This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.
Video 3.1 - Boltzmann Probability
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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.
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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.
