Analytical ultracentrifugation (AUC)

Analytical ultracentrifugation observes concentration profiles during rotation, rather than only recovering a separated pellet. Sector-shaped sample cells rotate past optical detectors. Absorbance scans measure attenuation at selected radii. A Rayleigh interferometer compares a sample sector with a solvent reference, converting concentration-dependent refractive-index changes into fringe shifts. Repeating scans follows a sedimentation boundary or the approach to a stationary distribution.^[ref]

For a dilute solute of buoyant mass $m_b$ and translational friction $f_h$, radial force balance gives $s=m_b/f_h$, where $s$ is the sedimentation coefficient. Diffusion supplies $D=k_{B}T/f_h$. A velocity experiment resolves boundary motion and broadening; a sedimentation equilibrium experiment instead measures $d\ln c/d(r^2)=m_b{\omega }^2/(2k_{B}T)$. This follows from the Boltzmann distribution in centrifugal potential $-m_b{\omega }^2{r}^2/2$. The two approaches separate frictional size information from buoyant mass information; shape, hydration and solvent properties must be specified to infer a unique geometric dimension.

Let $M$ be molar mass in $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, $\bar{v}$ partial specific volume, ${\rho }_s$ solvent density, and $N_A$ the Avogadro constant. The buoyant molecular mass is $m_b=M(1-\bar{v}{\rho }_s)/N_A$. Combining sedimentation coefficient $s=m_b/f_h$ and the thermal diffusion-friction relation $D=k_{B}T/f_h$ eliminates the unknown friction and proves the displayed formula, with $R_{\mathrm{gas}}=N_A{k}_B$. Once friction is known, $R_h=f_h/(6\pi \eta )$ defines a hydrodynamic radius using Stokes's law. Mass and radius are different inferences: an extended molecule need not occupy the volume of that friction-equivalent sphere.