Diffusion – useful equations

Diffusion coefficient, D

D = (1/f)kT
f - frictional coefficient
k, T, - Boltzman constant, absolute temperature
f = 6p h r
h - viscosity
The value for f calculated for a sphere is a minimal value; asymmetric shape of molecule or non-elastic interaction with solvent (e.g. hydration) will increase f.

Diffusion coefficient depends on size and shape of molecule, interaction with solvent and viscosity of solvent.

Diffusion over a distance

The relationship below is generally valid:

<x2> = qiDt
<x2> - mean-square displacement
(x is the mean distance from the starting point that a molecule will have diffused in time, t)
qi - numerical constant which depends on dimensionality: qi = 2, 4, or 6, for 1, 2, or 3 dimensional diffusion.
D - diffusion coefficient (usual units are cm2 s-1).
t - time.

Diffusion to a target

For target diameter y, and diffusion distance x, the diffusion of a particle to a target depends strongly on dimension. The time for diffusion is linear in y/x for 3 dimensions; proportional to log(y/x) for 2 dimensions; and independent of y/x for 1 dimension.

For example, when y/x = 0.1 (e.g., target diameter 1 nm, diffusion distance 10 nm), q3 = 0.35 and q2 = 1.22. As y/x decreases, the relative enhancement in time to target for two as compared to three dimensions becomes dramatic, since q2 only varies from about 1.22 to 0.24 as y/x goes from 0.1 to 10-4.

Frequency of collision

From the above it will be obvious that the frequency of collision between molecules will depend on several factors, including:

Diffusion coefficient (and hence h , r, T, etc)

Diffusion distance (concentration)

Target size (also r)

When reactions between molecules occur at every collision, the reaction is said to proceed at the diffusion limit. Such reactions have NO ENERGY OF ACTIVATION, and are called diffusion-controlled reactions.

from the Arrhenius equation

k = A exp[-Eact/RT]

(where k is the rate constant, R is gas constant).

When Eact = 0, k = A.

For a diffusion-controlled reaction, A is given (approximately) by:

Adiff = 4p(rm + rn) (Dm + Dn) No / 1000

(units M-1 s-1)

here rm, rn, Dm, and Dn are the radii (in cm) and diffusion coefficients (in cm2 s-1) of reaction species m and n, and No is Avagadro’s number. Division by 1000 cm3 gives k in units M-1s-1.

Some diffusion coefficients

System                             D (cm2 s-1)
-------------------------------------------------
Small molecule in water     1-1.5 x 10-5

Small protein in water             10-6

Phospholipid in membrane      10-8

Protein in membrane              10-10

Adiff for small molecules in solution         ~1010 M-1 s-1

Adiff for protein in membrane                 ~106 M-1 s-1