Video 3.5 - van der Waals Equation of State Redux

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From the course by University of Minnesota

Statistical Molecular Thermodynamics

163 ratings

University of Minnesota

Statistical Molecular Thermodynamics

163 ratings

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

From the lesson

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.

Video 3.5 - van der Waals Equation of State Redux 6:37

Video 3.6 - The Ensemble Partition Function15:27

Video 3.7 - The Molecular Partition Function 7:27

Meet the Instructors

Dr. Christopher J. Cramer
Distinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics
Chemistry

In this video, I want to revisit the Van Der Waals equation of state.

So we've just seen that there is a relationship between an ensemble

partition function. And an equation of state by having worked

with the trial partition function shown here.

Which involves a molecular or atomic partition function combined with a Nth

power divided by N factorial assembly. Of the ensemble partition function, where

these taken to the nth power and dividing by N factorial are associated with

non-interacting indistinguishable particles.

And we showed that when we took that ensemble partition function and solved

for the pressure, we recovered the ideal gas equation of state.

So in today's video I'd like to look at a different partition function.

And that's the one shown here. And so it continues to be a function of

number of particles, volume, beta. There's a 1 over N factorial.

There is a term 3N over 2 power that continues to include within it things

like the mass of the gas, here's beta, here's Planck's constant.

V minus N times r all to the Nth power and the exponential of s beta N squared

over V. And these two new parameters if you will

new constants r and s are positive constants.

And so that defines an ensemble partition function.

And you see the dependence on N and on V and on beta and I'll give you a moment

here to think a little bit. And work with that partition function and

see some limiting behavior. And then we'll come back and work some

more with it. All right.

So you've seen how this partition function reduces under certain limiting

behavior. And one thing I might point out here.

It's tempting to notice that here you have a term that involves something

raised to the Nth power, the 3N over 2 but there's an N if you like, in the

exponent. Here's something raised to the Nth power.

And hiding in this exponential is an Nth power.

It's N squared but I could take exponential of the N, take all that to

the N. Remember that a power of a power is like

something to the product of those two powers.

so it looks in a sense as though you could write this ensemble partition

function as some primitive partition function all raised to the Nth power.

Not unlike the ideal gas case. That's incorrect, though.

You shouldn't view it that way. There is not a molecular partition

function here and that's because it would not depend only on volume and beta.

There would still be a dependence on N in this term.

There would still be a dependence on N in this term and of course you can't have a

molecular partition function that depends on how many particles there are on in an

ensemble. That, that doesn't make logical sense.

So this particular ensemble partition function doesn't decompose into a

molecular component. It's just good for the ensemble.

But let's continue to work with it in particular.

Let's explore the expectation value of the pressure, which will be derived by

taking the partial derivative of the log of the partition function.

With respective volume and multiplying by kT.

So first, let's expand the log of this ensemble partition function.

And so if I take the log and I separate out all the various terms, the products,

the powers, they multiply the log, I'll get 3 halves as I pull this 3 half,

sorry, 3N over 2, 3 halves N, multiplying all the material inside that logarithm.

So 2 pi m in the numerator minus h squared and beta in the denomenator plus

again a power will come down N log this term.

Plus the log of an exponential is just the argument of the exponential and then

in ally here is the minus log N factorial here in this term.

So let's just move that to the next slide to continue working with it.

And if we differentiate that now with respect to volume.

Well, happily nothing in this term depends on volume, that's going to go

away. Here are the only two terms that depend

on volume and those are pretty simple differentiations to do.

So we'll end up with partial log Q, partial V is N over V minus N times r

minus S beta and squared, all divided by V squared.

So now that we have that in hand we can finish solving for pressure.

That is we're going to multiply times kT. That derivative and we end up then with

pressure is equal to NkT over V minus Nr minus sN squared over V squared.

So let me rearrange that equation a bit and in particular, I'll take the minus sN

squared over V squared, I'll bring it over on the other side and add it to

pressure. And then I'll multiply both sides, times

V minus NR. So here's the multiplier over here and

that removes the term and the denominator over here.

And finally, let's take N, the number of particles equal to Avogadros number so

that we'll be working with molar quantities.

And we'll express a and b in molar units in which case, I will get P plus a over V

bar squared times V bar minus b. Equals what is Avogadro's number times

Boltzmann's constant, that's the universal gas constant, RT.

And so you see that we have recovered the Van de Waals Equation of State.

So that partition function is the ensemble partition that is consistent

with the Van de Waals Equation of State. So with that in hand, seeing that

derivation of course we can work with the partition function in other ways as well.

So I think I'll pause here for a moment and let you consider the internal energy

of a Van de Waals gas. Good we've had the opportunity now to see

the relationship between an ensemble partition function and an associated

equation of state for two cases. The ideal gas and the Van der Wall's gas.

It gives you some indication of how powerful an entity, the ensemble

partition function can be. So let's continue to work with it more in

the next video. [SOUND]

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