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
