Computer simulation of metal complex dissociation duringfree metal determination using anodic stripping

voltammetryp

Shu Taoa, *, Kin.Che Lamb, Jun Caoa, Bengang Li a

aDepartment of Urban and Environmental Sciences, Peking University, Beijing 100871, People's Republic of ChinabDepartment of Geography, The Chinese University of Hong Kong, Hong Kong, People's Republic of China

Received 22 April 1998; received in revised form 5 June 1998

Abstract

As a most widely used technique, anodic stripping voltammetry often provides overestimated results of free metal

concentration or underestimated values of complexation capacity. A conceptual mode was proposed suggesting thatthe dissociation of the metal complex occurring in the di�usion layer close to the electrode surface during thedeposition period contributes to the measurement error. A computer program was coded to simulate the

transformation and mass transfer of metal ions, free ligands, and metal±ligand complexes in the di�usion layer. Itwas con®rmed that the dissociation of the complexes within the di�usion layer causes the overestimation error forfree metal determination using the anodic stripping voltammetry technique. The magnitude of the overestimation

depends on the complex dissociation coe�cient, concentration of the ligand, and transfer rate constant of the freemetals. The results of the computer simulation were partially con®rmed by experimental data based on a Cu-EDTAsystem. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: ASV; Free metal; Computer simulation; Di�usion layer; Complexation capacity

1. Introduction

Trace metals may exist in a variety of physicochem-

ical forms in natural waters. The speciation of trace

metals in aquatic systems is of great signi®cance to

their mobility, bioavailability, and toxicity. Although

the concentrations of free ionic metals in natural

waters are usually very low at sub-ppb levels, they are

the most important forms among all species (Larsen et

al., 1991). In general, complex forms of metals appear

to be non-toxic or, at least, considerably less toxic

than free metals ions (Tao et al., 1998). For ecotoxicity

evaluation of natural water, information on the con-

tents of major individual species is necessary in ad-

dition to the total contents of all species.

Another important application of free metal

measurement is to determine complexation capacity,

which is a measure of the total ligands available

(Petersen, 1982). Natural water can reduce the toxic

e�ects of trace metals notably due to complexation by

all kinds of organic or inorganic ligands. This capacity

is generally referred to as the complexation capacity of

the water. All methods for complexation capacity de-

termination involve measurement of the free metal

content using a complexation titration procedure.

A number of methods have been developed in the

past to measure either free metal content or the com-

plexation capacity. They include ion selective electrode

potentiometry (ISE), ion exchange on resins or MnO2,

bioassays, copper salt solubilization, voltammetry and

Computers & Chemistry 23 (1999) 61±68

0097-8485/98/$19.00 # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0097-8485(98 )00015-1

PERGAMON

pFunding was provided by National Excellent Young

Scientist Fund of China [49525102]

* Corresponding author. Tel.: 00-86-10-627-51938; fax: 00-

86-10-627-51938; e-mail: taos@sun.ihep.ac.cn

several others (Jardim and Allen, 1984). Of the com-

mon methods in use, only the ISE technique is a directanalytical probe of tree metal ions. The main problem

associated with this method is its low sensitivity (Hart,1981). Thus far, anodic stripping voltammetry (ASV)

has been the most widely used technique for this pur-

pose (Florence, 1992). When determining complexationcapacity, amperometric titration employing ASV as the

method for determining the uncomplexed metal can beadopted. This method is sensitive, rapid and applicable

over a wide range of pH values.

However, the ASV technique is not without its pro-blems. Perhaps the most serious of these relates to the

fact that the method was found to give generally over-estimated values of free metal concentration and

underestimated values of complexation capacity (Bu�eet al., 1984). Tao et al. measured the complexation ca-

pacity of an EDTA-copper system with known concen-trations of EDTA and copper using the ASV

technique. The results demonstrated that the measured

complexation capacity of the system is considerablylower than the calculated value (Tao et al., 1987).

