# Volume of gas at which relative fluctuation of gas density occurs

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?

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## 1 Answer

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.

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Can you please explain in detail where did the original values go? (Or how did we get to 0.01 and 4*10^-24.) The page you are linking to is above my level of understanding :- / – Andrew123321 Apr 30 at 14:04
${(V - V_0) \over V} = 0.1$ So approximately $\langle {(V - V_0)^2 \over V^2} \rangle = 0.01$ – Azad Apr 30 at 14:39
@Andrew123321 May I ask what grade are you in and where you see the question? – Azad Apr 30 at 14:42
I see. Can you please also expand on 4*10^-24?  I am an undergraduate student of computer science and this is an excercise from physics class I've taken. – Andrew123321 Apr 30 at 14:52
Well, $k$ is the Boltzmann constant $k = 1.38 \times 10^{-23} J/K$ , $T = 293.15 K$ , $P=10^5 Pa$ , $\delta = 0.01$. Just put them in the last formula. – Azad Apr 30 at 16:22
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