5.1. Friction in fluids[[Student version, December 8, 2002]] 143
roughly a micrometer. TheHandbook of Chemistry and Physicslists the mass density of butterfat
asρm,fat=0. 91 gcm−^3 (the density of water is about 1gcm−^3 ). Findc(h)/c(0) in equilibrium.
Is homogenized milk an equilibrium colloidal suspension?Returning to myoglobin, it may seem as though sedimentation is not a very useful tool for
protein analysis. But the scale height depends not only on properties of the protein and solvent,
but also on the acceleration of gravity,g.Artificially increasinggwith a centrifuge can reducex∗
to a manageably small value; indeed, laboratory centrifuges can attain values ofg′up to around
106 ms−^2 ,making protein separation feasible.
Tomake these remarks precise, first note that when a particle gets whirled about at angular
frequencyω,afirst-year physics formula gives its centripetal acceleration asrω^2 ,whereris the
distance from the center.
Your Turn 5b
Suppose you didn’t remember this formula. Show how to guess it by dimensional analysis,
knowing that angular frequency is measured in radians/s.
Suppose that the sample is in a tube lying in the plane of rotation, so that its long axis points
radially. The centripetal acceleration points inward, toward the axis of rotation, so there must be
an inward-pointing force,f=−mnetrω^2 causing it. This force can only come from the frictional
drag of the surrounding fluid as the particle drifts slowly outward. Thus, the drift velocity is given
bymnetrω^2 /ζ(see Equation 4.12 on page 107). Repeating the argument that led to the Nernst
relation (Section 4.6.3 on page 124) now gives the drift flux ascvdrift=cf /ζ=cf D/kBT,where
c(r)isthe number density. In equilibrium, this drift flux is canceled by a diffusive flux, given by
Fick’s law. We thus find, analogously to the Nernst–Planck formula, that in equilibrium
j=0=D(
−dc
dx
+rω(^2) mnet
kBT
c
)
Tosolve this differential equation, divide byc(r)and integrate:
c=const×emnetω(^2) r (^2) /(2kBT)
. sedimentation equilibrium, centrifuge (5.2)
5.1.2 The rate of sedimentation depends on solvent viscosity
Our discussion so far has said nothing about therateat which the concentrationc(x)arrives at
its equilibrium profile. This rate depends on the drift velocityvdrift,which equalsmnetg/ζ(by
Equation 4.12). The drift velocity isn’t an intrinsic property of the particle, since it depends on the
strength of gravity,g.Toget a quantity that we can tabulate for various particle types (in given
solvents), we instead define thesedimentation time scale
s=vdrift/g=mnet/ζ. (5.3)Measuringsand looking in a table thus gives a rough-and-ready particle identification. The quantity
sis sometimes expressed in units ofsvedbergs;asvedberg by definition equals 10−^13 s.
What determines the sedimentation time scales? Surely sedimentation will be slower in a
“thick” liquid like honey than in a “thin” one like water. That is, we expect the viscous friction
coefficient,ζ,for a single particle in a fluid to depend not only on the size of the particle, but also
on some intrinsic property of the fluid, called the “viscosity.” In fact, Section 4.1.4 already quoted
an expression forζ,the Stokes formulaζ=6πηafor an isolated, spherical particle of radiusa.