Morrison and Florence (1989) compared several tech-niques for measuring the copper complexation capacity

of natural and synthetic organic ligands in freshwaters.They found that analysis by ASV coupled with a hang-

ing mercury drop electrode or mercury ®lm electrodegave values of more than an order of magnitude lower

than other techniques.

A widely accepted assumption in explanation of thisphenomenon is that ASV measures the concentration

of labile metal instead of solely free metal. Therefore,the results obtained by ASV measurement very much

depend on the nature of ligands in the system.

It was also suggested that metal complexes may dis-sociate during the period of metal accumulation at the

mercury electrode and will contribute to the strippingcurrent and hence to an overestimation of free metal

concentration (Hart, 1981). Varney et al. (1984) indi-

cated that the dissociation of the complex might occurin the di�usion layer close to the electrode. Tao et al.

(1987) have observed a positive correlation betweenthe measured complexation capacity and the depo-

sition time when a Cu-EDTA system was investigated,further suggesting that kinetic dissociation occurredduring the deposition.The objectives of this study are to investigate theor-

etically, the complex dissociation process near the elec-trode surface; to simulate the process using computertechnique; to compare the simulated result with exper-

imental data; and to explicitly verify the dissociationassumption for the overestimation of ionic metalspecies using the ASV technique.

2. A conceptual model

As a non-consumable electrolysis with thorough stir-ring, the concentration of free metal in the bulk sol-

ution of the electrolyte cell remains almost constantover the entire period of deposition. The equilibriumbetween free metal and metal complex is not supposed

to change practically. The only place where the shift ofthe equilibrium may occur is within a thin layer nearthe surface of the electrode.

Fig. 1. Dissociation of metal complex within the di�usion

layer during deposition.

Fig. 2. Transform within the segments and the transfer between the segments.

T. Shu, et al., / Computers & Chemistry 23 (1999) 61±6862

When a ¯uid ¯ows in a turbulent manner past a

solid surface, the velocity is zero at the surface. There

must be a viscous layer or ¯uid ®lm very near the sur-

face due to viscocity. Beyond this thin ®lm, the ¯ow is

turbulent. The layer with more or less laminar ¯ow is

called the di�usion layer. The rate of mass transfer

from the bulk solution to the electrode surface depends

on the rate of molecular di�usion through the di�usion

layer. A concentration gradient of free metal will be

created within the ®lm. The free metal concentration in

the bulk solution outside of the viscous layer remains

unchanged during the deposition process. However,

the complexation equilibrium within the di�usion layer

may shift due to a signi®cant reduction in the concen-

tration of free metals that transfer towards the surface

of the electrode. A dissociation of the complex conse-

quently occurs within this ®lm. A conceptual model il-

lustrating the dissociation process is shown in Fig. 1.

As shown in Fig. 1, with a through stirring, the con-

centration of free metal ions (M +) in the bulk sol-

ution remains constant during the non-consumable

electrolysis. With or without ligands in the solution, a

concentration gradient of free metal (dash line in the

®gure), either curvilinear (unsteady state) or linear

(steady state), evolves within the di�usion layer when

metal ions di�use towards the surface of the electrode.

If there are ligands in the system, the equilibrium

between the metal ions and complex remains

unchanged in the bulk solution, but shifts within the

di�usion layer. The shift in the equilibrium will cause

a dissociation of the metal complex (ML) releasing

free metals (M +) and free ligands (L ÿ ) which di�use

towards either electrode surface (M +) or bulk sol-

ution (L ÿ) depending on their concentration gradi-

ents.

In a system without ligands, the metal ¯ux towards

the electrode originates solely from the transfer of the

free metal ions. With ligands as well as complex in the

system, however, the total metal ¯ux to the electrode is

the sum of two fractions; the contributions of masstransfer of the free metal plus that of the metal com-

plex. The later also di�use along its concentration gra-dient that evolves due to dissociation. For this reason,even when the free metal concentrations are identical

in two systems with and without ligands, the totalmetal ¯uxes from the bulk solution to the electrodewill be di�erent. This is the primary reason for either

the overestimation of free metal concentrations or theunderestimation of complexation capacities of naturalwater samples determined by the ASV technique.

