The mass transport equation:
(A6.1) |
may be converted into a dimensionless form:
(A6.2) |
where:
, and | (A6.3) |
The electrode surface concentration is given by:
a_{0} = [1 + exp(-Q)]^{-1} where | (A6.4) |
hence the concentration is a function of two space variables, a time variable and the normalised potential (which itself is a function of time):
(A6.5) |
In terms of dimensionless variables, the LSV waveform is:
(A6.6) |
where:
(A6.7) |
Hence the concentration is a function of four dimensionless variables:
(A6.8) |
The current is given by:
(A6.9) |
In terms of R and Z this becomes:
(A6.10) |
Hence:
(A6.11) |
where f_{disc} denotes an unknown function which depends only on t and v.
The mass transport equation:
(A6.12) |
may be rewritten in term of dimensionless space and time variables:
(A6.13) |
where:
and | (A6.14) |
Considering the electrode surface concentration, as for the microdisc electrode:
(A6.15) |
The current is given by:
(A6.16) |
or in terms of R:
(A6.17) |
Hence:
(A6.18) |
where f_{hemi} denotes an unknown function which depends only on t and v.
At the peak of the voltammogram :
(A6.19) |
The time dependence of the normalised potential is given by equation (A6.6). Differentiating gives:
(A6.20) |
Hence the peak is also defined by:
(A6.21) |
Applying this constraint to equation (A6.11) for a microdisc gives:
(A6.22) |
Similarly for the hemisphere, applying constraint (A6.21) to equation (A6.18) gives:
(A6.23) |
The ratio of peak currents is therefore:
(A6.24) |
Since:
(A6.25) |
then:
(A6.26) |
Hence the peak current ratio for 'equivalent' electrodes (where r_{hemi} = 2r_{disc}/p) is solely a function of dimensionless scan rate:
(A6.27) |
where f_{1}, f_{2} etc. denote arbitrary unknown functions.