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?