3. Computer simulation of the process

An attempt to con®rm the assumption of complexdissociation within the di�usion layer was made bysimultaneous simulation of two sorts of processes: the

mass transfer and the complex dissociation. To do so,the di�usion layer is divided into n segments as shownin Fig. 2. It was assumed that these segments are thin

enough that the di�erence in concentration within eachsegment is negligible at any time.Within each segment, the species of free metal, free

ligand, and complex are in equilibrium before depo-

sition begins (t=0) just like that in the bulk solution.After the deposition beings (t>0), the free metalstransfer toward the electrode surface. The decline in

the free metal concentration breaks the equilibrium inthe ®rst segment and the complex in the segment dis-sociates to release more metal ions. The transfer and

dissociation in the ®rst segment create concentrationdi�erences in all species between the ®rst and the sec-ond segments. Consequently, the free metal and the

complex in the second segment start to transfer intothe ®rst one while free ligands di�use in the oppositedirection. All these processes, then, occur in the sec-ond, third, fourth, and the rest of the segments in

sequence. Except at the very beginning, all these pro-cesses actually occur at the same time, creating concen-tration gradients of all species along the direction of

their di�usion.To be speci®c, the chemical dissociation and the

mass transfer processes that occur in the ith segment

of the di�usion layer are as follows:

MLi4M�i � Lÿi �dissociation of complex within

the ith segment�; �1�

M�i )M�iÿ1 �mass transfer of free metal from the

ith segment to the previous one� �2�

MLi )MLiÿ1 �mass transfer of complex from the

ith segment to the previous one�; �3�

Fig. 3. Plot of metal ¯ux against the number of cycles in the

simulation.

T. Shu, et al., / Computers & Chemistry 23 (1999) 61±68 63

Lÿi�1 ) Lÿi �mass transfer of free ligand to the ith one�:�4�

The model shown in Fig. 2 is a discrete one. As thesegment number increases toward in®nity, the model

tends to become continuous. As such, as long as thenumber of segments is large enough, the di�erence

within each segment can be neglected and the modelshows little di�erence from a continuous one.

The kinetics of the processes in the ith segment can

be presented in terms of a mass change due to the dis-sociation and transfer. The changes as a consequence

of dissociation are

d�M�i �=dt � kd�MLi� ÿ kc�M�i ��Lÿi �; �5�

d�Lÿi =dt � kd�MLi� ÿ kc�M�i ��Lÿi �; �6�

d�MLi�=dt � kc�M�i ��Lÿi � ÿ kd�MLi�; �7�while the changes due to mass transfer are

d�M�i �=dt � kt1�Mi� ÿ kt1�Mi�1�; �8�

d�Lÿi �=dt � kt2�Li�1� ÿ kt2�Li�; �9�

d�MLi�=dt � kt3�ML1� ÿ kt3�MLi�1�; �10�where ka and kc are rate constants of the dissociation

and the complexation, kt1, kt2, and kt3 are transfer rateconstants of free metal, free ligand, and complex, re-

spectively. The net mass change for each species is,

therefore, the sum of the change due to the chemicalreaction and the mass transfer. For example, the net

change in the quality of free metal in a segment is thesum of Eqs. [5] and [8].

A computer program was coded using Turbo Pascal

to simulate the dissociation and mass transfer pro-

cesses. The processes were simulated in following

sequence:

1. mass transfer from the ®rst segment to the elec-

trode;

2. dissociation within the ®rst segment;

3. mass transfer between the ®rst and the second seg-

ments;

4. dissociation within the second segment;

5. mass transfer between the second and the third seg-

ments;

6. dissociation within the third segment;

. . . . . .

2n dissociation within the nth segment; and

2n+1 mass transfer between the nth segment and the

bulk solution.

The procedure was simpli®ed by assuming that the

new equilibrium could be reached within a very short

period of time. The equilibrium calculation, therefore,

was based on the complex dissociation constant alone

without considering dissociation kinetics. After the

completion of all 2n+1 steps in sequence, the algor-

ithm started over again from the very ®rst step. In a

preliminary run, it was found that the steady state

could be reached after 3,000,000 to 30,000,000 cycles

depending on the parameters adopted (Fig. 3).

