The Four Laws of Thermodynamics

So, I'm reading "The Four Laws" by Peter Atkins and I'm having a little trouble understanding some parts of the first chapter. I figure I'll have more trouble later on so the purpose of this thread is to get help with some questions.First, he discusses Boltzmann's distribution regarding energy levels. It has the equation:(Population of state of energy E)/(Population of state of energy 0) = e^(-BE)Where beta is a constant inversely proportional to temperature.The graph is a histogram of the form 1/x^2. It shows that the majority of molecules occupy the lowest energy state of 0. All higher energy levels have a smaller population of molecules than the ground state 0.So far, so good. But then he talks about using this equation to get the Maxwell-Boltzmann distribution for the various speeds of molecules. Namely, this: The problem I'm having is that the graphs aren't isomorophic. Shouldn't they be similar given that velocity and energy are directly proportional?How do you get from the first equation of the form 1/x^2 to the Boltzmann-Maxwell distribution?

First, I apologize for the sloppiness in the first post. I wasn't very clear and I made a lot of mistakes.

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The number of molecules with a given velocity is the product of the probability of a molecule being in a given state with that velocity (that's the one that goes like e^(-BE)) multiplyed by the number of states possible with that exact velocity.

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I think I understand what you are saying here, but I have a few more questions if you will indulge me.The graph of the Boltzmann distribution in the book has energy (E) on the y access and the number of molecules in that energy state on the x access. The energy increases in discrete units and the lowest energy level has the most molecules in it. It looks something like: < !--UB -->< !--UB -->< !--UE-->So, if I were to naively use this to make a graph of the velocities of those molecules, I'd count the number of molecules in the bottom state, use the kinetic energy formula to figure out the velocity, and plot it on a seperate graph. Then i'd continue to go up in energy levels doing the same thing.But this wouldn't produce the "Maxwell-Boltzmann Distribution." Now, you're saying that different energy states can have molecules with the same velocity. That's what I don't understand, considering I don't see that in the Boltzmann Distribution. I just see the discrete energy levels rising with the relative number of molecules in that enery state going down.Thanks alot for the help! I need it. Edited by humoshi, : No reason given.Edited by humoshi, : No reason given.Edited by humoshi, : No reason given.

I kind of understand what you are saying. I tried to clarify my troubled thinking in my reply to fallacy cop. Maybe you can understand my problem a little better.Thanks for the help!

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However, suppose only one state has energy E1, but there are 4 states with energy E2. Then there will be 100 molecules with energy E1, but there will be 4*50 = 200 molecules with energy E2.

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I understand the math alright, but I really can't get a pictorial analog in my head.You are saying a state is defined by more than just its energy (E) and that different states can have the same E. So when you factor that in the ground energy state is no longer the most populated.The best picture I can get in my head is regarding energy levels in atomic orbital theory. Is it anything like how the there are three 2p orbitals and only one 2s orbital even though the three 2p orbitals are at a higher energy level.thanks.