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
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Charged Surface
1. Cations distributed thermally
with respect to potential
2. Cations shield surface and
reduce the effective surface
potential
X=0
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Surface Potentials
Cation distribution has
to account for all species,
i
PoissonBoltzman equation
Charge near electrode depends
upon potential and is integrated
over distance from surface  affects
the effective surface potential
Dielectric constant of solution
Permitivity of free space
Solution to the PoissonBoltzman equation can be simple if the
initial surface potential is small:
Potential decays from the surface potential exponentially with distance
Because Y goes to zero as x goes to infinity
B must be zero
Because Y goes to Y0 as x goes to zero (e0 =1)
A must be Y0
thus
Potential decays from the surface potential exponentially with distance
When k=1/x or x=1/k then
The DEBYE LENGTH x=1/k
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Petrus Josephus
Wilhelmus Debye
18441966
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What is k?
Charged Surface
Y=0.36 Yo
X=1/k
X=0
Debye Length with distance
Units are 1/cm
SimeonDenis Poisson with distance
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Ludwig Boltzman
18441904
In the event we can not use a series approximation to solve the
PoissonBoltzman equation we get the following:
Check as
Compared to tanh
By Bard
Set up excel sheet ot have them calc effect with distance
Of kappa on the decay
Example Problem with distance
A 10 mV perturbation is applied to an electrode surface bathed in
0.01 M NaCl. What potential does the outer edge of a Ru(bpy)33+
molecule feel?
Debye length, x?
Units are 1/cm
Since the potential applied (10 mV) is less than 50 can use
the simplified equation.
Radius of Ru with distance
The potential the Ru(bpy)33+ compound experiences
is less than the 10 mV applied.
This will affect the rate of the electron transfer event
from the electrode to the molecule.
Surface Charge Density with distance
The surface charge distance is the integration over all the charge
lined up at the surface of the electrode
The full solution to this equation is:
C is in mol/L
Y with distanceo
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Charged Surface
Y=0.36 Yo
d
X=1/k
X=0
Can be modeled as a capacitor:
d
differential
For the full equation with distance
d
d
At 25oC, water
Differential capacitance
Ends with units of uF/cm2
Conc. Is in mol/L
Can be simplified if with distance
Specific Capacitance is the differential
space charge per unit area/potential
Specific Capacitance
Independent of potential
For small potentials
Flat in this region with distance
GouyChapman Model
Henrik Jensen with distance,David J. FermnandHubert H. Girault*
Received 16th February 2001 , Accepted 3rd April 2001
Published on the Web 17th May 2001
Real differential capacitance plots appear to roll off instead of
Steadily increasing with increased potential
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O. Stern
Noble prize 1943
Hermann Ludwig
Ferdinand von Helmholtz
18211894
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Linear drop
in potential
first in the
Helmholtz or
Stern specifically
adsorbed layer
Exponential
in the thermally
equilibrated or
diffuse layer
Charged Surface
X=0
x2
Cdiffuse
CHelmholtz or Stern
Capacitors in series with distance
Wrong should be x distance of stern layer
For large applied potentials and/or for large salt concentrations
1. 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
Capacitance
Due to Helmholtz
layer
Capacitance due to diffuse
layer
Deviation concentrations
Is dependent upon
The salt conc.
The larger the “dip”
For the lower
The salt conc.
Create an excel problem concentrations
And ask students to determine the smallest
Amount of effect of an adsorbed layer
Experimental data does not concentrations
Correspond that well to the
Diffuse double layer double capacitor
model
(Bard and Faulkner 2nd Ed)
Siv K. Si concentrationsandAndrew A. Gewirth*
Fig. 5 Capacitance�potential curve for the Au(111)/25 mM KI in DMSO interface with time.
Received 8th February 2001 , Accepted 20th April 2001
Published on the Web 1st June 2001
Model needs to be altered to account
For the drop with large potentials
This curve is pretty similar to predictions except where specific
Adsorption effects are noted
Graphs of these types were (and are) strong evidence of the specific
Adsorption of ions at the surface of electrodes.
Get a refernce or two of
deLevie here
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Introducing the Zeta Potential
Imagine a flowing solution
along this charged surface.
Some of the charge will be carried
away with the flowing solution.
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Charged Surface
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Introducing the Zeta Potential, given the symbo lz
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Shear Plane
Flowing solution
Charged Surface
Sometimes
assumed
zeta
corresponds
to Debye
Length, but
Not necessarily true
Yzeta
The zeta potential is dependent upon how the electrolyte specific
concentration compresses the double layer. a, b are constants
and sigma is the surface charge density.
