# Planck distribtion

1. Apr 30, 2009

### najima

1. The problem statement, all variables and given/known data

I cant find the proof of "the ratio of the number of oscillators in their (n+1) )th quantum state of excitation to the number in nth quantum state is:
k is boltzman costant
N_(n+1)/N_(n)=exp(-hω/2π(kT)"
2. Relevant equations

3. The attempt at a solution:-( I dont have any idea

Last edited: Apr 30, 2009
2. May 1, 2009

### malawi_glenn

Can you write down the planck distribution and explain it in words?

3. May 1, 2009

### najima

for obtaining planck distribution first we use this equation

N_(n+1)/N_(n)=exp(-hω/2π(kT))

and then the fraction of total number of oscilators in nth quantum stae is

N_n/∑_(s=0)^∞▒ N_s =exp(-hω/2π(kT))/∑_(s=0)^∞▒〖exp(-shω/2π(kT))〗

<n>=∑_(s=0)^∞▒〖s exp(-shω/2π(kT))〗/∑_(s=0)^∞▒〖exp(-shω/2π(kT))

<n>=1/[exp(-hω/2π(kT))-1] " n" is average excitation quantum number of an oscillator

But I dont know how can get this equation" N_(n+1)/N_(n)=exp(-hω/2π(kT))"

4. May 1, 2009

### Matterwave

Have you learned about Boltzmann factors? This is basically a direct application of Boltzmann factors: $$\frac{n_i}{n_j}=e^{\frac{-\Delta E_{ij}}{kT}}$$

Do you know how to get the Boltzmann factors? (Hint: it has to do with entropy)

Last edited: May 2, 2009
5. May 2, 2009

### najima

I've seen boltzmann factor, but I dont know how I can proove it,could u tell me some hints?

6. May 2, 2009

### malawi_glenn

prove and prove, why do you need to prove the boltzman factor?

Sorry for my hint going via the plank distribution, working with boltzmann factors are much easier ;-)

7. May 2, 2009

### najima

thanx any way :-)

8. May 2, 2009

### Matterwave

So, the Boltzmann factor can be proved using Entropy of a reservoir and a particle in state i. The gist of it is, if you change the state of the particle, you change the energy of the particle and the entropy (multiplicities) of the reservoir. If you use some entropy and multiplicity relations, you can get the Boltzmann factor.

I don't remember the exact proof, but it's provided here:

http://www.physics.thetangentbundle.net/wiki/Statistical_mechanics/Boltzmann_factor [Broken]

Edit: oops, I realize I forgot a - sign in my first post. I've fixxed it.

Last edited by a moderator: May 4, 2017
9. May 2, 2009

### guruji

for the derivation of Boltzmman factor you can see the book SEARS AND SALINGER of thermodynamics.

In this book much simpler method is used.

10. May 2, 2009

### malawi_glenn

derivation of boltzman factor is done in almost any book/resource on statistical thermodynamics..