Electrochemistry for Engineers LECTURE 4 Lecturer: Dr. Brian Rosen Office: 128 Wolfson Office Hours: Sun 16:00

THE SURFACE POTENTIAL IS NOT THE SAME AS THE STANDARD REDUCTION POTENTIAL!

ELECTRODE POTENTIAL (CAN BE CONTROLLED)

Potential across Full Cell CATHODE ELECTROLYTE ANODE +E -E

The Electrical Double Layer

Review: RC Circuits When a capacitor is charged in a circuit, it acquires a potential difference between the plates. A typical RC circuit: S chg is on at t = 0, the charge on the capacitor is time dependent: q(t) chg = Q o (1- e -t/RC ) = Q o (1 - e -t/  )..where (Q o = EC)

A RC circuit for charging a capacitor in a single “charge” step: The charge - time profile: q(t) = Q o (1-e -t/RC ), at R =10 , C = 20  F, Q 0 = 1 mC, E = 50 V The Time Constant, 

At t = , the capacitor is charged to 0.63 (63%) of its final value (Q o ), q(t) = Q o (1-e -1 ) = 0.63Q o. At a time equal to 3 , the capacitor is charged to 95% of Q o. Upon charging to the voltage E after the switch (S chg ) is on, the current in the resistor R decreases, and eventually stops. When the capacitor is discharged, the current in the resistors starts at E/R and then decreases to zero: q(t) dis = Q o e -t/RC = Q o e -t/ 

At , the charge on the capacitor is reduced to 0.37 of the initial value; the discharging curve: q(t) = Q o e -t/RC : For charging (back to):

(a) charge vs. time, and (b) current in time. Circuit parameters are: R =10 , C= 20  F. [Q 0 = 20  C,  = 10  x 20  F = 0.2 ms]. a b = E/R. e -t/RC i (total) = i (faradaic) + i (non-Faradaic) From Cottrell From DL charging Therefore, in a POTENTIAL STEP EXPERIMENT

i (total) = i (faradaic) + i (non-Faradaic) From Cottrell From DL charging Therefore, in a POTENTIAL STEP EXPERIMENT = E/R. e -t/RC Since the non-faradaic current contributes to our experimental signal, we must figure out how to model C, the capacitance, of an electrochemical system! To do this, we must be able to model the interface! Which model should we use?

electric charge resides on a surface of an electronic conductor, e.g. metal (in an electrochemical cell) The Helmholtz model: a counterion resides at the surface 2 sheets of charge densities (  ) of opposite sign, separated by a distance of molecular order gives the “double layer” (dl) I. The Helmholtz Model  is the stored charge density,  is the dielectric constant of the medium,  0 is permittivity of the free space, V is the voltage drop between the plates. Charge density in units of charge/area

I.The Helmholtz Model VISUALIZED METAL (M) SOLUTION (S)

predictions (incorrect): no bulk concentration dependence of C d : The differential capacity (at constant parameters): I.Helmholtz Model - Problems

Interplay of two tendencies at the metal solution interface: - the tendency of the electrostatic interactions between ions and the charged plane - the tendency of thermal motions The result: the idea of a diffuse layer used a statistical mechanical approach to the diffuse dl description The model did much better (than the Helmholtz (H) model) but did not yield a satisfactory description of real system II. Gouy and Chapman

a solution sub-divided into laminae, parallel to the electrode and of thickness dx, all laminae are in thermal equilibrium with each other ions of any species i are not at the same energy in the various laminae, because the electrostatic potential f varies Components of the G-C theory

Hence: the number concentrations of species in two laminae has a ratio determined by a Boltzmann factor take a reference lamina far from the electrode, where every ion is at its bulk concentration n i 0 then the population in any other lamina is: e is charge of electron in C Φ is potential relative to bulk electrolyte k is boltzman constant z is sign charge

II. Gouy and Chapman VISUALIZED METAL (M) SOLUTION (S)

 is the potential at a given point with respect to the bulk solution, e is the electronic charge, k is the Boltzmann constant, T is the absolute temperature T, and the z i is the ion charge. The total charge per unit volume in any lamina is then: i runs over all ionic species.  (x) is related to the potential at distance x by the Poisson equation: By combining these equations, one obtains the Poisson-Boltzmann equation: Substituting the property:

=,, gives: Separate and integrate: gives: far from electrode the field is zero

Half angle relation: Symmetric electrolytes, n i 0 (anions) = n i 0 (cations) = n 0, the number concentration of each ions: “2:2”, “1:1”, z:z Substituting:, and using the number concentration of each ion in the bulk, n 0. Hyperbolic cosine: Anions Cations

A square root, both sides: : the electric field or the field strength; the gradient of potential at a distance x.

After separating d  and dx and integrating over    0 and 0  x Where  is a constant at constant T; For dilute aqueous solutions at 25 °C, ɛɛ 0 = 78.4, Boltzman K is x J/K :  = (3.29 x 10 7 )zC* 1/2 where C is in mol/L and  is in cm -1. At small  0 (<50/z mV at 25 °C):  =  0 e -  x 1/  : the characteristic thickness of the diffuse double layer

At small  0 (<50/z mV at 25 °C):  =  0 e -  x 1/  : the characteristic thickness of the diffuse double layer

One More Step! How to get Capacitance? Electric field = = 0 for all surfaces Except the electrode itself. On the surface = (dɸ/dx) at x=0 (notes to be posted)

Recall from earlier:

Still not completely correct Guoy ChapmanReal Systems

III. The Stern Model Stern combined the H and G-C theory and offered the most complete model for electrical dl without surface phenomena: chemisorption (specific adsorption), surface oxidation, reconstruction, relaxation, phase transitions, etc..

The inner layer: the compact Helmholtz or Stern layer; the inner Helmholtz plane (IHP), at the distance x 1. Specifically adsorbed ions. Solvated ions can approach the metal at the distance x 2 (the locus of the center of such non specifically adsorbed ions is called OHP).