This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.
Video 3.6 - The Ensemble Partition Function
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From the course by University of Minnesota
Statistical Molecular Thermodynamics
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From the lesson
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.
Meet the Instructors
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.
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