The simulation was performed for a number of scen-

arios with various step numbers, ligand concentrations,

dissociation constants, transfer rate coe�cients, and

deposition time. The parameters studied with their

levels are tabulated in Table 1. Because the study

Table 1

Parameters and their levels for the simulation

Parameters Levels

Number of steps 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 35, 40, 45, 50, 55, 60, 70, 80, 90, 100, 110, 120, 130,

150, 180, 200, 240

Complex dissociation 0.00001, 0.00002, 0.00005, 0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005,

constant 0.01, 0.2, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10,000, 20,000,

50,000

Total ligand 0, 0.001, 0.002, 0.004, 0.007, 0.01, 0.02, 0.04, 0.07, 0.1, 0.2, 0.4, 0.7, 1

concentration 2, 4, 7, 10, 20, 40, 70, 100, 200, 400, 700, 1000, 2000, 4000, 7000, 10,000

Transfer rate constant 0.00001, 0.00002, 0.00004, 0.00008, 0.00015, 0.0003, 0.0006, 0.0008,

of free metal 0.001, 0.002, 0.003, 0.004, 0.006, 0.008, 0.010, 0.015, 0.02, 0.03, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3, 0.4,

0.6, 0.8, 1.0, 2.0, 2.5

Transfer rate constant 0.00001, 0.00002, 0.00004, 0.00008, 0.00015, 0.0003, 0.0006, 0.0008,

of free ligand 0.001, 0.002, 0.003, 0.004, 0.006, 0.008, 0.01, 0.015, 0.02, 0.03, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3, 0.4,

0.6, 0.8, 1, 2, 2.5

Transfer rate constant 0.00001, 0.00000, 0.00000, 0.00001, 0.00002, 0.00004, 0.00006,

of complex 0.00008, 0.0001, 0.0001, 0.0003, 0.0004, 0.0006, 0.0008, 0.001, 0.002, 0.004, 0.006, 0.008, 0.01,

0.015, 0.03, 0.06, 0.08, 0.1, 0.2, 0.4, 0.6, 0.8, 1

Deposition time 1, 21, 41, 61, 81, 101, 121, . . . , 501

T. Shu, et al., / Computers & Chemistry 23 (1999) 61±6864

focused on the overall features of the processes, the

units of all parameters were omitted. The pattern,rather than exact quantity, of metal ¯ux was derived,discussed, and compared with the experimental results.

4. Results of the computer simulation

Two examples of the simulated results are presentedin Fig. 4. The three diagrams at the top correspond to

a system with free metal alone while the three at thebottom represent a system with free metal, free ligandas well as a metal±ligand complex. The free metal con-

centrations in the bulk solution of the two systems areidentical to each other. The diagrams from left to rightshow the concentration gradients of various species inthe di�usion layer at t=0 (beginning of the depo-

sition), t= t (during the deposition, unsteady state),and t41 (approaching in®nity, steady state), re-spectively.

With only free metal ions in the solution (top threediagrams in Fig. 4), the metal concentrations on thesurface of the electrode dropped to zero immediately

after the deposition began. The concentration pro®lesof the metal in the di�usion layer followed eitherFick's ®rst (t41, steady state, linear) or Fick's sec-ond (t= t, unsteady state, curvilinear) laws. The

amount of metal deposited on the electrode was posi-tively proportional to the slope of the concentrationpro®le at the electrode surface. With the ligand intro-

duced into the system (bottom three diagrams inFig. 4), however, the equilibrium established amongcomplex (ML), free metal (M +) and ligand (L ÿ )remained unchanged during the deposition only in thebulk solution. The complex dissociated within the dif-fusion layer in response to the decrease in free metal

concentration. Both free metal and complex trans-

ferred towards the electrode surface while free ligandsdi�used into the bulk solution following its own con-centration gradient. The total quantity of metal reach-

ing the electrode surface during the depositiondepended on the sum of the ®rst derivatives of the con-centration pro®les of metal (M +) and complex (ML)

at the point on the electrode surface. The ¯ux from themetal-only system was de®nitely smaller than that withligands and complex.

