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8. Fundamentals of Charged Surfaces. Moving the reagents Quickly and with Little energy Diffusion electric fields. +. +. +. +. Y o. Y* o. Charged Surface. 1. Cations distributed thermally with respect to potential 2. Cations shield surface and reduce the effective surface - PowerPoint PPT Presentation
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8. Fundamentals of Charged Surfaces
Moving the reagentsQuickly and with Little energy
Diffusionelectric fields
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
o
Cha
rged
S
urfa
ce
+
+
+
+
X=0
N
N o
G
kT
*
ex p
o
+
+
+
dx dx
o
* ** dx
+
+
***
o
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
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
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
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
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/
o
Cha
rged
S
urfa
ce
=0.36 o+
+
+
+
+
X=0 X=1/
+
+
+
+
What is
Petrus Josephus Wilhelmus Debye1844-1966
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
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
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
Set up excel sheet ot have them calc effectOf kappa on the decay
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
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
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
o
Cha
rged
S
urfa
ce
=0.36 o+
+
+
+
+
X=0 X=1/
+
+
+
+
Can be modeled as a capacitor:C
d
ddifferential
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
0
2000
4000
6000
8000
10000
12000
-15 -10 -5 0 5 10 15
y x co sh
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
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
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
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
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
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
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
Create an excel problemAnd ask students to determine the smallestAmount of effect of an adsorbed layer
Experimental data does notCorrespond that well to the Diffuse double layer double capacitormodel
(Bard and Faulkner 2nd Ed)
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
This curve is pretty similar to predictions except where specificAdsorption effects are noted
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
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.
Introducing the Zeta Potential, given the symbo l
oC
harg
ed
Sur
face
+
+
+
+
+
+
+
+
+
Shear Plane
Flowing solution
zeta
Sometimesassumedzeta correspondsto DebyeLength, butNot necessarily true
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.
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
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
Sample problem here
Reiger- streaming potentialapparatus.
Can also make measurements on blood capillaries
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
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
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
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
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
Insert a sample calculation
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
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
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
Measuring Mobilities (and therefore Diffusion)from Conductance Cells
- +
+
+
++
++
+
-
-
-
- -
To make measurement need to worry about all the processesWhich lead to current measured
Ac Voltage- +
O R-+
+
++
+
Charging
ElectronTransfer
Solution Charge Motion = resistance
--
-
--
- ++
R-O
Zf1 Zf2Rs
CtCt
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
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
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
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
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.
Solution Resistance Depends uponCell configuration
RA
length
A
Resistivity of soln.
Sample calculation in a thin layer cell
Resistance also depends upon the shapeOf an electrode
Disk Electrode Spherical electrode Hemisphericalelectrode
Ra
4a is the radius
Ra
4
Ra
2
From Baranski, U. Saskatchewan
Scan rate 1000 V/s at two different size electrodes for Thioglycole at Hg electrode
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
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
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
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
The resistance is computed from
Ra
m
cm xm
cm
4
0 3 6 8
4 0 20 1
4 6.
..
.
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.
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
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.
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
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
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