Physical Chemistry of Foods

i.e., the mean of the distances covered by a large number of identical
molecules—the linear displacement xof a molecule is zero (if no flow
occurs), but the average ofx^2 is finite, and Einstein derived that theroot-
mean-square distancecovered in a given direction,D 1 , will follow the relation

D 1 :<x^2 >^1 =^2 ¼

ffiffiffiffiffiffiffiffi

2 Dt

p

ð 5 : 15 Þ

wheretis the time—although with significant statistical variation. It may be
noted that the equation considers the absolute value of the distance in a
given direction, i.e., the projection of the real distance covered on a straight
line of given orientation (one-dimensional). The projection on a given plane
(two-dimensional), as in Figure 5.12, yields D 2 ¼H 2 ?D 1 , and the full
distance covered in three dimensionsD 3 ¼H 3 ?D 1.
The proportionality constantDis called thediffusion coefficient(S.I.
unit m^2 ?s
^1 ). Einstein also derived thatD¼kBT=f, and taking Stokes’s
expression for the friction factorffor spheres, the relation becomes

D¼

kBT

6 pZsr

ð 5 : 16 Þ

whereZsis the viscosity of the solvent andris the radius of the molecule or
particle. For nonspherical species we need the hydrodynamic radius, which
mostly must be experimentally determined.
It follows from Eqs. (5.15) and (5.16) that a big molecule or particle
will travel over smaller distances—i.e., diffuse at a slower rate—than a small
one, although the pattern of the motion is just the same. For visible
particles, Eq. (5.15) can be verified by microscopic observation, and it is
found to be exact. Some values for the diffusion coefficient and for root-
mean-square distances traveled are given in Table 5.3.

Question

Can you calculate from values given in this chapter the hydrodynamic radius of a
sucrose molecule?

Answer

0.46 nm.