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

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From the course by University of Minnesota

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.

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.

- 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

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

Â concepts that were in this week's material.

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

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

Â exponential decay. Secondly, the canonical ensemble refers

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

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

Â refer to the canonical ensemble as the NVT ensemble.

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

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

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

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

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

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

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

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

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

Â smaller. So, the higher the energy, the less

Â probability. And the appearance of the temperature and

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

Â exponential decay takes place. Various macroscopic properties can be

Â computed as averages of properties weighted by ensemble properties.

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

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

Â weighted by their probability in the ensemble.

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

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

Â individual particles. And that's a considerable simplification.

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

Â ensemble average energy for an appropriate partition function.

Â So we found a relationship between macroscopic thermodynamics and derived

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

Â capacity. The macroscopic pressure of a monatomic

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

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

Â allowed us to derive the ideal gas equation of state.

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

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

Â principles. We also worked with another partition

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

Â equation of state. So, the connection between equations of

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

Â functions for distinguishable and indistinguishable non-interacting

Â particles. And in particular, for the latter, we

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

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

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

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

Â How big Avogradro's number factorial might be?

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

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

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

Â themselves expressible as products of partition functions.

Â Namely, the partition functions associated with translational,

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

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

Â partition functions expressed over levels through inclusion of the degeneracy.

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

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

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

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

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

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

Â ideal monatomic gas and derive from first principles its translational partition

Â function.

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