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

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Statistical Molecular Thermodynamics

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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 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

All right. Let's finish our treatment of polyatomic

molecules. So just a review that, for the

polyatomic, we've got our four components of the energy: translation, rotation,

vibration, and electronic. The particle in a box gives us

translational energy levels. It only depends on the mass of the

particle. The polyatomic molecule that is.

And on the volume that we just make a convention choice for, which dictates

what the number is that we have for a partition function.

For the diatomic gas we had a ground electronic state that was a

disassociation energy for the polyatomic. It's the sum of all of the disassociation

energies. And we've just finished working with the

rotational contribution, and that depended on the three distinct moments of

inertia, some of which could be equal to one another, but there was no requirement

that they be equal to one another. So let's take a look now at the

vibrations. And let me remind you when we looked at

sort of the quantum mechanics of vibrations, in the first, oh actually,

the second week I think it was. we divided up all the intramolecular

motions, that's what a vibration is, into normal modes.

And each of those normal modes could be solved using the quantum mechanical

harmonic oscillator, Schrodinger equation.

And when you do that then, the energy is determined as a sum over modes of the

energy associated with each individual mode.

And so here then, the total vibrational energy is a sum over modes, j equals 1 to

alpha, so there will be alpha normal modes.

And recall, for a linear molecule, alpha is 3n minus 5 where n is the number of

atoms, and for a non-linear molecule, it's 3n minus 6.

Every one of those normal modes contributes Planck's constant times its

characteristic vibrational frequency. So I've indexed it by j, where j is the

normal mode that we're looking at. Times n plus a half, where n is the

quantum number associated with mode j. So is it in its ground state?

Is it in its first excited state? Is it in it's ninth excited state?

Whatever it is. So, n is the quantum number and u is the

vibrational frequency. Well, remember that a sum over energies

gives rise to a product of partition functions.

And so, the vibrational partition function is expressed as a product.

So, capital pi means product just as capital sigma means sum in mathematics.

And so I take a product over the alpha normal modes of the partition functions

of each of the individual modes. All right, and so, if you recall from

when we did a diatomic, this is what the partition function looks like.

And when I carry out the differentiation of the log of the partition function with

respect to temperature, in order to get the expectation value for the energy,

I'll get a complete, the equivalent expression as I got for a diatomic,

except that I'll have a contribution from each one of the normal modes.

So it's a sum over modes. And here's what you should remember from

the energy expression for vibration for a diatomic.

Similarly if I do that for heat capacity, I will again have a sum of individual

contributions to heat capacity associated with each one of the vibrations.

So, let's actually take a look at an example molecule and see how these

contributions add up. So here we have carbon dioxide.

And so physical chemists for years have loved to illustrate the vibrations of

carbon dioxide. It involves two fists and your head and

you get to move the three particles in various ways as you do the vibrations.

I can't resist doing it myself whenever I have an opportunity But for today's

purposes let's take a look at how these various normal modes contribute to the

vibrational heat capacity. So this is a linear molecule, and so

there are 3 N minus 5 normal modes, and N is 3 so 3 times 3 is 9 minus 5.

There should be 4 normal modes and what are they.

Well. There is a vertical bend and a horizontal

bend and they are being animated here. And they are degenerate with one another,

just which plane it's occurring in dictates whether we call it the

horizontal or the vertical. That would occurs with the lowest

frequency and hence the lowest vibrational temperature, 954 Kelvin.

There's also a symmetric stretch, so that's the one that chemist like, you get

to use your fists and head. The symmetric stretch has a higher

vibrational frequency and associated vibrational temperature, 1890 Kelvin and

finally the assymetric stretch is the highest frequency and it has a

vibrational temperature of 3360 Kelvin. And remember what happens as the

temperature goes from low to high and passes through a vibrational temperature.

The contribution of the heat capacity should go from zero to a factor of R.

And so what I'm showing in this plot is simply color-coded by which of these

various normal modes we're interested in.

What is the contribution as we go from absolute zero, or actually it'd be hard

to have this gas, but on a graph it's not that hard to do.

up to a temperature of 2,000 Kelvin. So what you see as expected then is that

for the lowest vibrational frequency it gets up to it's asymptotic value of r

relatively quickly. The next lowest vibrational frequency

does not ramp up as quickly but we certainly can see it asymptotically

approaching r. And as we only go out to two thousand

kelvin here, but the vibrational temperature for the highest frequency

motion is above three thousand, it's still rising steadily and it will

eventually asymptotically reach r but we're not quite there yet.

And so, the heat capacity of this gas changes with temperature.

As we go to higher and higher temperatures, the heat capacity is going

up and up as these contributions are being added to the constant, rotational

and translational, which we'll remember is, you're thinking to yourself, right, 5

halves R for a linear molecule. All right.

