8. Fundamentals of Charged Surfaces
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Transcript of 8. Fundamentals of Charged Surfaces
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8. Fundamentals of Charged Surfaces
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Moving the reagentsQuickly and with Little energy
Diffusionelectric fields
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o
Cha
rged
S
urfa
ce
+
+
+
+
X=0
N
N o
G
kT
*
ex p
1. Cations distributed thermallywith respect to potential2. Cations shield surface and reduce the effective surfacepotential
o
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o
Cha
rged
S
urfa
ce
+
+
+
+
X=0
N
N o
G
kT
*
ex p
o
+
+
+
dx dx
o
* ** dx
+
+
***
o
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n
ne
o
zF
R Tx
o i ii
z F
R Td
dxz F C e
i x2
2*
Surface Potentials
Poisson-Boltzman equation
Charge near electrode dependsupon potential and is integratedover distance from surface - affects the effective surface potential
Cation distribution hasto account for all species,i
Dielectric constant of solution
Permitivity of free space
Simeon-Denis Poisson1781-1840
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ze
kTo 1 o m V 5 0
x o
xe
o i ii
z F
R Td
dxz F C e
i x2
2*
Solution to the Poisson-Boltzman equation can be simple if the initial surface potential is small:
Potential decays from the surface potential exponentially with distance
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d
dx
z F C
e
z F Cz F
R T
z F
R T
i ii
o
z F
R Ti i
i
o
i i i i
i i2
2
2
11
2
* *
. . . . . .
Largest term
d
dx
F z C
R T
i ii
ox
2
2
2
*
Let
2
2 2
1
x
F z C
R Ta
i ii
o
*
Then:d
dx xx
a
2
2 2
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General Solution of:
x
x
x
x
xA e B ea a
d
dx xx
a
2
2 2
Because goes to zero as x goes to infinityB must be zero
x
x
x xA e A ea
Because goes to as x goes to zero (e0 =1)A must be
thus x o
xe
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Potential decays from the surface potential exponentially with distance
x o oe 1 0 3 6 7( . )
When =1/x or x=1/ then
The DEBYE LENGTH x=1/
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o
Cha
rged
S
urfa
ce
=0.36 o+
+
+
+
+
X=0 X=1/
+
+
+
+
What is
Petrus Josephus Wilhelmus Debye1844-1966
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2 2 21
2n z e
kTo
*
z x C( . )( )* /3 2 9 1 0 7 1 2
Debye Length
Units are 1/cm
26 0 2 1 0 1
1 0
1 0 0 1 6 0 2 1 8 1 0
7 8 4 98 8 5 4 1 9 1 0
1 0 0
1 3 8 0 6 5 1 02 9 8
2 3
3 3
22
1 9 2
2 5
1 2 2
2
2 3
1
2m oles
L
x
m ole
L
cm
cm
mch e
x C
ch e
un itlessx C
N m
m
cm
N m
J
x J
KKo C
.a rg
.
arg
.. .
2 1 6 0 2 1 8 1 0
7 8 4 9 1 3 8 0 6 5 1 0 2 9 8 6 0 2 2 1 0
21 9 2
2 5
2 3
1
2
2 3
#.
. . .cm
x
x
m ole
x ionso C
2 2 21
2C N z e
kTonc A
o
Does not belong
=1/cm
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zF
n z e
kTo
2 2 2
1
2*
z x C( . )( )* /3 2 9 1 0 7 1 2
Table 2: Extent of the Debye length as a function of electrolyte
C(M) 1/κ ( )
1 3
0.1 9.6
0.01 30.4
0.001 96.2
0.0001 304
Debye Length
Units are 1/cm
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In the event we can not use a series approximation to solve the Poisson-Boltzman equation we get the following:
ex p
ex p ex p
ex p ex p
x
ze
kT
ze
kT
ze
kT
ze
kT
2 2
2 2
1 1
1 1
0
0
Ludwig Boltzman1844-1904
Simeon-Denis Poisson1781-1840
Check as Compared to tanhBy Bard
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Set up excel sheet ot have them calc effectOf kappa on the decay
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Example Problem
A 10 mV perturbation is applied to an electrode surface bathed in0.01 M NaCl. What potential does the outer edge of a Ru(bpy)3
3+
molecule feel?
Debye length, x
z x C
XA
xA
( . )( )
/( . )( . )
.
* /
/
3 2 9 1 0
11 0
1 3 2 9 1 0 0 0 13 0 4
7 1 2
8
7 1 2
Since the potential applied (10 mV) is less than 50 can usethe simplified equation.
Units are 1/cm
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x o
xo
x
xe e ez 1 0 7 4 3
9
3 0 4. .
The potential the Ru(bpy)33+ compound experiences
is less than the 10 mV applied.
This will affect the rate of the electron transfer eventfrom the electrode to the molecule.
Radius of Ru
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Surface Charge Density
The surface charge distance is the integration over all the charge lined up at the surface of the electrode
oa a a
dxd
dxdx
d
dx
0
2
2 0
The full solution to this equation is:
o oo o
o o
kT nze
kT
C z
(8 ) s in h ( )
. ( * ) s in h ( . )
1
2
1
2
2
11 7 1 9 5
C is in mol/L
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o
Cha
rged
S
urfa
ce
=0.36 o+
+
+
+
+
X=0 X=1/
+
+
+
+
Can be modeled as a capacitor:C
d
ddifferential
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For the full equation
Cz e n
kT
ze
kT
oo
2
2
2 20
1
2 co sh
C z C z o 2 2 8 1 9 51
2* co sh . At 25oC, water
d
d
Differential capacitanceEnds with units of uF/cm2
Conc. Is in mol/L
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0
2000
4000
6000
8000
10000
12000
-15 -10 -5 0 5 10 15
y x co sh
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o o o
Can be simplified if (o ~ 25 mV),
Specific Capacitance is the differential space charge per unit area/potential
C
A
dq
A d
d
dspecific
C
A o Specific CapacitanceIndependent of potentialFor small potentials
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o
Flat in this regionGouy-Chapman Model
Cz e n
kT
ze
kT
oo
2
2
2 20
1
2 co sh
0
20
40
60
80
100
120
-500 -400 -300 -200 -100 0 100 200 300 400 500
E-Ezeta
Capacitance
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Real differential capacitance plots appear to roll off instead ofSteadily increasing with increased potential
Physical Chemistry Chemical PhysicsDOI: 10.1039/b101512p
Paper
Photoinduced electron transfer at liquid/liquid interfaces. Part V. Organisation of water-soluble chlorophyll at the water/1,2-dichloroethane interface�
Henrik Jensen , David J. Fermn and Hubert H. Girault*
Laboratoire d'Electrochimie, D partement de Chimie, Ecole Polytechnique F d rale de Lausanne, CH-1015, � � �Switzerland
Received 16th February 2001 , Accepted 3rd April 2001 Published on the Web 17th May 2001
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o
Cha
rged
S
urfa
ce
+
+
+
+
+
X=0
+
+
+
+
Linear dropin potentialfirst in theHelmholtz orStern specificallyadsorbed layer
Exponentialin the thermallyequilibrated ordiffuse layer
CdiffuseCHelmholtz or Stern
x2
Hermann Ludwig Ferdinand von Helmholtz1821-1894
O. SternNoble prize 1943
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Capacitors in series
Cz e n
kT
ze
kTD iffuse
oo
2
2
2 20
1
2 co sh
C
A H elm ho ltz or S terno
C
C C C
series
N
11 1 1
1 2
. . . . . .
1 1 1 1
1 2CC
C C Cseriesseries
N
. . . . . .
Wrong should be x distance of stern layer
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For large applied potentials and/or for large salt concentrations1. ions become compressed near the electrode surface to
create a “Helmholtz” layer.2. Need to consider the diffuse layer as beginning at the
Helmholtz edge
1 1
2
2
2
0 2 20
1
2C
x
z e n
kT
ze
kT
oo
co sh
CapacitanceDue to Helmholtzlayer Capacitance due to diffuse
layer
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DeviationIs dependent uponThe salt conc.
The larger the “dip”For the lower The salt conc.
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.7
0.71
-500 -400 -300 -200 -100 0 100 200 300 400 500
E-Ezeta
Capacitance
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Create an excel problemAnd ask students to determine the smallestAmount of effect of an adsorbed layer
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Experimental data does notCorrespond that well to the Diffuse double layer double capacitormodel
(Bard and Faulkner 2nd Ed)
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Fig. 5 Capacitance potential curve for the Au(111)/25 mM KI in DMSO interface with time. �
Physical Chemistry Chemical PhysicsDOI: 10.1039/b101279g
PaperComplex formation between halogens and sulfoxides on metal surfaces
Siv K. Si and Andrew A. Gewirth*
Department of Chemistry, and Frederick Seitz Materials Research Laboratory, Uni ersity of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
Received 8th February 2001 , Accepted 20th April 2001 Published on the Web 1st June 2001
Model needs to be altered to accountFor the drop with large potentials
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This curve is pretty similar to predictions except where specificAdsorption effects are noted
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Graphs of these types were (and are) strong evidence of the Adsorption of ions at the surface of electrodes.
Get a refernce or two of deLevie here
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Introducing the Zeta Potential
oC
harg
ed
Sur
face
+
+
+
+
+
+
+
+
+
Imagine a flowing solutionalong this charged surface.Some of the charge will be carriedaway with the flowing solution.
