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 3

This module delves into the concepts of ensembles and the statistical probabilities associated with the occupation of energy levels. The partition function, which is to thermodynamics what the wave function is to quantum mechanics, is introduced and the manner in which the ensemble partition function can be assembled from atomic or molecular partition functions for ideal gases is described. The components that contribute to molecular ideal-gas partition functions are also described. Given specific partition functions, derivation of ensemble thermodynamic properties, like internal energy and constant volume heat capacity, are presented. 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 continue to work with the partition function for an ideal gas and see what

else we can learn from it. Last time, we discussed the internal

energy of an ideal gas. I'd like to begin this time by looking at

the ideal gas heat capacity for a monatomic ideal gas, like helium, we were

working with helium last time. And so the molar heat capacity of a

substance expresses the energy that's required to raise the temperature of 1

mole of that substance by 1 Kelvin. And another way to think about that is,

multiplication of the molar heat capacity by a measured temperature change, so I

have a thermometer and a mole of my substance and I look at how much the

temperature goes up. If I multiply the molar heat capacity by

that temperature change, I will know how much energy had to be put in to get that

change. Because the molar heat capacity is how

much is needed for one degree. I multiply by how many degrees it is and

there I have it. So let's pause for a moment and make sure

thats clear and then we'll come back. [SOUND] So, now that you're comfortable

that molar heat capacity is in units of energy per Kelvin, let's consider that we

can express that as a derivative. So the molar heat capacity at constant

volume in particular, so molar because there's a bar over the top, that's what a

bar means, a molar quantity just as we did molar volume earlier in the course, v

bar. And constant volume here because that's a

variable we are holding constant within our ensemble.

Is equal to the partial derivative of the internal energy, so here's the energy

units in the numerator. With respect to temperature.

So here's the temperature units in the denominator, holding volume constant.

So just the same v subscript on each of these two terms.

So, from our prior work, on the other hand, we know that the partial derivative

of the molar, internal energy, excuse me, is equal to the partial derivative of the

expectation value of the energy, right? That statistical, mechanical concept, is

equivalent to the classical thermodynamic concept.

So I can replace it. And we computed in the last video that

for a mole worth of substance, that value of the expectation value is three halves

RT. So this is now become an extremely simple

partial differentiation. So we're going to differentiate three

halves RT, with respect to T. And you end up with 3 halves R.

And so statistical thermodynamics, then, has given us a route to understanding

where does the energy go when a system is heated, and why the capacity to store

energy is different for different substances.

Because it's the form of the partition function, which we can work with in order

to get energies and heat capacities by a differentiation of that energy.

And we're going to see in subsequent examples to come, why heat capacities

maybe different for a diatomic gas compared to a monotomic gas compared to a

polyotomic gas. I want to continue with this manipulation

of the partition function in order to derive important properties for the ideal

gas. And this is just reminding you of some of

the differential calculus we've already done.

But now, I'm going to consider the partial derivative of the partition

function with respect to the volume. So up till now we've already seen this

one, partial log Q, partial Q is just 1 over Q.

Partial Q, partial V. Well, we don't have an explicit form for

v in the partition function itself, we just know that the energy depends on it,

and so. Given that I have the exponential of a

function of the variable I'm differentiating with respect to.

I will pull down the multiplier of that function so thats minus beta and the

minus beta goes there. And then, by the chain rule, I have to

include the partial derivative of the function, with respect to the variable.

So here it is. So that's the differential of this

expression, with respect to volume. So, putting that all together then, if I

take partial log Q, partial volume, that'll be this times this because of

chain rule, this multiplying this. So I get minus beta over Q, that's the

preceding factor. Here's this partial derivative that I

don't yet know exactly how to expand, and here's the exponential of the energy.

But this Q out front, this 1 over Q. Again, can be moved inside the sum and

make this a probability. Because it'll be an exponential divided

by the partition function. So I'll call this the probability of

being in state j. And as always it depends on N, V, and

beta. Or, of course I don't have to write it in

terms of of beta. I can also include a T here.

So I've got beta here, I move it over to another side and make it KT.

I have some over J minus this partial derivative, probability.

I've taken beta and I've moved it over here and multiplied this quantity.

So I get KT. Partial log Q, partial V.

And I want to continue to work with that equation for just a moment.

So I've just reproduced it at the top of this slide.

So, again, mostly, I'm interested in understanding the utility of the

partition function this week. Something we will derive in the future.

But for right now, I ask you to just accept on faith.

Is that pressure is minus the partial derivative of energy with respect to

volume for fixed number of particles. So if pressure is like energy, that is

it's an average over weighted ensembles, recall that was a fundamental

postulative, statistical thermodynamics. That an expectation value is an average

over possible values. If that's true, that the observed

pressure is probability weighted possible pressures, well I can replace this

pressure with this expression minus partial E, partial V.

Here's my probability and that's what I have up here.

Alright, so I'll just equate it to what it also is.

It says that the pressure that I aught to observe is kT partial log Q, partial V.

So, something we've derived by manipulating the equations.

And now let me go back to my partition function for the ideal monotomic gas,

helium, that we were talking about. Just the equations we've seen before for

the atomic partition function and for the ensemble partition function.

And so once again I'm going to need log Q so I'll just expand it.

I've shown you this equation before so I won't spend any time on it.

And now I'll evaluate for the pressure, I get kT, so here's my kT.

What is the partial of the logarithm of Q with respect to volume?

This term doesn't depend on volume, this term doesn't depend on volume, this term

doesn't depend on volume, Here's a term that depends on volume.

Well, partial log V with respect to V will be 1 over V.

It's multiplied times N, so I'll get N over V, and that doesn't depend on

volume. So, I'm all done with that, so I have

pressure equals kTN over V. And if N is Avogadro's number, if I'm

using a mole of particles, then N over V becomes the molar volume.

And N time, the Avogadro's number time Boltzman constant is the universal gas

constant, so I get R. And I have P equals RT over molar volume.

Alright and so I've just I've dropped my little expectation value brackets to

imply observation. So the observed pressure is equal to RT

divided by molar volume. Well that's the ideal gas equation of

state. We've actually derived the ideal gas

equation of state. From thermodynamic relationships of

pressure, energy volume and partition function given certain partition

functions for a monatomic ideal gas. So a, a tremendously powerful result.

So what's the big picture so far. The partition function encompasses all

possible states of an ensemble. Thermodynamic functions can be computed

from the partition function. A central one being the internal energy

and because the internal energy, it's dependence on other state variables gives

rise to other functions. Its a very useful one to have.

For simple partition functions, that is, like the ones I've so far told you were

valid for helium and that we will derive in the future, the relevant calculations

are actually terribly straightforward. That was not difficult calculus we did,

of that logarithm, and that give results that agree with, and more importantly,

rationalize classical thermodynamics as observed through experiment.

Now we haven't yet derived that ideal monatomic gas partition function, but at

least we've shown that it's consistent with the ideal gas equation of states.

Which certainly looks like it's probably a good partition function.

Alright. Well, that was the ideal gas.

But, we've already seen in last week's series of lectures, most gases are not

ideal only at very high dilution. So, is this partition function useful for

other things. Well, next actually, lets take a look at

the van der Waals Equation of State and see if there is a way to relate it to a

certain partition function.

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