Question

So I understand the Gibbs paradox and that without the N!

factor entropy isn't extensive.

But why are these seemingly unrelated things related to each other, why did gibbs in a sense stumbled upon the physical concept of indistinguishability of particles in this way.

If entropy is a measure of microstates; why would distinguishable particles remove extensivity of the microstates? Is there a clear way of understanding this, without big calculations involving Sackur Tetrode equations and such?

Answer 1

The Gibbs paradox and the relationship between indistinguishability of particles and entropy can be understood without complex calculations. The number of microstates for a system depends on whether particles are distinguishable or indistinguishable. Without the N! factor, entropy would not be extensive, meaning it would not scale linearly with the number of particles.

Step-by-step explanation:

The Gibbs paradox and the role of indistinguishability of particles in relation to entropy can be understood without complex calculations. Consider a system comprised of N particles distributed among n boxes. The number of microstates possible for such a system is n! (factorial), which accounts for the indistinguishability of particles.

If the particles were distinguishable, the number of microstates would be n^N. Without the N! factor, the entropy would not be extensive, meaning it would not scale linearly with the number of particles. This is why Gibbs stumbled upon the physical concept of indistinguishability of particles in relation to entropy.

Gibbs stumbled upon the concept that without considering the indistinguishability, entropy would increase erroneously when mixing two gases of identical particles, since it would treat them as if they were different, thus considering more microstates than actually exist.

Entropy, being a measure of disorder within a system, should not increase upon mixing identical particles as no new microstates are created.