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

We've reached the last members of the chemical menagerie.

Â Now that we've taken care of atoms and diatomic molecules, let's talk about

Â polyatomic molecules. Already shown you this slide, how is

Â energy stored in a molecule? There's electronic energy, which is both

Â potential and kinetic associated with the electrons, and then, there is kinetic

Â energy of the molecule itself. There's the translational energy, which

Â is movement of the entire molecule in space, comes from particle in a box

Â solutions. There's the rotational energy of the

Â molecule rotating about its center of mass that comes from the Rigid-rotator

Â equation. And finally, there is vibrational energy

Â associated with the nuclei moving relative to one another.

Â It's handy to divide up the various contributions to the kinetic energy that

Â comes from molecular motion into what are known as degrees of freedom.

Â So, if I want to specify completely the position in space of a molecule that has

Â n nuclei, I must need three n coordinates, right?

Â There's an x, a y and a z coordinate for each atom, and if there are n atoms,

Â that's 3 times n. So, we often will refer to that then as

Â 3n degrees of freedom which dictate specification of every atomic position.

Â And degrees of freedom can be thought of as being subdivided into translational,

Â rotational, or vibrational components. And so, if I think about translation,

Â that's really the motion of the center of mass in three-dimensional space, and so,

Â there are three degrees of freedom. It can move in the x direction, it can

Â move in the y direction, and it can move in the z direction.

Â There's rotation, so that is movement about the center of mass.

Â And for a linear molecule, if I imagine a linear molecule laid out along some axis,

Â I can rotate this way. I guess that's end over end in a sense.

Â I can rotate this way, end over end in a different plane.

Â But, there is no rotation about the axis that's defined by the bond of the

Â molecule. So there's two ways to rotate a diatomic.

Â Once I am nonlinear, so that's really the, the issue there.

Â A diatomic molecule per force is linear. Two points determine a line.

Â once I go to three or more atoms, I could be linear.

Â There are examples of linear molecules with more than two atoms, in which case,

Â I still only have two ways to rotate, but once I'm nonlinear, now I can rotate

Â about any of the Cartesian axes, so that's three degrees of freedom.

Â Well, so all that's left then is vibrations, and when determines the

Â number of vibrational degrees of freedom by taking the total 3n and subtracting

Â out the translational, always 3 and the rotational, 2 for linear, 3 for

Â nonlinear. And that gives you these these values, 3n

Â minus 5 vibrations for a linear molecule, 3n minus 6 vibrations for a nonlinear

Â molecule. And so, when we sum them together

Â appropriately, always linear or always nonlinear, you always get 3n as you must.

Â The diatomics, which we dealt with up till now, of course only have our per

Â force linear and only have 3n minus 5 vibrations.

Â So let's, let's just remember that. 3 times 2 would be 6, minus 5 is 1.

Â One vibration. We looked at the vibration in a diatomic

Â molecule. In a polyatomic, there would be more.

Â So actually, let's pause here for a moment, and this is sort of trivial

Â arithmetic and and thinking about the physics, but maybe take a moment and

Â think about the numbers of degrees of freedom available in different molecules.

Â Well, let's take a look now at rotational energy levels in polyatomic molecules.

Â So it turns out, if the polyatomic molecule is indeed still linear, then,

Â exactly the same Schrodinger equation applies.

Â You get exactly the same solutions. Namely, that the allowed energy levels

Â are given by this expression, with these quantum numbers, and this degeneracy.

Â The only difference is the moment of inertia is defined by all of the atoms,

Â not just by two atoms, and so this is simply a generalization of the formula

Â for moment of inertia, namely that it's the sum over all the atoms, their mass

Â times the square of their distance from the center of mass.

Â Nonlinear molecules, things become a bit more complicated.

Â So actually, there are three rotational axes in a nonlinear molecule, and for

Â each of those axes, there is an associated moment of inertia.

Â There are categories of molecules and the categories are named depending on the

Â relationship between the moments of inertia.

Â So for certain special molecules, all three moments of inertia along the

Â different Cartesian directions are the same, such molecules are called spherical

Â tops. And so, an example of a spherical top

Â would be a baseball, that's not a molecule, but it's a macroscopic object.

Â Or, in the chemical arena, methane is a molecule with three moments of inertia

Â all equal to one another. The next step down, clearly, would be

Â instead of having all three be equal to one another, have two be equal to one

Â another, and so, such molecules are called symmetric tops.

Â And again, a macroscopic object that has that characteristic would be an American

Â football, and so its got one long axis. But otherwise, it looks pretty symmetric

Â and that gives rise to two identical moments of inertial, but one that's

Â different. and example in the case of a molecule

Â would be ammonia. So ammonia, two equal moments of

Â inertial. And then finally, obviously, the most

Â general possible case would be all three moments of inertial are different from

Â one another, that is called an asymmetric top.

Â Seems, sort of odd maybe to keep using the word tops since we've removed any

Â kind of agreement between moments of inertia, but thatâ€™s what the terminology

Â is. So an example of an asymmetric top would

Â be macroscopically, a boomerang and a molecule that looks a little bit like a

Â boomerang water. So water with its three atoms bent with

Â oxygen in the middle. When you take account of those those

Â different moments on inertia, you get complicated expressions for the energy

Â levels. I'm not going to show you actual

Â formulas. You can look up certain of them if you

Â really needed them, but for the moment, we are going to focus on the more simple

Â linear levels that provides enough basis to make progress.