5. In¯uence of various parameters on the overestimation

The e�ects of the number of fragments, ligand con-centration, complex dissociation constant, transfer ratecoe�cients of all species, and deposition time on the

metal ¯ux were studied by simulating various scenarios(Table 1).It appears that when the number of fragments is

large enough, the error embedded in the discontinuityassumption of the model is negligible. The variation in

Fig. 4. Simulated concentration pro®les of free metal ion (M +), free ligand (L) and complex (ML) within the di�usion layer, with-

out (on the top) or with (at the bottom) ligand. (EÐElectrode; SÐDi�usion layer; BÐBulk solution).

Fig. 5. In¯uence of number of steps on the metal ¯ux.

T. Shu, et al., / Computers & Chemistry 23 (1999) 61±68 65

metal ¯ux with the number of fragment or steps was

calculated and the results are plotted in Fig. 5 as dis-

crete circles(o).

With an increased number of fragments, the steady

state ¯ux (t41) decreased quickly at ®rst and then

leveled o�. The simulated result can be ®tted with thehyperbolic equation shown in Fig. 5 as a solid line.

Based on this result, a step number of 120 was applied

to all other scenarios in the study.

The in¯uences of ligand concentration (Lf ) and com-

plex dissociation constant (kd) on deposition ¯ux were

similar to each other. Actually, the quotient of the two

is equal to a ratio of free metal and complex concen-trations. Fig. 6 and 7 present the calculated ¯ux at var-

ious ligand concentrations and dissociation constants.

The results ®tted with the linear equations are shown

as solid lines in the ®gures.

With an extremely low concentration of ligand

(0.001) or small value of dissociation constant(0.000001), the calculated ¯ux reached a constant value

of 0.259. This is exactly the same as the result when

the system contains no ligand at all. Linear increases

in ¯ux with increases in either ligand concentration ordissociation constant were observed as shown in Figs. 6

and 7.

Fig. 8 presents the relationship between the mass

transfer rate constant of free metal and the ¯ux. The

¯ux dropped sharply as the constant went up from

0.000001 and reached its minimum when kc equalled0.008. Further increases in kc caused an additional

increase in metal ¯ux. An increase in the transfer rate

of free metal may have double e�ects in the opposite

direction. It may cause a direct increase in the amount

of metal di�used toward the electrode (increase in

¯ux). It may also give rise to a lessening of the com-

plex dissociation (decrease in ¯ux). For the particular

case simulated in this study, 0.008 happened to be a

turning point above which the ®rst e�ect overcame the

second one. As can be seen in Fig. 8, a hyperbolicequation ®ts the simulated results well.

The in¯uences of the transfer rate constants of free

ligand and complex on the metal ¯ux at steady state,

however, were very much di�erent from that of free

metal. The plot of the ¯ux against the transfer rate

constant of free ligand is presented in Fig. 9 as an

example. The shape of the curve is quite similar tothat shown in Fig. 8, though, the metal ¯ux varies

only from 0.3114 to 0.3118 over the same range of

transfer rate constants from 0.0001 to 2.5. It appears

that the transfer rate of either free ligands or complex

has no signi®cant in¯uence on the metal ¯ux.

Fig. 6. In¯uence of the ligand concentration on the metal

¯ux.

Fig. 7. In¯uence of dissociation constant on the metal ¯ux.

Fig. 8. In¯uence of transfer rate constant of free metal on

metal ¯ux.

Fig. 9. In¯uence of transfer rate constant of free ligand on the

metal ¯ux.

T. Shu, et al., / Computers & Chemistry 23 (1999) 61±6866

It was demonstrated that the extent of the overesti-

mation of ASV for the free metal concentration

increased with deposition time (Tao et al., 1987). It

happens to be part of the evidence con®rming the com-

plex dissociation. The phenomenon was simulated in

this study by comparing the metal ¯uxes reaching the

electrode surface in systems with and without ligands.

