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 2

This module begins our acquaintance with gases, and especially the concept of an "equation of state," which expresses a mathematical relationship between the pressure, volume, temperature, and number of particles for a given gas. We will consider the ideal, van der Waals, and virial equations of state, as well as others. The use of equations of state to predict liquid-vapor diagrams for real gases will be discussed, as will the commonality of real gas behaviors when subject to corresponding state conditions. We will finish by examining how interparticle interactions in real gases, which are by definition not present in ideal gases, lead to variations in gas properties and behavior. 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

Well, let's review the concepts from Week 2 that are particularly important.

We discussed gases, and in particular, real gases this week.

But we did start with the ideal gas equation of state.

And recall that that's PV equals nRT. Where P is pressure.

V is volume. n is the number of moles.

R is the universal gas constant. And T is the temperature.

It's a little bit more convenient to work with molar volume which is an extensive

property, not an intensive property. And that would be expressed as PV bar

equals RT. And indeed, the ratio of PV bar to RT

itself has a name. That's called the compressibility and

it's usually indicated by capital Z. Gases are in fact rarely ideal.

That is, they do not obey the ideal gas equation of state except at very low

pressure. The van der Waals equation of state, on

the other hand, is a better predictor of non-ideal gas behavior, and there are

other equations of state that do well, as well, and they include the Redlich-Kwong

and the Peng-Robinson equations, which we looked at briefly.

All of those three equations of state for non-ideal gases are cubic in the molar

volume. And as a result, they are able to

describe the liquid and gaseous regions of pressure volume isotherms.

Recall that, on the pressure-volume isotherms, there are some important

points. One of them is the critical point.

At the critical point, the van der Waals equation of state has only a single root.

A single value, value of molar volume that satisfies the cubic equation.

And, once we know that critical volume, we can determine the van der Waals

constants, a and b from that position, the critical volume, the critical

temperature, and the critical pressure. The Law of Corresponding States says that

all gases have the same properties if compared at their corresponding states.

Where under corresponding conditions that is, where corresponding conditions means

that the conditions for that gas, relative to its own critical conditions

are the same as the conditions for a different gas relative to its critical

conditions. So that is a corresponding state or

corresponding conditions. We also saw another equation of state,

the virial equation of state. And the virial expansion that appears in

that equation of states has coefficients that are directly related to

intermolecular interactions. The second virial coefficient, B2v,

measures the deviation of the volume of a real gas compared to an ideal one under

the same temperature and pressure conditions.

So, if B2v is a positive number, that describes how much more volume the real

gas occupies than an ideal one would under the same conditions and vice versa.

When B2v is negative, that means that the gas is occupying less volume than an

ideal one at the same conditions. We looked at the sorts of attractive

interactions that could occur between two different molecules in a real gas.

And in particular, we looked at the Lennard-Jones potential where the

intermolecular u is expressed in terms of the strength of a molecular interaction

epsilon and the size of the molecules themselves sigma.

And the Lennard-Jones potential has an attractive term that drops off as r to

the 6th, that is, as it goes as r to the minus 6, and a repulsive terms that goes

as r to the minus 12. So at very short distances, the repulsion

rises very, very steeply. The Lennard-Jones parameters themselves,

epsilon and sigma, can be determined through analysis of experimental second

virial coefficient values. We also talked about the physical

underpinnings of the r to the minus 6 attractive term and the most dominant

contribution to it, namely dispersion. We talked about London's development of a

quantum mechanical explanation for dispersion.

And his equation that produces its magnitude based on atomic or molecular

ionization potentials and the polarizabilities of the individual

molecules. Finally, we looked at some simpler

potentials than the Lennard-Jones potential that allowed us to solve for

the second virial coefficient analytically in order to gain some

physical insight. One potential was the square-well

potential, where there is a region of attractive interaction followed by a hard

wall. And determined that the predicted second

virial coefficients are not bad. And compare reasonably favorably to the

full Lennard-Jones potential over reasonable ranges of temperature and

pressure, and I showed an example for nitrogen gas.

The simplest potential, the hard-sphere potential, where there is no attractive

force, and at the point of two molecules touching, there is an infinitely

repulsive potential. is not very appropriate under most

conditions, but at very high temperatures, it does describe reasonably

accurate gas behavior, and it says that the second virial coefficient is positive

and independent of temperature. So it's that region in a plot of the

second virial coefficient versus temperature, where the, the plot has

turned over and is almost a straight line horizontally.

The square-well potential that has a free parameter lambda in it, includes the

length over which a constant potential is effective, and this emphasizes again,

that we can get good agreement with experimental data.

All right. Well, that completes the summary of the

key concepts from this week. you'll have an opportunity to explore the

concepts in more detail on the homework associated with this week.

Good luck with that. And I look forward to continuing as we

move to next week. And, to talk about what we will be

looking at next week, we're going to start with the concept of Boltzmann

probability. See you then.

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