The microscopic explanation of entropy has never been fully accepted since there is
incomplete proof for equation 2 and there are counter examples where the increase of
disorder cannot be justified. Among these counter examples the spontaneous
crystallization of a super-cooled melt and the crystallization of a super saturated solution
are the most prominent challenges for the Boltzmann’s model If a super-cooled melt is
allowed to crystallize under adiabatic conditions then the entropy of the system increases.
In a supersaturated solution there is a possibility of the deposition of crystalline solute.
The deposition of crystalline solute is a spontaneous process with an increase of entropy.
A crystallization of a liquid or a deposition of a crystalline solute does not imply any
increase in disorder. I would like to add one additional examples to the above mentioned.
The phase transformation of solid helium to liquid helium II does not require energy.
The lack of latent heat results the same entropy for both of these phases [Fig. 1],
indicating the same level of disorder in solid and in liquid. These counter examples
suggesting that the microscopic explanation of entropy is dubious.