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.

从本节课中

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

Great. We have now derived, from first

principles, partition functions for an ideal gas, an ideal monotomic gas.

And I think that's a, it's a pretty exciting concept, what we've managed to

do. We've managed to go from first principles

to explaining macroscopic properties. And I'm excited about that.

I hope you're excited but what I want to do here is continue to work with that

ensemble partition function and look at the properties we derive from it.

So we derive for that monatomic ideal gas that the ensemble partition function is

equal to a molecular partition function to the nth power, divided by n factorial.

This is valid for an ideal gas. That means non interacting particles.

That's what allows us to take a molecular, or in this case, atomic

partition function, and raise it to the nth power.

And the particles being indistinguishable, we divide by N

factorial. Now, the individual components of that

monatomic ideal gas molecular partition function are translational components

that depends on volume and temperature. And an electronic component, and for

purposes of this lecture, we'll consider it to include the degeneracy of the

ground state, that's G1 and we'll consider the possibility that there is a

first excited state that may be accessible.

So we'll include the degeneracy of the second state g2e to the minus beta e2,

that would be the next term appearing in the, partition function for the

electronic component. So, let's look first at the internal

energy of the ideal gas. And you'll recall that internal energy is

equal to k, Boltzmann's constant, T squared.

Partial log of the ensemble partition function with respect to T.

That constant number of particles and volume.

And if I make the substitution, then, that the ensemble partition function for

my indistinguishable non interacting atoms, in this case is little q, the

atomic partition function, to the nth power over n factorial.

I can substitute that in for capital Q. I don't need to keep the N factorial

term. It has no dependence on temperature.

So it won't be there. But I'll then have NkT squared, partial

log q, little q, partial T, at constant volume, right.

And so the N that was there as a exponent, when I have a logarithm, will

come out and be a multiplier. And finally, I can make the insertion

what is this little q, it's just what I had in the last slide, namely it's the

product of a translational partition function and an electronic partition

function, when the gas in question is montaomic.

And so, if I write that out, that is insert all the individual terms, first I

will have the translational component. And that involves 2pi, the mass of the

atom, Boltzmann's constant times temperature over Planck's constant square

at all to the 3 2nd power times volume. And followed by the electronic partition

function where we're keeping the first two terms and if there were more terms,

of course we could add them but, for now, let's just keep an eye on the first two

terms. And when I then go on to look at the

relevant partial derivative I'll bring down the three halves power, then it's

partial partial T log of a whole bunch of things.

They will all separate out because I'm taking a log rhythm.

The only one that will depend on T is T itself, so I'll get partial partial T log

T, that's just 1 over T, and that gives rise to this term then.

3 halves N, so the 3 halves, Nkt squared, over T from this piece, 3 halves, Nkt,

and so that is the derivative associated with the translational partition

function. Now, what about the derivative associated

with the electronic contribution to the overall partition function?

So, I think I'm going to let you work on a piece of that derivative, and then

we'll come back and see it in more detail.

Alright, so obviously the degeneracy g1 has no dependence on temperature, and

hence it doesn't appear in a derivative. This term on the other hand, the second

term in the overall partition function expansion, does, and if you adopt the

chain rule, you'll get a 1 over q electronic and then this is the piece I

asked you to work on, N, degeneracy of the second, of the first excited state,

so the second term in the partition function, epsilon two, e to the minus

beta epsilon two, and had there been more terms we would have had to be continue

with this differentiation. Now generally that contribution from the

electronic partition function is small, right this depends on the second term

actually having some relevance. Had we really been working with something

entirely in the ground stage. It doesn't depend on temperature.

There would be no contribution overall to the internal energy.

And in that circumstance when that term is indeed small we would have that the

internal energy is simply this first term.

3 halves NkT. So internal energy dominated by the

translational contribution because the fraction in excited electronic states is

usually very small at low at every day temperatures is what I mean by low.

Is a room temperature here we are at a temperature that is not all that hot.

Roughly 300 kelvin. So we may as well actually put some

numbers on that though just to get a feel for the importance of that electronic

term, so let me take two example gases, monatomic ideal gases.

