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

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分子热力学统计

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

从本节课中

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

Now that we've rationalized behaviors of gases based on interactions between gas

molecules. Let's spend a little time looking at such

interactions in more mathematical detail. So the viral expansion I presented to you

as a given and it comes from statistical mechanics and statistical mechanics and a

bit more depth than we want to do for a course that focuses on thermal dynamics.

But, it derives from knowledge of exact relationships between virial coefficients

and intermolecular interactions. So lets think about such an interaction.

If I have two molecules and they interact according to some potential energy

function and that potential energy functions depends only on the distance

separating those two molecules. I'll call it r.

And so I've illustrated here two, apparently they are monatomic species.

Maybe it's a noble gas. They are separated by sum distance r.

If the energy of the interaction depends only on that distance.

Then B2v can be expressed as minus 2 pi times avogadro's number.

So, 6.02 times 10 to the 23rd. The integral from 0 to infinity and that

integral is ranging over r, that is why I goes from 0 to infinity.

So from overlapping particles which probably is not a favorable situation

because r is 0 all the way out infinite separation and the argument over the

integral is the exponential of minus u of r.

So this is the potential energy function. It says, given a distance r, what is the

energy? It could be positive if it's repulsive.

It could be negative if it's attractive. But there's some functional form for that

interaction. Divided by kBT.

So what's kB? We'll see a lot more of kB coming up.

For now, I'll just name it. That is Boltsman's constant, and it is a

constant that when multiplied times T, has units of energy because you should

never exponentiate anything that has units if you can avoid it.

So we'd like this whole thing to be unitless.

If this is energy and here's temperature, then this must have units of energy

divided by temperature. You can look it up if you'd like to in

tables anywhere. For now, we'll just call it a constant.

That entire exponential minus 1, all quantity multiplying r squared dr.

So, where does u of r come from? Well, in principle you could compute it

using quantum mechanics. So given a gas, maybe helium, that's a

nice simple gas. We could put helium at various distances,

two helium atoms, one from another, compute their energy of interaction, fit

that to some sort of curve, your favorite curve.

And that would define a function as a function of distance.

it might not be so bad for helium but more generally such calculations can

still be a challenging undertaking. A lot of observation indicates though

that in a simple way if you have two things interacting [NOISE] in general.

That at very long range, there is an attractive interaction.

Not at infinitely long range, if you take two things too far apart, of course they

don't feel each other at all. But as they begin to come closer to one

another and begin to feel one another, they interact with a 1 over r to the

sixth dependence. And a negative symbol here means it's

attractive. So negative energy is downhill in energy.

And there is some characteristic coefficient called the c6 coefficient.

Six because it's associated with the 1 over r to the sixth term that dictates

how large that attractive interaction is. And then meanwhile as they get closer and

closer and closer sooner or later, these atoms with their clouds of electrons.

The electrons start to inter-penetrate these two clouds, that's a very bad

situation. You get a whole of repulsion that's by

nuclear fusion it doesn't happen spontaneously and that repulsion is

observed to go up as roughly the 12th power of the distance between the two

species. So I'll express that then as a positive

term because it's repulsive. Positive energy c12 and again the

subscript just indicating it's on a power term, that's 12 power inverse over r to

the 12. And that is a repulsive term.

And so just to see that this, you know, really works, if r is getting very, very

large, then r to the 6th is going to infinity.

And so the attractive piece will go to zero.

R to the 12th goes to infinity even faster.

So you're dividing by infinity, that goes to zero at very large distances.

No interaction. As they get closer and closer together

the attraction turns on first but then r to the 12th as I go closer to zero.

So zero, a number near zero to the 12th power is much, much smaller than a number

near zero to the sixth power. And so I'll be dividing by something much

closer to zero in this term than in this term.

And so the repulsion will overwhelm the attraction.

So, if I add them together, as I've sort of eluded to in my spoken discussion, I

could write, u of r, the potential energy function, as c12 over r12, to the 12th

that is, minus c6 over r to the 6th. So that kind of functional form was

really first explored in a lot of detail by John Lennard Jones.

And it is ref, and he wrote it in a slightly different way.

We'll look at that in a moment. and it's called the Lennard Jones

Potential. Generations of scientists have begun

their careers believing that there was someone named Lennard and someone named

Jones. In fact that is not true.

There was simply a single Lennard Jones. And so let's take a look at his equation

in this case. It says that u of r is equal to 4 times

epsilon times the quantity sigma over r, all raised to the 12th power.

So you see hiding in there, the r 12th terminated denominator minus sigma over r

so, same quantity sigma r, but now raised to the sixth power.

So you see the r to the sixth hiding in the denominator, its positive and

negative in front of the 12th and 6th power terms respectively as we saw

previously. And the relationship then between the C12

and C6 coefficinets and these new epsilon and Sigma coefficients, those are shown

here. So C12 is four epsilon sigma to the 12th

and C6 is four epsilon sigma to the sixth.

If you graph this function, I've kind of alluded to it in words but let's actually

look at a graph of it, defining the zero of the potential energy as being an

infinite separation, and the units that I will graph on are reduced units, that is,

they're unitless. So it's u itself, which would normally

have units of energy but epsilon has units of energy.

And so I'm graphing this as u divided by epsilon.

So as I come in, an attractive force first turns on.

And I begin to go down, down, down. The form of the function is such that the

bottom point of the well that's created by the attraction, then being balanced by

the repulsion is Epsilon d. And so, epsilon over epsilon would be 1,

and it's down, so it's at negative 1 on this plot, we see the bottom of the well.

Then, the repulsive force takes over, it crosses through zero again.

And where would that be? Well, when Sigma over r is one One to the

12th is equal to one to the 6th. They're both equal to one.

I get 1 minus 1 is 0. So I should plot, actually, r divided by

sigma. So again a reduced distance expressed in

units of sigma, if you will. So sigma has units of distance, and at r

over sigma equal to 1, that's where the potential crosses back through zero and

then it rises again steeply. So that functional form has some other

interesting properties that allow you to look at other critical points.

I think let's pause for a moment and I'll let you play a bit with the equation, and

perhaps identify another one of those interesting points.

Well we've had a chance to look at some specific parameters before.

And some parameters have had more intuitive feel than others.

So A and B parameters for the Van der waal's equation of state for instance,

not necessarily speaking obviously to people, well at least not necessarily to

me. And then there was B to V which had a

very clear and intuitive explanation having to do with volume.

Happily, the Leonard-Jones parameters are generally reasonably intuitive as well.

Because they have units that we think in. Energy and distance.

So, let's take a look at some tabulated Leonard-Jones parameters for various

gases. And I've got here all the noble gases, up

through xenon. As well as three diatomics, hydrogen,

nitrogen, and oxygen. So here are the epsilons.

And actually, in this case expressed in units of temperature, because we divided

by Boltzmann's constant. But still, it's sort of the variations

that are interesting. And so what we see is that the well

depth, as a way to think about this, right?

The epsilon told you How far down does attraction take you in the potential.

Is only 10 kelvin in for helium, then 36 for neon, 120 for argon, 164 for krypton,

by the time we're up to xenon, 229, so it's 23 times, roughly.

As attractive to bring two Xenon ato-, atoms together than it is to bring two

Helium atoms together. And that's consistent, actually with what

we know about the properties of the Noble gases.

So, the polarizability, we'll actually see this in a little more physical detail

soon. But the polarizability of the huge Xenon

atom with all it's electrons leads to favorable London forces between the two

atoms, or a dispersion. It can be called or induced dipole,

induced dipole, so that they can attract one another.

In helium, that's present but to a much lesser degree and so, you don't get a

very deep well. In the diatomics, hydrogen is least

attracted to itself, nitrogen by more, and oxygen by more.

So, the oxygen welled up is deeper than the nitrogen welled up.

The oxygen molecules are more attracted to one another.

Sigma, which is here tabulated in picometers can be thought of as a measure

of molecular size. And so not surprisingly, as we go from

helium to neon, all the way down to Xenon, the noble gases get bigger.

They get a whole lot more electrons and, and their valance shells expand.

And then hydrogen being formed to the lightest element compared to the other

two diatomics, is smaller than the other two diatomics.

Nitrogen and oxygen are quite close to one another, I'm not sure I want to

interpret 12, a 12 picometer difference in this case.

Suffice it to say they're similar size. Well, I want to take a closer look now, I

eluded at at the outset that the virial expansion is interesting because there is

a relationship between the potential energy between to particles and virial

coefficients. And I showed you the relationship, that

B2v was equal to this integral. Well now, instead of having a generic

function u, we've actually got a specific function to play with.

The Lennard-Jones potential. So let's insert the Lennard-Jones

potential. So this is a little bit of an exercise in

equation drawing, which can be an enjoyable exercise.

I'm going to replace ur over kt. I'll replace ur with this.

So here you see it. I've got the exponential.

My minus sign is still in there, minus ur, so I put in a minus the 4 epsilon is

going to be over kBT. Here it is, 4 epsilon over kBT.

They all multiply, this quantity in brackets.

It's still in there. Minus 1 all times r squared dr.

That's an imposing looking equation, it's got a whole lot of Greek letters and

other letters. Let's simplify it a little bit at least,

just to work with. And in particular, let me define t star.

So, t star is going to be Boltzmann's Constant times temperature divided by

epsilon. So, notice what I've done, this is a

little bit like, reduced unit in a way again.

Because, I've divided by something that's specific to a given gas, every gas has

its own epsilon, so I'm going to take temperature and I'm going to transform it

to tstar in a substance sensitive way. But in any case from a substitution

standpoint that means this e over KBT just becomes 1 over T star, so that'll

simplify the expression. And then let me replace, make, make this

substitution of x is equal to r over sigma.

So here I have sigma r, sigma over r, so I'll take the inverse of that to get r

over sigma. So I get all these x to negative powers.

There also out here was an r squared and a dr, so if I rethink about this r is

equal to sigma times x. So I would replace this r with a sigma

squared x squared. Sigma squared is a constant.

Let me pull it out here. And there was another sigma that we'll

see the other sigma in a minute. So, I've got a sigma squared out here.

And if r is equal to sigma times x, then dr is equal to sigma times dx.

So, I'll leave the dx here and I'll bring that other sigma out front.

So, that's why there's an appearance of a sigma cubed here.

So now I have minus 2 pi sigma cubed, Avogadros number, integral from zero to

infinity e of something that looks, I mean a friendly integral by any means but

it certainly looks cleaner. I've got simple powers of x and I've got

an x squared and a dx. Well You won't look this interval up in a

table, it turns out. It needs to be solved numerically.

But it can be solved numerically. And I'm just reproducing it here on this

slide. I want to do one last thing to it,

though. I want to introduce the other necessary

reduction, creation of a reduced unit by getting rid of the other molecularly

specific term that still appears. So, there's still a sigma here.

And remember, sigma is substance specific.

So, let me define B star 2v. And in this case, it is a function of t

star. So, I'll have a reduced argument that is

a function of a reduced property. and I will define B2, B2V star is B2V.

And we'll get rid of this kind of ugly term up front, 2 3rds pi sigma cubed

Avogodro's number. So that puts the substance specificity

into B2, 2V star. So I'll call that out.

And, and notice incidentally keep in mind that sigma had units of distance.

So if I have distance cubed, that's like a volume.

So to the extent this is substance specific, it can be thought of as

equivalent to a, a characteristic molar volume.

And that finally simplifies the expression as b star 2 v function of t

star is equal to minus three and this same integral expression we've seen

previously. Now the reason I did this, why did I go

through this long derivation of what looked like a complicated integral?

Well it gives rise to a pretty interesting additional law of

corresponding states. You expect to find corresponding states

here because I've built all the molecularity into the reduced variables.

So I can measure B2v of t star. I do it as a function of t.

And then I look up epsilon. And I turn all my t's into t stars.

And then I just divide by 23rds pi, Avogadro's number sigma cubed because

I've looked up whatever sigma is. And I will now plot B2v star against t

star and you get this plot. Where once again there are a whole bunch

of different symbols on there, little bit small maybe, I don't know if they all

show up perfectly. There's crosses, there's circles, there's

squares, all gases fall on the same curve.

And remember that the Leonard-Jones parameters themselves are determined from

experimental B2v values, alright? So this is a, a universal property of the

gases if you will. It's, once more it's not that it's just

some fitting game that somehow gave rise to that.

I also want to call your attention to an interesting feature.

This is B2v star, it has this molar volume deviation from ideality

interpretation. So at T star, about 3.2, all gases pass

through B2v star equal to 0. That is, there is no difference between

their molar volume, their real gas molar volume and the ideal gas molar volume.

They behave ideally. So that temperature, which is called the

Boyle temperature, is an in, an interesting temperature for a given gas.

It is the temperature at which the real gas behaves as though ideal.

Now it's different for every gas because remember T star has built into it actual

temperature and the molecular property epsilon.

But you can look up those epsilons and derive at what temperature would you

expect an individual gas to behave in an ideal fashion.

So that's a kind of a fun temperature to work at because you've got an ideal gas

even though you have no right to an ideal gas.

It's a real gas. Well that's the, the last law of

corresponding states we'll look at for a little while.

in the next lecture in the series what I do want to take a look at, the Leonard

Jones potential is, is interesting and useful.

And we've seen its ability to predict interesting things about corresponding

states of gases. But I want to consider some other

intermolecular interactions as well. So I'll take a look at physics behind

them and maybe some functional forms. And in addition, before doing that, I'll

recall for you when we looked at the values of epsilon for diatomic gases, I

called your attention that oxygen has a larger epsilon than nitrogen.

That is, oxygen is more attracted to other oxygen molecules than nitrogen

molecules to other nitrogen molecules. You might imagine, I haven't necessarily

proved it for you but you might imagine that that attractiveness would have some

influence on boiling point. So the less attracted gas molecules are

to each other, the lower the boiling point.

So, the implication would be that molecular nitrogen should boil lower than

molecular oxygen. Or if we're moving the temperature in the

other direction, that molecular oxygen should liquify as we cool things down

faster than gaseous nitrogen. So that can lead to an interesting

demonstration actually. Let's see if we can maybe make a little

bit of liquid oxygen and determine what interesting properties that substance

might have. So we'll do a demonstration and then

we'll come back to these other intermolecular potentials.

[NOISE] As our study of real gases has progressed, we've discussed various

properties of gases and how they vary by composition.

The two most common gases in our planet's atmosphere are nitrogen, N2, at 78% and

oxygen, O2 at 21%. Nitrogen has a boiling point of 77

kelvin, while oxygen has a boiling point of 90 kelvin.

Thus, at liquid nitrogen temperatures, we would expect oxygen to condense from the

air. Let's actually do that.

I'm going to fill this can with liquid nitrogen.

[SOUND] The steam you see as I'm doing this.

Is actually water vapor from the air condensing into water and ice from the

cold of the nitrogen. But we won't worry about that.

As the can cools down. It's going to reach a temperature on the

outside. It already has of 90 Kelvin at which

point, oxygen in the atmosphere begins to condense onto it.

And rolls down to drip into this styrofoam cup, which is sufficiently

insulating that we should be able to collect it enough to play with if we wait

for a little while. As I tip this for the camera to record,

you can see it's pale blue color. Oxygen also has a magnetic moment because

of the unpaired electrons in the molecule.

And, if I dip this magnet into the liquid, do you see how the drops cling to

the magnet? However, let's consider whether being

blue and paramagnetic is sufficient proof that this is liquid oxygen.

How else might I convince you that I'm not fooling you and I've really

condensed, just a lot of liquid nitrogen instead.

Perhaps by using an even colder liquid in the can.

Well, one thing oxygen does is support combustion and while its only 21% of the

atmosphere. In this liquid, it's sustensibly pure.

So I wonder what would happen if I were to dip this salad crouton, toasted bread

into the liquid oxygen and then touch a flame to it.

Shall we try?

[NOISE]

Wow, that crouton burnt with a vengeance, didn't it?

That's the effect of having so much oxygen immediately available.

Even if you believed me right from the beginning, that was certainly a fun way

to further prove the chemical nature of the condensed gas.

The difference in boiling points for nitrogen and oxygen, derives from their

different intermolecular interactions. While ideal gases by definition do not

have such interactions real gases do and we'll be studying how those interactions

effect gas behavior in detail using various quantitative models.

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