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

Alright, it's time to dig in. So, we are going to take a look at atomic

energy levels. What are the allowed energies in an

atomic system? And in order to answer that question,

first we have to ask ourselves, how is energy stored within an atom?

So, when we connect, macroscopic thermodynamics, to a molecular

understanding, we need to understand that energy distribution on a microscopic

scale. And if we think about atoms, as the most

fundamental particles, there are two ways to store energy in an atom.

One is the electronic energy. So, the electrons, which are distributed

about a nucleus, will have associated with them kinetic energy, they're in

motion, and potential energy. They are attracted to the positive

nucleus, and they are repelled from one another.

An atom will also as a physical object, have translational energy, that is, as an

atom moves through space, it has kinetic energy, associated with it's velocity and

it's mass. So that's kinetic energy only, unlike the

electronic energy, which is potential as well as kinetic.

One of the simplest systems chemically, simplest system in the universe, I

suppose, is the hydrogen atom. So the hydrogen atom is one electron

surrounding one proton. It turns out that the Schrodinger

equation that describes the motion of an electron about a proton can be solved

exactly, analytically, and from that solution, we learn the following.

There are quantized energy levels, one, two, three, four, up to an infinite

number actually. And, in terms of terminology, we refer to

the lowest allowed energy level as the ground state of a system.

So, a ground state hydrogen atom has an energy, binding the electron to the

proton of minus 2.18 times 10 to the minus 18th joules.

The next allowed energy level up, n equal two, has an energy of minus 5.44 times 10

to the minus 19th joules. And there's another level after that,

number three, and then number four, and you'll notice that spacing between these

levels is getting closer, and closer. So they're not equally spaced, and if we

look, at the physical properties hydrogen atoms, in these various energy levels.

We discover that's, what is happening is that the electron is moving further and

further away, in terms of the average distance from the proton.

And so for the ground state, remember that's the lowest quantum number, so n is

called a quantum number, it is the integer that indexes what level, what of

the allowed energy levels is occupied? The distance, the average distance, this

is actually an expectation value, is what we would call that in quantum mechanics.

So what is the most likely distance you would get after averaging over a large

number of experiments, is 5.3 times 10 to the minus 11th meters.

And so, if you remember an angstrom is about 10 to the minus 10th meters.

Its about half an angstrom. This unit of distance actually defines

the atomic unit of distance which is called the Bohr.

So one Bohr is defined by the hydrogen atom in its ground state and as I raise

the energy higher higher higher, ultimately I get the electron to the

infinitely separated from the proton. That is there is no interaction between

them and we define that to be zero there is no interaction so the energy is zero.

And that's why we see here at n equals infinity that zero, that would be an

ionized hydrogen atom. And relative to that zero, that allows us

to assign this number to the ground state.

Now if you focus on these energy levels, you would note that they have a certain

progression which can be well described by a simple equation.

It says that the energy of a given allowed level indexed by its quantum

number is equal to some constant divided by the square of the quantum number.

So, if n is one, then n squared is one and we would get the ground state energy,

which, sure enough, is right here, it just isn't expanded to as many digits, so

rounded to 2.18. We can also convert joules to a different

unity of energy. Here's the wave number.

So if you prefer reciprocal centimeters, it's 109,680 divided by n squared

reciprocal centimeters. So, this simple formula to compute the

energy levels would also let us compute the difference in energy between

different levels. And the energy required to ionize the

hydrogen atom, by letting n go to infinity at which point energy is zero.

So I'll let you take a moment here to actually do a little self assessment,

making use of this equation. And then we'll come back to continue to

look at atomic energy levels. Most of you will certainly have seen this

analysis of hydrogen before, and probably, the way you've seen it is in

the context of hydrogenic orbitals. So, what are the wave functions

associated with these allowed energy levels?

Well, the ground state energy level, there is one way to function with that

energy and we usually call that the 1S orbital.

