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Looking at the diffusion regime in more detail

• We will analyse the problem in terms of current and not current density

• The effects of convection and migration will be eliminated

• We will see how this influences our electrochemical reaction

Diffusion

• Consider the cathodic reduction of O at a dropping mercury electrode (DME) – steady state current will be achieved

O + ne = R

• In the absence of migration and convection, the rate of mass transfer (and the rate of reaction at the electrode) is diffusion-controlled:

v = -DO(dcO/dx)

(Fick’s law) where DO is the diffusion coefficient (cm2 s-1)

dcO/dx is the concentration gradient of O.

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Nernst Diffusion Layer

• Assume a linear variation in concentration between the bulk and the electrode surface.

Diffusion

• If the concentration of O is cO(0) at the surface of the electrode, and cO* in the bulk solution, distance from the electrode, then

dcO/dx = {cO* - cO(0)}/

and, with the convention of a cathodic current being positive

v = i/nFA = DO {cO* - cO(0)}/or

i = nFA DO {cO* - cO(0)}/

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Diffusion-limited Current, iL

• The maximum or limiting rate, iL, occurs when cO(0) = 0 as O is being reduced as fast as it can be brought to the electrode surface - the rate is diffusion-limited.

iL = nFADOcO*/ (2)

• Combining equations (1) and (2),

cO(0) = (iL - i)/nFADO (3)

• Similarly, for a net cathodic reaction

i = nFA DR {cR(0) - cR*}/

• When R is not present in the bulk, cR* = 0, and

cR(0) = i/nFADR (4)

Half-wave Potential, E1/2

• As the kinetics of mass transfer are rapid, the concentrations of O and R at the electrode surface will be at the equilibrium values governed by the Nernst equation

E = E0 + (RT/nF) ln {cO(0)/cR(0} (5)

• Combining (3), (4) and (5):

E = E0 - (RT/nF) ln {DO/DR} + (RT/nF) ln {(iL - i)/i}

E = E1/2 + (RT/nF) ln {(iL - i)/i}

where E1/2 = E0 - (RT/nF) ln {DO/DR}

and E = E1/2 when i = iL/2

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Half-wave Potential, E1/2

Half-wave Potential, E1/2

• A plot of E versus ln{(iL - i)/i} – is a straight line of slope RT/nF, and an intercept of E1/2 on the

vertical axis.

• A plot of E versus log10{(iL - i)/i} – is a straight line of slope 2.303RT/nF = 59.1/n mV at 250C, and an

intercept of E1/2 on the vertical axis

• When DO = DR, E1/2 = E0.

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Problem

• Estimate the limiting current density at 298 K for an electrode in a 0.10 M Cu2+(aq) unstirred solution in which the thickness of the diffusion layer is about 0.3 mm. The ionic conductivity (of Cu2+ is 107 S cm2 mol-1 where = n2F2D/RT.

Irreversible Processes

• Voltammetric waves for irreversible processes are more drawn out than reversible ones and their half-wave potentials, E1/2, are more extreme than E0.

• A quick test for reversibility is the Tomes criteria:

|E3/4 - E1/4| = 56.4/n mV at 250C,

where the potentials E3/4 and E1/4 are those for which i = 3iL/4 and iL/4, respectively.

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Irreversible Processes

• It can be shown that

E = E1/2 + (0.0591/n)log10{(iL - i)/i}

• Remember: A plot of E versus log10{(iL - i)/i}should be linear with a slope of 0.0591/nV at 250C.

• Also the Tomes criteria states:

|E3/4 - E1/4| = 56.4/n mV

• Therefore the slope and Tomes criteria will be significantly larger than for a reversible system with an equivalent value of n.

Experimental electrochemistry - Voltammetry

• Voltammetry - the potential of applied to an electrode is varied as a function of time, and the current flow recorded.

• Voltammogram - plot of current, I, versus potential, E.

• Polarography - voltammetry using dropping mercury electrode.

CE REF WE

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Practical details

Nearly every experiment requires the presence of a supporting electrolyte –minimises solution resistance

For CV experiments we use a 3 electrode setup

WE : working electrode : process of interest occursTypically Pt, Au, carbon, ITO, boron doped diamond

CE : counter electrode : Pt wire/coil/mesh, graphite rod

REF : Reference electrode Dependent on solvent system

A potential is applied between WE and REF Current is recorded between WE and CE.

Therefore a stable REF electrode is essential (eg Ag/AgCl, Calomel…..)

