Answer
$3.45\;nm$
Work Step by Step
From the very general considerations, the mean inter-particle distance is proportional to the size of the per-particle volume
$d_{avg}\approx\frac{1}{\sqrt[3] n}$
where $n=\frac{N}{V}$ is the particle density.
Using gas equation, we may write
$d_{avg}\approx\frac{1}{\sqrt[3] \frac{p}{kT}}$
or, $d_{avg}=\sqrt[3] \frac{kT}{p}$
or, $d_{avg}=\sqrt[3] \frac{1.38\times 10^{-23}\times 300}{1.01\times 10^{5}}\;m$
or, $d_{avg}=3.45\times 10^{-9}\;m$
or, $d_{avg}=3.45\;nm$
