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

1. Apr 13, 2014

### Flucky

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 = 6.022x10$^{23}$
∴ N = 2.46x10$^{25}$

I let m = 4.652x10$^{-26}$ 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$_{b}$ = 4.44x10$^{26}$

∴Ω = e$^{4.44x10^{26}}$

which brings about a "math error".

Cheers

2. Apr 14, 2014

### Flucky

Shameless bump

3. Apr 15, 2014

### Flucky

Maybe a mod could move this into the Advance forum?

4. Apr 15, 2014

### dauto

That seems correct to me, except for the missing units. S=6122 (what units?).
You only get a math error if you try to plug that huge number into a typical calculator. Why wold you do that?

5. Apr 16, 2014

### Flucky

Ah that's good then. I just thought it was an unreasonably large number.

JK$^{-1}$