Question

The general statement of the Boltzmann distribution law is that for a system ofNNparticles, each having access to energy statesε1,ε2,…,εkε1,ε2,…,εk, the ratio the number of particles with energyεjεj,NjNjto the total number of particles is NjN=gje−βεj∑igie−βεiNjN=gje−βεj∑igie−βεiwheregigiis the degeneracy of the energy stateii. However, when about the kinetic theory of gases, I found the formula for the probability density of the distribution of energies to bef(E)=2π(πkT)3/2e−EkTE1/2dE.f(E)=2π(πkT)3/2e−EkTE1/2dE.This formula has extra factors includingE1/2E1/2andT−3/2T−3/2which are not present in the Boltzmann distribution formula so I'm wondering why these two formulas don't contradict each other. Note that in general energy of the stetsϵjϵjcould be very close to each other, in that in any interval(E,E+dE)(E,E+dE)there lie a very large number of energy states. Now the question is:What is the probability of having a particle with energy between(E,E+dE)?What is the probability of having a particle with energy between(E,E+dE)?

Answer 1

The Boltzmann distribution method and the kinetic theory of gases yield different probability density formulas due to the 3-dimensional distribution of particle speeds, each incorporating factors relevant to their context.

Step-by-step explanation:

The Boltzmann distribution provides a general way to calculate the probability of finding a particle with a certain energy within a system in thermal equilibrium. The expression for the probability density of the distribution of energies in the context of kinetic theory of gases includes additional factors because it accounts for the distribution of particle speeds in three dimensions. This requires additional factors related to the kinetic energy and temperature dependence of the particle speed distribution.

When considering the probability of having a particle with energy between (E, E+dE), this is computed by multiplying the probability density function f(E) by dE, where f(E) incorporates both the Boltzmann factor e-E/kT and the square root of energy E1/2, reflecting the relationship between energy and speed distribution in three-dimensional space.