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

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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 3

This module delves into the concepts of ensembles and the statistical probabilities associated with the occupation of energy levels. The partition function, which is to thermodynamics what the wave function is to quantum mechanics, is introduced and the manner in which the ensemble partition function can be assembled from atomic or molecular partition functions for ideal gases is described. The components that contribute to molecular ideal-gas partition functions are also described. Given specific partition functions, derivation of ensemble thermodynamic properties, like internal energy and constant volume heat capacity, are presented. 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

Alright, now that we've had a chance to explore the utility of partition

functions for predicting gas properties, let's take a deeper dive into the

partition function itself. And in particular, the ensemble partition

function. First let me note, sort of

philosophically, that the ensemble partition function capital q.

Is to statistical mechanics what the wave function, usually abbreviated as capital

Psy is to quantum mechanics. That is, it plays a central role and it

allows one to determine all of the properties that you might be interested

in, for a macroscopic system. Just as psi, the wave function contains

all the information necessary to determine quantum mechanical properties

for microscopic systems. So the Q, that we've actually been

working with so far, and I've just called it a partition function, which is fine

But we can be a bit more specific. It should in fact be called the conical

partition function. And the ensemble that we've been working

with, you remember the very large water cooler, is the canonical ensemble.

And what that means, that adjective canonical, is it's telling you which

variables were we holding fixed in our ensemble.

So we were holding fixed the number of particles.

The volume and the temperature. We will look at some additional ensembles

later on, but the canonical ensemble is a very important and useful one.

So, Q is needed to compute these macroscopic properties.

We've played around with it a bit when we've been handed certain ensemble

partition functions. But for an arbitrary system, what you

need in order to have all the relevant energy levels is all of of the

eigenvalues, that is all of the solutions to its N-body Schroedinger equation.

Where N is the number of particles which is presumably a very, very large number.

So that is rarely practical to compute, where rarely practical means essentially

never. It's, it's just about impossible.

Happily Q, capital Q, can be approximated based on results for individual molecular

energy levels as opposed to needing to know about all the energies for an

interacting system. Under favorable circumstances.

So let's look more carefully at this issue of the ensemble partition function

in terms of a molecular partition function, little q.

So given a system of distinguishable, that is I can apply labels to them and

tell them apart. Non-interacting.

And so that's like an ideal gas. Doesn't int, interact; the individual

molecules have no interaction with each other.

Identical. So every molecule is the same molecule.

Given that system of distinguishable, non-interacting, identical particles the

ensemble partition function can be written as a product of the individual

molecular partition functions. Let's look at why mathematically.

So the ensemble function is sum over all accessible energy states e to the minus

the energy of that state. However, what are the accessible energy

states? Well, the molecules don't interact with

one another. So the total energy is just the sum of

all their individual energies. So I've got that here, I've got the

energy for molecule one. And I'm able to label them, molecule one,

molecule two, molecule three, out to molecule N, divided by k t.

Okay, well this is the exponential of a sum, so that's equal to the product of

exponentials, and if I consider all the possible energies that every one of these

molecules can take on That'll give me the product of a sum now over, my subscript

here actually contains the label for the molecule.

So this is for molecule 1. I'll run over index j e to the minus all

the possible energies for that molecule. And the same for two, and the same for

three, and so on, out to N. So products out of this sum will make

every one of these things appearing in this sum.

And they're all the same molecules, so this term in brackets is exactly equal to

this term in brackets, and so on. They're all the same, they're just differ

by the label on the molecule. So it is the partition function, because

that's what this is. Right?

It's a sum over accessible energy states, e to the minus energy, divided by kT.

That is a partition function. Just happens to be the partition function

for the molecule. So it is, molecular partition function,

which depends on V and T raised to the nth power.

So here's where the n dependance comes in.

The molecule doesn't depend on n. There's one molecule.

The ensemble partition function depends on N because we're to the Nth power.

Right, a key feature here is, the non interacting aspect means that this energy

is just the sum of individual energies. And the identical character of the

molecules let's us go from this product to this exponential.

Finally, again, emphasizing little q only requires you to know about the allowed

energies of an individual atom or molecule, not about interactions between

them. Alright, well, let's pause for a moment

and I'll give you a chance to work on a problem, and then we'll return.

Let's talk about indistinguishability. So this result for the ensemble partition

function, that it's simply the molecular partition function raised to the Nth

power. It's very pretty and nice and reasonably

easy to work with. But it's only sometimes correct and the

problem is, that atoms and molecules are typically indistinguishable.

So, gas molecules, liquid molecules, they're in a homogeneous mixture, they're

moving around, there's no way to label them, Unless you want to change the

number of neutrons in their nuclei, but then you'd need a different kind of

partition function we're talkin about everything being the same, identical

means indistinguishable. And so just to provide a concrete example

to allow us to talk about things, imagine that you have two particles.

Each has an energy, and there are only three energy levels available to each of

these two particles, call them e1, e2. I've actually got epsilon here.

I'll use a Greek letter for an individual atom or molecule, epsilon 1, epsilon 2,

and epsilon 3. So how many different ways can they be

arranged if they're distinguishable? Well if there distinguishable molecule

one can take on any of three energy levels, and molecule two can take on any

of three energy levels, so there's 3 times 3, there's 9 possibilities, right?

