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

This week we'll be spending some time on truly fundamental statistical mechanical

concepts that underlie our coverage of statistical thermodynamics.

So I want to pause for a moment before diving in to say a little bit about the

history of the field as it developed. And, let me begin by offering a quote

from Albert Einstein who said, a theory is the more impressive the greater the

simplicity of its premises is, the more different kinds of things it relates, and

the more extended is its area of applicability Therefore, the deep

impression which classical thermodynamics made upon me, it is the only physical

theory of universal content concerning which I am convinced that, within the

framework of the applicability of its basic concepts, it will never be

overthrown. And so that's a fairly strong statement

from someone who ought to know pretty well.

And an indication of how powerful thermodynamics was viewed to be by

physicists in the 19th and 20th centuries.

And so thermodynamics, as it was originally developed, what Einstein was

referring to as classical thermodynamics. Is simply the study of energy and it's

transformations. And a great deal of progress was made on

thermodynamics prior to any atomic theory of matter being accepted.

So, classical thermodynamics then, encompasses a powerful set of laws, many

of which, we will in fact. Develop and encounter and employ, but

none offering any kind of molecular insight.

Concomitant with the development of an atomic and molecular understanding of

matter which did not come easily, there were many people who opposed the very

idea of molecules and atoms. So called positivists who felt, if you

couldn't see it, you couldn't measure it, and tangibly interact with it, and

molecules and atoms were much too small to do that in the 20th century, the 19th

century. that you were not allowed to posit them

to assume their existence. And Ernst Mock, actually, in Vienna was

a. Massive proponent of this ides and fought

with [UNKNOWN], who will see in a moment for an enourmos amount of time over the

very idea. Now what seems curious to us, we accept

it but at the time atoms and molecules were quite controversial.

But as our understanding of matter grew with an atomic and molecular theory,

statistical thermodynamics was developed to connect the microscopic properties of

those species, atoms and molecules, to the already established macroscopic

behavior of substances. And so most simply put then, statistical

thermodynamics relates the averages of molecular properties to bulk

thermodynamic properties like pressure and temperature and enthalpy, things

we've talked about already. And many more that we'll see as we

continue. And so here is Ludwig Boltzmann.

He lived from 1844 to 1906. Committed suicide in 1906.

He suffered from bouts of depression and modern scholars imagine that he probably

suffered from bipolar disorder. But during his enormously productive

career. He developed, many of the key concepts

under pinning statistical thermodynamics and the Boltzmann equation s equals k log

w is one of the most famous equations of physical chemistry, statistical mechanics

what have you. So, in 1877, he wrote that down where S

is entropy. A phenomenon that we'll deal with in a

few weeks. And the K,B is Boltzmann's constant and

you see it here on the slide in units of Joules per Kelvin.

And it takes on that particular value. And W stood in German for

Wahrscheinlichkeit and it can mean a number of things in fact, in that

equation the number of micro-states, the disorder, the degeneracy.

That German word actually is well translated as Likelihood, probability,

but maybe less probability is the standpoint of mathematical, although

obviously, this is a mathematical equation and more in the standpoint of

likelihood. But in any case the equation then shows a

relationship between the likelihood of a number of things and a concept known as

entropy. And the actual value of the Boltzmann

constant is such that when multiplied times avogadro's number it gives the

universal gas constant. And just to emphasize how important that

equation is after Boltzmann death, if you look carefully here, you will see it

engraved on the monument that that adorns his grave.

And so you know you care a lot about an equation when you take it into eternity

with you. So let me discuss a concept that I hope

will be tangible enough to not be purely mathematical, and hopefully give us some

insight into the statistics that we're about to discuss.

And so I'd like you to imagine. A very large water cooler.

A water cooler that is nearly infinitely large, arbitrarily large, but not

necessarily infinite. And it's a water cooler, it's held at a

constant temperature. It's full of bottles, so I will call the

temperature t And let's use bottles of 330 milliliters a sort of standard

bottle, which contain about 10 to the 25th molecules in about that much volume

if we're thinking about water for example.

And those molecules are in a bottle and they're interacting with one another.

So we know from quantum mechanics, that there's an enormous set of allowed

energies associated with all those molecules of accessible energy states.