The concentrations of free metals in the two systems

were identical. The di�erence between the two is the

overestimation error generated due to complex dis-

sociation. The variation of the error was plotted

against deposition time in Fig. 10.

The simulated results were ®tted with a power func-

tion and illustrated in Fig. 10 along with the ®tted

curve. As expected, the overestimation error increased

with deposition time. The rate of increase (the ®rst de-

rivative), however, decreased at the same time,

suggesting that a steady state was approaching with a

longer deposition time.

6. Comparison between simulated and experimental

results

The simulated results were partly validated by exper-

imental results. The anodic stripping voltammetry(ASV) technique was applied to measure Cu in the pre-

sence of EDTA. The total Cu content in the solution

was 28.35 mmol/l, EDTA concentrations were 0, 5.3,10.6, 15.9, 21.2, 26.5 and 31.8 mmol/l, respectively. A

PAR-384 voltammeter coupled with 303 hanging mer-cury electrode was used. The bu�er system consisted

of NaAc(0.2N)±HAc(0.2N) with a 4.5 pH level.

Deposition was carried out at ÿ0.3 V for 2, 5, 10, and20 s. The scan range and rate were ÿ0.3 V±1.0 V and

10 mV/s, respectively.

The in¯uence of deposition time on the overestima-tion error was con®rmed by the experimental results.

The measured free Cu contents at four di�erent depo-

sition times (2, 5, 10, and 20 s) in a solution with totalCu content of 28.35 mmol/l and EDTA content of

5.30 mmol/l are shown in Fig. 11.

The trend of variation in overestimation errorshown in Fig. 11 is similar to that simulated in Fig. 10.

A power function can again be applied to ®t the

measured data well. The regression equation, as wellas the ®tted curve, is presented in Fig. 11. The function

is identical to that derived from computer simulation(Fig. 10) except for the units that were not included in

the simulation study.

In¯uence of ligands on the overestimation was also

con®rmed by the experiment. With the same concen-tration of Cu (28.35 mmol/l) and di�erent concen-

trations of EDTA (0, 5.3, 10.6, 15.9, 21.2, 26.5 and31.8 mmol/l), free metal concentrations in the solutions

were measured suing ASV methods. The experiments

were carried out at various deposition times and theresults were similar to one another. The experimental

error at 5 s-deposition time are illustrated in Fig. 12 as

an example. The overestimation errors (di�erencebetween measured and calculated free Cu concen-

Fig. 10. Overestimation error as a function of deposition time

(T).

Fig. 11. In¯uence of deposition time on the overestimation

error detected by ASV determination for free Cu with EDTA

presence (total Cu 28.35 mmol/l, EDTA 5.30 mmol/l).

Fig. 12. In¯uence of EDTA on the overestimation error

detected by ASV for free Cu.

T. Shu, et al., / Computers & Chemistry 23 (1999) 61±68 67

trations) of the measurement are plotted against theconcentration of EDTA.

A brief examination of the ®gure indicates that over-estimation errors increase when more EDTA is addedto the solution. Except for the last one, all data points

fall along a straight line. A simple linear regressionwas performed to ®t the data and the derived functionis presented in Fig. 12 as well. Omitting the units that

were not considered in the computer simulation, thepattern of in¯uence detected by the experiment is iden-tical to that found by computer simulation. For the

last data point, the equivalent concentration of EDTAexceeded that of Cu.

7. Conclusion

The results of the computer simulation con®rmed

the assumption that the overestimation of free metalconcentrations by the ASV technique with the presenceof ligand is generated from the dissociation of metal

complex within the di�usion layer near the electrode.It also demonstrated that the magnitude of the overes-timation depends on the complex dissociation coe�-

cient, ligand concentration, transfer rate constants, aswell as deposition time. Part of the simulation resultswere con®rmed by experimental data.

Acknowledgements

The authors wish to express their grateful thanks toProf Bruce Ralston and Prof Richard Dawson fortheir valuable comments on the manuscript.

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