Volume of gas at which relative fluctuation of gas density occurs

Question

I have the following question:

In what volume of gas occurs 10 % relative fluctuation of gas density under pressure of $10^5\text{ Pa}$ and temperature of $293.15\text{ K}$?

I don't understand the topic but I assume this is about ideal gas. Can you please explain this to someone who has just high school physics knowledge?

Answer 1

You can compute relative fluctuation of gas volume (which is the same as fluctuation of gas density here) by computing probabilities using entropy (or equivalently Gibbs energy) difference. The following page has explained all the steps.

The final formula would be: $\delta = \langle {(V - V_0)^2 \over V^2} \rangle = {1 \over N} $

In which $N$ is number of atoms which we can compute from ideal gas state equation: $N = {PV \over kT}$

So $\delta = \langle {(V - V_0)^2 \over V^2} \rangle = {k T \over P V}$

$\delta = \langle {(V - V_0)^2 \over V^2} \rangle = 0.01$

$V = {k T \over P \delta} = 4\times 10^{-24} ~\text m^3 = 4000 ~\text{nm}^3$

This volume is small; you need very few atoms to have such a huge fluctuations.