Echem 21

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Mass transfer equation

Mass transfer in solution occurs by diffusion,migration, and convection

Diffusion and migration result from a gradient

in electrochemical potential,.

Convection results from an imbalance of forces

on the solution

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Mass transfer cont.

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Mass transfer cont.

For linear mass transfer

Consider, aj~ Cj, the Nernst-Planck equ is;

(4)

(5)

(6)

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Migration

In the bulk solution (away from the electrode),concentration gradients are generally small, and the

total current is carried mainly by migration.

All charged species contribute.

For speciesjin the bulk region of a linear mass-

transfer system having a cross-sectional area ,ij= im,j or,

(7)

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Migration

Resistance,

tj, the transference number of j

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Mixed migration and diffusion

Note that imand idmay be in the same or opposite

directions, depending on the direction of the

electric field and the charge on the electroactive

species

i = im+ id

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The migrational component is always in the same

direction as id for cationic species reacting at

cathodesand for anionic species reacting at anodes

It opposesidwhen anionic species are reduced at

cathodesand when cations are oxidized at anodes.

When neutral species are reduced at cathodeonly

idis present.

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Diffusion

It is possible to restrict mass transfer of an

electroactive species near the electrode to the

diffusive mode by using a supporting electrolyte and

operating in a quiescent solution.

Most electrochemical methods are built on the

assumption that such conditions prevail

Diffusion is a process of central importance

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Diffusion: A macroscopic view

Diffusion, which normally leads to the

homogenization of a mixture, occurs by a

"random walk" process

moving in steps of length, l, with one step

being made per unit time,

Where will the molecule be after a time, t?

Answer:will be found at different locations

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Diffusion

Equivalently, we can envision a large number of

molecules concentrated in a line at t = 0 and ask what

the distribution of molecules will be at time t.

This is sometimes called the "drunken sailor problem"

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Diffusion

Where, we envision a very drunk sailor emerging

from a bar and staggering randomly left and right

What is the probability that the sailor will get

down the street a certain distance after a certaintime t?

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Diffusion

At time , it is equally likely that the molecule is at

+land -l; andat time 2, the relative probabilities of being at +2l,

0, and -2l, are 1, 2, and 1, respectively.

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Diffusion

The probability, P(m, r), that the molecule is at agiven location after m time units (m = t/) is given

by the binomial coefficient

Where, x = (-m + 2r)l, with r = 0, 1 , . . . m.

Mean square displacement,

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Diffusion

Diffusion coefficient, D, identified as is a

constant related to the step size and step frequency

This equation provides a handy rule of thumb for

estimating the thickness of a diffusion layer (e.g., howfar product molecules have moved, on the average,

from an electrode in a certain time)

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Next class

Fick's Laws of Diffusion

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Echem 21