures and it is effectively constant.

So, this is a, a good approximation for that at very high temperature.

Now, let's take a look at the square well model.

And so, in this case, when we plug in these three different conditions.

Infinite potential inside the hard wall, a region of attractiveness, and a region

of no interaction, will have three different integrals.

Still this last one will go to 0, just as it did in the hard sphere case.

The first one will also be the same as in the hard sphere case, so here is the hard

sphere result multiplied by 1 And then if you plug in minus epsilon, for the

argument of the exponential in the second integral you'll end up with this

expression. Lambda cubed minus 1, E to the positive

epsilon over KT, minus 1. So, in essence, this is a three parameter

model, then, because we've got this new parameter lambada that tells us over what

distance does the attractive interaction persist.

And so this is just an indication of how this expression for the second variable

coefficient fits to experimental data under certain conditions.

So these are data for nitrogen gas. Measured over a range of a little less

than 100 k. It looks like maybe this got all the way

down to liquid nitrogen. that the boiling point of nitrogen is 77

kelvin. And then up, up, up to quite high

temperatures 700 it looks like on here. And the experimental data points or the

open circles. And then if we treat the three perimeters

in this expression. Sigma.

Epsilon and lambda as free parameters and just ask how, how we can best fit it.

We end up with 328 picometers for the diameter of the molecule, 95.2 kelvin for

the the depth of the well, expressed in these units, epsilon over kB.

And finally, the multiple of the diameter over which the attractive force is felt,

1.58. So, not quite two diameters away, it

ceases to be an attractive interaction. And you see, if we actually compare that

to the Leonard-Jones potential, which again involves sort of fitting to second

varial coefficient, we get very, very similar numbers for the well depth.

And, just a slightly different diameter. The Lennard Jones potential is not a hard

wall potential. It rises as r to the 12th.

So that's a little bit different. Well, so hopefully, those provide some

more insight into why the second virial coefficient behaves the way that it does.

And help to reinforce this idea that molecules at short distances are

attracted to one another. As we go to higher temperatures, that

attraction matters less as they start to interact more like hard spheres bouncing

off on another, and we will be exploring more gas behavoir from first principles

as we continue to go forward. That actually brings us to the end of the

second week of material having to do with gases, and what I want to finish up with

for this week is a review of the most important concepts.

So we'll move to that next.