Cottrell equation

Files:
Coulometry.EXP
Coulometry 2Q.CRV

The "electrochemical surface" of a polished platinum electrode is determined from a chrono coulometry operated on the electrochemical reduction of complexed ferric ions, in a phosphate buffer (pH7) solution. The analysis uses the integration of Cottrell equation which establishes [2] the time evolution profile of the current versus the surface, the concentration and the diffusion coefficient of the species for a large potential step at planar electrode. The integration of the Cottrell equation establishes that the slope of Q versus sqrt (t) is directly proportional to the electroactive surface. This is why the electroactive surface can be determined from a chrono coulometry. The electroactive surface is calculated and the rugosity, defined as the ratio between the electrochemical surface and the geometrical surface, is also determined, close to one.

Solution{Potassium Hexacyanoferrate III (Ferricyanide) plus
Potassium Hexacyanofferate II (Ferrocyanide) at
10E-2 mol/l in phosphate buffer, pH 7.00}
prepared from redox buffer type BS870, diluted 10 times
in phosphate buffer type PH7-6B
WORKPlatinum disc electrode
S = 0.196±0.01 cm² (Diameter = 5.0±0.05 mm)
REFCalomel electrode (XR100)
AUXPlatinum wire (XM100)
CP06Electrochemical cell, room temperature.

An OCP and a Potentiostatic pause at OCP are performed prior to the coulometry at imposed potential = 0 mV versus REF.

1) Display : Type = normal X = sqrt(Time) Y1= Quantity(C) Y2 = Current

Q = -2nFSC sqrt{(D*t)/Pi}

This is the “integration” of Cottrell equation. It is valid for or a large potential step at planar electrode. Considering a reduction process the slope of Q versus sqrt (t) equals a,
Slope= a = -2nF*S*C*[sqrt(D/Pi)]
and the surface can be calculated as S = - a / {2nF*C*[sqrt(D/Pi]}

With:
a = slope of Q versus sqrt(t)
Q = Total charge from t=0
n = number of electron exchanged = 1
C = Ferricyanid bulk concentration [C= 0.01 mol/l]
F = 96500 C
Pi = Pi number
D= Diffusion coefficient D =0.735e-05 cm². s-1 from [1]
S = electrochemical surface

The slope is determined from a linear regression with Q versus sqr(t):
X min.: 3.975
X max.: 5.79
Mode : y=f(x)
Coefficient : 0.999868
Result : y(mC/cm²) = -0.609*x(sqrt ) + 0.703
x(y=0) = 1.15307
The measured slope = -0.609
The electrochemical surface is calculated, S = 0.206 cm²

The electroactive surface determined (0.206 cm2) is just slightly larger than the geometrical surface of the polished platinum electrode (0.196 cm2). The rugosity is very low.

[1] D =0.735e-05 cm². s-1 from handbook of Chemistry and physics 79th Ed. 1998-99 p 5-94
[2] Integration of the Cottrell equation versus time in "Electrochemistry, principles, methods and applications" Christopher M.A. Brett & Ana Maria Oliveira Brett - Oxford science publication p206 Oxford University Press, 1993