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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

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8 fundamentals of charged surfaces

Moving the reagents

Quickly and with

Little energy

Diffusion

electric fields


8 fundamentals of charged surfaces

+

+

+

+

Yo

Y*o

Charged Surface

1. Cations distributed thermally

with respect to potential

2. Cations shield surface and

reduce the effective surface

potential

X=0


8 fundamentals of charged surfaces

+

+

+

+

+

+

+

+

+

Yo

Y*o

Charged Surface

Y**o

Y***o

X=0

*

**

dx

***

dx

dx


8 fundamentals of charged surfaces

Simeon-Denis Poisson

1781-1840

Surface Potentials

Cation distribution has

to account for all species,

i

Poisson-Boltzman 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


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

Largest term the

Let

Then:


8 fundamentals of charged surfaces

General Solution of: the

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


8 fundamentals of charged surfaces

Potential decays from the surface potential exponentially with distance

When k=1/x or x=1/k then

The DEBYE LENGTH x=1/k


8 fundamentals of charged surfaces

+ with distance

+

+

+

+

+

+

+

+

Petrus Josephus

Wilhelmus Debye

1844-1966

Yo

What is k?

Charged Surface

Y=0.36 Yo

X=1/k

X=0


8 fundamentals of charged surfaces

Debye Length with distance

Does not belong

=1/cm

Units are 1/cm


8 fundamentals of charged surfaces

Debye Length with distance

Units are 1/cm


8 fundamentals of charged surfaces

Simeon-Denis Poisson with distance

1781-1840

Ludwig Boltzman

1844-1904

In the event we can not use a series approximation to solve the

Poisson-Boltzman equation we get the following:

Check as

Compared to tanh

By Bard


8 fundamentals of charged surfaces

Set up excel sheet ot have them calc effect with distance

Of kappa on the decay


8 fundamentals of charged surfaces

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.


8 fundamentals of charged surfaces

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.


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

Y with distanceo

+

+

+

+

+

+

+

+

+

Charged Surface

Y=0.36 Yo

d

X=1/k

X=0

Can be modeled as a capacitor:

d

differential


8 fundamentals of charged surfaces

For the full equation with distance

d

d

At 25oC, water

Differential capacitance

Ends with units of uF/cm2

Conc. Is in mol/L


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

Flat in this region with distance

Gouy-Chapman Model


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

+ with distance

+

+

+

+

+

+

+

+

O. Stern

Noble prize 1943

Hermann Ludwig

Ferdinand von Helmholtz

1821-1894

Yo

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


8 fundamentals of charged surfaces

Capacitors in series with distance

Wrong should be x distance of stern layer


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

Deviation concentrations

Is dependent upon

The salt conc.

The larger the “dip”

For the lower

The salt conc.


8 fundamentals of charged surfaces

Create an excel problem concentrations

And ask students to determine the smallest

Amount of effect of an adsorbed layer


8 fundamentals of charged surfaces

Experimental data does not concentrations

Correspond that well to the

Diffuse double layer double capacitor

model

(Bard and Faulkner 2nd Ed)


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

This curve is pretty similar to predictions except where specific

Adsorption effects are noted


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

+ specific

+

+

+

+

+

+

+

+

Introducing the Zeta Potential

Imagine a flowing solution

along this charged surface.

Some of the charge will be carried

away with the flowing solution.

Yo

Charged Surface


8 fundamentals of charged surfaces

+ specific

+

+

+

+

+

+

+

+

Introducing the Zeta Potential, given the symbo lz

Yo

Shear Plane

Flowing solution

Charged Surface

Sometimes

assumed

zeta

corresponds

to Debye

Length, but

Not necessarily true

Yzeta


8 fundamentals of charged surfaces

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.


8 fundamentals of charged surfaces

Shear Plane can be talked about in specific

two contexts

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Yo

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


8 fundamentals of charged surfaces

Streaming Potentials specific

From the picture on preceding slide, if we shove the solution

Away from the charged surface a charge separation develops

= potential



8 fundamentals of charged surfaces

Reiger- streaming potential specific

apparatus.

Can also make measurements on blood capillaries


8 fundamentals of charged surfaces

In the same way, we can apply a potential and move ions and specific

solution

Anode

Yo

Jm

Charged Surface

Vapp

+

+

+

+

+

+

+

+

+

+

Jo

Jm

X=0

Cathode


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

At terminal velocity specific

The mobility is the velocity normalized for the electric field:


8 fundamentals of charged surfaces

Sir George Gabriel specific

Stokes 1819-1903

Stokes-Einstein

equation

r = hydrodynamic

radius

(Stokes Law)

Typical values of the electrophoretic mobility are

small ions 5x10-8 m2V-1s-1

proteins 0.1-1x10-8 m2V-1s-1

Reiger p. 97



8 fundamentals of charged surfaces

When particles are smaller than the Debye length you get specific

The following limit:

Remember: velocity is mobility x electric field

Reiger p. 98


8 fundamentals of charged surfaces

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



8 fundamentals of charged surfaces

Measuring Mobilities (and therefore Diffusion) specific

from Conductance Cells

+

-

+

-

+

+

+

-

-

-

+

+

+

-

To make measurement need to worry about all the processes

Which lead to current measured


8 fundamentals of charged surfaces

- specific

+

Ac Voltage

Solution

Charge

Motion = resistance

+

Charging

R-

O

+

+

-

R-

Electron

Transfer

O

-

+

-

-

-

+

-

+

+

Zf1

Zf2

Ct

Ct

Rs


8 fundamentals of charged surfaces

An aside specific

diffusion

Related to ket

Electron transfer at electrode surface can be modeled as the

Faradaic impedance, Z2


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

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



8 fundamentals of charged surfaces

For the capacitive term to drop out of the electrical circuit

We need:

The frequency will have to be very large.


8 fundamentals of charged surfaces

Solution Resistance Depends upon circuit

Cell configuration

A

length

Resistivity of soln.



8 fundamentals of charged surfaces

Resistance also depends upon the shape circuit

Of an electrode

Disk Electrode

Hemispherical

electrode

Spherical electrode

a is the radius


8 fundamentals of charged surfaces

Scan rate 1000 V/s at two different size electrodes for circuit

Thioglycole at Hg electrode

From Baranski, U. Saskatchewan


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

Compute the resistance of a disk electrode circuit

Of 0.2 cm radius in a 0.1 M CaCl2 solution



8 fundamentals of charged surfaces

Remember – we were trying to get to mobility circuit

From a conductance measurement!!!!

Also remember that mobility and diffusion coefficients are

related


8 fundamentals of charged surfaces

We can use this expression to calculate circuit

Diffusion coefficients


8 fundamentals of charged surfaces

Fe(CN) circuit 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 around

It which slows it down.


8 fundamentals of charged surfaces

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


8 fundamentals of charged surfaces

Free energy change circuit

work required to change bonds

And bring molecules together


8 fundamentals of charged surfaces

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: