Derive an expression for the internal energy of a collection of harmonic oscillators.?

Question

Use
q=1/(1-e^(hv/kT))
and
E=Sum over all states i (Ne^(-Ei/kT))/q*Ei

Answer 1

Dear Jenny,
This is a very strange request for enlightenment on a difficult area.
You will find it fully covered in Physics and Physical Chemistry Books under 'Plank's Distribution Law'
BUT here is an outline:
Use Boltzmann's Distribution Law: n = n0 * e^(-E/kT).

For Oscillators,
the Total Number of Oscillators is given by:
N = N0{0 +e^(-E1/kT) + e^(-E2/kT) ........}

and the Total Energy is
E = N0{0 + E1*e^(-E1/kT) + Ee^(-E2/kT) ........}


For Harmonic Oscillators,
the Total Number of Oscillators is given by:
N = N0{0 +e^(-1hv/kT) + e^(-2hv/kT) ....+e^(-jhv/kT)....}
N = N0*SUM(-jhv/kT) from 0 to infinity

and the Total Energy is
E = N0{0 +1hve^(-1hv/kT) + 2hve^(-2hv/kT) ...+jhve^(-jhv/kT).....}
E = N0*SUM{jhve^(-jhv/kT)} from 0 to infinity

which gives E average as E/N = hv/(e^(-jhv/kT) - 1 )
and in 3 Dimensions this gives Planck's Law as:
E(v)dv = (8pihv^3/c3) * dv/(e^(-jhv/kT) - 1 )

I hope this is of SOME help - but use these notes to look it up in a book!!!!!!!

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