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Introducing the Zeta Potential, given the symbo l
oC
harg
ed
Sur
face
+
+
+
+
+
+
+
+
+
Shear Plane
Flowing solution
zeta
Sometimesassumedzeta correspondsto DebyeLength, butNot necessarily true
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C C1
21
2ex p
The zeta potential is dependent upon how the electrolyteconcentration compresses the double layer. are constantsand sigma is the surface charge density.
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Shear Plane can be talked about in two contexts
o
Cha
rged
S
urfa
ce
+
+
+
+
+
+
+
+
+
Shear Plane
+
+
+
+ +
+
+
+
++
++
ShearPlane
Particle in motion
In either case if we “push” the solution alonga plane we end up with charge separation whichleads to potential
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Streaming Potentials
From the picture on preceding slide, if we shove the solutionAway from the charged surface a charge separation develops= potential
P
o
so lu tion resis ce m
zeta po ten tia l
v is itykg
m s
tan
co s
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Sample problem here
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Reiger- streaming potentialapparatus.
Can also make measurements on blood capillaries
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o
Cha
rged
S
urfa
ce
+
+
+
+
+
X=0
+
+
+
+
Cathode
Anode
Vappapp
+
Jo Jm
Jm
In the same way, we can apply a potential and move ions and solution
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Movement of a charged ion in an electric field
Electrophoretic mobility
app lied electric fie ld
f frictiona l drag r
v electropho retic velocity
6
The frictional drag comesabout because the migratingion’s atmosphere is movingin the opposite direction, draggingsolvent with it, the drag is related to the ion atmosphere
f v z eii
i
The force from friction is equal to the electric driving force
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Electric ForceDrag Force
Direction of Movement
Ion accelerates in electric field until the electric forceis equal and opposite to the drag force = terminal velocity
f z eelectrica l i
f r
vis ity
r ion ic rad ius
ion velocity
fr ic tiona l
6
co s
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f f
r z e
fr ic tiona l electric
i
6
At terminal velocity
z e
ri
6
The mobility is the velocity normalized for the electric field:
uz e
ri
i 6
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v z e
f
z e
ru
i i iep
6
Typical values of the electrophoretic mobility aresmall ions 5x10-8 m2V-1s-1
proteins 0.1-1x10-8 m2V-1s-1
F rictiona l drag r 6(Stokes Law)
r = hydrodynamicradius
Stokes-Einsteinequation
Reiger p. 97Sir George Gabriel Stokes 1819-1903
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Insert a sample calculation
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u epo
2
3
When particles are smaller than the Debye length you getThe following limit:
Remember: velocity is mobility x electric field
Reiger p. 98
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What controls the hydrodynamic radius?- the shear plane and ions around it
Compare the two equations for electrophoretic mobility
uf
epo o
2
3
uz e
rep
i 6
f z e
ro i
6
rz e
fi
o
6
Where f is a shape term which is 2/3 for sphericalparticles
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Relation of electrophoretic mobility to diffusion
DkT
f
kT
r
6
Thermal “force”
F rictiona l drag r 6
DkT
f
uz e
ri
i 6
DkT
f
kT
zeu electropho retic m igra tion
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Measuring Mobilities (and therefore Diffusion)from Conductance Cells
- +
+
+
++
++
+
-
-
-
- -
To make measurement need to worry about all the processesWhich lead to current measured
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Ac Voltage- +
O R-+
+
++
+
Charging
ElectronTransfer
Solution Charge Motion = resistance
--
-
--
- ++
R-O
Zf1 Zf2Rs
CtCt
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Z R
C
f c t
sC s
11
2
2
1
2
1
2
1
2
Electron transfer at electrode surface can be modeled as the Faradaic impedance, Z2
diffusion
Related to ket
An aside
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Zf1 Zf2Rs
CtCt
Solving this circuit leads to
RZ
Z
C
RZ
Z
C
R
Z C
R
Z C
Tf
f
t
sf
f
t
T
f t
s
f t
1
1
2
2
1 2
1 1
11 1
11 1
( ) ( )
Applying a high frequency, w, drops out capacitance and FaradaicImpedance so that RT=Rs
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What frequency would you have to useTo measure the solution resistance betweenTwo 0.5 cm2 in 0.1 M NaCl?
C
A
d
d
d
dspecific o
o
( )
z x C xm
( . )( ) .*3 2 9 1 0 1 0 4 1 017 1 / 2 7
C C A Aspecific o CheckCalculationTo show thatIt is cm converted to m
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C C A Aspecific o
C A xm
x cm xm
cmx
C
J mo
1 0 4 1 01
2 0 51 0 0
7 8 5 4 8 8 5 4 1 07 22
1 22
. . . .
