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.

从本节课中

Module 4

This module connects specific molecular properties to associated molecular partition functions. In particular, we will derive partition functions for atomic, diatomic, and polyatomic ideal gases, exploring how their quantized energy levels, which depend on their masses, moments of inertia, vibrational frequencies, and electronic states, affect the partition function's value for given choices of temperature, volume, and number of gas particles. We will examine specific examples in order to see how individual molecular properties influence associated partition functions and, through that influence, thermodynamic properties. 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

For the last few lectures, we've focused on monatomic ideal gases.

And actually, a couple of examples I offered you were lithium and fluorine.

These are unusual gases to say the least. You wouldn't find them very typically.

ideal or not, monatomic gases are relatively unusual.

I guess there's helium in balloons, that's not so uncommon.

And if you're a chemist in the lab, you might occasionally put argon in a balloon

as an inert gas, but really, most interesting gases are not monoatomic.

But, there are a lot of interesting diatomic gases.

Oxygen, nitrogen, chlorine and fluorine, as actual, gases, are diatomic, they're

not very friendly gases, but in any case, there's a lot of diatomic gases out

there. So let's take a look at the partition

functions for diatomic gases and how we can use those to predict their

properties. Well, what goes into the energy of an

ideal diatomic gas? Things get more interesting than was the

case for the monatomic system. Because, in addition to translational and

electronic degrees of freedom, which we had before for the monatomic case, a

diatomic can also rotate, so here is a little rigid rotator rotating in space,

and it can vibrate. So the two atoms can move relative to one

another along their bond axis. Each of those two kinds of motion,

rotation and vibration, can be treated within a quantum mechanical approach.

It's called the rigid rotator problem, and the harmonic oscillator problem.

And actually in week one, I presented you with the results from those quantum

mechanical analyses, with respect to allowed energy levels, and degenerasi of

those energy levels. So, the diatomic energy then is going to

be composed of the total energy. It's composed of a translational

component, a rotational component, a vibrational component, and finally, also

an electronic energy component all summed together.

So, we'll start, as always from our most general expression.

It is an ideal gas, so the particles are non interacting, which means the ensemble

partition function can be written as a product of molecular partition functions.

I can finally stop saying atoms and molecule here, because we are talking

about diatomics and highers here. Always molecules, molecular partition

function. So, that molecular partition function

itself is a product, because there's a sum of energies involved, of the

component partition functions: translation rotation, vibration,

electronic. Let's actually consider those degrees of

freedom separately. The translational partition function is

identical in all respects. It's the same problem.

It's a particle in a box problem. So identical in all respects to that we

saw in the monatomic case. The only difference is the total mass of

the system, the particle if you will. It is now the sum of the two atomic

masses in the diatomic molecule, compared to just the mass of the atom.

So I am going to label my atoms with these little, little tiny font letters m1

and m2, but they look better here. Those are the masses of the two

individual atoms. Obviously if it's a homonuclear diatomic,

that's the same mass twice, but it doesn't have to be.

Carbon monoxide is a perfectly nice gas. Well it's not that nice if you're

breathing it but it's a perfectly acceptable gas as a diatomic.

And carbon is obviously different than oxygen in properties and mass.

What I want to focus on in this lecture is something we haven't seen yet and that

is the vibrational partition function. So, let me begin by drawing a series of

energy diagrams as a function of the distance R separating the two masses.

Alright and I'm going to skip for a moment, I'm going to save for a later

lecture the question of rotations. I will remind you that we saw in the

first week what the rotational energy levels are.

They're equal to hr squared, over twice the moment of inertia, times a rotational

quantum number, times the quantity of that quantum number, plus one.

So we've seen that and we will work with it in a future lecture but now I want to

focus on the vibrations. Vibration occurs within a potential well.

So here is a potential energy as two molecules approach one another, the

energy goes down as they interact in a favorable way, but ultimately you get

them close enough to one another that they don't like interacting anymore.

They begin to repel one another. And so this looks a little like the

dispersion attraction and then the hard wall repulsion we saw say, in the

Vanderwall's sorry the Leonard Jones equation and potential.

However for a real chemical bond, usually the depth of that well is much greater.

Now before talking about the vibration in the well, let me focus on a special

point. And that is the asymptote we would call

this at very long distance. So at very long distance the 2 atoms do

not interact at all. They're too far apart to feel one

another. It is convenient to declare that to be

the zero of electronic energy. The two atoms in their respective ground

states not interacting. So in that case, the electronic energy of

the bound molecule, which is shown here, is called minus D e.

So de showed here by convention, we call that the dissociation energy of the

molecule. And by convention we take that to be a

positive number. You can think of it as the energy you

need to pump into the molecule to blast it apart.

And so if we want to think about the binding it's minus de.

And so the electronic energy that would be used in the partition function for the

electronic component is going to be the degeneracy of the ground state.

We'll usually be working with closed shell so called singlet molecules that

degeneracy will be one, but to be general we could write it.

G1, e to the minus the energy, but the energy is itself a minus de, so we just

get a positive value here. e to the de over kT.

