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

2.5 Diffusion Across a Membrane with Changing Concentrations 45
2.5 Diffusion Across a Membrane with Changing Concentrations
In the previous examples, the volumes (V) of the receiver and donor solutions were
not considered. However, Vdonor and Vreceiver must not be equal, and in fact they are
often adjusted to extend the duration of the steady state. In deriving an equation
allowing Cdonor and Creceiver to vary, the volumes of the compartments are included
explicitly (Hartley and Graham-Bryce 1980).
The total amount (M) in the system is assumed constant, and the amount in the
membrane is taken to be negligible. Hence, the solute is either in the donor or in the
receiver:
Mtotal = Mdonor + Mreceiver = CdonorVdonor +CreceiverVreceiver. (2.19)
Initially, when t = 0 the donor concentration is C0 and Creceiver = 0. The system is
in equilibrium when concentrations in receiver and donor are equal (C∞):
C0Vdonor
C∞ =
Vdonor+Vreceiver
. (2.20)
As long as Cdonor ≫ Creceiver, we can write for the mean flow rate (F)
F = Vdonor(Cdonor,1 −Cdonor,2)
(t2 −t1)
= Vdonor∆Cdonor
, (2.21)
∆t
where numerical suffixes denote times (t). For experiments allowing major change
of ∆C to occur, we must use the differential form of (2.21)
dCreceiver dCdonor
Vreceiver = −Vdonor
dt
dt = F = PA(Cdonor −Creceiver). (2.22)
The integral solution can be expressed as
�
1
−PA +
Vdonor
1
�
t = ln
Vreceiver
Cdonor −Creceiver
. (2.23)
C0
Equation (2.23) assumes more convenient forms under the following situations. If
Cdonor is held constant by having Vdonor ≫ Vreceiver or by other means, Vdonor becomes
unimportant and we can write
−PAt
Vreceiver
= ln C0 −Creceiver
. (2.24)
C0
If Creceiver is held zero or nearly so by having Vreceiver ≫ Vdonor, (2.23) becomes
−PAt
Vdonor
= ln Cdonor
. (2.25)
C0