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

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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

Prior to giving you a chance to apply what you've learned thus far on this

week's graded homework. Let's spend a moment and look at the key

concepts that were in this week's material.

So first, the probability of a state in an ensemble, being populated decreases

exponentially with it's energy and temperature dictates the rapidity of that

exponential decay. Secondly, the canonical ensemble refers

to an ensemble that has a fixed number of particles N, a fixed volume V, and a

fixed temperature T. And so, sometimes, you'll hear people

refer to the canonical ensemble as the NVT ensemble.

The partition function capital Q, can be viewed as a measure of the number of

accessible states at a given temperature. Very large value for Q, very large number

of accessible states, and vice versa. The ratio of the exponential of the

energy of a given state divided by KT, relative to the partition function,

provides the probability of a member of the ensemble being in that state.

And so, while I haven't got the equation here, maybe we'll remember it.

Probability is E to the minus that state's energy divided by KT.

And so, referring back to that first bullet point, notice that if the energy

gets larger and larger, E to the minus that energy is getting smaller and

smaller. So, the higher the energy, the less

probability. And the appearance of the temperature and

the denominator, of the argument of the exponential, dictates how quickly that

exponential decay takes place. Various macroscopic properties can be

computed as averages of properties weighted by ensemble properties.

So remember, that's the fundamental postulate of statistical thermodynamics.

That the observed energy reflects an average over all the possible energies

weighted by their probability in the ensemble.

When we consider non-interacting particles, the partition function of the

ensemble can be written as the product of the partition functions for the

individual particles. And that's a considerable simplification.

The macroscopic internal energy of a monatomic ideal gas is related to the

ensemble average energy for an appropriate partition function.

So we found a relationship between macroscopic thermodynamics and derived

statistical thermodynamics, and we did the same thing with the molar heat

capacity. The macroscopic pressure of a monatomic

ideal gas is related to the ensemble average pressure for an appropriate

partition function as well. And, we showed that it was consistent and

allowed us to derive the ideal gas equation of state.

So far, those partition functions have just been handed to us.

Actually, one of our goals next week is going to be to derive them from first

principles. We also worked with another partition

function handed to us and showed that it was consistent with the van der Waals

equation of state. So, the connection between equations of

state and partition functions is clear. There's a difference between partition

functions for distinguishable and indistinguishable non-interacting

particles. And in particular, for the latter, we

need to divide the ensemble partition function of the former, which is just the

product of all the molecular partition functions by N factorial, where N is the

total number of particles in the system. Incidentally, for those of you who have

wondered how big, if N is say Avogadro's number.

How big Avogradro's number factorial might be?

You won't be able to plug that one into your favorite spreadsheet, I'm afraid.

We'll just have a notion about this very, very large number N factorial.

But we'll come back to that later. Molecular partition functions are

themselves expressible as products of partition functions.

Namely, the partition functions associated with translational,

rotational, vibrational, and electronic energy levels within the molecule.

And finally, partition functions expressed over states can be related to

partition functions expressed over levels through inclusion of the degeneracy.

So there's a unique level for every energy, but there maybe multiple states

that have the same energy, and that multiplicity is called the degeneracy.

And that's just a bookkeeping tool that may prove useful in the future.

All right, those are the key concepts. Good luck on this week's homework, and

next week, we will allow the train of statistical thermodynamics to pull us

further down the tracks. And we will begin looking in fact at the

ideal monatomic gas and derive from first principles its translational partition

function.