Â Now, what about the vibrations in a polyatomic molecule?

Â So, we refer to the, the various vibrations in a molecule, a polyatomic,

Â now that there's more than one, as soon as you go past a diatom, you have more

Â than one vibration. So, each one of them has a, a

Â characteristic motion and that motion is called a normal mode.

Â So, the individual vibrations are also called the normal modes of the molecule.

Â And so as an example, water has three normal modes.

Â That is, it's non-linear, so the number of vibrational degrees of freedom will be

Â 3n minus 6, 3 translations, 3 rotations, so 3 times 3 is 9 minus 6 is 3.

Â Three normal modes, three vibrational modes.

Â If we look at those modes, the actual vibrations themselves correspond to, and

Â they're shown here with their corresponding vibrational frequencies.

Â The lowest vibrational frequency is a bending motion and generations of

Â physical chemists have done this, so bear with me.

Â Imagine that my two fists are hydrogen atoms and my head is an oxygen atom.

Â The lowest frequency is a bend, so it consists of the molecule doing something

Â like this, just a little bit of a chicken move sort of, that's a bending motion.

Â There's also a symmetric stretching motion at 3,686 wave numbers.

Â And so, if we do the anthropomorphism of water again, that's the two hydrogen

Â atoms moving outward from the oxygen in concert with one another.

Â And then, finally, the last mode is the asymmetric stretch, and that's where one

Â molecule, one hydrogen atom moves in while the other moves out.

Â So here is the asymmetric stretch, and actually, the oxygen will move a bit so

Â it's kind it's like doing a little exercise in the middle of a lecture

Â video. Thank you for bearing with me for that,

Â it's a, it's a classic routine. Coming back to the to the energetics,

Â each of these normal modes act as an independent harmonic oscillator, so each

Â one contributes. The normal harmonic oscillator

Â contribution, but they've got associated with them these unique frequencies that

Â are shown over on the right-hand side. So, I would index for vibration 1,

Â Planck's constant times the vibrational frequency for vibration 1 times the

Â quantum number, that describes which level the first normal mode is sitting

Â in. And then I would add to that h times the

Â frequency for two times its quantum number plus a half, and so on, so I just

Â add them up. So what's the total energy then in a

Â polyatomic molecule, or indeed, in, in an atom or a diatomic for that matter?

Â Well, it's the sum of the energies overall the degrees of freedom.

Â So remember, that, for a molecule in a three-dimensional box, you would get this

Â from solving a particle in a box equation.

Â So there are quantum numbers and box lengths, side lengths required, as well

Â as a mass. Rotation for linear molecules is given by

Â this rotational energy formula and these quantum numbers, and I won't write down

Â as I said, the nonlinear, sometimes it has a nice form, sometimes less nice.

Â We'll deal with that if we need to . The vibrations are determined from the

Â quantum mechanical harmonic oscillator expression, these energy levels.

Â One associated with each vibration, and finally, the electronic energy, which has

Â a pretty simple form for the hydrogen atom.

Â Hydrogen atom, not that exciting. We won't be doing a whole lot more with

Â it. for diatomics, there's a number we can

Â associate it with, with it, DE. Otherwise, it's probably something we're

Â going to be looking up. And what I want to leave you with is a

Â feeling for the spacing of these energy levels.

Â How far apart are they? And that really matters in terms of how

Â you can store energy in a system? So there's a general trend in energy

Â spacing. And that is, that the electronic energy

Â spacing is much, much greater than the vibrational spacing, which itself is

Â greater than rotational, which itself is much, much greater than translational.

Â So let me, rather than just memorizing words, let's actually have a picture at

Â least to associate with that. So, shown here is a plot of two

Â electronic states of a system. So here's some low energy potential

Â associated with a, let's call it the ground state.

Â And then, somewhere up here is an excited electronic state, a long way up in

Â energy, this separation. I'll just, just dictate that.

Â We'll call that big. Big relative to what?

Â Everything else. So, within this potential is a series of

Â vibrational levels. So they are much more closely spaced, one

Â to another, than the bottom of the ground state electronic energy to the first

Â excited state electronic energy. And so, if I zoom in on the vibration

Â energy levels and move them over here. Now, I could see above each of the

Â vibrational levels the rotational spacing.

Â So the rotational levels are much closer to one another than the vibrational

Â levels are. And finally, if I zoom in, yet again,

Â just take this little red circle here, the first few rotational levels.

Â And expand those a great deal, then, on top of each rotational level, superdense

Â are the translational levels. So they're extremely close to one

Â another. And keeping in mind that spacing will be

Â useful as we proceed forward and think about what that means for how energy can

Â partition in to these various kinds of motion.

Â So, that wraps up what we're going to do in terms of studying the quantized energy

Â levels of atoms and molecules. And indeed, we're almost done with the

Â material for this week of the course. The next video, we'll try to provide a

Â sort of review summary picture of what we've talked about that's important so

Â far and that will complete Week 1.

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