So, the next level up n equals two, has four different wave functions.

All of which have the same energy. That's called degeneracy.

So when there are multiple solutions to the Shrodinger equation, all with a

common energy, we say that those solutions are degenerate.

So given that there are four possible solutions, the degeneracy is four.

And we often index degeneracy by g, and again a subscript to say which level.

If we go up to the third allowed energy level or the third quantum level

sometimes we would say now it turns out that there are nine orbitals or wave

functions that satisfy the Schrodinger equation that all have that energy.

So the degeneracy is now nine. And looking at these orbitals of course

you recognize here's the classic one s. Here's a two s, it has has a node

somewhere in it but we're drawing it solid so we can't see it.

And here are the two p orbitals. Here's a three s, two nodes hidden in

there somewhere. Here are the two p's and for some of

them, this one for instance, you can actually see the nodal structure.

And here you see it as well for the three p orbitals.

And these are the three d orbitals, there are five of those.

And these are the classic atomic orbits, more carefully, hydrogenic orbitals that

one is presented with very early in the study of chemistry.

Now, some of you may be experiencing a moment cognitive dissonance perhaps

because you've seen these orbitals arranged in energy ordering before and

someone has told you that the two s is lower in energy than the two p's.

Or the three s's is lower than the three p's is lower than the three d's.

That's actually not true, it's not true for this system.

As I've shown it, these degeneracies are correct.

All nine of the solutions for the third quantum level, the three s, the three p,

the three d, they all have the same energy.

In a one electron atom, normally we don't work with just one electron.

It's not just hydrogen in an excited state.

It is a more complicated atom. It turns out once you add additional

electrons to the system, that changes the energies of s and p and d solutions.

So everything that you've been taught hasn't been a lie.

But at least for the hydrogenic system, the degeneracy as shown here is what it

is, and you've probably noticed, it looks like it's n squared, right.

For n equal one, the degeneracy is one, for n equal two it's four, three goes to

nine, and sure enough that's correct. The degeneracy of hydrogenic solutions is

n squared where n indexes the quantum level.

So, what do we do with many electron atoms?

Well, it turns out there's no simple formula for the electronic energy of such

atoms. We can either do electronic structure

calculations with a big digital computer. Or, you know atoms have been around a

long time, and they have been studied pretty carefully.

So you can look up in tabulated data, what are energy levels?

So here's an example drawn from so called Moore's tables, which are these big

tables of atomic energy levels that are available.

And it's for the Sodium atom, and it's just showing you for a variety of

different states, different ways to organize the electrons around the atom

what's the degeneracy and what's the allowed energy associated with that

state. If you look here you see that the ground

state, which will be used to define zero here, you have to go up 16,956 it's a

very precise .183 wave numbers to the first excited state.

If you were to work out, I won't make you do a self assessment, but if you were to

work out, what wavelength of light that is, you would discover it's sort of

roughly yellowish. And that's why sodium, lighting, on

highways for instance, or in cities. Often has kind of a yellowish tinge to

it, that's sodium vapor that's been heated hot enough that it emits light

from it's first excited state down to it's ground state, and the color of that

light is dictated by the allowed energy level.

Alright, happily, for most of the thermonamics that we will be working on

we will not be at temperatures that are hot enough that these kinds of states are

accessed very often. That will matter when we compute

partition functions, which we'll be doing in not so long.

But for now we will just recognize that complicated atoms or molecules, we may to

just have to look up things. Translational energy, so that's the other

energy available to an atom. It's moving through space, it has kinetic

energy. So, how do we go about computing that?

We construct a Schrodinger equation, and the equation has to do with a particle of

mass M constrained in a box having variable side links.

So in one dimension then, you would have a little particle here and it can move

left or right and the box, it's only one dimension and we'll think of a one

dimensional box, you're allowed to be anywhere from say zero to A, where A is

the length. Particle can be anywhere in there and it

can't be outside there. And if you solve the relevant trade in

your equation, you will discover that the allowed energy levels, so indexed by a

quantum number, and the number goes, one, two, three, dot dot dot.

