Physical Chemistry of Foods

encounter frequency. The basic theory is due toSmoluchowski: it gives the
encounter rate of diffusing particles or molecules, assuming that every
encounter leads to reaction. For the reaction AþB?AB the result is

d½AŠ

dt

¼

d½BŠ

dt

¼

d½ABŠ

dt

¼ 4 pðDAþDBÞðrAþrBÞNANB ð 4 : 13 Þ

whereD¼diffusion coefficient,r¼collision radius of the molecule, andN
¼number of molecules per unit volume. If we have a reaction of the type
AþA?A 2 , we have to divide the rate by a factor of 2, since we otherwise
would count every encounter twice; this also applies to any derived
equation. The rate constant is the rate divided byNANBand, converting
from molecules per m^3 to moles per liter, we obtain

kS¼ 4 pðDAþDBÞðrAþrBÞ 103 NAV ð 4 : 14 Þ

The Stokes–Einstein relation gives, for the diffusion coefficient of spherical
particles,

D¼

kT

6 pZr

ð 4 : 15 Þ

whereZis the viscosity of the solvent andris now the hydrodynamic radius
(see Section 5.3 for more on the relations betweenD,Z, andr). For perfect
spheres, collision radius and hydrodynamic radius are equal. Assuming,
moreover,rA&rB, we obtain by inserting (4.15) into (4.14)

kS& 8? 103

RT

3 Z

ð 4 : 16 Þ

Comparing now Eqs. (4.12) (Eyring) and (4.16) for dilute aqueous
solutions at room temperature, whereZ& 10
^3 Pa?s, and takingDG{¼ 0
(because it is assumed in the Smoluchowski treatment that every encounter
leads to reaction), we obtain rate constants of 1: 5? 1014 and
6 : 6? 109 L?mol
^1 ?s
^1 , respectively. The encounter rate would thus be
very much smaller than the rate according to the Eyring theory, implying
that the reaction is diffusion controlled (albeit still quite fast, unless the
reactant concentrations are very small). However, zero activation free
energy is in many cases unrealistic. Taking, for instance,
DG{¼25 kJ?mol
^1 , which is a fairly small value, we obtain from Eq.
(4.12) a rate constant of 7? 109 , i.e., about the same as from (4.16). For an
activation free energy of 50 kJ?mol
^1 , we obtain 3? 106 L?mol
^1 ?s
^1 , and
now the reaction rate presumably is controlled by activated complex
formation. On the other hand, if the viscosity were much higher, and a value