LON-CAPA Diffusion � useful equations

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
r - radius of sphere
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> = q_iDt <x²> - mean-square displacement
(x is the mean distance from the starting point that a molecule will have diffused in time, t)
q_i - numerical constant which depends on dimensionality: q_i = 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[-E_act/RT]

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

When E_act = 0, k = A.

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

A_diff = 4p(r_m + r_n) (D_m + D_n) N_o / 1000

(units M^-1 s^-1)

here r_m, r_n, D_m, and D_n are the radii (in cm) and diffusion coefficients (in cm² s^-1) of reaction species m and n, and N_o 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

A_diff for small molecules in solution ~10¹⁰ M^-1 s^-1

A_diff for protein in membrane ~10⁶ M^-1 s^-1