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

2.1 Models for Analysing Mass Transfer 35
Flux (mol m −2 s −1 )
2.5e-9
2.0e-9
1.5e-9
1.0e-9
5.0e-10
slope: 4.16 x 10 −7 m/s
0.0 0 1 2 3 4 5 6
Donor concentration (10 −3 mol/m 3 )
Fig. 2.3 The effect of donor concentration on steady state flux of urea
Hence, model 1 is suitable in all cases when membrane thickness is not known or
difficult to estimate accurately.
2.1.2 Model 2
We next want to find out how the flux varies when we change membrane thickness
(ℓ). With biological membranes thickness cannot be manipulated, but for the present
purpose we use a gelatine membrane which can be prepared at different thicknesses
by casting 10% hot aqueous gelatine between two glass plates separated by spacers.
After cooling to room temperature, stable membranes are obtained that can be used
in experiments. As above, we determine the flux of urea at a given urea concentration
of the donor, but we use membranes of different thicknesses (ℓ). When we plot J
vs ℓ, we find that the flux is inversely related to membrane thickness. The flux is
reduced by one half when the membrane thickness is doubled (Fig. 2.4). At a given
donor concentration, the product of flux and membrane thickness is constant
Jℓ = D(Cdonor −Creceiver). (2.3)
Equation (2.3) is a form of Fick’s law of diffusion, and the new proportionality
coefficient is the well-known diffusion coefficient (D) having the dimension m 2 s −1 .
D may not be constant, and it often depends on concentration. However, this is
easy to find out by conducting experiments using a membrane with constant thickness
but different donor concentrations. In our example, the plot J vs 1/ℓ is linear
and has the slope of 1.25 × 10 −12 molm −1 s −1 (Fig. 2.4 inset). D is obtained by
dividing this slope by the concentration difference between donor and receiver (here
1 × 10 −3 molm −3 ). In our example, this leads to a D of 1.25 × 10 −9 m 2 s −1 .