C A xm
x cm xm
cmx
C
J mo
1 0 4 1 01
2 0 51 0 0
7 8 5 4 8 8 5 4 1 07 22
1 22
. . . .
C xC
Jx
C
C Vx
C
Vx F 7 2 1 0 7 2 1 0 7 2 1 0 7 2 1 07
27
27 7. . . . . . . .
The predicted capacitance of both electrodes in 0.1 M NaCl wouldBe 0.72 microfarads
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For the capacitive term to drop out of the electrical circuit We need:
11
1 1
7 2 1 01 4 1 0
76
C
C xx
t
t
.
.
The frequency will have to be very large.
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Solution Resistance Depends uponCell configuration
RA
length
A
Resistivity of soln.
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Sample calculation in a thin layer cell
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Resistance also depends upon the shapeOf an electrode
Disk Electrode Spherical electrode Hemisphericalelectrode
Ra
4a is the radius
Ra
4
Ra
2
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From Baranski, U. Saskatchewan
Scan rate 1000 V/s at two different size electrodes for Thioglycole at Hg electrode
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kR A
1
Conductivity is the inverse of Resistance
Resistivity and conductivity both depend uponConcentration. To get rid of conc. Term divide
kC C R C A
1
A plot of the molar conductivity vs Concentration has a slopeRelated to the measurement device, and an intercept related toThe molar conductivity at infinite dilution
m olar conductiv ity
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o s dard m olar conductiv ity tan
This standard molar conductivity depends upon the solutionResistance imparted by the motion of both anions and cations Moving in the measurement cell.
t
t
o
o
Where t is a transference number which accounts for the Proportion of charge moving
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TransferenceNumbers can beMeasured by capturingThe number of ionsMoving.
Once last number needsTo be introduced:The number of moles of ionPer mole of salt
o v v
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Compute the resistance of a disk electrodeOf 0.2 cm radius in a 0.1 M CaCl2 solution
o v v
oC a C l
mm ol
mm ol
mm ol
2 1 2 0 0 0 7 6 3 1 0 0 11 9 0 0 2 7 1 62 2 2
. . .
0 0 2 7 1 61 1
0 11 0
1 0 0
2
3 3
3.
.
m
m ol C m ol
L
L
cm
cm
m
1
0 0 2 7 1 6 0 11 0
1 0 00 3 6 8
2
3 3
3
. .
.m
m ol
m ol
L
L
cm
cm
m
m
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The resistance is computed from
Ra
m
cm xm
cm
4
0 3 6 8
4 0 20 1
4 6.
..
.
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Remember – we were trying to get to mobilityFrom a conductance measurement!!!!
uz F
i
oi
i
Also remember that mobility and diffusion coefficients are related
DkT
zeu
kT
ze zF
kT
z eFx
z
J m ol
Cio
io
io
27
2 22 6 6 1 0.
D xz
J m ol
Cio
2 6 6 1 0 7
2 2.
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We can use this expression to calculateDiffusion coefficients
D xz
J m ol
Cio
2 6 6 1 0 7
2 2.
D xx
m
m ol J m ol
Cx
m J
C3
7
42
2 21 0
2
22 6 6 1 0
3 0 2 7 1 0
38 9 2 1 0
.
.
( ).
m J
C V s
C
V C
J
m
s
2
2
2
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D xx
m
m ol J m ol
Cx
m
s4
7
42
2 21 0
2
2 6 6 1 04 4 2 1 0
47 3 4 1 0
.
( ).
Fe(CN)63- diffusion coefficient is 9.92x10-10 m2/s
Fe(CN)64- diffusion coefficient is 7.34x10-10 m2/s
The more highly charged ion has more solution solutes aroundIt which slows it down.
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How does this effect the rate of electron transfer?
k Zet e l
G
kT
ex p
Probability factor Collisional factor
ZkT
m~
2
1
2
Where m is the reduced mass.
Z is typically, at room temperature,104 cm/s
Activation energy
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G
G o
2
4
Free energy change
work required to change bondsAnd bring molecules together
in ou t
ou t
o D A D A op s
e
a a r
2
4
1
2
1
2
1 1 1
a donor rad ii
a accep tor rad ii
op tica l d ie lectric cons t
regu lar d ie lectric cons t
e electron ch e
D
A
op
s
tan
tan
arg
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G e E w wo o p r ( )
( )w w Uz z e e
a
e
aep r
ra p
a
D
a
A
rD AD A
2
04 1 1
Formal potential
Work of bringing ions together
When one ion is very large with respect to other (like an electrode)Then the work term can be simplified to:
( )w w U zep rr
The larger kappa the smaller the activation energy, the closerIons can approach each other without work