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

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分子热力学统计

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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

Now that we've done the ensemble partition function, let's look more

closely at the molecular partition function.

So, remember that the probability that a member of an ensemble is in quantum state

j. Is given by P sub j, that's the

probability. The exponential of minus beta times the

associated energy, divided by the partition function, that is the sum of

those exponentials over all possible states.

So, similarly, the probability pi, I'll use the Greek letter here for a molecule,

so pi sub j, hat a molecule is in its j molecular energy state, is given by a

completely analogous formula. But instead of using a capital E for

energy, I'll use a lower case epsilon, just to emphasize it's molecular.

But it still has energy units, it's for the molecule.

So, the difference then would be an ensemble energy versus a molecular

energy. Now by analogy, the probability that a

molecule is in some vibrational state, because there are many contributors to a

molecule's energy. So we can talk about the probability with

a superscript for the particular kind of energy level, vibration here.

That'll be e to the minus beta and all I'm doing is calling out that this energy

is associated with a particular component of molecular energy.

And of course you can write completely analogous expressions, where the

superscript would not be vibe, it might be elec for the electronic energy levels,

or trans for the translational, or rote for rotational.

Furthermore, we can make all the analogies for the thermodynamic state

functions that we had made previously. Thus for instance, the expectation value

of the vibrational energy will be the probability averaged accessible energies,

and that'll be minus the partial derivative of the log of the partition

function now indexed with a subscript that it's the vibrational partition

function we're talking about with respect to beta.

And if you'd rather work with temperature instead of beta, then it's kT squared

partial log q, partial T. And the only difference then between all

these various partition functions is just the subscript.

Of course, the energies are different as well that are available.

But the formalism of working with the partition functions is the same.

There's an electronic energy. There's a vibrational energy.

There's a rotational energy. And there's a translational energy.

So, let's take a moment to think about the relationship between energies and

partition functions. I'll let you take a look at that, and

then we'll come back. Okay.

So, just a quick review. That the energy of a molecule is a sum of

its translational energy, its rotational energy, its vibrational energy, and its

electronic energy. And that leads, then, to a molecular

partition function. So when I take e to the minus beta times

the total energy, which is this sum here, then, because it's an exponential of a

sum, I can break it up into a product of exponentials.

So here's the partition function for translation, e to the minus beta, all the

possible translational levels. All the possible rotational levels.

And I'm emphasizing with different indices here, I, j, k, l.

It certainly can be any combination of these various energy levels.

Because in this partition function it can be any combination of energy levels.

So the molecular partition function is the product of the individual

translational, rotational, vibrational, and electronic partition functions.

Let's also talk about degeneracy. So, when we look at partition functions.

We've, up 'til now, discussed them as a sum over states.

I've talked about state being a possible energy.

Each state is associated with a wave function having that energy, but states

that have the same energy are called levels.

The number of different wave functions that have the same energy for a given

level is called the degeneracy. And we might write that as g.

So if there are multiple solutions to a given Schrodinger equation, and they give

the same energy eigenvalue, we would call all of those solutions degenerate.

It is sometimes more convenient, instead of writing the partition function as

being a sum over states. In which case, degenerate terms would be

repeated multiple times. G sub j times where g is the degeneracy

of state j. Instead to write as a summation over

levels, and so if it's repeated, let's say the degeneracy is three.

If it's repeated three times in this sum, we'll actually only keep it once in this

sum because we're running over levels. But we'll multiply times 3.

Now that may look pretty trivial and indeed it is trivial mathematically.

But again, it's sometimes more convenient from a standpoint of working with the

relevant equations as we may see. So, let's just do a particular example.

So if you recall from our looking at the solutions to various Schrodinger

equations, the rotational partition function for a linear molecule.

We found that the allowed energy levels are h bar squared over twice the moment

of inertia and they're indexed by quantum number J.

J times J plus 1. And J could take on values 0,1,2 and we

noted that these levels are degenerate And they're degenerate 2J plus 1 times.

So the ground state, J equals 0, not degenerate, g equals 1.

But the first excited states, 2 times 1 plus 1, is threefold degenerate, and so

on. So if I were to write out the rotational

partition function summed over states. I'd get E e the 0th energy level, plus e

to the minus e over kT for the first rotational level, except it'll appear

once, twice, three times. That's the degeneracy.

And then I'd go to the second level, which would appear, 2 times 2 plus 1,

five times, and then seven times, and so on.

The alternative way to write it is to sum over levels.

Here's the degeneracy, so I get one for the first level, e to the minus ej equals

0, plus 3e to the minus j equals 1, and the next term will be plus 5e to the

minus ej equals 2 over kT, and so on. Alright, again pretty trivial

mathematically but it's just more convenient to explicitly denote the

degeneracy. So we'll use this later in certain

manipulations. Alright, we've come to the end of the new

content in this week of the course. I will spend some time on a review of

what I consider the key concepts to be in the next video.

And then, we'll have wrapped up.