The%20parameter%20a%20is%20called%20the%20transient%20coefficient%20and%20lies%20in%20the%20range%200%20to%201.

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The parameter a is called the transient
coefficient and lies in the range 0 to 1.
Based on the above new expressions, the net
current density can be expressed as
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• Example 25.1 Calculate the change in cathodic
  current density at an electrode when the
  potential difference changes from ?? to ??
• Self-test 25.5 calculate the change in anodic
  current density when the potential difference is
  increased by 1 V.

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Overpotential

• When the cell is balanced against an external
  source, the Galvani potential difference, ?F ,
  can be identified as the electrode zero-current
  potential, E.
• When the cell is producing current, the electrode
  potential changes from its zero-current value, E,
  to a new value, E.
• The difference between E and E is the
  electrodes overpotential, ?.
• ? E E
• The ?F ? E,
• Expressing current density in terms of ?
• ja j0e(1-a)f? and jc
  j0e-af?
• where jo is called the exchange current
  density, which is when there is no net current
  flow, i.e. ja jc

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• The Butler-Volmer equation
• j j0(e(1-a)f? - e-af?)
• The low overpotential limit
• ? lt 0.01 V
• The high overpotential limit
• ? 0.12 V

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The low overpotential limit

• The overpotential ? is very small, i.e. f? ltlt1
• When x is small, ex 1 x
• Therefore ja j01 (1-a) f?
• jc j01 (-a f?)
• Then j ja - jc j01 (1-a) f? - j01 (-a
  f?)
• j0f?
• The above equation illustrates that at low
  overpotential limit, the current density is
  proportional to the overpotential.
• It is important to know how the overpotential
  determines the property of the current.

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Calculations under low overpotential conditions

• Example 25.2 The exchange current density of a
  Pt(s)H2(g)H(aq) electrode at 298K is 0.79
  mAcm-2. Calculate the current density when the
  over potential is 5.0mV.
• Solution j0 0.79 mAcm-2
• ? 5.0mV
• f F/RT
• j j0f?
• Self-test 25.6 What would be the current at pH
  2.0, the other conditions being the same?

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The high overpotential limit

• The overpotential ? is large, but could be
  positive or negative !
• When ? is large and positive
• jc j0e-af? j0/eaf? becomes very small
  in comparison to ja
• Therefore j ja j0e(1-a)f?
• ln(j) ln(j0e(1-a)f? ) ln(j0) (1-a)f?
• When ? is large but negative
• ja is much smaller than jc
• then j - jc - j0e-af?
• ln(-j) ln(j0e-af? ) ln(j0) af?
• Tafel plot the plot of logarithm of the current
  density against the over potential.

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Applications of a Tafel plot

• The following data are the anodic current through
  a platinum electrode of area 2.0 cm2 in contact
  with an Fe3, Fe2 aqueous solution at 298K.
  Calculate the exchange current density and the
  transfer coefficient for the process.
• ?/mV 50 100 150 200 250
• I/mA 8.8 25 58 131 298
• Solution calculate j0 and a
• Note that I needs to be converted to J

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• Self-test 25.7 Repeat the analysis using the
  following cathodic current data
• ?/mV -50 -100 -150 -200 -250
• I/mA -0.3 -1.5 -6.4 -27.61 -118.6
• In general exchange currents are large when the
  redox process involves no bond breaking or if
  only weak bonds are broken.
• Exchange currents are generally small when more
  than one electron needs to be transferred, or
  multiple or strong bonds are broken.

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The general arrangement for electrochemical rate
measurement
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25.10 Voltammetry

• Voltammetry the current is monitored as the
  potential of the electrode is changed.
• Chronopotentiometry the potential is monitored
  as the current density is changed.
• Voltammetry may also be used to identify species
  and determine their concentration in solution.
• Non-polarizable electrode their potential only
  slightly changes when a current passes through
  them. Such as calomel and H2/Pt electrodes
• Polarizable electrodes those with strongly
  current-dependent potentials.
• A criterion for low polarizability is high
  exchange current density
• due to j j0f?

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Concentration polarization

• At low current density, the conversion of the
  electroactive species is negligible.
• At high current density the consumption of
  electroactive species close to the electrode
  results in a concentration gradient.
• Concentration polarization The consumption of
  electroactive species close to the electrode
  results in a concentration gradient and diffusion
  of the species towards the electrode from the
  bulk may become rate-determining. Therefore, a
  large overpotential is needed to produce a given
  current.
• Polarization overpotential ?c

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• Consider the reduction half reaction
• Mz ze ? M
• The Nernst equation is
• E E? (RT/zF) ln(a)
• When using a large excess of support electrolyte,
  the mean activity coefficients stays
  approximately constant,
• E E? (RT/zF)ln(?) (RT/zF)ln(c)
• E Eo (RT/zF)ln(c)
• The ion concentration at OHP decreases to
• c due to the reaction, resulting
• E Eo (RT/zF)ln(c)
• The concentration overpotential is
• ?c E E (RT/zF)ln(c/c)
• (typo in the 8th edition)

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• The thickness of the Nernst diffusion layer
  (illustrated in previous slide) is typically 0.1
  mm, and depends strongly on the condition of
  hydrodynamic flow due to such as stirring or
  convective effects.
• The Nernst diffusion layer is different from the
  electric double layer, which is typically less
  than 1 nm.
• The concentration gradient through the Nernst
  diffusion layer is dc/dx (c c)/d
• This concentration gradient gives rise to a flux
  of ions towards the electrode J -
  D(dc/dx)
• Therefore, the particle flux toward the electrode
  is
• J D (c c)/ d

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• The cathodic current density towards the
  electrode is the product of the particle flux and
  the charge transferred per mole of ions (zF)
• j zFJ zFD(c c)/ d
• The maximum rate of diffusion across the Nernst
  layer is when c 0 at which the concentration
  gradient is the steepest.
• jlim zFJ zFDc/ d
• Using the Nerst-Einstein equation (D
  RT?/(zF)2),
• jlim cRT?/(zFd)
• where ? is ionic conductivity

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• Example 25.3 Estimate the limiting current
  density at 298K for an electrode in a 0.10M
  Cu2(aq) unstirred solution in which the
  thickness of the diffusion layer is about 0.3mm.
• Solution one needs to know the following
  information
• molar conductivity of Cu2 ? 107 S cm2
  mol-1
• d 0.3 mm
• employing the following equation
• jlim cRT?/(zFd)
• Self-test 25.8 Evaluate the limiting current
  density for an Ag(s)/Ag(aq) electrode in 0.010
  mol dm-3 Ag(aq) at 298K. Take d 0.03mm.

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• From j zFD(c c)/ d,
• one gets c c - jd/zFD
• ?c (RT/zF)ln(c/c) (RT/zF)ln(1 - jd/(zFDc))
• Or
• j (zFDc)(1 ezf?)/d