1.1
1 Potential Difference between two metals in a cell without liquid junction
2 Potential difference between two metals in a cell with liquid junction
3 4 5 1.2
• t
6 7 8 9 10 2.Introduction to potentiomentry
2.1. Galvani potential and electrochemical potential
Zero energy – charged particle in vacuum at infinite separation from a charged phase
Basic concepts _ 1. Electrochemical potential - μ
work done when a charged particle is transferred from infinite separation in vacuum to the interior of a charged phase
2. Chemical potential - μ work done when a charged particle is transferred from infinite separation in vacuum to the interior of a phase stripped from the charged surface layer 3. Inner potential ( Galvani potential ) - φ work done when a charged particle is transferred from infinite separation in vacuum across a surface shell which contains an excess charge and oriented dipoles 11 12 e =+
_ μμ φ The inner potential may be regarded as a sum of:
1) Potential created by the excess charge, the so called outer potential or Volta potential, denoted as Ψ. For a charge q on a sphere of radius a: q ψ = a
2. Potential created by oriented dipoles with dipole moment p referred to as surface potential, denoted as χ and equal to : χ = πNp /4 ε Therefore the inner potential is equal to:
φ =ψ +χ
13 Thermodynamic relationships
In a charged system the inner potential φ is an additional independent variable of the total Gibbs free energy:
G = f (p,T,nj,φ)
Basic thermodynamic definitions: _ 1. μ = (/)dG dn i ipTnn,,ji≠ ,φ
2. μ = (/)dG dn iip,,Tnji≠n ,φ = 0
Equilibrium condition: two phases α and β are in thermodynamic equilibrium when:
__ βα
= μμii 14 2.2 Electrochemical equilibria and electromotive force (emf) of a cell without liquid junction 2.2
2e
Note that vertical bars in the scheme of the cell denote a phase boundary 15 The cell at equilibrium
__ _ _ μμ+++=++2()2Cu μμμμμ− 2 2 2(Cu ') Zn2+ eClAg Zn AgCl e After rearrangement
⎛⎞__ ()(μμ2+−−−Zn μ222AgCl μμ − Ag − μ =− μ 2()(')⎜⎟eeCu − Cu ) Zn Cl ⎝⎠
Electromotive force of the cell _ recall μ e =−μφe F ⎛⎞__ −−=2()(')2()(')2⎜⎟μμeeCu Cu φφ F() Cu −= Cu FE ⎝⎠
16 Electromotive force of a cell
ECuCu= φ()−φ (')
or
()μμ2+−−−− μ(222 μμ ) Zn Zn − AgCl Ag Cl = E 22FF
()μ 2+ − μ ()222μμμAgCl−− Ag − E = Zn Zn E = Cl 1 2F 2 2F However this separation is formal or conventional in reality: μ − μμμ+−22 =−Δ=GFE 2 ZnCl2 Zn Ag AgCl 17 Examples of electrochemical cells and their emf
1. Cu’│Ag│AgCl│Cl-, Zn2+│Zn│Cu
()μμ2+−−−− μ(222 μμ ) E =−Zn Zn AgCl Ag Cl 22FF 2. Cu’│Ag│AgCl│Cl-, Mn2+│Mn│Cu
()μμ2+−−−− μ(222 μμ ) E =−Mn Mn AgCl Ag Cl 22FF 3. Cu’│Ag│AgCl│Cl-, Sn4+,Sn2+ │Pt│Cu
()μμ42++−−− μ(222 μμ −) E =−Sn Sn AgCl Ag Cl 22FF The second term in all three equations is the same hence we can write
E= Etest − const 18 Reference electrode convention
(222μμμ−−− ) AgCl Ag Cl = 0 2F
1. Cu’│Ag│AgCl│Cl-, Zn2+│Zn│Cu
(μ 2+ − μ ) E = Zn Zn 2F 2. Cu’│Ag│AgCl│Cl-, Mn2+│Mn│Cu
(μμ2+ − ) E = Mn Mn 2F
3. Cu’│Ag│AgCl│Cl-, Sn4+,Sn│Pt│Cu
(μμ42++− ) E = Sn Sn 2F Electrode potential – emf of a cell built up from a given test electrode and a reference electrode 19 Reference electrodes
+ H3O │H2│Pt Hydrogen electrode
- Cl │Hg2Cl2│Hg Calomel electrode
Cl-│AgCl│Ag Siver chloride electrode
Electrode sign convention
Write electrode reactions such that electrons flow through an external circuit from the reference to the test electrode e
