...^1.1

Think of a semi-permeable membrane in osmosis. It allows some molecules, typically small ones, to pass while not allowing large ones to do so.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^1.2

This is a bit of a cheat. In fact, one would have to wait forever in order to be sure that the properties of a system were not changing, as changes can be very slow. On the other hand, since we live in a dynamic universe, it is guaranteed that no system will remain unchanged forever! The cure is to assume that the system is observed for a time long compared to any ``ordinary'' process that might take place.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... state.^1.3

In the real world it is often hard to do this exactly. Frequently the constraint is imaginary and we do the experiment as a thought experiment.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^1.4

At worst, for a mole of particles, one would have to, in principle, specify 3 moles each of coordinates and momenta!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^1.5

This is no problem. Devices for just this purpose are manufactured and sold. They are called constant temperature baths and are typically water baths with heating and cooling facilities to maintain a constant temperature.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.1

Folks occasionally talk about several fairly exotic additional states of matter such as plasmas. Under normal conditions of temperature, pressure, and density, we almost never end up observing these exotic states.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.2

Again a small warning: some liquids are sufficiently viscous as to hardly move at all. They are still liquids however. Solids also have a definite crystal structure (most of them anyway) while liquids do not.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.3

Of course, if the container is tall enough gravitational effects come into play. We will exclude that case, at least for now.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.4

That's likely because folks like to use numbers of the order of 1 to describe things. And common pressures vary from exceptionally small to exceptionally large, depending on your field of study.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.5

This assumes that the gas is ideal. In other materials there are various potential energies to be considered as well

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.6

Molecules have very little brain power. What they "know" is what they feel, and what they feel are attractive and repulsive forces from other molecules, the molecules in a wall, etc. Ideal gases feel none of these and, in fact, if we were totally consistent, could not be contained inside of any wall.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.7

Liquids are not very compressible. Thus even a small change in volume implies a large change in pressure

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... forces^2.8

Electrostatic forces being those coming from ions, dipoles, etc.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.9

They fall off something like the sixth power of the reciprocal of the distance between molecules, if not faster.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.10

Thus according to this equation, different gases will behave (slightly) differently, which is in accord with experiment. The ideal gas law, on the other hand, claims that all gases behave in exactly the same way, which is not in accord with experiment.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...b.^2.11

This isn't 100 percent true and we'll return to this in just a minute.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.12

This reasoning is actually flawed. The wall is made up of molecules too. And they also have attractive forces. We will ignore this complication...

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.13

It is also cubic in the number of moles, but one only rarely needs to solve for that since most often one works with the molar volume and not the number of moles.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.14

This point will be taken up later. Suffice it to say that a positive slope would imply a violation of the law of conservation of energy.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Maxwell^2.15

Yes, he of the Maxwell Equations in electromagnetic theory.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.16

But you knew that already...

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...eq:vdwab3.^2.17

I.e. multiply the left hand-side of Equation 2.5.4 by the left-hand side of the reciprocal of Equation 2.5.5, and do the same for the right-hand sides.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... useful^2.18

Especially in more advanced courses such as statistical mechanics.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...p:^2.19

Technically, it's a Maclaurin series, but never mind. It is also a Taylor series.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.20

A good part of science is knowing when to apply the tricks and when not to!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^2.21

Students may amuse themselves figuring out why.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^3.1

Of course you already knew all this.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^3.2

Real people don't go around asking each other such questions, but this is a mathematical interlude...

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.1

Unless, of course, you are looking at this on a black and white printed page. In that case the upper path is the Red path and the lower one the Blue path.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.2

We will soon meet three exceptions (there are more), the heat, the work, and the heat capacity. These do depend on the path taken.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.3

That is, came to have the same temperature.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.4

Think about a ball being thrown. The atoms and molecules in the ball are not only moving randomly, but in addition they are all also moving with a common speed in a given direction. The actual motion of the atoms and molecules is the sum of these two motions, one random, the other directed.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... all.\^4.5

I'm glossing over a major question: Can one go between any two states adiabatically? The answer is no, one cannot. But if one can't go from 1 to 2 adiabatically, it turns out that one then can go from 2 to 1 adiabatically. And as we've already seen, this allows us to determine the change in $\Delta U$ just as if we'd actually gone from 1 to 2 instead.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... piston.\^4.6

Such objects can be obtained from the Chemistry Stockroom where they are stored right next to the bottles of ideal gas.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... ideal)\^4.7

Let's not make this any more complicated than it has to be...

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.8

If we exceed this pressure, the piston won't go up at all and we will do no work!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... volume,\^4.9

One thinks of heating an unopened can of beans in a campfire. The water inside vaporizes, the pressure goes up enormously, and the can explodes, usually creating an awful mess if not serious injury.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... pressure^4.10

You knew they were coming, didn't you. After all, what's the point of making a fuss over the constant volume restriction if there weren't other types of restrictions?

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.11

There is a relationship between these two heat capacities. We'll get to that later.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... like^4.12

You won't be surprised to learn that indeed, heat capacities are often given by such equations.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...enthalpy.^4.13

Why H for enthalpy? J. Willard Gibbs, the practically unknown genius who did much to develop thermodynamics, used Greek letters for many of the thermodynamic functions. For the enthalpy he used the Greek letter eta. An H is a capital eta.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.14

It is worth taking a minute or two and demonstrating to yourself that the units of pV are indeed joules.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.15

You will have noted that I'm covering each of our more or less standard processes.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...^4.16

This is a real case where the heat capacity is infinite!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.