Okay, and then there would be additional terms that would involve the first

excited state, the second excited state, et cetera, and that's actually what I've

shown up here. So if this is the molecule in its ground

electronic state, I could go way, way, way up in energy from epsilon 1, the

first electronic state, the ground state, to epsilon 2, the first excited state.

Maybe it's got a potential that looks like this.

So the atoms are still stuck to each other, in this electronic state.

Maybe it takes the same amount of energy to pull them apart, maybe it takes more,

maybe it takes less, you would just have to look at the molecule to know.

But in any case, that's what's up here. A very high energy electronic excited

state. So it would have an energy that would be

dictated by this difference added to D e. Okay, well that sets rotation and

elctronic, now let's really zero in on vibration.

So remember that there is zero point energy associated with molecular

vibrations. That is the ground state energy level is

not zero, instead there is h nu over 2 so one half h nu, h is Planck's constant and

nu is the frequency of the vibration. So there's that much energy in the ground

state vibrational wave function level. So in that case, the allowed energy is

the accessible energies of the system, are h times nu times n plus a half, where

n is the quantum number. So even when n is equal to zero, you'll

get one half h nu. Those of you with a really good memory

will recall that in the first week of the course, when I showed these allowed

energy levels for the harmonic oscillator, instead of n, I used V.

And I told you that you would just have to put up with the fact that physicists

and chemists for years have been using V, and even though V looks a lot like nu,

you would just have to suck it up and live with it.

But, I decided that, let's break from convention, and let's just try to be more

clear. So I'm going to use n instead of V to

index these quantum levels. It's going to be a lot easier to spot the

difference, and I think we'll all be happier as a result.

Of course, you're not allowed to confuse n here with n, say, the hydrogenic atomic

energy levels, but I don't think you'll do that.

I think it'll all work out well for everyone involved.

So just again recapitulating that. The harmonic oscillator approximation

says, energy levels h, nu times n plus a half is are accessible.

All the levels are non-degenerate. That's a, a feature of the harmonic

oscillator. Nu is the vibrational frequency.

And n is the quantum number. And probably you didn't need to know how

to pronounce n but that's just to distinguish it from the v that was on a

slide a long time ago. Great well lets actually then put this

into the partition function expression. So I want to sum over quantum numbers, e

to the minus beta times allowed energy levels.

What are the allowed energy levels? n plus a half h nu.

It's just a, I've written the h nu over on the other side here, in this part.

Now, let me note that I've got an exponential of a sum, so I can pull out

the various parts. I'll take a beta h nu times n, so here's

that piece. And a beta h nu times a half, and here's

that piece, and the reason I want to do that is this is just a constant.

It depends on temperature, but it doesn't depend on the quantum number n, alright?

So nu is the vibrational frequency. h is Plonck's constant.

Beta is 1 over kT. I can pull that whole thing out in front.

It multiplies every single term. So, really I'm summing over this

expression, e to the minus beta h nu and then here's n.

Well, so, here's the good news. That partition function does not converge

so quickly that we only need to evaluate the first term or two.

It also is not so dense in energy levels that we can approximate it with an

integral. However, the universe has a way of

rewarding the perserverant. It turns out that that sum has an

analytic solution. And, to, to recognize that solution, I'll

recall for you that if you look at the sum from n equals 0 to infinity, of x to

the nth power, as long as x is less than 1, that is equal to 1 over 1 minus x.

And so, e to the minus a positive number times a positive number times a positive

number will always be less than 1. And as a result we can treat e to the

minus beta h nu, that's the thing being raised to the nth power, right?

Because, if you take an exponential to the nth power it's like, taking the

exponential of the argument multiplied times n.

So, it, exactly fits this expression, and as a result, the vibrational partition

function is this thing that was multiplying this sum.

So that's still out there. Here's my e to the minus beta h nu all

raised to the nth power, so I'll put this down in the denominator.

Here's the constant that came from on top.

So here's our partition function: e to the minus theta h nu over 2, all divided

by 1 minus e to the minus beta h nu. So that's a nice closed form expression,

and just requires us to know the vibration associated with the particular

diatomic molecule. So I"ll let you get a little more

familiar maybe with that partition function by doing a problem designed to

employ this nice closed form and illustrate some features of the

vibrational partition function. Well let's wrap up our work with the

vibrational partition function, with, a little bit more, notational

simplification. But hopefully that'll also offer us some

qualitative insight into the behavior of the molecule.

And so, let me note for you that, it's common to define something called the

vibrational temperature. And we indicate that with theta, with a

subscript vibe for vibration. And so the vibrational temperature is h

times nu, the vibrational frequency, divided by Boltzman's constant.

And so notice that it has units of temperature because Plonck's constant is

Joules seconds, nu, a frequency is in per seconds so that would give us Joules in

the numerator. Boltzmann's constant is Joules per

Kelvin, so we have a per Kelvin in the denomenator, which is like, Kelvin.

Right? So there's a temperature that can be

related to a vibrational frequency by manipulation with the appropriate

constants. And so just for example at 298 Kelvins,

that's room temperature. If we ask about what's sort of the

available energy that is what's kT when expressed in wave numbers.

