In the electrode kinetics section we have seen that the rate of reaction
can be influenced by the cell potential difference. However, the rate of
transport to the surface can also effect or even dominate the overall
reaction rate and in this section we look at the different forms of mass
transport that can influence electrolysis reactions.
We have already seen that a typical electrolysis
reaction involves the transfer of charge between
an electrode and a species in solution. This whole process due to the interfacial nature of the
electron transfer reactions typically involves a series of steps.
In the section on electrode kinetics we saw how the electrode voltage can effect the rate of the
electron transfer. This is an exponential relationship, so we would predict from the electron
transfer model that as the voltage is increased the reaction rate and therefore the current will
increase exponentially. This would mean that it is possible to pass unlimited quantities of current.
Of course in reality this does not arise and this can be rationalised by considering the expression
for the current that we encountered in the electrode kinetics section
Clearly for a fixed electrode area (A) the reaction
can be controlled by two factors. First the rate constant k_{red} and second the surface concentration
of the reactant ([O]_{o}). If the rate
constant is large, such that any reactant close to the interface is immediately converted into
products then the current will be controlled by the amount of fresh reactant reaching the interface
from the bulk solution above. Thus movement of reactant in and out of the interface is important
in predicting the current flowing. In this section we look at the various ways in which material
can move within solution  so called mass transport.
There are three forms of mass transport which can influence an electrolysis reaction
*  Diffusion 
*  Convection 
*  Migration 
In order to predict the current flowing at any particular time in an electrolysis measurement we
will need to have a quantitative model for each of these processes to
complement the model for
the electron transfer step(s).
Diffusion
Diffusion occurs in all solutions and arises from local uneven
concentrations of reagents.
Entropic forces act to smooth out these uneven distributions of concentration and are therefore
the main driving force for this process. One example of this can be seen in the animation below. Two materials are held
separately in a single container separated by a barrier. When the barrier
is removed the two reagents can mix and this processes on the
microscopic scale is essentially random. For a large enough sample
statistics can be used to predict how far material will move in a certain
time  and this is oftern referred to as a random walk model.
Animation
Diffusion is particularly significant in an electrolysis experiment since
the conversion reaction only occurs at the electrode surface. Consequently there
will be a lower reactant concentration at the electrode than in bulk solution. Similarly a higher
concentration of product will exist near the electrode than further out into solution.
The rate of movement of material by diffusion can be predicted
mathematically and
Fick proposed two laws to quantify the processes. The
first law:
relates the diffusional flux J_{o} (ie the rate of
movement of material by diffusion) to the concentration gradient and the
diffusion coefficient D_{o}.
The negative sign simply signifies that material moves down a
concentration gradient ie from regions of high
to low concentration. However, in many measurements we need to know how
the concentration of material varies as a function of time and this can be
predicted from the first law. The result is Fick's second law
In this case we consider diffusion normal to an electrode surface (x direction). The rate of change of the concentration
([O]) as a function of time (t) can be seen to be related to the change in the
concentration gradient. So the steeper the change in concentration the greater the rate of
diffusion. In practice diffusion is often found to be the most significant
transport process for many electrolysis reactions.
Fick's second law is an important relationship since it permits the
prediction of the variation of concentration of different species
as a function of time within the electrochemical cell. In order to solve
these expressions analytical or computational models are usually employed.
Convection
Convection results from the action of a force on the solution. This can be a pump, a flow of gas
or even gravity. There are two forms of convection the first is termed natural convection and is present in any solution. This
natural convection is generated by small thermal or density differences and acts to mix the solution
in a random and therefore unpredictable manner. In the case of electrochemical measurements
these effects tend to case problems if the measurement time for the experiment exceeds 20
seconds.
It is possible to drown out the natural convection effects from an electrochemical experiment by
deliberately introducing convection into the cell. This form of convection is termed forced convection. It is typically several orders of
magnitude greater than any natural convection effects and therefore effectively removes the
random aspect from the experimental measurements. This of course is only true if the convection
is introduced in a well defined and quantitative manner.
We will see in
later sections
examples of such systems, including the rotating disc and wall jet electrodes. In each of these
devices the convection is introduced so that a laminar flow profile
created.
The figure above shows the cross section of liquid flowing through a pipe. Solution is introduced
from the right handside and pumped through the pipe. If the flow is controlled, after a small lead
in length, the profile will become stable with no mixing in the lateral direction, this is termed
laminar flow.
If however the solution is pumped through the cell at a high rate then the
transport can become turbulent, where the solution movement is
essentially a random and unpredictable. In order to
predict this change over between turbulent and laminar behaviour, work was
carried out by Hagan in the mid 1840's and later by Reynolds who was the
first to put forward a predicitive model.
In this particular example there is a maximum velocity in the centre and minimum
velocity at the side walls. For laminar flow conditions the mass transport equation for (1
dimensional) convection is predicted by
where v_{x} is the velocity of the
solution which can be calculated in many situations be solving the appropriate form of the
NavierStokes equations. An analogous form exists for the three dimensional convective
transport. When an electrochemical cell possesses forced convection we must be able to solve the
electrode kinetic, diffusion and convection steps, to be able to predict the current flowing. This
can be a difficult problem to solve even for modern computers and yet we still have one final form
of mass transport to address!
Migration
The final form of mass transport we need to consider is migration. This is essentially an
electrostatic effect which arises due the application of a voltage on the electrodes. This effectively
creates a charged interface (the electrodes). Any charged species near that interface will either
be attracted or repelled from it by electrostatic forces. The migratory flux induced can be
described mathematically (in 1 dimension) using
However due to ion solvation effects and diffuse layer interactions in solution, migration is
notoriously difficult to calculate accurately for real solutions. Consequently most voltammetric
measurements are performed in solutions which contain a background electrolyte  this material
is a salt (eg KCl) that does not undergo electrolysis itself but helps to shield the reactants from
migratory effects. By adding a large quantity of the electrolyte (relative to the reactants) it is
possible to ensure that the electrolysis reaction is not significantly effected by migration. The
purpose of introducing a background electrolyte into a solution is not however solely to remove
migration effects as it also acts as a conductor to help the passage of current through the solution.
Mass Transport in Electrochemical Cells
To gain a quantitative model of the current flowing at the electrode we must account for the
electrode kinetics, the 3 dimensional diffusion, convection and migration, of all the species
involved. This is currently beyond the capacity of even the fastest computers  and will be for
some time. However, as we will discover later electrochemical cells and experimental conditions
can be employed to cheat the mass transport equations. We can effectively remove much (but
not all) of the mass transport complexity by carefully designing and controlling the
electrochemical experiment.
