= Solution
For a dilute suspended sphere in <Stokes flow>, the translational friction coefficient is $f_h=6\pi\eta R$, with dynamic viscosity $\eta$ and hydrodynamic radius $R$. To derive the diffusion-friction relation, put particles in a weak conservative force $F=-U'$. Their drift velocity is $F/f_h$, while the diffusive flux is $-D\,dc/dx$. At equilibrium the <Boltzmann distribution> gives $c'=-cU'/(k_BT)=cF/(k_BT)$. Zero total flux therefore gives $cF/f_h-DcF/(k_BT)=0$, hence the <Stokes–Einstein relation>
$$
\boxed{D=\frac{k_BT}{f_h}=\frac{k_BT}{6\pi\eta R}.}
$$
Here $k_B$ is the <Boltzmann constant>, $T$ absolute temperature and $D$ the translational <diffusion coefficient>. The sphere, no-slip continuum-fluid and dilute thermal-equilibrium assumptions enter through the drag law and flux balance.
At $T=293.15\,\mathrm K$ and the supplied viscosity, the predicted product is $D R=2.1472\times10^{-19}\,\mathrm{m^3\,s^{-1}}$. The derived predictions and radii inferred from the measured diffusion coefficients are:
|| Protein
|| Predicted $D/(10^{-9}\,\mathrm{m^2\,s^{-1}})$
|| Inferred <hydrodynamic radius> / nm
| <Ribonuclease>
| $0.119$
| $2.15$
| <Beta-lactoglobulin>
| $0.0795$
| $2.68$
| <Haemoglobin>
| $0.0693$
| $3.07$
| <Collagen>
| $0.00693$
| $30.7$
The last three agree closely at the precision of the supplied data. The ribonuclease diffusion coefficient is about $16\%$ below the spherical prediction, corresponding to a larger frictional radius than the stated radius. Hydration, shape, a different operational radius and coarse measurement precision can matter; molar mass is not an independent parameter in the spherical diffusion law. A compact constant-density globule would have $R\propto M^{1/3}$, whereas collagen's large frictional radius relative to its mass is consistent with an extended nonspherical molecule. The inferred <Stokes radius> is a friction-equivalent radius, not automatically a geometric half-length.
In <analytical ultracentrifugation>, scan the radial protein concentration while a rotor spins at known angular speed $\omega$. Optical absorbance or refractive-index interference records a moving sedimentation boundary and its diffusive broadening.<Optical Systems Used for AUC by Beckman Coulter Life Sciences>[Radial scans during rotation, absorbance spectrophotometry and Rayleigh interference detection]{ref} With solvent density $\rho_s$, partial specific volume $\bar v$ and molar mass $M$ in $\mathrm{kg\,mol^{-1}}$, the buoyant mass is $m_b=M(1-\bar v\rho_s)/N_A$. Force balance gives $f_hv=m_b\omega^2r$, so the <sedimentation coefficient> is $s=v/(\omega^2r)=m_b/f_h$. A boundary trajectory gives $s=\omega^{-2}d\ln r/dt$; diffusion gives $f_h=k_BT/D$.
Combining them gives the <molar mass from sedimentation and diffusion>:
$$
\boxed{M=\frac{R_{\rm gas}Ts}{D(1-\bar v\rho_s)},\qquad R_h=\frac{k_BT}{6\pi\eta D}.}
$$
Known mass and buoyancy can instead determine friction from $s$ alone, and comparison with a compact sphere tests shape or hydration. In <sedimentation equilibrium>, the stationary radial profile obeys $d\ln c/d(r^2)=m_b\omega^2/(2k_BT)$, providing buoyant mass without a friction model. This is a sedimentation-diffusion balance in a uniform solvent, rather than density-gradient banding at an isopycnic point.
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