It's about 200 wave numbers. Wave numbers is also perfectly good unit

to express a vibrational frequency. And so, a vibrational frequency of 210

wave numbers, to pick a, very near value, has a vibrational temperature of 302

Kelvin. And so, it's just a linear relationship,

it's not too challenging. So here is the frequency expressed in

wave numbers. Here is the vibrational temperature,

expressed in Kelvin. And you can just read anywhere along this

graph to relate the two. And so here's the particular case of 210

wave numbers related to, excuse me, here we have 210 wave numbers related to about

300 Kelvin. And the reason that's moderately

convenient is it lets us write the vibrational partition function in a,

slightly less imposing form. So instead of e to the minus beta h nu

over 2. I'll divide it by quantity one minus e to

the minus beta h nu. Well everywhere that h nu divided by

Boltzman's constant. So remember Boltzmann's Constant is

hiding in beta. Beta is 1 over kT.

So here's an h nu over k. All that's left is the 1 over T.

So that transforms the vibrational partition function, which depends on

temperature. This really makes the temperature

dependence explicit. It's e to the minus the vibrational

temperature divided by 2T, all divided by the quantity 1 minus e, minus the

vibrational temperature over T. So, with that partition function in hand,

we can now do all the things that we do, with partition functions.

For instance, we can compute the contribution of the vibrations to the

average energy. So the same thing we've been doing again

up till now, we're going to take nkT squared times the partial derivative of

the log of the the partition function with respect to t.

I'll now write out that partition function, and I'm going to let you do the

differential calculus at home today, it's actually a worthwhile exercise.

This is not the most trivial, but neither is it the most difficult, so sort of fun

to do on a bus while travelling somewhere.

Everybody does derivatives on a bus. Don't you?

I know I do. But in any case, the contribution to the

internal energy nK quantity vibrational temperature over two, plus vibrational

temperature over e to the vibrational temperature divided by T minus 1.

So here's a term that depends on the temperature.

So this internal energy is temperature sensitive, and if we now differentiate

again with respect to temperature, that will give us the constant volume heat

capacity. Again, I leave the differential calculus

for the interested viewer. But I'll present you the result.

It is, this is a molar quantity I've tabulated here, so with Avogadro's number

involved for n, this will become R, theta vib over T squared, all times this

quantity. So, looking at those functions, it's

probably not, entirely intuitive exactly how they would behave.

Let's actually plot them. So let's look at the diatomic vibrational

heat capacity. First off, I'll call your attention over

to the left here. So this is temperature divided by the

vibrational temperature. That is I'm going to vary the temperature

of my gas, compared to the constant vibrational temperature for whatever the

gas is. So, whatever vibrational frequency that

diatomic has, that's a constant that defines a temperature.

But I'll change the bath temperature in which my gas exists in order to either be

less than the vibrational temperature. That's over here from zero to one.

Or greater than by a factor of two, by a factor of three and this line defines

when I have my external temperature exactly equal to the vibrational

temperature. And look what happens to the heat

capacity. So at very, very low temperature, there

is no contribution to the heat capacity. And as I raise the temperature, it goes

up, up, up, up and it eventually plateaus and I'm actually plotting the constant

volume vibrational component of the heat capacity, divided by R.

And so the fact that it's plateauing here, at a value of about 1.

So, I will just draw that in specifically.

Says at very high temperature, a diatomic contributes R to the constant volume heat

capacity. So that is the classical limit, that is I

am, I am able to access all the vibrational states because I'm at a

really high temperature. And that requires me to put extra energy

into the system to raise the temperature. So just like in the electronic case I

discussed for florine, when you put in energy, it will try to explore all the

accessible levels. When it goes into the translational

levels, that raises the temperature, and you have a heat capacity associated with

that. But if there are other places where

energy can be absorbed, electronic we looked at for fluorine, here's a case

where it's vibrational, levels that can be accessed with increasing energy.

And so you need to put extra energy in R worth for a vibration in order to

continue raising the temperature. Now, if we also take a look, this is just

a different plot, I'll continue to plot the vibrational contribution to the

constant volume heat capacity, divided by R.

So again, everything plateaus at one. But this is for different vibrational

frequencies. So what would one expect?

If you have a really low vibrational frequency, that implies that the levels

must be spaced relatively densely and we'd expect to be able to access those

levels relatively quickly. So we would expect to see this plateau

behavior occur at pretty low temperature. On the other hand, by the time I get up

to higher and higher frequencies, 750 wave numbers is the largest one on this

plot, well now my energies are spaced further apart, and as a result, it's

harder to populate them, and I'll put more of my energy into translation before

I get into those levels. And so sure enough the blue curve, the

low frequency, plateaus very quickly, the green curve less quickly and the red

curve only relatively slowly, although it will eventually hit the same asymptote.

So that gives you a feel for how the vibration can store energy and how the

vibrational frequency dictates how readily it stores energy as a function of

temperature. Alright that completes our investigation

of vibrations as they contribute to the diatomic molecules ensemble partition

function. We're going to continue next time and

look at the other components as they contribute to that ensemble partition

function. See you then.