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Lecture 2.05: Hydrogen equation of state
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来自 Caltech 的课程
太阳系科学
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从本节课中
Unit 2: The insides of giant planets (week 1)
与讲师见面
Mike BrownProfessor
Planetary Astronomy
If you were comfortable with the, the level of physics and
math in the last lecture that's great, we're going to keep going.
If you were uncomfortable with it, you can do one of a couple things.
You can look up some basic physics differential equations
on, on many different sites, will provide some of them.
Or you can just follow along in a sort of higher conceptual level, don't sweat the
details of the physics and the math so much, but just still try to get the point.
The point last time was Hydrostatic equilibrium, you have a balance between
compressing a gas, and how much it wants to re-expand, and that balance is
between the pressure of that gas and all of the weight on top
of it, and the other important point to remember is the equation of state.
That's such an important point we're going to work on it again, today.
Equation of state, really is a huge part of the name of the game
for understanding what's going on inside of
a giant planet, really, inside any planet.
I'll remind you equation of state simply means what is
the connection between the pressure and the density, and can we
do it a little better than our simple PV equals nRT
that we learned in high school, for the ideal gas law?
We're going to make an approximation today that's the
opposite end of the spectrum from PV equals nRT.
We're going to do something called a Fermi gas.
Now it's going to look a little bit crazy, a little bit complicated, but
in the end, it's relatively straightforward
to at least conceptually understand going on.
So, bear with me because we are now delving into quantum mechanics.
And we're delving into quantum mechanics because in
the limit of high density, the reason that
things don't just collapse on top of themselves,
is because of quantum mechanics, pretty much anything.
If I take a solid thing like this pen, why
can I not push my fingers together on this pen?
It's because, essentially, the electrons inside of the atoms, inside
of this pen, do not want to be pushed together.
It's not quite as simple as that, but it's almost as simple as that.
Why don't the electrons want to be pushed together?
Has nothing to do with electrostatics like negatively charged
electrons like to repel each other because for every electron
inside of here, there is a proton inside of here,
so overall this is neutral, so it's not electrostatic repulsion.
It's purely quantum mechanical repulsion, and the reason it happens is because
electrons are a fundamental particle of a type called fermions.
Fermions have a critically important property that is the
reason for this lack of ability to compress this pen.
Fermions, no two Fermions can occupy the
same quantum mechanical state at the same time.
Usually when we think about quantum mechanical states, if you ever think
about quantum mechanical states, you think about an electron going around a nucleus.
And it has an energy level, and an electron can
then jump up to a different energy level or down.
And you can only have a certain number of electrons in each energy level.
Well, actually I'm not even going to talk about electrons and energy levels.
We're just going to now make a, a simplifying approximation that
we have electrons, a sea of electrons in free space.
And in that sea of electrons, we magically put enough
positive charges, so we overall have a, a net neutral charge.
Even in this sea of electrons, the electrons
don't want to be in the same state.
Now, I keep saying that word, don't want to be in the same state.
Let me draw you a picture of what that means quantum mechanically.
If, if I had a box, and I put an electron in a
box, and I asked myself the question, where is the electron in that box?
Well, one of the weird things about quantum mechanics
is that there's no answer to where the electron is.
Really there's a, a probability that it's in any particular place.
And that probability inside of a box goes something like this.
There's a high probability it's, it's close to the center, at any point
in time there's a low probability that it's at its, it's at the end.
That's for one electron.
If we put in two electrons, well, electrons have this
funny property that they, they can either be electrons that
are spinning up, spinning in this direction, or they can
be electrons that are spinning down, and spinning in that direction.
And those two electrons can actually be in the same state at the same time.
So, the second electron will be found in the same range of places.
Put a third electron there.
Where does it go?
Well, it can't have the same probability distribution as those first two electrons.
This is what I mean when I say it's in a different state.
In fact, the sec, the third electron will be humped.
It will either be here or it'll be over here or some range
over here, but it will be less likely to be in the center.
Fourth one, same place.
Fifth one, you might be able to guess what's going to happen.
It can be here or here or here, it has three humps.
It's a little hard to see that drawing, let me erase all the
other electrons, and just draw you where those fifth and sixth electrons are now.
Seventh and eighth electrons, you guessed it.
It's a very strange thing, why you might ask, why because quantum mechanics.
Quantum mechanics does this.
This is when I say that electrons can't occupy the same state.
Every time I put new electrons into the box, those new electrons have
to be in a new state that has, has not been occupied before.
We're going to label these states by the number of peaks they have.
This one has one, two, three, four peaks.
We're going to name it as k equals 4 for this one.
This one had a k equals 3, and we're going to try to figure
out how much energy each electron has in each one of its states.
And I'm going to do this by using a very, very poor analogy.
That really is a pretty bad representation of how you really calculate
the energy of the state, and yet it, it works moderately well.
So, so if you know your quantum mechanics better, I apologize.
If you don't know quantum mechanics, think of it
this way, it's a good way to think about.
The first and second electrons, which are somewhere in
the box with, with a probability something like this.
