3. Diffusion in an Ultracentrifuge with Density Gradient (40 pts). The ultracentrifuge is often used to measure the mobility of macromolecules or larger aggregate particles in a solvent medium. In this instrument, solutions or suspensions of particles are placed in a rotor, which spins many times per second around an axle. With high enough rotation velocity, @ (rad/sec), thousands of g forces can be generated. For a particle of volume Vp and density pp dissolved or suspended in a medium of density pm, Archimedes' principle states that the displacement force of the particle as it moves through the medium is F = ro (pp Pm)V p where r is radial distance of the particle from the axle's center. Measuring the velocity, v of displacement, e.g. by fluorescence tracking, one can obtain the mobility u according to u = NF Nro (p Pm)Vp P where NA is Avogadro's number. The diffusion coefficient in the medium can then be determined using the Einstein relation, D=uRT. The ultracentrifuge can also be used to determine a particle's density if the medium itself has a density gradient. After being introduced into the medium, particles migrate to where the medium's density is the same (p, = Pm), since there is no force there. However, there is still diffusion due to thermal agitation, which spreads out the band of particles at equilibrium. Let ro be the point where the sedimentation force vanishes, and assume that the density gradient near ro is constant. Denote this gradient by (op / dr). Then we may write - Pm (r) = pp + (@pm / r), (r ro) ro m Assuming that the diffusion coefficient is essentially constant in the band where the particles settle, show that particle distribution in the band is Gaussian in shape, and relate the variance of the particle distribution to the physical parameters prescribed here. You may ignore the effects of cylindrical geometry and assume, to good approximation, that |r ro| < < r. (Hint: use either the Smoluchowski equation or the Boltzmann distribution.)