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.