Reference electrode , Test electrode ΔG>0 E<0
ΔG<0 E>0 20 21 22 o (μ 2+ − μ ) Eo = Zn Zn 2F or (μμo − ) Eo = OxRe d nF 23 24 25 2.3 Conductance, transference numbers, mobility, flux and current
Transference number
When current flows through a cell a fraction t+ is carried by cations and a fraction t- by anions such that:
tt+−+ = 1 or in general ∑ti =1 i
ti is the transference or transport number
Conductance
Conductance of an electrolyte equal to the reciprocal or resistance is given by: 1 κ A L == Rl Where κ is the conductivity, A is the area and l is the length of a segment of electrolyte conducting current 26 Ionic mobility
Ionic mobility ui is a limiting velocity of an ion in an electric field of unit field strength. Ionic mobility is measured in cm2 V-1 s-1 or (cm/s per V/m). It is related to the ionic conductivityκ by the formula:
κ = F∑ zuCiii i
The transference number of ion i is the contribution to conductivity made by that Ion divided by the total conductivity:
κ zuC t ==i iii i κ ∑ zuCj jj j
27 28 Equivalent conductivity Defined by: κ Λ= Cz
since and κ = F∑ zuCiii κiiii= FzuC i
Λ =+F()uu+ − or
Λ =+()λ+ λ−
where λii= Fu is the equivalent conductivity of individual ion 29 In the case of a single electrolyte :
λ t = i i Λ
or ui ti = uu+ + −
or zCiiiλ ti = ∑ zCj jjλ j
30 2.4 Electrochemical cell with a liquid junction potential
ECuCuZnCu=−φ()φφφ (')() ={ − (')()()} +−{ φαφφβφαφφβ Zn} +−+{ ()()} { ()Cu − ()}
{φβ() φα−= ()} ELJ
31 Different ways to make the liquid junction potential
32 Boundary between two electrolyte solutions
MA
M+, A-
33 Liquid junction potential between two different solutions of the same electrolyte
34 Example: concentration cell
35 36 Henderson equation
Activity coefficients independent of concentration and the liquid junction a continuous mixture of the two electrolytes
37 How to eliminate liquid junction potential? Consider the following interface
uC−− uC AA AA−− t − = t = A A− uC− −−−+++++++ u C uC u C uC+++ u C uC u C A A Cl Cl K K M M AA− −−−++++ ClClKK M M
uC+ uC>> uC + uC assume that CCKCl>> MA then Cl−− Cl K ++ K A −− A M ++ M
in predominant fraction of the liquid – junction region: u u AM− ⎛⎞C A MM+ ⎛⎞C A t − = ⎜⎟ 0 and t + = ⎜⎟ 0 A uuC+ M uuC+ Cl−+ K ⎝⎠KCl Cl−+ K ⎝⎠KCl 38 Consequently:
11ββt Edt=− i μμμ=− dt− d LJ ∫∫∑ i ()K++ K Cl − Cl − Fzααi i F 1 To a good approximation ddμ ==μμ d and since KCl+−2 KCl tt≈ K + Cl−
11ββ1 Etdtdttd=−−=−−≈μμ μ0 LJ ∫∫()KKClCl++ − − ()KCl+ −KCl FFαα2
39 Elimination of the liquid junction potential by a salt bridge
40 2.5 Donnan equilibrium and Donnan potential( IUPAC definitions)
41 Donnan equilibrium and Donnan potential
42 2.6 Selectively permeable membranes
C+ C+(p)
C+(q)
C1 Cx
C C 2 C-(p) -
1 C-(q) 2
pqx
Δ=φφ(2)(1)(2)()()() φ − φ φ φ ={ φ φ φ −qqpp} +{ −} +{ ()(1) − }
Δφ =ΔφφφDL(2) +ΔJD +Δ (1) 43 or AgBr
44 45 46 S1 S2
K + + + + H (s1) + Napp(p) = H (p) + Na (s1) + + + + H (s2) + Na (q) = H (q) + Na (s2)
57 A B
A
B
48 49 valinomicine
50 51 52 2.8. Membrane potential – overview
)
53 54 }
55 56 2.8
57 58 59 60 Review
1. Solid state membrane ( AgCl pelet)
ΔφLJ = 0
2. Membrane with dissolved ion exchanger
ΔφLJ = 0 3. Membrane with a fixed ion exchanger
ΔφLJ ≠ 0
61 General expression for the membrane potential
⎧ zi ⎫ RT ⎪ pot z j ⎪ EMM=Δφ =ln ⎨aKa iijj + ⎬ Fz i ⎩⎭⎪ ⎪
62 2 ICz= 0.5∑ ii i
Ci 63 t