One would be a gas of lithium atoms and one would be a gas of fluorine atoms, so

those are mildly unusual gases. You might have trouble actually

assembling a flask full of lithium atom gas and fluorine atom gas, but that's

okay, this is a demonstration that's a thought experiment.

So we'll just work with the data, which are known for these atoms.

And in particular, what's known for lithium is that its first excited state

is quite high in energy relative to its ground state.

14,900 let's round to 4, reciprocal centimeters.

And that excited state has a degenerisity of 2.

For fluorine on the other hand, there's actually a rather low line first excited

state. It's only 404 reciprocal centimeters

above the ground state. And it too has a degenerisity of 2.

And in order to put specific numbers on things.

Let's actually pick values for the number of particles, and the temperature.

And in particular, let's take Avagadro's number of particles.

So we'll be working with a mole. And, let's use standard room temperature,

298.15 Kelvin. And I'll express my energy in kilojoules.

And so this low lying electronic state for the Florine atom means that at room

temperature, so if, if you think of room temperature multiplied times the

universal gas constant, gives you a feel for sort of what's the ambient energy

that's just floating around in the room available for harvesting.

If you were to express that in wave numbers, it would be about 200 wave

numbers, more or less. And so here we have a excited state, a

first excited state, that's only about double that up in energy.

So it's relatively accessible given the thermal bath in which these fluorine

atoms acting like an ideal gas would be residing.

And, the electronic partition function instead of being one, just remember one

refers to everything being in the ground state is got some population of the first

exited state, so it's 1.142. So, if we now go down we evaluate the

contribution of the internal energy from translation, that's 3 halves, N is

Avogadro's number times Boltzmann's constant, will be r.

So we'll get 3 halves, rT, and in kilojoules, 3 halves rT for this

temperature is 3.719. And for fluorine, of course, it doesn't

matter what the atom is here, there's no dependence on the atom, it's just a

constant. So it's 3.719 for fluorine as well.

If we go to lithium, and we plug in to this expression, so q electronic for

lithium is effectively 1, so that's what in the denominator here.

14903.66, and then we express kT in the relevant units of reciprocal centimetres

as well. You discover that the contribution that's

made to the internal energy is 2.081 times 10 to the minus 26.

So that is 26 orders of magnitude smaller than the contribution from translation,

which is to say it just doesn't matter at all.

On the other hand, if we plug in the appropriate values for Fluorine, with its

much lower lying excited state, there is a non-negligible contribution.

It's 1.204. So, it's about a third again as much as

what's being derived from translation. And that's An interesting phenomena

associated with that low laying electronic state.

Now what about the heat capacity? So that's an interesting property of a

gas. how much heat can it store and how much

will added heat cause its temperature to rise?

So, given this expression for the internal energy and, recalling that the

heat capacity at constant volume is the partial derivative with the internal

energy, with respect to temperature, we then need to differentiate this

expression, with respect to temperature, and when I do that, well, the first term

here is pretty trivial. So, when I take three halves NkT, and I

differentiate with respect to T, I get 3 halves Nk, that's straightforward.

The next term, not quite as friendly. We've got temperature dependence and the

argument of the exponential, the partition function itself has temperature

dependence, so you would need to use the quotient rule here to complete the full

differentiation. I'm going to express one of the terms

that would come out of doing that quotient, and that is the differential of

the thing on top times 1 over q electronic, and so that will give this,

and I will let you work that out for yourself, if you just love doing

differentiation. There will be an additional term but I've

basically run out of room on this slide, and I'm not going to write down that

additional term. It is there.

and again if you love differentiation I'll let you do that for yourself.

You can see that if there were ongoing terms because of a second excited state,

a third excited state, this all gets a little bit messy relatively quickly.

But what I want to do is focus on, the importance of a low-lying excited state,

or lack thereof to the heat capacity. And so let's continue to work with our

slightly unusual ideal gases of lithium atoms and fluorine atoms.

And now again I'll keep track of the translational part and the electronic

part, and I'll actually keep that extra term that's not being show in the slide.

So if you try to work these numbers out for yourself, that second one is there,

but again, I'm a little short on space, so I won't write it out.

