# Distinguishable particles in Maxwell-Boltzmann distribution

In the derivation of Maxwell-Boltzmann (MB) probability distribution function, why did he choose distinguishable particles? I mean, suppose you consider CO gas. and you are applying the probability function, but you know that the molecules are indistinguishable. Now in Bose-Einstein and Fermi-Dirac statistics we consider indistinguishability of particles due to quantum approach because the wavefuntions of the particles overlap with each other. But I mean in common sense you can feel that the molecules must be indistinguishable. But why did Maxwell choose distinguishable particles? And what are the problems you have if you consider indistinguishable particles in MB statistics or what are the conditions for which the particles of the gas becomes distinguishable?

## 1 Answer

It is true that atoms and molecules are fundamentally indistinguishable, but when they are enough far apart you can distinguish them.

To be a bit more quantitative, indistinguishable particles become distinguishable when the average distance $d$ between them is much larger than the thermal De Broglie wavelength $\lambda$. If we consider an ideal gas, this means that

$$d \approx \left(\frac V N \right)^{1/3} \gg \lambda = \frac h p = \frac{h}{\sqrt{2 m E}} \approx \frac{h}{2 m k T}$$

Typical values of $\lambda$ at room temperature are of the order of $10^{-11}$m: this means that a classical description is perfectly fine at room temperature as long as the number density $\rho$ satisfies

$$\rho = \frac N V \ll \lambda^{-3} \approx 10^{33} \text{m}^{-3}$$

Notice that this is an humongous density. So it is only at extremely high densities (or extremely low temperatures) that the quantum nature of the gas will become apparent. For the standard temperatures and densities probed experimentally at the time of Maxwell and Boltzmann, the MB distribution was in perfect agreement with the experimental data.

Note that this doesn't necessarily mean that at high temperature/low pressure the molecules become different: simply, they are far enough to be able to tell one from the other (the overlap between the respective wavefunctions becomes negligible).