64 2.9 Quantitative analysis – Standard addition method
Standard addition method: 1) Use an excess of an “inert electrolyte” (0.15M NaCl in Ca2+ analysis) to keep the ionic strength constant (γ = constant). 2) Take the measurement of the unknown: RT EE10=+ ln CAγ FzA 3) Spike unknown with a standard solution ( volume Vs, , concentration Cs )
RT ⎛⎞CVA Ass+ CV EE21=+Δ=+ EE 0 ln ⎜⎟γ FzAAs⎝⎠ V+ V
4) Calculate the difference E21− EE=Δ RT ⎛⎞CV+ CV zFA Δ E CV+ CV Δ=E ln ⎜⎟A Ass or 10 2.3RT = A Ass Fz C() V+ V AAAs⎝⎠ CVA ()As+ V and after rearrangement Cs CA = zFA Δ E⎛⎞VV 102.3RT 1+−A A ⎜⎟ 65 ⎝⎠VVs s 66 67 ISE – sources of errors
68 2.11 Biosensors
69 b) Tissue electrode E/mV
-log C(glutamine)
70 c) Bacterial electrode
71 d) Chemically modified field effect transistor
pn-junction
72 Field effect transistor
73 Chemically modified field effect transistor
74 3. Introduction to electrometric methods
3.1 Structure of the metal interface neutrality condition
σ M =−(σσid + )
interface – two capacitors in series
111 =+ CCCid σ
OHP- outer Helmholtz plane; distance of the closest approach of solvated ions IHP – inner Helmholtz plane; plane in which specifically adsorbed ions are located Diffuse layer – layer within which ionic cloud screening the charge on the metal is located 75 Capacity of an Au electrode adsorbed anions
60
50 Nonadsorbing electrolyte
40 2
30 F/cm μ
C 20
10 adsorbed neural organic molecules 0 -1.2 -0.8 -0.4 0.0 0.4 E V vs. Ag/AgCl 76 Capacity of an electrode changes with potential – the changes reflect changes in the Coverage by specifically adsorbed ions or molecules
σ M = fE(,)Γ
⎛⎞∂∂σσMM ⎛⎞ ddσ M = ⎜⎟Ed+Γ ⎜⎟ ⎝⎠∂∂E Γ ⎝⎠ΓE
dσσMM⎛⎞⎛⎞∂∂ σ MdΓ C ==⎜⎟⎜⎟ + dE⎝⎠⎝⎠∂∂Γ E Γ E dE }
true capacity pseudo capacity
77 Potential drop across the metal solution interface no specific adsorption specific adsorption of anions
78 3.2 Electrochemical cell in the presence of a current flowing through the cell Two electrode cell
79 when potential E is applied from an external source current flows through the cell
From Ohm’s Law: E i = Z
ZZZ= ie++ ZRe f + Z ext
Zi – impedance of the indicator electrode Ze – impedance of the electrolyte ZRef – impedance of the reference electrode Zext – impedance of the external circuits 80 In an electrochemical experiment Ze, ZRef, Zext must be eliminated a. Elimination of ZRef ZfRC= (, )
ρ - specific resistivity; l l - length; ZRRe f ==ρ A A -area 1 Z = C - capacity C ωC ω = 2πν -angular frequency −1 CA∝ hence ZA∝
Consequently if : AAiR<< efthen ZZiRe>> f The surface area of an indicator electrode should be much smaller than the surface area of the reference or auxiliary electrodes 81 b) Elimination of Ze
Zee= R Use 3-electrode system and Luggin capillary
82 Luggin capillary and uncompensated resistance Ru
x For a planar electrode : R = ρ u A
1 ⎛⎞x For a spherical electrode: R u = ρ ⎜⎟ 4π rr00⎝⎠+ x
r0 - radius of the spherical electrode; x - distance from the electrode surface
83 Change of potential across the capacitor as a function of time q EiR=+ Cd t ∫idt q 0 Ec == CCdd
since: t E − ie= RCs d Rs E across the capacitor one obtains: C
tt t E t −−⎛⎞ EedtEe==−RCsd ⎜⎟1 RCsd c RC ∫ ⎜⎟ sd0 ⎝⎠ 95 Linear potential sweep or voltage ramp Evt= where v is the sweep rate in V s-1 q EiR=+ C or d dq q vt=+ Rs dt Cd if q=0 at t=0
t ⎛⎞− ivC=−⎜⎟1 eRCsd d ⎜⎟ ⎝⎠
if t>>RsCd
ivC= d 96 Cyclic-linear potential sweep E = vt
t ⎛⎞− ivC=−⎜⎟1 eRCsd d ⎜⎟ ⎝⎠
