The rate of sedimentation
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
661-662
2025-12-17
50
The rate of sedimentation
A solute particle of mass m has an effective mass meff = bm on account of the buoyancy of the medium, with b =1−ρvs , where ρ is the solution density, vs is the partial specific volume of the solute (vs = (∂V/∂mB) T, with mB the total mass of solute), and ρvs is the mass of solvent displaced per gram of solute. The solute particles at a distance r from the axis of a rotor spinning at an angular velocity ω experience a centrifugal force of magnitude meffrω2. The acceleration outwards is countered by a frictional force proportional to the speed, s, of the particles through the medium. This force is written fs, where f is the frictional coefficient (Section 19.3). The particles therefore adopt a drift speed, a constant speed through the medium, which is found by equating the two forces meffrω2 and fs. The forces are equal when
The drift speed depends on the angular velocity and the radius, and it is convenient to define the sedimentation constant, S, as
S =
Then, because the average molecular mass is related to the average molar mass Mn through m=Mn/NA,
S =
On substituting the Stokes relation for spherical molecules (eqn 19.12), we obtain
S =
and Smay be used to determine either Jn or a. Again, if the molecules are not spherical, we use the appropriate value of f given in Table 19.3. As always when dealing with macromolecules, the measurements must be carried out at a series of concentrations and then extrapolated to zero concentration to avoid the complications that arise from the interference between bulky molecules.
At this stage, it appears that we need to know the molecular radius a to obtain the molar mass from the value of S. Fortunately, this requirement can be avoided by drawing on the Stokes–Einstein relation (eqn 19.11) between f and the diffusion coefficient, D. The average molar mass is then:
M=
where we are not specifying which mean molar mass because the average obtained depends on technical details of the experiment. The result in eqn 19.19 is independent of the shape of the solute molecules. It follows that we can find the molar mass by combining measurements of S and D by ultracentrifugation and dynamic light scattering, respectively.
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