And the energies themselves will be a function in each bottle of N, the number

of molecules in the bottle, and V, the volume of the bottle.

Alright, so I'll just indicate here an allowed set of energies.

And now I want to ask a question. I reach into my infinite water cooler,

and I pick out a bottle of water. And the question is, what's the

probability that the one I pick out will have a specific energy?

So I've indexed it here by I. It'll be in state I.

That's what I'll call it. That has energy, E sub I.

So that's a worthwhile question to think about.

What's the likelihood I get a certain energy.

Well, let's use another number, A sub I. A sub I is how many bottles in the

ensemble, in the infinite water cooler. Have that energy, E sub i, given fixed

number of particles. There's always the same number of

molecules in the bottle, and fixed volume, it's always the same-sized

bottle. And, in particular, let me think about

two different energies, perhaps. So I'd like to know the number of bottles

having one energy versus the number of bottles having a different energy.

So I'd like to know the ratio of the number of bottles in two different

states, j and k. And that'll be given as a function of the

energy. Namely, and I'll just write it

mathematically here, aj over ak is some function of ej and ek.

Now it seems pretty clear that this shouldn't depend on where I set my

arbitrary 0 of energy. So there's a ground state perhaps that a

bottle could have. I could call that 0 but I could call many

things 0. I would expect the ratio between of 1.

Set of bottles to a different set of bottles, is independent of the actual

number I used to describe the energy. Instead, what ought to be true is, it's

got the depend on the difference in energy, the difference in energy is

independent of how I assign zero because it drops out when I take the difference

between two energies. So I'll just repeat that equation then

that the ratio of aj over ak, number of bottles in state j divided number of

bottles in state k, some function of the difference between the two energies.

Well, let's think about another energy then, in another state of the system.

I'll index it by L. So, if I want to know the ratio of J to L

and K to L, number of bottles in their respective states.

Well, that, I'll just, all I'm doing is changing the index on this equation.

So I have AJ over AL as a function of EJ minus EL.

And now it's K over L so it's a function of EK minus EL but AJ over AL, is

going to be equal to AJ over AK the ratio of J to K times the ratio of K to L.

Lets all just replace these two ratios J over K and K over L.

By their functions, whatever they are, functions that depend on the difference

in energy. But that allows me then to equate.

Here;s aj over al. That's this.

Function of ej minus el is equal to the product of these 2 functions.

So let me move to a new slide so that I can have a little space to do a little

more mathematical work. And, let me give these quantities inside

the parentheses simpler names. I'll call Ej minus Ek x, I'll call Ek

minus El y, well notice that Ej minus El is equal to X plus y.

And so, what function is it true for that the function of x plus y is equal to the

function of x times the function of y? It's actually the exponential function

that has that property. All right, so the product of two

exponentials is equal to the exponential of the sum.

And that implies that our original number of states for a given energy must be

expressed as an exponential. So exponential minus the energy of the

state, some factor beta, which is a constant that we haven't determined yet

because I want to have the most general equation I can have.

Some other constant that may have to do with just how big the water cooler is and

notice that these constants c and beta they must be positive constants.

Why must beta be positive? Well if beta isn't positive, then as the

energy goes higher and higher and higher, remember there is more and more

accessible states as you go up we would be getting e to a positive value and just

go off to infinity and I don't have an infinite water cooler I've got a really,

really large one, but it's not infinite. And, for the same reason, notice that the

energy must be bounded from below. That is, once beta is taken to be a

positive constant, I can't let my energy go to negative infinity, because then

I'll have negative of negative infinity is positive infinity, times a positive

number, again, I would explode. I'd go off to an infinite system.

And so that's an interesting interesting proof, to some extent, that everything's

got a ground state and you can assign a value to that however you like but there

must be some ground state. and similarly, c must be positive because

we don't have negative numbers of bottles.

Alright, well, that's the beginning of some statistics.

The important thing to note there is that the ratio of energies that is that is

dictated by our statistics is such that there is an exponential dependence on the

number of things at a given energy. And the number becomes smaller and

smaller as the energy goes up because of the form of that equation.

So we're going to work with that a little bit more in the next lecture and that

will focus on Boltzmann population.