# [Statistical Physics] Microstates in a large system/Boltzmann entropy

1. The problem statement, all variables and given/known data

For a box containing 1m$^{3}$ of nitrogen at S.T.P., estimate the number of microstates which make up the equilibrium macrostate.

2. Relevant equations

S = Nk$_{b}$(ln$\frac{V}{N}$ + $\frac{5}{2}$ + $\frac{3}{2}$ln$\frac{2πmk_{b}T}{h^{2}}$)

where the entropy of a volume, V , of an ideal gas, containing N molecules of mass m at temperature T

S = k$_{b}$lnΩ

3. The attempt at a solution

First off I don’t know which mass it is asking for in the equation. Is it the mass of each individual molecule? Or the mass of all the molecules? Or the molar mass? Either way I tried them all but still couldn’t get an answer.

I first worked out what N was.

40.82 mols in 1m$^{3}$ of an ideal gas
1 mol = 6.022×10$^{23}$
∴ N = 2.46×10$^{25}$

I let m = 4.652×10$^{-26}$ kg (the mass of a nitrogen molecule)

Plugging those numbers into the first equation gives entropy, S = 6122

Now I know the number of microstates is going to be huge, but from the second equation:

lnΩ = S/k$_{b}$ = 4.44×10$^{26}$

∴Ω = e$^{4.44×10^{26}}$

which brings about a "math error".