= Solution
For radius $a=10\,\mathrm{nm}$ and gold <density> $2.0\times10^4\,\mathrm{kg\,m^{-3}}$, $m=8.38\times10^{-20}\,\mathrm{kg}$. The supplied Stokes model gives $\zeta=3.96\times10^{-12}\,\mathrm{kg\,s^{-1}}$ and $D=1.05\times10^{-9}\,\mathrm{m^2\,s^{-1}}$ at $300\,\mathrm K$. Therefore the standard one-second diffusive rms distances are \b[$45.7\,\mu\mathrm m$ in one coordinate] and \b[$79.2\,\mu\mathrm m$ in three dimensions].
Using the source's supplied shorthand literally with $m\langle v^2\rangle=3k_BT$ instead gives $\sqrt{3k_BTt/\zeta}=56.0\,\mu\mathrm m$. That shorthand lacks the factor two in the standard long-time mean-square displacement; it should not be identified with either of the preceding standard rms conventions. None of these rms values is a cumulative path length.
An ideal drag-free fall from rest gives \b[$gt^2/2=4.91\,\mathrm m$]. Actual gravitational settling in the assumed viscous model is much smaller: $\tau_p=m/\zeta=2.12\times10^{-8}\,\mathrm s$, $u_t=mg/\zeta=0.208\,\mu\mathrm{m\,s^{-1}}$, and displacement in one second is approximately \b[$0.208\,\mu\mathrm m$]. Thus <diffusion> overwhelms settling, even though ideal vacuum fall exceeds the rms <diffusion> distance. For <rarefied-gas nanoparticle drag>, at a 20-nm size in argon, continuum no-slip drag may require a rarefaction/slip correction; pressure and gas mean-free-path data are absent, so these numerical transport estimates are conditional on the supplied Stokes model.
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