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

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From the course by University of Minnesota

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.

From the lesson

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

Let's continue working with the partition functions that contribute to the ensemble

partition function for an ideal Monatomic Gas.

Remember that we're working with an atomic system and as a result they're

only two contributors to the energy. The translational component, and the

electronic component. We just finished up with the

translational, so on this lecture, let's focus on the electronic contribution to

the energy. We'll start with the usual expression for

a partition function. But, in this case, I want a sum over

levels, as opposed to states. And that's dictated by the degeneracies

that may arise in electronic energy levels.

And in that case I'll have the electronic partition function a sum over levels the

degeneracy of the level e to the minus beta the energy of that level.

So epsilon being used to indicate the energy as always.

And g to indicate the degenercy. So, we can choose to set the ground state

electronic energy as the zero of energy. So thermodynamics is essentially always

about changes in properties, changes in energy, changes in enthalpy.

And as a result, where you set zero is an arbitrary choice.

But some 0s are more convenient for mathematical purposes than others.

And so we're going to adopt a convention for this system that the zero of energy

is the ground electronic state energy. And in that case, the electronic

partition function would be written, well as we sum over the first level.

It's the degeneracy of the ground state times e to the minus beta times zero, so

that'll just be one. And so the first term in the partition

function is just the ground state degeneracy.

The second term will be the degeneracy of the second state.

We could call that the first excited state.

Times the exponential of minus beta times it energy relative to the ground state

energy. So its actually the energy separation

between the first excited state and the ground state.

Notice that unlike the translational partition function there's no dependence

on volume here. Atoms have the same electronic state

nergies whether they're in a big volume or a small volume.

So the only thing that this partition function depends on is temperature,

because temperature appears in beta, beta is 1 over kt.

Well, the good news is unlike the translational partition function where we

needed to do a little bit of mathematical manipulation to evaluate an infinite sum.

The electronic partition function is a sum, which typically converges extremely

rapidly. So we only need to evaluate the first few

terms and at that stage all the remaining terms make negligible contribution.

Indeed, often, we only need to consider the first one or two terms in the series.

Namely the degeneracy of the ground state, perhaps the first excited state,

perhaps the second excited state and, and so on.

In any case they rapidly become smaller as we consider these terms.

And just as a general rule to remember, something always to bear in mind when

thinking about these things, at room temperature, roughly 300 kelvin.

One only needs to keep terms in a partition function expansion where the

energy of the term is somewhere below, about 1000 wavenumbers.

Alright, so, at that stage, if I plug in a, a relative energy to the ground state

of 1,000 wavenumbers, I'll get e to the minus beta times 1,000 wavenumbers.

So, I would express boltzmann's constant in the necessary wavenumber-like units.

And you discover it makes just about one percent worth of difference, alright?

And as you go to higher and higher energies you will finally drop below this

roughly one percent contribution. And that's not really much of a

contribution so you can cut it off there effectively.

And so if we look at actual electronic energy levels and here's a table for a

few gases and we looked at some of these in actually the very week.

And so tabulated our hydrogen and helium and lithium and florine.

what you generally find is there's some trends here if you look at the noble gas

atoms. So here's helium, that's a noble gas.

Its first excited state is 160,000 wave numbers of the ground state.

So I said at room temperature you only needed to go up about, what did we say we

should look back at the last slide, about 1,000 wavenumbers.

But here, it's 100,000 plus wavenumbers. So that first excited state makes no

contribution whatsoever. If we look at alkaline metal atoms, so

that's lethium, for example. For state, roughly 15,000 wave numbers

above the ground state. No contribution at room temperature.

Finally the halogen atoms, it does turn out that there is a first excited state

at 404 wave numbers above the ground state.

That comes from something known as the spin orbit splitting in the flooring

atom. We don't need to worry about the quantum

mechanical basis for that, but it does create a relatively low lying first

exited state. So in that case, we might look at the

second state, but now the second state is up at 100,000 plus wavenumbers, so much

too high in energy to matter. So we would only keep the first, and now

the second term would be useful for quantitative accuracy.

And so in fact I'll let you spend a moment here to work out what the

contribution to the partition function. Is for the first excited state in the

fluorine atom at a particular temperature.

So really, the electronic partition function is the simplest partition

function. Because instead of a sum of many terms,

we typically really only need to keep one, or at most two, terms.

And that will generally be true as long as we're not dealing with metals and

other special systems where low lying exited states may, may prevail.

But we're not going to see any of those as we move forward, and talk about gases.

Hence, we've got the electronic partition function, can be approximated as

degeneracy of the ground state. Plus degeneracy of the first excited

state exponential of minus beta times that excited state's relative energy

relative to the ground state. However, we should keep in mind, even

though we're going to make use of this approximation in most of our work.

If you did go to extremely high temperatures like the surface of the sun,

interesting chemistry might happen there. or very small values of excited state

energy's, which can happen in metals. Then additional terms could contribute,

and so, if when working with this particular proximation.

You were to find that the magnitude of this second term was substantially larger

than a percent or so. You'd likely want to look at the third

term and be sure it could, in fact, be neglected.

In order to find those terms incidentally, certainly for atomic energy

levels, we're talking about an atomic ideal gas here.

One kind find all the atom's spectroscopic data, the electronic energy

levels tabulated to many, many digits as you saw in a table a few slides ago.

And for molecular calculations sometimes one can find tabulated data.

And one can also resort to high level quantum chemical calculations where

they're needed. But we won't be exploring that in this

course. We are, though, at a stage where we can

finally put together the ensemble partition function.

We've got the two components we'd need. And so given that this is the

translational partition function, derived from particle in a box energy levels.

And this is the electronic partition function, derived from the analysis we

just went through. Then the ensemble partition function can

be written as a product of this times this.

All raised to the nth power, where n is the number of atoms, and divided by n

factorial. So now that we've actually got an

explicit form for the full ensemble partition function.

In the next lecture we'll take a look at using that ensemble partition function.

In order to make predictions of properties of macroscopic amounts of a

Monatomic Ideal Gas.

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