Strange maxwell boltzmann statistics, what is it actually?

[jessicaw]

Originally, it is derived from the weight of a confuguration,i.e. how electrons are distributied in different energy level:n=gexp(a+be)
It gives the number of electrons in each energy level.However suddenly it can be apply to a cloud of gas?? By dividing a box of gas into different energy and see how many gas atoms in each energy. My professor said it is pseudo energy level in phase space.

So now an atom becomes a box of gas? Each energy level of atom becomes a division of a cloud of gas? Or am i missing anything(as i think the all gas atoms in the cloud of gas has the same energy)? I cannot find any rigorous proof of this analogy.

Help explaining this. My professor just runs away after each lecture so i do not have chance to ask her and we are all afraid of her. Also she does not use textbook and she teaches MB stat, boseeinstein stat, fermi dirac stat without any introduction on what does the distribution mean(to atom/ to gas),just a lot of equations and derivations and i do not know what to do now excpet memorizing.

 

[Andrew Mason]

Originally, it is derived from the weight of a confuguration,i.e. how electrons are distributied in different energy level:n=gexp(a+be)
It gives the number of electrons in each energy level.

You have misunderstood your professor. Maxwell-Boltzmann statistics apply to macroscopic particle systems ie those containing a large number of particles whose interactions can be modeled without regard to quantum effects. The energy distribution of particles in such systems in thermodynamic equilibrium (eg. a gas) follows the Maxwell-Boltzmann distribution.

An electron in an atom obeys quantum statistics. A single atom does not have a temperature. Energy levels in an atom obey quantum statistics not Maxwell-Boltzmann statistics.

AM

 

[jessicaw]

You have misunderstood your professor. Maxwell-Boltzmann statistics apply to macroscopic particle systems ie those containing a large number of particles whose interactions can be modeled without regard to quantum effects. The energy distribution of particles in such systems in thermodynamic equilibrium (eg. a gas) follows the Maxwell-Boltzmann distribution.

An electron in an atom obeys quantum statistics. A single atom does not have a temperature. Energy levels in an atom obey quantum statistics not Maxwell-Boltzmann statistics.

AM

But maxwell Boltzmann statistics is derived from energy level concept, right?

 

[Andrew Mason]

But maxwell Boltzmann statistics is derived from energy level concept, right?

No. Maxwell-Boltzmann statistics were developed long before anyone understood electrons in atoms. A Maxwell-Boltzmann distribution shows how energy is distributed among a large number of particles that are in thermodynamic equilibrium (such as a volume of gas). It has nothing to do with energy levels in an atom.

AM

 

[cortiver]

Just to clarify, Maxwell-Boltzmann statistics can be used to predict, given a large number of atoms in thermal equilibrium (i.e. not just a single atom), what proportion will be in a given energy level (e.g. how many in the ground state, how many in the first excited state, etc.). This is what the formula you quoted in your original post means.

The idea when applying Maxwell-Boltzmann statistics to a container full of an ideal gas is that the relevant energy levels are those obtained by solving the time-independent Schrödinger equation for an atom confined inside the container. I presume if you're studying statistical mechanics you've probably done some basic quantum mechanics, so you'll be familiar with the energy levels for a particle in a one-dimensional box (if not, you can Google for "particle in a box"):

$$E_n = h^2 n^2/(8mL^2)$$

Thus, for gas atoms in a one-dimensional box, the probability that a given atom will be occupying the n-th state is proportional to $\exp\left[-E_n/(k_B T)\right]$. You can do the same kind of thing for a three-dimensional box, you just have to take into account the three possible degrees of freedom.

As was mentioned above, Maxwell-Boltzmann statistics was originally developed before quantum mechanics. The classical version of Maxwell-Boltzmann statistics involves predicting the distribution of atoms in a continuous phase space rather than discrete energy levels. Surprisingly enough (or maybe not -- quantum mechanics should reduce to classical mechanics in the appropriate limit, after all!) this calculation gives the exact same results as the quantum mechanical version except at extremely low temperatures where the discrete nature of the quantum energy levels becomes important (at such low temperatures, you should be using Fermi-Dirac or Bose-Einstein statistics rather than Maxwell-Boltzmann statistics anyway).

If you're looking for a textbook to supplement your lectures, I can only suggest the one that was used for the introductory thermodynamics/stat-mech course I took recently, which was Thermal Physics by Daniel Schroeder. It has a pretty gentle introduction to statistical mechanics at the end.