Diffusion coefficient, D

k, T, - Boltzman constant, absolute temperature

r - radius of sphere

The value for

**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:

(x is the mean distance from the starting point that a molecule will have diffused in time, t)

q

D - diffusion coefficient (usual units are cm

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), q_{3} = 0.35 and q_{2} = 1.22. As y/x decreases,
the relative enhancement in time to target for two as compared to three
dimensions becomes dramatic, since q_{2} 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

(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:

(units M^{-1}
s^{-1})

here r_{m}, r_{n}, D_{m}, and D_{n}
are the radii (in cm) and diffusion coefficients (in cm^{2} s^{-1})
of reaction species m and n, and N_{o} is Avagadro’s number. Division by 1000 cm^{3} gives k in units M^{-1}s^{-1}.

**Some diffusion coefficients**

System
D (cm^{2} 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
~10^{10} M^{-1} s^{-1}

A_{diff} for protein in membrane
~10^{6} M^{-1} s^{-1}

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