Clearly, the concentration at the reference plane decreases to zero after

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Fig. 2.12 Concentration distribution for linear diffusion and constant concentration gradient at the reference plane x = 0 (Eq. 2.5.11), where - - - " ' D = 1 0 - 5 c m 2 - s - \ c = 5 x 10~5 mol cnT 3 (T = 1.83 s)

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This case is an example of a time-restricted process as it cannot proceed after the instant t=x (the concentration would assume negative values). Time r is termed the transition time. Consider a tube of length / containing a solution of concentration c 2 , with one end (x = 0) rinsed with a solution of concentration cx and the other (JC = /) with a solution of concentration c2. In contrast to the previous example, after a period of time steady state will be reached, characterized by the relationship

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(2.5.14)

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for f oo. The boundary conditions for the ordinary differential equation (2.5.15)

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The solution of this system is then

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(2.5.17)

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The material flux at the steady state 7st is constant at every point in the system: ^ ^ (2.5.18)

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As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c = 0) while the initial concentration of the solution is different (for example c = c ), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation

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2 ) + (d2/dz2) = (d2/dr2) + (d/dr)(2/r), where r is the distance from the origin (located at the centre of the sphere).

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(d2c t = 0, t>0, t>0,

2 3c c = c c=0 c = c (2.5.19)

where r0 is the radius of the sphere. The concentration gradient for r = r0 is given by the equation (dc\ c c =-7=^+(2.5.20)

The concentration gradient will obviously assume the steady-state value for t rl/nDy

dr/r=ro r0

Thus the time during which the transport process attains the steady state depends strongly on the radius of the sphere r0. The steady state is connected with the dimensions of the surface to which diffusion transport takes place and does, in fact, not depend much on its shape. Diffusion to a semispherical surface located on an impermeable planar surface occurs in the same way as to a spherical surface in infinite space. The properties of diffusion to a disk-shaped surface located in an impermeable plane are not very different. The material flux is inversely proportional to the radius of the surface and the time during which stationary concentration distribution is attained decreases with the square of the disk radius. This is especially important for application of microelectrodes (see page 292). Further examples of diffusion processes characterized by boundary conditions connected with specific electrode processes will be considered in Section 5.4. 2.5.2 Simultaneous diffusion and migration

Consider a dilute electrolyte solution containing s components (nonelectrolytes and various ionic species) in which concentration gradients of the components and an electric field are present. The material flux of the ith component is then given by a combination of Eqs (2.3.11) to (2.3.20): J, = -ut(RT grad ct + c^F grad <j>) This relationship can be expressed in the form J,-=-!*, <:, grad ft (2.5.23) (2.5.22)

Ill where the driving force for the simultaneous diffusion and migration is the gradient of the electrochemical potential /i, = jU, + z{F(^. The concept of the electrochemical potential is discussed in detail in Section 3.1.1. Equations (2.3.11), (2.3.16) and (2.3.18) can be combined to give J, = -Dt grad c, - - ^ UiCi grad 0 or, for transport with respect to coordinate x alone,