Maxwell Boltzmann

You are given N atoms that are distinguishable with two possible energy levels $E1 = 0$ and $E2 = E$. What is the Maxwell-Boltzmann distribution (number of atoms n1 and n2) for N = 10000 atoms at 

(i) $kT = 0.3E$, 

(ii) $kT = E$, 

(iii) $kT = 5E$, and 

(iv) in the limit $kT→∞$

 

The Maxwell Boltzmann statistics for the number of particles found as having energy $Ei$ is

$N_i/N=(exp⁡(-E_i/KT))/(∑_j exp⁡(-E_j/KT))$

where the summation is done over all possible energies.

Thus for $E1 =0$ and $E2= E$ one has

$N_1=N*1/(1+exp⁡(-E/KT))$ and

$N_2=N*(exp⁡(-E/KT))/(1+exp⁡(-E/KT))=N*1/(1+exp⁡(+E/KT) )(=N-N_1)$

For $KT =0.3*E$ one has

$N_1=10000/(1+exp⁡(-0.3))=5744.42=5744$ and

$N2= 10000/(1+exp⁡(+0.3))=4255.57=4256$

For $KT = E$ one has

$N_1=10000/(1+1/e)=7310.58=7311$ and

$N_2=10000/(1+e)=2689.41=2689$

For $KT =5*E$ one has

$N_1=10000/(1+exp⁡(-5) )=9933.07=9933$ and

$N_2=10000/(1+exp⁡(5) )=66.93=67$

For $KT=∞$ one has (inifinite activation energy)

$N_1= 10000/(1+0)=10000$ and

$N_2=10000/(1+∞)=0$

Rerefence

Maxwell Boltzmann Statistics