= Solution
The <sedimentation scale height> simplifies because the same drag coefficient controls both drift and diffusion:
$$
z_0=\frac D{v_s}
=\frac{k_BT}{m_bg}
=\frac{3k_BT}{4\pi a^3(\rho_p-\rho_f)g}.
$$
Applying the given criterion $a\leq z_0$ gives
$$
a^4\leq\frac{3k_BT}{4\pi(\rho_p-\rho_f)g},
\qquad
a_{\max}=
\left[\frac{3k_BT}{4\pi(\rho_p-\rho_f)g}\right]^{1/4}.
$$
Equivalently, the gravitational <Péclet number> on the particle-radius scale is $a/z_0=m_bga/(k_BT)$ and must be at most one.
At $300\,\mathrm K$ the limiting \b[radius] is $a_{\max}\simeq8.42\times10^{-7}\,\mathrm m=0.84\,\mu\mathrm m$, corresponding to diameter approximately $1.68\,\mu\mathrm m$. Viscosity cancels from this equilibrium criterion, although it affects the time to approach equilibrium.
For the $100\,\mathrm{nm}$ radius particle in part (i), $D\simeq2.20\times10^{-12}\,\mathrm{m^2\,s^{-1}}$ and $z_0\simeq0.50\,\mathrm{mm}$, about $5.0\times10^3$ particle radii. It is therefore firmly diffusion-dominated on its own size scale. This does not mean uniform concentration over an arbitrarily tall vessel: the equilibrium density still decays on the finite scale $z_0$. The supplied size criterion is an approximate gravity-versus-diffusion criterion, not a universal definition of every stable <colloid>.
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