The measurement of diffusion coefficients
The solutions of the diffusion equation are useful for experimental determinations of diffusion coefficients. In the capillary technique, a capillary tube, open at one end and containing a solution, is immersed in a well stirred larger quantity of solvent, and the change of concentration in the tube is monitored. The solute diffuses from the open end of the capillary at a rate that can be calculated by solving the diffusion equation with the appropriate boundary conditions, so D may be determined. In the diaphragm technique, the diffusion occurs through the capillary pores of a sintered glass diaphragm separating the well-stirred solution and solvent. The concentrations are monitored and then related to the solutions of the diffusion equation corresponding to this arrangement. Diffusion coefficients may also be measured by the dynamic light scattering technique described in Section 19.3 and by NMR.
We saw in Impact I21.2 how electrolytes are transported across cell membranes. Here we use the diffusion equation to explore the way in which non-electrolytes cross the lipid bilayer. Consider the passive transport of an uncharged species A across a lipid bilayer of thickness l. To simplify the problem, we will assume that the concentration of A is always maintained at [A] = [A]0 on one surface of the membrane and at [A] = 0 on the other surface, perhaps by a perfect balance between the rate of the process that produces A on one side and the rate of another process that consumes A completely on the other side. This is one example of a steady-state assumption, which will be discussed in more detail in Section 22.7. Then ∂[A]/∂t = 0 and the diffusion equation simplifies to
D
=0
where D is the diffusion coefficient and the steady-state assumption makes partial derivatives unnecessary. We use the boundary conditions [A](0) = [A]0 and [A](l) = 0 to solve eqn 21.74 and the result, which may be verified by differentiation, is
[A](x) = [A]0(1−
)
which implies that the [A] decreases linearly inside the membrane. We now use Fick’s first law to calculate the flux J of A through the membrane and the result is
J =D
However, we need to modify this equation slightly to account for the fact that the con centration of A on the surface of a membrane is not always equal to the concentration of A measured in the bulk solution, which we assume to be aqueous. This difference arises from the significant difference in the solubility of A in an aqueous environment and in the solution–membrane interface. One way to deal with this problem is to define a partition ratio, KD (D for distribution) as
KD=
where [A]s is the concentration of A in the bulk aqueous solution. It follows that
J =DKD
In spite of the assumptions that led to its final form, eqn 21.78 describes adequately the passive transport of many non-electrolytes through membranes of blood cells. In many cases the flux is underestimated by eqn 21.78 and the implication is that the membrane is more permeable than expected. However, the permeability increases only for certain species and not others and this is evidence that transport can be mediated by carriers. One example is the transporter protein that carries glucose into cells. A characteristic of a carrier C is that it binds to the transported species A and the dissociation of the AC complex is described by
AC→5A+C K=
where we have used concentrations instead of activities. After writing [C]0 = [C] + [AC], where [C]0 is the total concentration of carrier, it follows that
[AC]=
We can now use eqns 21.80 and 21.78 to write an expression for the flux of the species AC through the membrane:
J =
= J max
Where KD and D are the partition ratio and diffusion coefficient of the species AC. We see from Fig. 21.27 that when [A] << K the flux varies linearly with [A] and that the f lux reaches a maximum value of J max = DKD[C]0/l when [A] >> K. This behaviour is characteristic of mediated transport.
Fig. 21.27 The flux of the species AC through a membrane varies with the concentration of the species A. The behaviour shown in the figure and explained in the text is characteristic of mediated transport of A, with C as a carrier molecule.
