The Determination of The Determination of Consecutive Formation Consecutive Formation

Constants of Complex Ions Constants of Complex Ions from Polarographic Datafrom Polarographic Data

Paper by: Donald D. DeFord and David Paper by: Donald D. DeFord and David N. HumeN. Hume

Presentation by: Molly FinsterPresentation by: Molly Finster

02/04/9902/04/99

ContentsContents

IntroductionIntroduction Experimental TechniqueExperimental Technique Principal ReactionsPrincipal Reactions Experimental ResultsExperimental Results Fundamental EquationsFundamental Equations Current ApplicationCurrent Application ConclusionsConclusions QuestionsQuestions

IntroductionIntroduction

This paper describes a method of This paper describes a method of mathematical analysis of the shift of mathematical analysis of the shift of half-wave potential with ligand half-wave potential with ligand concentrationsconcentrations

This analysis makes possible:This analysis makes possible:•the identification of the successive the identification of the successive complex ions formed between a metal and complex ions formed between a metal and a liganda ligand•the evaluation of the corresponding the evaluation of the corresponding formation constantsformation constants

Experimental TechniqueExperimental Technique

Principles of Principles of voltammetryvoltammetry• a potential is applied between a a potential is applied between a

reference and working reference and working electrode for a definite timeelectrode for a definite time

• a redox reaction takes place on a redox reaction takes place on the working electrodethe working electrode

• the electrons exchanged the electrons exchanged produce a current that is produce a current that is measured by an auxiliary measured by an auxiliary electrodeelectrode

Voltammetry is utilized to obtain the Voltammetry is utilized to obtain the necessary polarographic data, which is a necessary polarographic data, which is a measure of current vs. potentialmeasure of current vs. potential

Principal ReactionsPrincipal Reactions

The reduction of a metal complex to the metallic The reduction of a metal complex to the metallic state (amalgam) at the working electrode:state (amalgam) at the working electrode:

elec

trod

e

MLn

M+z + nL-z/n

MLn

M+z + nL-z/n

diffusiondissociation

Solution

ze- + M+z

redox

M

reaction layerdiffusion layer

compact layer

Experimental ResultsExperimental Results

Typical results:Typical results:

• As reduction commences, the current increasesAs reduction commences, the current increases• After a steep increase in current, the current is limited After a steep increase in current, the current is limited

by the rate at which the metal ions can diffuse from the by the rate at which the metal ions can diffuse from the solution to the working electrode surfacesolution to the working electrode surface

• The magnitude of the diffusion current is directly The magnitude of the diffusion current is directly proportional to the analyte concentration in solutionproportional to the analyte concentration in solution

Half-wave Potential

E1/2

Diffusion Current, Id

I

E

Fundamental EquationsFundamental Equations

Electrical PotentialElectrical Potential- assuming the - assuming the electrode reactions are reversible:electrode reactions are reversible:

Formation ReactionFormation Reaction- assuming the - assuming the formation of the complexes is rapid and formation of the complexes is rapid and reversible:reversible:

MM

aaae C

C

nF

RTEE

0

00 ln

iL

iLMMjMLjMLj CCKC )()(

Fundamental Equations Fundamental Equations (con’t)(con’t)

Summing the formation reactions for the Summing the formation reactions for the individual complexes and rearranging:individual complexes and rearranging:

CurrentCurrent- concentrations of the complex - concentrations of the complex and metal amalgam can be directly and metal amalgam can be directly related to the current:related to the current:• diffusion current:diffusion current: electrode current:electrode current:

j MLj

jL

jLMMj

jMLj

MM CCK

C

C

)()(

0

0

j

MLjcd CkIi 0aMe CkIi

Fundamental Equations Fundamental Equations (con’t)(con’t)

Substituting the above equations in to Substituting the above equations in to the electrical potential, one gets the the electrical potential, one gets the half-wave potentials of:half-wave potentials of:• the reducible ion in the presence of the the reducible ion in the presence of the

complex forming ligand:complex forming ligand:

• the metal ion (assuming the metal ion (assuming =1):=1):

j MLj

jL

jL

M

cac

Ck

I

I

nF

RTEE

)()(

ln)( 0

21

M

sas I

I

nF

RTEE ln)( 0

21

Fundamental Equations Fundamental Equations (con’t)(con’t)

Finally, by combining and rearranging Finally, by combining and rearranging the two half-wave potential equations, the two half-wave potential equations, one can get an expression consisting of one can get an expression consisting of experimentally measurable quantities:experimentally measurable quantities:

The the first derivative of The the first derivative of F0(X) with with respect to respect to CLL is given by: is given by:

c

scs

j MLj

jL

jLj

I

IEE

RT

nFCKXF log)()(435.0log)( 0

21

0

21

10

LL

s

C

KXF

XF

00

1

)(

)(

Current ApplicationCurrent Application

Luther et al. Luther et al. (1996)(1996)

Similar to Similar to before, the before, the n+1n+1th th derivative derivative of of F0(X) is:is:

Or:Or:

LL

s

nn

n C

KXF

XF

)(

)(1

s

nLLnn

KCXFXF

)()( 1

ConclusionsConclusions

The mathematical form of the F-The mathematical form of the F-Functions is very useful in:Functions is very useful in:• establishing the number of complexes establishing the number of complexes

formed in a given systemformed in a given system• providing a qualitative check on the validity providing a qualitative check on the validity

of the dataof the data• determining the consecutive formation determining the consecutive formation

constants of metal-ligand complexesconstants of metal-ligand complexes

QuestionsQuestions

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