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 1

This module includes philosophical observations on why it's valuable to have a broadly disseminated appreciation of thermodynamics, as well as some drive-by examples of thermodynamics in action, with the intent being to illustrate up front the practical utility of the science, and to provide students with an idea of precisely what they will indeed be able to do themselves upon completion of the course materials (e.g., predictions of pressure changes, temperature changes, and directions of spontaneous reactions). The other primary goal for this week is to summarize the quantized levels available to atoms and molecules in which energy can be stored. For those who have previously taken a course in elementary quantum mechanics, this will be a review. For others, there will be no requirement to follow precisely how the energy levels are derived--simply learning the final results that derive from quantum mechanics will inform our progress moving forward. 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 review what we've learned so far. I'll try to outline here the concepts I

consider most fundamental, most critical to try to bear in mind, and feel as

though understanding has been achieved. So, first at the microscopic level,

energy levels are discrete and quantized. So the Universe obeys the rules of

quantum mechanics. And, there is a relationship between the

frequency of radiation and energy. And the quantization phenomenon is

associated with the proportionality between these two.

And it's Planck's constant. It multiplies frequency to get the

energy. Solution in the Schrodinger equation is a

quantum mechanical technique that allows us to determine wave functions and

allowed quantized energy levels. Once we define a system by some sort of a

potential and set a boundary conditions. Energy levels in atoms are associated

with electronic and translational energy. Those are the only two things that

contribute to the energy of an atom. The electronic energy has within it the

kinetic energy of electrons, and their attraction to the nuclei, and their

repulsion from one another. And actually if there's more than one

nuclei, they repel each other as well. So that's all there.

It's a kinetic and a potential term. Translation, on the other hand, is just

pure kinetic energy of something moving through space.

For a one electron atom, like hydrogen, or helium plus, or lithium two plus.

You can generate these in the lab. Although, but you can't put them in a

bottle, necessarily. The, that cations, that is.

Actually, you can't put hydrogen atom in a bottle either, but that's a story for a

different course. the exact energy levels can be computed

for the electronic energy, and they depend on a quantum number n, which can

be one, two, three, and so on. Then we saw the formula for that.

For many electron atoms on the other hand usually we can't compute that by a simple

formula and we need to get them experimentally or by actually doing

quantum mechanical calculations, which are not the subject of this course.

Translational energy comes from solving the Schrodinger equation for a particle

in a box. And the allowed energy levels are found

to have a dependence on quantum numbers. And I labeled them here as n sub q.

Where q is just how many dimensions are there.

So if there's only one dimension. There'd only be one quantum number.

If there's two dimensions. There'd be two quantum numbers.

Every set of quantum numbers is ranging over one, two, three, whatever, they're

integers again. It also depends on the side length of the

box. If there is one dimension to the problem,

translational levels have no degeneracy but they can be degenerate in more than

one dimension. There is zero-point energy associated

with translation. And so if you look at the energy for the

ground state, you'll see it's not 0. The qua-, it depends on the square of the

quantum number, and the first quantum number is 1.

So there's zero-point energy in translation just as there is in

vibration. Diatomic molecules add to translation and

electronic energy, two new kinds of energy, both kinetic, rotational and

vibrational. The rotational energy is determined from

solving the Schrodinger equation for a rigid rotater.

And the allowed rotational energy levels depend on the moment of inertia

associated with the molecule, and the quantum number J, where J ranges,

beginning at 0 again over integer values zero, one, two and so on.

The degeneracy of those levels is 2 J plus 1.

And rotation does not have zero-point energy.

The ground state rotational energy is zero.

Vibrational energy is computed by solving the Schrodinger equation for a harmonic

oscillator. The allowed vibrational energy levels

depend on the vibrational frequency and a quantum number v, which begins at zero,

zero, one, two, and so on. Vibrational energy levels are not

degenerate. There is zero-point energy associated

with vibration. It's equal to one half Planck's constant

times the frequency of the vibration. For a diatomic molecule, the difference

between the ground state electronic energy and the negative of the bond

association energy. So that, it's negative, because our

convention is that dissociation energy is positive when you rip it apart.

the difference between those two is the zero-point vibrational energy.

Degrees of freedom, so, molecules have 3 times n degrees of freedom, where n is

the number of atoms that comprise the molecule.

And those degrees of freedoms are broken up into three translational degrees of

freedom. Rotational degrees of freedom, two for

linear molecules, three for non linear molecules.

And the remainder associated with vibration.

The rotational energy levels for polyatomic molecules are determined just

like for diatomics when they're linear. And they're more complicated, and depend

on multiple moments of inertia when they're non linear.

The vibrational energy levels for polyatomic molecules are associated with

their normal modes, so every normal mode has its own levels that can be occupied.

Same quantum mechanical expression of the harmonic oscillator, so series of quantum

numbers, and we simply sum together their energetics to get the net vibrational

energy. The total energy of a system whether it

be atomic or molecular is derived by summing the energies associated with all

the constituent pieces. Electronic translational rotational and

vibrational and finally the spacing of those energy levels is electronic energy

levels usually. Much larger spacing, than is true

vibrational energy levels, which are larger, are spaced by more than

rotational energy levels, which generally, are still, much, much more

spaced one from another than translational energy levels.

So translational levels are very very dense in the energy spectrum, you might

say. Whereas by the time you're up to

electronic energy, very diffuse we might say in a spectrum.

So, now that we know how to compute the energy for a molecule as a function of

the quantum numbers describing the, the states that it occupies.

Translational, rotational, vibrational, electronic.

We are prepared to begin using those concepts to see how they influence

thermodynamics. And we're going to do that, but I

actually want to spend a little time beforehand going back, perhaps a bit in

time, and looking at what the people who developed thermodynamics did.

They did a lot of experiments and they did them on gases.

Gases proved to be convenient for a number of reasons, and we'll talk about

that. But I think it's very helpful to look at

the data, to think about the data, to think about things the way those early

experimentalists did. To provide a context for then taking the

tools of quantum mechanics and of statistical mechanics and connecting them

to the thermodynamics that originally many, many of these thermodynamic

concepts were arrived at empirically. And it's the connection to the quantum

world that is fascinating and interesting if you're a chemist.

So, we will start looking at gases and we will look at ideal gases to begin.

And so, the next video in the series will address the ideal gas equation of state.

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