At the conclusion of section 19.2 on page 506 it states emphatically:
As long as the interactions between different parts of a system randomize the distribution of energy, each part of the system tends to have an equal share of the system's energy.
As the saying goes, it's bad luck to prove things that aren't true.
As it turns out, there is an "equipartition of energy theorem" ... but that's not what it says. Not even close. As a particularly simple and well-known counterexample, if we have a mixture of He and CO gas at room temperature, we do not observe equal energy per molecule, nor equal energy per atom. On a per-molecule basis, the energy ratio is approximately 3-to-5 ... not 1-to-1. So talking about "each part of the system" is not right.
It would be marginally better to talk about "each degree of freedom" but that's not right either. Under the same conditions, comparing He, H2, CO, and CO2, the ratios are 3-to-4.87-to-5-to-6.66 ... and I don't think CO2 has 6.66 degrees of freedom.
That could perhaps be swept under the quantum-mechanical rug, but that doesn't solve the problem either. Consider a column of gas several miles high, in the earth's gravitational field. For simplicity pretend it's approximately isothermal. The energy in the "part" of the system near the top is dramatically less than in the "part" near the bottom.
You might suspect the theorem only applies to kinetic energy ... but no, that's not right either. We know that very nearly half of the heat capacity of ordinary solids comes from potential energy, not just kinetic energy.
What the theorem actually says, in its simplest form, is that there is 1/2 kT of energy _per QUADRATIC degree of freedom_. Some things in this world are quadratic, and some are not. The energy of a column of gas in the atmosphere is neither quadratic nor negligible.
Now we have a problem, because section 19.2 proves something that isn't true. It's an example of fallacious reasoning. There are a number of problems, but the crucial point is that an example is not a proof. What's true for a small box containing four equivalent particles might not be true for a mile-high column containing zillions of dissimilar particles.
On the other side of the same coin, the reasoning behind the real equipartition theorem is quite a bit more intricate than what we see in section 19.2.
Equipartition is related to the simpler yet profound and subtle idea that all microstates that have the same energy are equiprobable. This however applies to microstates of the system as a whole, comparing one microstate to another. This is not remotely the same as comparing one "part of the system" to another.
See item #155 for a catalog of entropy-related issues.
At the conclusion of section 19.2 on page 506 it states emphatically:
As the saying goes, it's bad luck to prove things that aren't true.
As it turns out, there is an "equipartition of energy theorem" ... but that's not what it says. Not even close. As a particularly simple and well-known counterexample, if we have a mixture of He and CO gas at room temperature, we do not observe equal energy per molecule, nor equal energy per atom. On a per-molecule basis, the energy ratio is approximately 3-to-5 ... not 1-to-1. So talking about "each part of the system" is not right.
It would be marginally better to talk about "each degree of freedom" but that's not right either. Under the same conditions, comparing He, H2, CO, and CO2, the ratios are 3-to-4.87-to-5-to-6.66 ... and I don't think CO2 has 6.66 degrees of freedom.
That could perhaps be swept under the quantum-mechanical rug, but that doesn't solve the problem either. Consider a column of gas several miles high, in the earth's gravitational field. For simplicity pretend it's approximately isothermal. The energy in the "part" of the system near the top is dramatically less than in the "part" near the bottom.
You might suspect the theorem only applies to kinetic energy ... but no, that's not right either. We know that very nearly half of the heat capacity of ordinary solids comes from potential energy, not just kinetic energy.
What the theorem actually says, in its simplest form, is that there is 1/2 kT of energy _per QUADRATIC degree of freedom_. Some things in this world are quadratic, and some are not. The energy of a column of gas in the atmosphere is neither quadratic nor negligible.
Now we have a problem, because section 19.2 proves something that isn't true. It's an example of fallacious reasoning. There are a number of problems, but the crucial point is that an example is not a proof. What's true for a small box containing four equivalent particles might not be true for a mile-high column containing zillions of dissimilar particles.
On the other side of the same coin, the reasoning behind the real equipartition theorem is quite a bit more intricate than what we see in section 19.2.
Equipartition is related to the simpler yet profound and subtle idea that all microstates that have the same energy are equiprobable. This however applies to microstates of the system as a whole, comparing one microstate to another. This is not remotely the same as comparing one "part of the system" to another.
See item #155 for a catalog of entropy-related issues.