And so the partition function. And this is the partition function

expanded in sort of a graphical way, so the ensemble partition function is the

sum over all possible energy states, but the possible energy states are just the

sums of the two individual energy states. So, I can write that as a exponential of

a sum, as a product of exponentials I'll just separate that into a product of 2

summations. Each of those is an individual partition

function. This is just what we did before, except

you know, step by step in, sort of gory detail.

Q squared, Q to the N. Alright, that's, that's this upper result

here. And so, what are the possible states?

1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 2, 3, 3, 1, 3, 2, 3, 3.

Not really any rocket science, yet. However, if we think about the energies

of some related states, one and two have the same energy as two and one.

All I've done is reordered the labels on the particles.

And the same for one and three and three and one.

And the same for two and three and three and two.

So if the particles are indistinguishable, I'm not allowed to put

those labels on, then I need to throw away the duplicates.

So that takes my nine terms down to six therms.

And so I'll keep the one where the first label's smaller than the second label.

So I got 1, 1, 1, 2, 1, 3, 2, 2, 2, 3, 3, 3.

Six things left. So let's bring those 6 things to the top

of the next slide. Now what if no 2 particles can be in the

same state. So that's known as fermion statistics or

fermionic behavior. So fermions are particles that our

universe is made up of. That follow a certain rule, and that rule

is no two fermions can be in exactly the same state, and sometimes people would

say characterized by the same quantum numbers.

So, if that's true, and what are typical fermions, electrons, for example are

fermions, but in any case, when that's true, more states drop out.

I can't have 1-1. I can't have 2-2, and I can't have 3-3.

So my original nine reduced to six are now down to three.

So here are the three states I'm allowed to have left.

One two, one three, two three. Unique for indistinguishable fermions.

And, incidentley, even for bosons And so the boson is the other kind of particle

in the universe besides the fermion, where it's okay for two bosons to be in

exactly the same state. However, even if they are allowed to,

it'll be very unlikely to find two particles in the same state if The number

of available states vastly exceeds the number of particles.

So we're going to see that that is typically true.

But for now just sort of accept, right if I've got a zillion boxes and 5 particles

to throw into the, the odds that any 2 land in the same box are going to be

very, very small. So in that case, pretty much all of the

over counting of the unallowed terms in this simpler expression for the partition

function comes from failure to consider the permutational symmetry in the

labeling of the particles. Right?

And let me make that more clear with an equation.

So, here I have the ensemble partition function exponential of a sum of a whole

bunch of energies for individual particles.

Now, if all the energy levels are different for the individual particles.

That is there is every one of these epsilons is a unique number.

Well, out of the n possible numbers, because there's n particles, this first

one can take on n values, can be any one of those.

Second one of course, you've now used up one of your n values, so it can take on n

minus 1 values. And finally of course by the time you are

down to the last particle. It only takes the energy level that's

left. There's only one way to do that.

So there are n factorial different ways then, to make the same contribution to

the ensemble partition funcion. And if we choose to take q as the product

of the molecular partition functions, that is capital q the ensemble as that

product. Each being allowed to take on the various

values will over count that term. Will make it again, and again, and again

with different permutations. How many permutations?

N factorial of them. So we should divide by n factorial in

order to remove that that non-allowed distinguishability.

And come up with an ensemble partition function capital Q.

And we actually employed this. I told you I'd derive it, and now we

have. in, and it is appropriate then, for

Fermions and bosons, when they are indistinguishable particles.

So let me return to this question of, is it possible for two particles to be in

the same state, and the same energy level.

And the rule that one can use is, if this expression here, number of particles

divided by volume, times Planck's constant squared, divided by eight times

the mass times Boltzmann constant, times the temperature, all raised to the three

halves power. If that expression is considerably less

than one then Boltzmann statistics are valid the division by n factorial to

compute the ensemble partition function is valid the particles in the system will

obey Boltzmann statistics. Let's just explore how often that might

happen that this inequality is satisfied. And let's just take a simple system.

Molecular hydrogen as a gas, at one bar pressure.

And 300 Kelvin. So, under those conditions.

We can replace N over V using the ideal gas equation of state as P over kT.

And that allows us, then, to derive. The pressure in SI units would be 10 to

the 5th Pascal. The Boltzmann's constant is expressed

here, and the temperature. And that all adds up to 2.414 times 10 to

the 25th per cubit meter. Meanwhile, if we evaluate the quantity in

parentheses, we have h squared over 8 m k t.

Here's Planck's constant squared, 8, the mass of molecular hydrogen is 3.35 times

10 to the minus 27th kilograms. Here is Boltzmann's constant, here is 300

kelvin again. And the net when raised to the three, has

power 2.486 times 10 to the minus 31st cubic meters.

Multiplying these two quantities together, we end up with the full

expression is equal to 6 times 10 to the minus 6th.

Right, so six one-millionths, which certainly is a great deal less than one.

Then hence, we would expect Boltzmann's statistics to be satisfied for hydrogen

gas at this pressure. And, indeed, we see, in order to get up

to a value closer to one, given that pressure appears here linearly.

And this is about hm, it's about a, let's round it and say it's one one hundredth

thousands. That would be ten to the minus fifth.

You would need to go to a hundred thousand times higher pressure.

That'd be one hundred thousand bar. So that's a whole lot of pressure, and

obviously everyday systems are not under that kind of pressure.

Alright we've worked with the ensemble partition function.

That's great and now have a better feel for that en-factorial in the denominator

and its dependence on the molecular partition function.

So next we're going to look at the molecular partition function and talk

about its construction.