Shear Plane can be talked about in specific
two contexts
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Shear Plane
Charged Surface
In either case if we “push” the solution along
a plane we end up with charge separation which
leads to potential
Shear
Plane
Particle in motion
Streaming Potentials specific
From the picture on preceding slide, if we shove the solution
Away from the charged surface a charge separation develops
= potential
Sample problem here specific
In the same way, we can apply a potential and move ions and specific
solution
Anode
Yo
Jm
Charged Surface
Vapp
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Jm
X=0
Cathode
Movement of a charged ion in an electric field specific
Electrophoretic mobility
The force from friction is equal to the electric driving force
The frictional drag comes
about because the migrating
ion’s atmosphere is moving
in the opposite direction, dragging
solvent with it, the drag is related to the ion atmosphere
Drag Force specific
Electric Force
Direction of Movement
Ion accelerates in electric field until the electric force
is equal and opposite to the drag force = terminal velocity
At terminal velocity specific
The mobility is the velocity normalized for the electric field:
Sir George Gabriel specific
Stokes 18191903
StokesEinstein
equation
r = hydrodynamic
radius
(Stokes Law)
Typical values of the electrophoretic mobility are
small ions 5x108 m2V1s1
proteins 0.11x108 m2V1s1
Reiger p. 97
Insert a sample calculation specific
When particles are smaller than the Debye length you get specific
The following limit:
Remember: velocity is mobility x electric field
Reiger p. 98
What controls the hydrodynamic radius? specific
 the shear plane and ions around it
Compare the two equations for electrophoretic mobility
Where f is a shape term which is 2/3 for spherical
particles
Relation of electrophoretic mobility to diffusion specific
Thermal “force”
Measuring Mobilities (and therefore Diffusion) specific
from Conductance Cells
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To make measurement need to worry about all the processes
Which lead to current measured
 specific
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Ac Voltage
Solution
Charge
Motion = resistance
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Charging
R
O
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R
Electron
Transfer
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Zf1
Zf2
Ct
Ct
Rs
An aside specific
diffusion
Related to ket
Electron transfer at electrode surface can be modeled as the
Faradaic impedance, Z2
C specifict
Ct
Rs
Zf1
Zf2
Solving this circuit leads to
Applying a high frequency, w, drops out capacitance and Faradaic
Impedance so that RT=Rs
What frequency would you have to use specific
To measure the solution resistance between
Two 0.5 cm2 in 0.1 M NaCl?
Check
Calculation
To show that
It is cm converted to m
The predicted capacitance of both electrodes in 0.1 M NaCl would
Be 0.72 microfarads
For the capacitive term to drop out of the electrical circuit
We need:
The frequency will have to be very large.
Resistance also depends upon the shape circuit
Of an electrode
Disk Electrode
Hemispherical
electrode
Spherical electrode
a is the radius
Scan rate 1000 V/s at two different size electrodes for circuit
Thioglycole at Hg electrode
From Baranski, U. Saskatchewan
Conductivity is the inverse of Resistance circuit
Resistivity and conductivity both depend upon
Concentration. To get rid of conc. Term divide
A plot of the molar conductivity vs Concentration has a slope
Related to the measurement device, and an intercept related to
The molar conductivity at infinite dilution
This standard molar conductivity depends upon the solution circuit
Resistance imparted by the motion of both anions and cations
Moving in the measurement cell.
Where t is a transference number which accounts for the
Proportion of charge moving
Transference circuit
Numbers can be
Measured by capturing
The number of ions
Moving.
Once last number needs
To be introduced:
The number of moles of ion
Per mole of salt
Compute the resistance of a disk electrode circuit
Of 0.2 cm radius in a 0.1 M CaCl2 solution
The resistance is computed from circuit
Remember – we were trying to get to mobility circuit
From a conductance measurement!!!!
Also remember that mobility and diffusion coefficients are
related
We can use this expression to calculate circuit
Diffusion coefficients
Fe(CN) circuit 63 diffusion coefficient is 9.92x1010 m2/s
Fe(CN)64 diffusion coefficient is 7.34x1010 m2/s
The more highly charged ion has more solution solutes around
It which slows it down.
How does this effect the rate of electron transfer? circuit
Activation energy
Collisional factor
Probability factor
Where m is the reduced mass.
Z is typically, at room temperature,
104 cm/s
Formal potential circuit
Work of bringing ions together
The larger kappa the smaller the activation energy, the closer
Ions can approach each other without work
When one ion is very large with respect to other (like an electrode)
Then the work term can be simplified to: