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

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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.

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Module 4

This module connects specific molecular properties to associated molecular partition functions. In particular, we will derive partition functions for atomic, diatomic, and polyatomic ideal gases, exploring how their quantized energy levels, which depend on their masses, moments of inertia, vibrational frequencies, and electronic states, affect the partition function's value for given choices of temperature, volume, and number of gas particles. We will examine specific examples in order to see how individual molecular properties influence associated partition functions and, through that influence, thermodynamic properties. 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

Well, we've done one atom, and we've done two atoms.

So now it's time to do three or more atoms.

So, let's take a look at polyatomic molecules.

And in addition to the usual translational and electronic degrees of

freedom, a polyatomic molecule can also rotate, as a diatomic can.

But it also has multiple vibrations that distinguishes it from a diatomic which

has a single vibration. Nevertheless, the total polyatomic energy

is the sum of the same four components as for a diatomic, translational,

rotational, vibrational, electronic. And, just as for the other two cases

we've done so far, the translational is determined from solving the particle in a

box, Schrodinger equation, that only depends on the mass of the particle, not

how many atoms contribute to that mass. So we get a partition function, and an

energy that is entirely equivalent to what we've seen before.

And also, as for a diatomic gas, when it comes to the electronic energy, we assign

the ground state energy value. Remember, in the diatomic, it was the

dissociation energy, or actually, the negative of the dissociation energy.

Assigned as the bottom of the energy well, the ground state.

So in a polyatomic it's the same idea but because there are multiple bonds that can

dissociate, it's the sum of all the dissociation energies.

As for the rotations, this is a little bit different.

So, one case not at all that different. If our polyatomic molecule remains

linear, as a diatomic must be with only two atoms.

Then in fact, you get the same solution to the rigid rotator equation.

The only difference is of course that the moment of inertia has to be determined

for all of the atoms in the molecule, not just for two atoms.

But in any case, you get the same rotational partition function.

It's T over the rotational temperature, and the symmetry number appears.

And once again the symmetry number tells you essentially if I rotate it end over

end a half a rotation, will I get something indistinguishable from what I

started with? That would be carbon dioxide, for

example. If that's the case, then the symmetry

number is 2. If, on the other hand, I have to proceed

with an entire rotation to get back where I started, then the symmetry number is 1.

And that might be replacing one of the oxygen atoms in carbon dioxide with a

sulfur atom. Which is carbonyl sulfide, incidentally,

as well, if you want to do a little chemical nomenclature.

Now, for non-linear polyatomics the situation is a bit more complicated.

And once we have atoms in space that are not aligned in a, a single line.

There will be three unique moments of inertia.

And you can think of those as being aligned on Cartesian axes if you like.

There's an ix, an iy, and an iz. Or maybe we should give them different

labels. We'll call them ia, ib, and ic.

And given that there are three of them, there are sort of three possibilities.

Either they're all equal to one another. So when that happens we have equivalent

rotational temperature, so if the moments of inertia are the same, the rotational

temperatures are the same. And the name for that is a spherical top.

Obviously the next possible case is that two of them are the same and one of them

is different and that too has a special name.

That's known as a symmetric top. And finally, the last case is that none

of them are equal one to another. In which case they all have different

moments of inertia. Different rotational temperatures.

And that is called a asymmetric top. May seem a little bit odd we call it a

top. When all three are different.

Never the less, that's the name in rotational spectroscopy.

And so if we look at the partition functions for these systems, I'm not

going to derive them, it's a bit more complicated to do the quantum mechanics,

but for the spherical top, for example, I'll present you the result.

It actually looks pretty similar in terms of energy levels.

What's changed here is, notice the degeneracy is no longer 2J plus 1.

It's actually 2J plus 1 squared. So the degeneracies are going up more

rapidly. And there are characteristic expressions

for the other tops as well. So, in all cases, of course, the

rotational partition function can be written in a general way.

And it will still be the case that as long as the rotational temperature

associated with each moment of inertia is well below the temperature at which we're

working, we can replace these sums with integrals and solve the integrals.

And when you do that, the solutions you end up with are shown here.

So for the spherical top we've got something that looks reasonably familiar.

It's something that dropped out of the diatomic temperature, divided by

rotational temperature but now raised to the 3 halves power and appearing out

front a pi to the 1 half and our symmetry number is still there.

And in this case the symmetry number may take on values other than 1 or 2 and it

just depends on the nature of the molecule.

A symmetric top has a T over theta rotation for the two rotational

temperatures that are equal to one another because the moments of inertia

are equal to one another. And that's raised to the first power we

might say, and then there's a 1 half power for T over the other rotational

temperature and again a pi to the 1 half power out front.

And finally in the asymmetric top, there is this expression: T cubed all, the

product of all the rotational temperatures all to the 1 half power if

you will. So I'm going to just put that on another

slide to continue to work with it. If these are the partition functions and

this is the ensemble partition function associated with these products.

Note that all of these have 3 halves temperature, a power dependence on the

temperature of 3 halves, that is. And so if, we carry out, the partial

differentiation of the log of the partition function, that 3 halves, so

here it's a 1 plus a half is 3 halves. Here, it's T cubed to the 1 half.

That's 3 halves. Here, it's just explicitly 3 halves.

So when I take the log of these partition functions, which is what I do when I

compute internal energy. The 3 halves will come down as a

multiplier, because it's the exponent on something I'm taking the log of.

So, I'll end up with a 3 halves log T, with respect to T, so that'll be 3 halves

times 1 over T. That 1 over T will cancel 1 of the Ts in

T squared, and I'll end up with 3 halves RT.

That's the contribution to the internal energy of the rotations in a polyatomic

that's not linear. And for the heat capacity, I just have to

take the derivative of the internal energy with respect to temperature.

I get 3 halves R. And, again, this is consistent then with

the way we thought about what happened in a diatomic.

In a diatomic, there were two unique ways to rotate.

You can't rotate about your diatomic axis, but we said each of those two ways

to rotate, contributes 1 half R. So, when we're no longer linear, there

are in fact, three ways to rotate about each of the unique axes, as you will.

Each of them contributes 1 half R and so the total now is 3 halves R instead of 2

halves R or just R. Okay, so, with that in mind, let's pause

for a moment and I'll let you play with those concepts a bit and we'll come back.

Alright, so that takes care of rotations, and I think you appreciate now the

difference between a linear molecule and non linear molecule in terms of how those

differing degrees of freedom contribute to the heat capacity and to the molar

internal energy. Next let's put together some of the other

components and work with the full partition function and wrap up our

treatment of polyatomic gases. [SOUND].

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