The degeneracy of every level is one, so every state is unique in its energy, goes

as n squared, here's Planck's constant again, h squared.

Divided by eight, times the mass, times the square of the length of the box.

So, given that you would know those parameters ahead of time, the mass, the

length of the box, poof. You have all of the allowed energy

levels. If we go to three dimensions, it's a

little bit more complicated. Obviously, our box can be a little more

complicated. I've got a little particle here, it's

bouncing around A for one length, b for another length, c for another length.

I've drawn what looks like a q, but they don't have to all be equal.

And when you do the solution, you find that it has roughly the same structure.

There's still this h squared over 8m that you find here, h squared over 8m where m

is the mass. But now, every dimension x, y, and z has

its own quantum number associated with it, still depends on the square, still

depends on the square of the lengths of the sides, but notice that states can be

degenerate. That is, for instance, let's say that a

and c just happen to be the same length. Well, in that case, quantum numbers,

indexing a state, I could talk about state one one two which is just the list

of the x y and z quantum numbers. If I consider a different state, two one

one. Since a and c are the same value, it

doesn't matter whether I take two squared here and one squared here, or vice versa.

I'll get the same total energy. And so the degeneracy will just depend on

the side lengths and the quantum numbers. It's hard to predict until you've given

all of the data. So lets pause for a second and I'll let

you play with those equations in a short self assessment and you can make sure

that you've grasped that concept, of the quantum numbers in the particle in a box.

Alright, we've completed, our examination of the available energy levels, for an

atomic system. Next, we're going to move on to consider

molecules and we'll consider the simplest molecule beyond an atom.

We really shouldn't call an atom a molecule, but two atoms can be a

molecule. So we will look at diatomics and the

allowed energy levels in diatomic molecules.

Before we move on to that lecture though, let's take a look at one more

demonstration to illustrate some of the key principles of quantum mechanics, and

in particular wave like behavior. In a prior demonstration, we saw that

electronic energy levels in atoms are quantized, and I pointed out how unusual

that seems compared to the continuum of energies that can be accessed by say,

throwing a rock. However, we are all familiar with

quantized phenomena at the macro-scopic level, although we don't necessarily

think of them that way. For instance, an organ pipe of a given

size, can only emit certain tones when air vibrates within it.

If you want a different tone, you have to make a pipe of a different size.

In this apparatus, I have a motor, that is going to drive this large string in

such a way to generate a wave. The other end of my string is fixed, so

that it must always be in the same position.

A mathematician, or a physicist, might call this a boundary condition.

That is, a condition that must be satisfied at the edge of a system.

Such boundary conditions can often lead to Quantized Phenomena and are present in

many ways at the microscopic level. But here, we'll see an illustration at

the macroscopic level. So, when I turn the motor on, I can

adjust the speed and notice that the motion of the string.

[SOUND]. [NOISE] It's rather chaotic.

Until I hit one certain point, I generate a so-called standing wave.

The system is seemingly indefinitely stable in its behavior.

Now, if I turn the speed up more [NOISE] the motion again becomes chaotic until a

certain rate of speed at which I hit a new standing wave.

[SOUND] See how this one has one position in the center where the band neither goes

up nor down? That is called a node.

And we can call this new wave an overtone or a first harmonic of the original wave

that had no nodes. In general, in wave mechanics, the more

nodes. [SOUND] The more energy, if we turn the

speed up still higher, we may be able to hit the second overtone.

[SOUND]

And there it is, see how it adds two nodes.

With a powerful enough motor we could continue to access higher, and higher

over tones. But for now let's just take this as a

another example of a quantized phenomenon.

And note that in atoms and molecules, such phenomena will ultimately play an

important role in how they store, and distribute energy.

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