Solutions of the diffusion equation
The diffusion equation, eqn 21.68, is a second-order differential equation with respect to space and a first-order differential equation with respect to time. Therefore, we must specify two boundary conditions for the spatial dependence and a single initial condition for the time-dependence. As an illustration, consider a solvent in which the solute is initially coated on one surface of the container (for example, a layer of sugar on the bottom of a deep beaker of water). The single initial condition is that at t = 0 all N0 particles are concentrated on the yz-plane (of area A) at x = 0. The two boundary conditions are derived from the requirements (1) that the concentration must everywhere be finite and (2) that the total amount (number of moles) of particles present is n0 (with n0 = N0/NA) at all times. These requirements imply that the flux of particles is zero at the top and bot tom surfaces of the system. Under these conditions it is found that
c(x,t) =
e−x2/4Dt
as may be verified by direct substitution. Figure 21.26 shows the shape of the concentration distribution at various times, and it is clear that the concentration spreads and tends to uniformity.
Another useful result is for a localized concentration of solute in a three-dimensional solvent (a sugar lump suspended in a large flask of water). The concentration of diffused solute is spherically symmetrical and at a radius r is
c(r,t) =
e−x2/4Dt
Other chemically (and physically) interesting arrangements, such as transport of sub stances across biological membranes can be treated (Impact I21.3). In many cases the solutions are more cumbersome.
Fig. 21.26 The concentration profiles above a plane from which a solute is diffusing. The curves are plots of eqn 21.72 and are labelled with different values of Dt. The units of Dt and x are arbitrary, but are related so that Dt/x2 is dimensionless. For example, if x is in metres, Dt would be in metres2; so, for D = 10−9 m2 s−1, Dt = 0.1 m2 corresponds to t = 108 s.
