Maxwell-Boltzmann Energy distribution

What is the general equation of the Maxwell-Boltzmann distribution with n_i/n_t (on the y-axis) and energy on the x-axis (i.e. as a function of energy)? The catch is, I need an equation encapsulating temperature dependence as well. And we should avoid the actual composition of the system (e.g. what molecules are in it exactly) as far as possible.
Any links would be welcome.

Reply 1

joostan

Original post by Big-Daddy

Note that for probabilities:

\displaystyle\int^\infty_0 f(E) dE = 1

Where f(E) is the distribution

The equation is given by

\dfrac{N_i}{N_t} = \dfrac{g_i e^{\frac{-E_i}{kT}}}{\sum_j g_j e^{\frac{-E_j}{kT}}}

http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution

(edited 11 years ago)

Reply 2

Big-Daddy

Original post by joostan

Note that for probabilities:

\displaystyle\int^\infty_0 f(E) dE = 1

Where f(E) is the distribution

The equation is given by

\dfrac{N_i}{N_t} = \dfrac{g_i e^{\frac{-E_i}{kT}}}{\sum_j g_j e^{\frac{-E_j}{kT}}}

http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution

Whilst I'm glad my post piqued your interest, this equation is not what I was looking for. I have seen the first equations that come up on Internet searches! This one is the general one to learn but not the one that does the job for me.

The problem with this is that a) it's too specific - degeneracy relies on the molecular composition, which as I said I don't want if it can possibly be avoided (as I want a more general approach, one that considers the system without concern for its molecular structure - and if this means the ratio is relative to other ratios rather than exact I'm fine with that), and b) it lacks temperature dependence which again I said I need.

(edited 11 years ago)

Reply 3

Bradshaw

There is temperature dependence in the above equation?!

Also, the degeneracy factors are necessary, as they allow the formula to cope with a system that has degenerate energy levels.

Reply 4

Bradshaw

Ah I notice you might want it as a function of energy, rather than temperature? It which case that is something called the microcanonical distribution.

Reply 5

Dalilsp

side question -> at what level of study is this sort of knowledge required? Im guessing graduate?

Reply 6

natninja

Original post by Big-Daddy

you might find some stuff in a stat mech/statistical thermal physics textbook, I'm sorry I can't be more help than that but I don't do thermo properly until next year of my degree course...

Reply 7

BJack

Original post by Dalilsp

side question -> at what level of study is this sort of knowledge required? Im guessing graduate?

I don't think I touched the details of the MB distribution before uni; at A level I'm sure it was enough to know the general form of it and how it changes with temperature. The thing is, A level chemistry is so devoid of any maths that you wouldn't be expected to be able to do anything with the explicit formula (which is as joostan gave it) even if you did have it!

Reply 8

Big-Daddy

Original post by Bradshaw

Ah I notice you might want it as a function of energy, rather than temperature? It which case that is something called the microcanonical distribution.

Hmm I was hoping for a function of both energy and temperature, in the sense that I could then plot a Maxwell-Boltzmann distribution (n_i/n_t on the y-axis, E on the x-axis) for a specified temperature.

Original post by Dalilsp

side question -> at what level of study is this sort of knowledge required? Im guessing graduate?

Original post by BJack

Whatever level you learnt it at, I'm looking for the answers. Assume undergrad then.

Reply 9

Bradshaw

Original post by Big-Daddy

I'm not sure why you think the formula given won't let you do this?

Reply 10

Big-Daddy

Original post by Bradshaw

I'm not sure why you think the formula given won't let you do this?

It may require a lot more information than a question I get would give (now I look back on it, it also may not, in which case I will owe joostan an apology!). I suppose actually the question is: what is g_j? Do I need to know the composition of the system in terms of molecules to work it out? If so, or even if not, how do I work it out? (A quick introduction please!)