Water and Solute Permeability of Plant Cuticles: Measurement and ...

240 8 Effects of Temperature on Sorption and Diffusion of Solutes and Penetration of Water
travelled by the molecule. Diffusion in liquids depends on viscosity of the medium
and on solute size (4.16). With increasing temperature, viscosity decreases and D
increases. Diffusion in solids, for instance polymer membranes, is more complicated.
Temperature dependence of D in membranes is analysed as an activated
process using the Arrhenius equation:
D = Do exp − E D
RT (8.6)
where ED is the activation energy of diffusion, R the gas constant, T the Kelvin
temperature and Do is the pre-exponential factor. In diffusion across polymer membranes,
the activation energy is a measure of the energy expanded against the
cohesive forces of the polymer in forming gaps through which solutes can diffuse.
This concept assumes that vacancies, holes or void volumes must be formed
between polymer chains by thermal motion, which can accommodate solutes. These
voids or holes must be large enough to accept the penetrant, and they must form
very close to the original position of the solute, such that they can be reached in
a single jump. Frequency of hole formation and the size of these holes determine
rates of diffusion, not the jumping frequency of the molecules themselves, which is
much larger. Both increase with temperature, and this is the reason why D increases
with temperature and why the increase is much larger in membranes than in solutions.
Based on the argument that the activation barrier in membrane diffusion is
the energy necessary to form a hole of proper dimensions, Glasstone et al. (1941)
proposed the transition state theory, and derived an expression for diffusion:
2 κT ∆G
D = λ · exp− RT (8.7)
h
where λ is the mean free path of the solute in the solid, κ is Boltzmann’s constant,
h is the Blanck constant and ∆G is the Gibbs free energy of activation, which is
∆G = ∆H − T∆S (8.8)
where ∆H and ∆S are the enthalpy and entropy change per mole during formation
of the transition state. The enthalpy of activation ∆H is related to the Arrhenius
activation energy (ED)
∆H = ED − RT (8.9)
and substituting (8.9) and (8.8) into (8.7) we obtain
2 κT
∆S
D = λ × exp− R ×exp
h − ED RT (8.10)
which is similar to (8.6), and the pre-exponential factor Do is
Do = λ 2 κT
h
× exp− ∆S
R . (8.11)