97 4.1 Electrode reaction as a series of multiple consecutive steps A charge transfer reaction provides an additional channel for current to flow through the interface. The amount of electricity that flows through this channel depends on the amount of species being oxidized or reduced according to the Faraday law: M m = A it nF Faraday law links electrical quantity such as current to the chemical quantity such as concentration or mass of the analyte:
mCVM= A This expression may be rearranged to give expression for current: m N nFi == nF AtM t n- number of electrons in a redox reaction, N-number of moles, M - molecular weight, F- Faraday constant, C-concentration, V-volume A 99 Currents which are described by the Faraday law are called faradaic currents m N nFi == nF AtM t This equation described average current flowing through the electrode During time – t. During infinitesimal period dt the number of electrolyzed moles is dN and the expression for the instantaneous current is: = / dtnFdNi
The rate of a chemical reaction is : = / dtdNv Hence faradaic current is a measure of a reaction rate dN nFi == nFv dt 100 Current flowing in the absence of a redox reaction – nonfaradaic current
In the presence of a redox reaction – faradaic impedance is connected in parallel to the double layer capacitance. The scheme of the cell is:
The overall current flowing through the cell is :
i = if + inf
Only the faradaic current –if contains analytical or kinetic information 98 Non-faradaic ( capacitive ) current constitutes a limitation to the detection limit of electrometric techniques
Experimental conditions for electroanalytical measurements:
1. ii>> ( large concentration of electroactive species, f non long times of experiment)
( correction for non-faradaic current) 2. iiif = − non
are different functions of time i f and inon
Under a steady state condition:
but i f ≠ 0 inon = 0 Current – voltage characteristics of an ideal polarizable and ideal non-polarizable electrode E i = Z Z =∞ f Z f = 0
90
For a multistep reaction
102
* ⎡ x ⎤ Cxt(,)= Cerf⎢ 1/2 ⎥ ⎣2(Dt ) ⎦
6.
se 6.1
6.3 Review
response From limiting chronoamperometric curves – analytical information 7.1 Polarography
7.2 Tast Polarography
1 1 − 6 3 itd ∝ itc ∝
Detection limit 7.3 Pulse Polarography 1 1 22 2 ⎛⎞7 * inFDCAt= ⎜⎟ 2 / π A = 0.85mt33 ⎝⎠3 1 1 22 2 ⎛⎞7 * Polarography inFDCmtt= ⎜⎟ 2 0.8533 / π ⎝⎠3 1 1 22 2 Pulse Polarography ⎛⎞7 *' inFDCmttt= ⎜⎟ 2 0.8533 /π (− ) ⎝⎠3 Schematics of a set up for pulse polarography Detection limit 10-7M 7.4 Differential pulse polarography Current is sampled twice a time t and t’. Quantity measured is a difference ∂i : δiitit= ()− (') Differential pulse Differential pulse δi δii=+δδdc i
δiititcc=−() c (')
Normal pulse Normal pulse iitt= dc(')()−+ it i
at peak potential
δiittdd (')− but
δiitcc<< () Detection limit of differential pulse polarography ~10-8M
7.5 Striping voltammetry
Electroanalytical techniques – Figures of merits
8. Linear sweep voltammetry
Perturbation – linear voltage sweep ( voltage ramp)
E = Evti −
v sweep rate in V/s Qualitative prediction of the shape of a CV curve – chronoamperometric experiments
There is no analytical solution to this problem – the equation has been solved numerically
The shape of the CV curve is determined by the shape of numerical function:
Cyclic voltammetry
-5 •DO = DR = 1 x 10 cm/s