Moving around with some velocity, we don't know what it is, we'll call it v-not.
So for k equals 1, the velocity is v-not.
So the electrons are moving around here
with some effective velocity, something like that.
Now the third and fourth electrons, think of it this way, either here or here.
There's some probability that they're over here, and
some probability that they're over here, and they
are, in some sense, having to move faster
to jump back and forth between these, these possibilities.
They have something like twice as much region to cover,
they have to be moving something like twice as fast.
I know, again, if you know your quantum
mechanics, you're probably about to revolt at this point.
But, but go with me.
Three humps, one, two, three.
going to make the same argument, if three times as
many peaks to have to jump around between to
get around all those three peaks with equal probability,
they have to move something like three times the speed.
[BLANK_AUDIO]
And you can see how this continues on.
All right what good does this do me?
I'm actually interested in energy, and I'm interested in the kinetic energy.
As you remember, kinetic energy is one half mv square so the energy of
these electrons, I'm going to ignore things like
one halves, I'm even going to ignore things like
ms, I'm going to even ignore things like V naughts, I'm going to say that the energy
of this electron is proportional to the
velocity squared, which is equal to k squared.
So that first electron, that only has one hump
in it, is the lowest energy electron that is.
Next, lowest energy has two humps, next lowest energy has three humps.
By the time you put in, a million electrons
into one box, and you have a million humps,
you have an energy that, last electron that you
put in there, has to have a very high energy.
It'd like to have a lower energy, everything likes to have a lower energy.
But there are no low energy states available to it.
So if I now just count electrons and I wanted
to say what the energy of what, of that last
electron I put in, well now I have to admit
that this is, I was just doing one dimensional box here.
Boxes are really three dimensional so, you can
put electrons with humps in these, this direction.
You could put electrons with humps in this direction, and you could put
electrons with humps in the direction in and out of the, the screen here.
Each one of those counts as a different state.
So think of it this way.
If I have a certain number of electrons that I'm
filling inside this box, what's my highest level of k?
Well, I can do fill it up k in
this direction, k in this direction, k in this direction.
So the total number of electrons is
proportional to that highest value of k cubed.
And it's nice to think of this not in
terms of total number of electrons, but in terms
of something like distance between electrons, or maybe volume
that each electron gets to occupy on its own.
Even though that's not a real concept, it's a nice easy concept to do.
So the number of electrons is proportional to one over
the volume that each electron occupies, which of course is
equal to, proportional to one over the radius cubed of
the distance to the next closest electron of each electron.
So we have the k is proportional to 1 over r.
Kind of makes sense, if that number of humps is huge, that means
that that distance to the nearest electron, 1 over r, is kind of small.
That's kind of what this is saying.
This is great, because now we know the energy is
proportional to 1 over r squared, and we know that
the pressure, a, one of the definitions of pressure, is
that it is the change in energy with respect to volume.
That makes sense if you compress it, make the volume go down,
the energy inside there goes up, and that's the pressure that you feel.
So dE dV, energy is proportional to 1 over r squared, change in energy
with respect to volume is going to be proportional to 1 over r to the 5th.
The density of the material, of course, is
going to be proportional to 1 over r cubed.
The closer you put those electrons together, the denser the thing is
going to be, and so you're left with a very simple equation of state.
I just made up an equation of state.
Pressure is proportional to density to the five thirds.
Let's think where all this came from, one more time.
It's the simple fact that the electrons cannot occupy the same
state inside of a box, and by a box I mean anything.
This pen is a box and so every time we add one more electron into this
box, we have to add another hump in
this distribution of where that electron must be.
What I'm really saying is that every time I add another
electron in there, it has to be an ever higher energy electron.
The energy of the electron that I add
is proportional to the number of humps squared.
That's because energy is one half Mv squared, and that number
of humps is also related to the average distance between electrons.
So, when you compress the material, you make
R smaller, you make Rho larger, but you're
making the pressure even larger, because you're increasing
the energy of all those electrons inside there.
Okay, this is probably the last time we're going to do this much level of
detail of quantum mechanics, but I want you to think of it this way.
I want you to think of walking down the road.
Every time your foot goes on the street, you are pressing
down on the road and you are pressing down on those electrons.
You're forcing those electrons a little bit close together and they're resisting.
They increase their density a little bit.
They increase their pressure quite a bit and they resist
being deformed by the weight of your foot coming down.
Drive down the road in your pickup truck, and it's slightly
different, you have more weight pushing down, but again, not too bad.
Put yourself inside of Jupiter, however, and you have enough pressure
pushing you down that the pressure can finally increase by dramatic amounts.
More importantly, though, we have an equation of state.
That is appropriate, close to appropriate for high pressure material.
It's not perfect, the inside of Jupiter is not a sea
of electrons inside of a bigger sea of positively charged thing.
It is not a fermi gas like this one is, but it is pretty close.
And this gets approximately the right form of behavior
over some of the ranges that we'll care about.
We'll use an equational state like this as an experimental equational
of state, and we will explore what's going on inside of Jupiter.