So continuing to work with one mole and at room temperature, this value 3 halves

N times k that's just three halves r. And so in these units of joules per

kelvin that's 12.47 and again that's independent to the nature of the gas,

there's no excited state energy appearing in the translational part.

If we plug in this large number to this exponential relative to the ambient kT,

we get 5.02 times 10 to the minus 27th. Again a completely neglegeiable

controbution to the heat capacity. However in the case of fluorine, when we

put in the much smaller 404 wave numbers, you end up with a contribution from the

electronic partition function, of 5.193. And so that's not quite, but it's close

to half again, as much, as is coming from translation.

And I'd like to make clear what, what that means.

Remember what the heat capacity describes.

It says, how much heat would I need to put into my ideal gas in order to raise

its temperature by 1 degree. And that requires us to think about what

is temperature, when I clasp that flask full of gas, why does it feel warm or

cold to me. And so temperature is a measure of the

kinetic energy of the molecules, so there zooming around and bumping into the walls

of the flask and that causes the wall of flasks to also have some motion, and it's

that motion, in a sense, that we sense as temperature.

So, if I put heat into the system, in the case of lithium, the only place it has to

go is into the translations. It can't populate anything else.

And as a result, it takes this much, 12.47 to raise the temperature by a

degree. On the other hand, what happens in the

fluorine system? When I put heat into it, it's got two

options. Just like the lithium, it has a

translational component. And it can put energy, heat in this case,

into the translational modes. But it also has available to it

relatively low lying electronic excited states.

Those don't contribute to the electronic energy.

Rather, it's just a place where the heat can be stored by increasing the

population of those excited states. And as a result, instead of taking 12.47

joules worth of heat It takes, hm, looks like 18.4 roughly.

So more heat to raise the temperature 1 degree because some of that heat isn't

going in to what I perceive as temperature of the kinetic energy of the

molecules, instead it's going into populating those excited states.

Of course I can harvest that out again later but the point is that energy is

being stored in a place that doesn't contribute to the rise in temperature.

So it's an interesting phenomenon and generally associated with availability of

states. Okay maybe the last thing to do is work

with one more property, namely the pressure.

So here we have the monatomic ideal gas, and recall the pressure is kT, partial

log q partial V, so I'll do exactly what I did before.

I'll replace q with molecular partition function of the nth power over N

factorial. Again, and the N factorial doesn't depend

on V, so I'll just lose that, and this equality is held.

And now I insert for the molecular partition function, the product of the

translational and electronic partition functions.

And, when I go and expand using the actual electronic, sorry here's the

electronic, and translational partition functions.

There is only one thing that depends on V, and that is V itself.

So, I just have partial, partial V of log V.

Well that's pretty straight forward, so I get 1 over V and NkT over V is the

ultimate result. P is equal to NkT over V and of course

that is the ideal gas law. PV equals nRT if n to number of moles.

Little n, that is, number of moles as opposed to capital N, number of

particles. So let's just summarize, then what we've

actually derived for this partition function for a monatomic ideal gas.

Where these are the individual components, the translational component

of the molecular partition function, and the electronic component.

If we restrict ourselves to those cases where the first excited state, the first

electronic excited state, that is, is too high in energy to really contribute, so

not unlike the somewhat exotic fluorine atom, but more like other systems, then

the energy is 3 halves NkT. The heat capacity is 3 halves Nk and the

pressure is NkT over V. And we more typically for convenience

work with molar quantities and molar units, in which case we'll have U bar,

the molar internal energy is 3 halves RT. The molar constant volume heat capacity

is 3 halves R and finally the pressure is equal to RT divided by the molar volume.

Alright, well, that takes care of the monatomic ideal gas.

There are not that many interesting monatomic gases, Lithium and fluorine, as

I mentioned, are extremely exotic if they're monatomic.

There are the noble gases, helium, neon, argon, they get more exotic as they get

heavier, but let's move on to the next thing that occurs after a monatomic ideal

gas and that is an ideal diatomic gas. More things will happen in the diatomic,

and so it will take us more than one lecture to get through them all but we'll

start with the first half of the ideal diatomic gas next.

[NOISE]