Dropping Mercury Electrode

• Clean, reproducible surface.

• Large overpotential for 2H+ + e H2(g) (Eo = 0 V) means Hg electrode can operate at more negative potentials than other electrodes. Good for studying reduction reactions: Mn+ + ne = M(Hg).

• Oxidation of Hg (~0.25 V vs SCE) means Hg electrodes not suitable for studying oxidation.

• Other ‘working’ electrodes - C, Pt, Au

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The Voltammogram

• The potential, E, of the working electrode (Hg, C, Pt, Au) is varied relative to a reference electrode (SCE, Ag|AgCl).

• When electroactive species are oxidised or reduced a current will flow -anodic or cathodic current respectively

The Voltammogram

• The current, I, is measured between the working electrode and the inert ‘counter’ or ‘auxiliary’ electrode (eg., Pt gauze).

• Voltammogram is I versus E.

• Polarogram for Cd2+ at DME (scan E = 0 to -1.3 V)

• DME is the traditional method for doing voltammetry

• Due to the toxicity of mercury this is no longer so common

• The electrochemistry of other electrode materials will be discussed later

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The Voltammogram

• At E < -0.6 V, only small residual currents flow.

• At E ~ - 0.6 V, reduction of Cd2+ occurs: Cd2+ + 2e Cd(Hg) and Faradaic current flows. Cd dissolves in Hg to form an amalgam.

• The current plateaus as the rate of reaction (and hence the current) is limited by the rate at which Cd2+ diffuses from the bulk solution to the surface of the electrode to replace those that have been reduced.

• At E > -1.2 V, reduction of H+ occurs.

Oscillations are due to growth and fall of Hg drops. As drop grows, area increases, more Cd2+

ions reach surface, and current increases.Current quickly decreases as drop falls off

0.5 mM Cd2+ in 1 M HCl

Current Sampling

• Drop of constant size suspended once per second.

• Voltage step typically 4mV more negative than the previous one applied.

• Current measured for short period just before drop dislodges.

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Current Sampling

• Removes oscillations.

• Improved signal-to-noise ratio lowers detection limit.

• Signal = Faradaic current due to reduction of metal.

• Noise = charging current.

Charging Current

• After a voltage is applied to an electrode, charge flows onto the electrode and ions (of opposite charge) in the solution flow towards the electrode. This produces charging current.

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Charging Current

• Charging current decays exponentially - more rapidly than Faradaic current.

• By waiting until the end of each voltage step to measure the current, the signal-to-noise ratio is increased.

Diffusion Current at DME

• The diffusion limited current at the top of an oscillating polarogram (at a DME):

IL = 708 n C D1/2 m2/3 t1/6

• n = no electrons; C = concentration (mol cm-3); D = diffusion coefficient (cm2 s-1); m = Hg flow rate (mg s-1); and t (s) = drop time.

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Diffusion Current

• If all experimental parameters are constant, IL C. – Calibration curve Cunknown

• IL = (limiting current) - (residual current)

• Residual current measured with electro-inactive supporting electrolyte only. – Faradaic component due to presence of impurities.– Non-Faradaic component due to migration of supporting electrolyte.

Shape of Polarogram

• The shape of the polarogram is described by:

• E1/2 is half-wave potential, the potential at which I = IL/2.

• A plot of E versus log[I/(IL - I)] will have intercept E1/2 and slope 0.0592/n as discussed previously

II

I

nEE

Llog

0592.02/1

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E1/2 and Eo for DME

• For reduction of a metal ion to an amalgam, Mn+ + ne = M(Hg)

E1/2 ≈ Eo + Es + 0.0591/n log10[M(Hg)]

where

– Eo is the standard reduction potential for Mn+ + ne = M(s),

– Es is the standard reduction potential for the cell M(s)|Mn+(aq)|M(Hg, sat), and

– [M(Hg] is the concentration of metal in the saturated amalgam.

Typical values

E1/2 (V) Ion

-0.38 Pb2+

-0.46 Tl+

-0.58 Cd2+

-0.99 Zn2+

-2.12 Na+

-2.14 K+

Original “polarographic spectrum” developed 1920-30 by Heyrovsky –

Note shift wrt table due to different RE, and E convention

reversed

Note problem detecting Na, K.

Modern instrumentation and further development of technique allows analysis with excellent limit of detection

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Chemical Analysis

• Qualitative - E1/2

• Quantitative - iL c

– Calibration curve

– Standard Addition

– Internal Standard

– See Harris, Chapter 5