And when we go and we do the differentiations necessary to move from a
full partition function to the internal energy, in this case the molar internal
energy, we get the expression shown here, so 5/2 RT for the linear case coming from
translations and rotations. But 3rt an extra half rt because of the
extra rotation in the non-linear cases coming from translation and rotation,
again a different sum over normal modes in the vibrational component.
And if we now carry out differentiation with respect to temperature of the molar
internal energy, we'll get the molar heat capacity at constant volume.
Again we'll see a 5/2 appearing in front of R for linear, a 3 appearing in front
of R for nonlinear, and different summation from the contributions of the
normal modes. So, if you find this slide mildly
imposing, I have to admit that this is one of the more beautiful slides in the
course. It certainly is loaded with Greek
letters, and Roman letters, and exponents and.
Really, it's, it's a gorgeous piece of equation writing.
No one memorizes these things, not the best statistical mechanism around, one
goes and looks them up in books. But knowing how to work with them and use
them effectively and remembering some of the conceptual aspects of how, say, a
linear differs from a non-linear is really sort of the crucial take-home.
Skill that one wants to get out of having done these derivations.
So let's pause for a moment now, and I want to see if you've acquired some of
those skills, and then we'll come back. Alright, I'd like to wrap up, by asking a
question that after you've seen a huge slide full of math will often come into
your head I think, if you're a, average everyday person and that is.
Does this all work? So, let's look at a specific example.
Let's look at a reasonably important molecule: water.
And in particular we're going to look at water at pretty high temperatures, so
here in Kelvin we're going to go from about room temperature, that's not that
interesting a temperature but then we're going to take it up to looks like that's
about 373 kelvin right there, so that's the boiling point of water.
And then we're going to keep taking the temperature up.
So we're working with steam, and what I want to know is what's the heat capacity
of my steam associated with the vibrational contribution to the heat
capacity. And I'll express that then in terms of r,
how many factors of r are being attributed.
So remember that water has 3n minus 6. It's a nonlinear molecule, normal mode.
So there's three normal modes. It can contribute, you know, a fair
amount to the total heat capacity. And the, rotational, sorry, excuse me,
the vibrational temperatures of those modes, are given here.
2290, 5160, 5360. So, the heat capacity, actually I think
the slide is labeled incorrectly here. This is the full heat capacity.
So we're starting at 3. So 3 is the 3R associated with
translations and rotations. But we're now going to add to that, as
the vibrations begin making their contributions.
Because we're moving the temperature up. 'Cuz at low temperature, they contribute
nothing. So we ought to get from three up to six.
Six would be an asymptote, which is off the scale of this slide.
But in any case, the calculation that comes.
From this expression that we have derived from first principles, if you like.
We did quantum mechanics. We worked out partition functions for
ideal gases. We made a prediction of what the heat
capacity should be. Well that's the line so I'm just plotting
this function. The Points, the open circles.
Those are actual measurements of the heat capacity of steam, so very hot steam.
And what you see is, they fall very, very closely on the line, effectively.
So, you might have thought to yourself, well, steam?
How ideal a gas would steam possibly be? Well, surprisingly, the ideal gas
approximation is. working pretty well here.
Maybe it shouldn't be surprising, maybe people thought it would be an ideal gas.
But I think most people would say, oh you know water molecules they interact with
one another. The ideal gas approximation says that
they don't interact with one another. The quality of the ideal gas
approximation is really remarkable. And finally, I want to come back again to
talk about building a hammer last time. I think we might have a screwdriver here
as well. Because what does this graph really give
you? Now I, somebody did the measurements.
That's where we got the points from but let's say you hadn't done the
measurements ahead of time. You wanted to know, the answer to a very
practical question. How much coal do I have to burn in order
to get my steam from 700 degrees to 800 degrees because maybe I want to power a
steam engine and I think hotter steam is going to give me more energy?
Well, you would have been able to compute it from first principles.
You would predict the heat capacity and then you'd know how much energy it takes
to raise the steam another hundred degrees.
And so that's an extremely powerful tool of thermodynamics that can be put to
practical use. So there is our steam engine, I didn't
pick this steam engine randomly. It is a representative of the power of
thermodynamics. It's pulling us further down the tracks.
And we've pretty much wrapped up this week's, material, new material, so we'll
spend one lecture reviewing what I think are the critical concepts before moving
on in the course.