Well, let's put all of it together and look at the ensemble partition functions

for the full polyatomic ideal gas. In the linear case, we're going to have

for the molecular partition function, the translational component, the rotational

component, the vibrational And the electronic in the nonlinear case, the

difference is I'll, I'll express it most generally for an asymmetric top.

So the rotational piece looks a little different and the sum, the number of

normal modes in the vibrational piece is different.

And when we go and we do the differentiations necessary to move from a

full partition function to the internal energy, in this case the molar internal

energy, we get the expression shown here, so 5/2 RT for the linear case coming from

translations and rotations. But 3rt an extra half rt because of the

extra rotation in the non-linear cases coming from translation and rotation,

again a different sum over normal modes in the vibrational component.

And if we now carry out differentiation with respect to temperature of the molar

internal energy, we'll get the molar heat capacity at constant volume.

Again we'll see a 5/2 appearing in front of R for linear, a 3 appearing in front

of R for nonlinear, and different summation from the contributions of the

normal modes. So, if you find this slide mildly

imposing, I have to admit that this is one of the more beautiful slides in the

course. It certainly is loaded with Greek

letters, and Roman letters, and exponents and.

Really, it's, it's a gorgeous piece of equation writing.

No one memorizes these things, not the best statistical mechanism around, one

goes and looks them up in books. But knowing how to work with them and use

them effectively and remembering some of the conceptual aspects of how, say, a

linear differs from a non-linear is really sort of the crucial take-home.

Skill that one wants to get out of having done these derivations.

So let's pause for a moment now, and I want to see if you've acquired some of

those skills, and then we'll come back. Alright, I'd like to wrap up, by asking a

question that after you've seen a huge slide full of math will often come into

your head I think, if you're a, average everyday person and that is.

Does this all work? So, let's look at a specific example.

Let's look at a reasonably important molecule: water.

And in particular we're going to look at water at pretty high temperatures, so

here in Kelvin we're going to go from about room temperature, that's not that

interesting a temperature but then we're going to take it up to looks like that's

about 373 kelvin right there, so that's the boiling point of water.

And then we're going to keep taking the temperature up.

So we're working with steam, and what I want to know is what's the heat capacity

of my steam associated with the vibrational contribution to the heat

capacity. And I'll express that then in terms of r,

how many factors of r are being attributed.

So remember that water has 3n minus 6. It's a nonlinear molecule, normal mode.

So there's three normal modes. It can contribute, you know, a fair

amount to the total heat capacity. And the, rotational, sorry, excuse me,

the vibrational temperatures of those modes, are given here.

2290, 5160, 5360. So, the heat capacity, actually I think

the slide is labeled incorrectly here. This is the full heat capacity.

So we're starting at 3. So 3 is the 3R associated with

translations and rotations. But we're now going to add to that, as

the vibrations begin making their contributions.

Because we're moving the temperature up. 'Cuz at low temperature, they contribute

nothing. So we ought to get from three up to six.

Six would be an asymptote, which is off the scale of this slide.

But in any case, the calculation that comes.

From this expression that we have derived from first principles, if you like.

We did quantum mechanics. We worked out partition functions for

ideal gases. We made a prediction of what the heat

capacity should be. Well that's the line so I'm just plotting

this function. The Points, the open circles.

Those are actual measurements of the heat capacity of steam, so very hot steam.

And what you see is, they fall very, very closely on the line, effectively.

So, you might have thought to yourself, well, steam?

How ideal a gas would steam possibly be? Well, surprisingly, the ideal gas

approximation is. working pretty well here.

Maybe it shouldn't be surprising, maybe people thought it would be an ideal gas.

But I think most people would say, oh you know water molecules they interact with

one another. The ideal gas approximation says that

they don't interact with one another. The quality of the ideal gas

approximation is really remarkable. And finally, I want to come back again to

talk about building a hammer last time. I think we might have a screwdriver here

as well. Because what does this graph really give

you? Now I, somebody did the measurements.

That's where we got the points from but let's say you hadn't done the

measurements ahead of time. You wanted to know, the answer to a very

practical question. How much coal do I have to burn in order

to get my steam from 700 degrees to 800 degrees because maybe I want to power a

steam engine and I think hotter steam is going to give me more energy?

Well, you would have been able to compute it from first principles.

You would predict the heat capacity and then you'd know how much energy it takes

to raise the steam another hundred degrees.

And so that's an extremely powerful tool of thermodynamics that can be put to

practical use. So there is our steam engine, I didn't

pick this steam engine randomly. It is a representative of the power of

thermodynamics. It's pulling us further down the tracks.

And we've pretty much wrapped up this week's, material, new material, so we'll

spend one lecture reviewing what I think are the critical concepts before moving

on in the course.

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