The absolute rate theory implicitly assumes that the addition of a
given amount of free energyðDG{Þwould be sufficient for a reaction
event to occur. In fact, there is another condition, which is that the
reactants must collide in a specific way, because their mutual
orientation will determine whether reaction can occur. This may
decreasekby one or two orders of magnitude as compared to the
predicted value. However, for reactions in solution another effect
plays a part. When two reactants encounter each other, they stay
close to each other for a relatively long time. According to Einstein,
the mutual displacementDof two molecules by diffusion over a time
t is given by hDi^2 ¼ 6 Dt. In water, the diffusion coefficient
D& 1 : 5? 10 ^9 m^2 ?s^1. Assuming the molecules to be in each other’s
interaction sphere if their separation distance< 0 :3 nm, and putting
this value equal toD, we arrive at a timet& 10 ^11 s before they
would diffuse away from each other. This may seem a very short
period, but it would allow the molecules to collide, say, 50 times for
every encounter. This effect seems often to roughly compensate for
the orientation effect mentioned. Nevertheless, Eq. (4.12) may be
fairly uncertain.
Smoluchowski’s equation (4.13) is reasonably rigid, but in the
derivation of (4.16) several assumptions were made. Putting the
hydrodynamic radii of the molecules equal to their collision radii is
especially questionable. As discussed above, Eq. (4.15) would only
apply in homogeneous dilute systems.
Strictly speaking, Smoluchowski’s equation only applies if each
encounter leads to reaction. If the chance of a reaction event is
small, as is the case for largeDG{, the concentration gradient of
reactant B near a molecule of reactant A will be smaller, and this
will upset the relations on which Eq. (4.13) is based.Altogether, and unfortunately, Eq. (4.17) cannot be seen as more than
indicating trends.
4.4 FURTHER COMPLICATIONS
In practice, numerous complications are encountered, concerning the order
of reactions, their rate, and their temperature dependence. Some aspects
important for the food scientist or technologist will be mentioned, if only to
warn against pitfalls in handling or interpretation of kinetic results.