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Journal of the Chilean Chemical Society

On-line version ISSN 0717-9707

J. Chil. Chem. Soc. vol.49 no.4 Concepción Dec. 2004 

  Chil. Chem. Soc., 49, N 4 (2004): págs: 345-350




1 Faculty of Chemistry, University of Concepción, Casilla 160-C, Concepción, Chile. Fax: 56-41-245974;e-mail:
2 Instituto de Química, Facultad de Ciencias, Universidad Austral de Chile, Casilla 567, Valdivia, Chile


The Liquid-phase polymer-based retention (LPR) technique allows calculating the formation constant of polymer-metal ion complexes (Kf). The dependence of the relative error in Kf (eK) on the error in the total retention coefficient a (ea) has been searched and it was found that its influence is higher as a increases, so that its measurement must be performed with higher precision. In addition, ea is higher for high a values due to the stronger influence of errors in the measurements of the different magnitudes that allow its calculation. In order to achieve a measurement of Kf with a relative error lower than 7 %, the experimental a found should not exceed the value 0.4 when relative errors for the independent variables ranging between 2 % and 5 % are considered.

Keywords: error, formation constant, polymer-metal complexes, simulation, ultrafiltration


Water-soluble polymers have been extensively studied during the last two decades, due to their ability to form complexes or exchange metal ions. These properties allow their use in industrial scale, in the treatment of residual waters, and in quantitative procedures at laboratory scale.1-6) As a result of the polymer-metal ion interactions, which can be only electrostatic or include the formation of coordinate bonds, a new chemical species may be produced, termed polymer-metal complex (PMC), whose physicochemical properties are very different with respect to the original substances.7) A metal ion complexing polymer is called polychelatogen.5)

When a PMC is formed in aqueous medium, an equilibrium is achieved, which is described in Scheme 1 (the charges are not included to simplify the notation):

M (aq) + n L (p) « MLn (p)

Scheme 1

where (p) emphasizes that the L functional groups (ligands or ion exchange groups) are linked to a polymeric backbone. The constant that determines the equilibrium is called the formation constant of the PMC. This is a fundamental parameter for system characterization as its knowledge allows quantifying the stability of the PMC and determining the selectivity of a polymer towards binding a specific cation. To define the formation constant of a PMC, it is accepted to apply an expression analog to that used for complexes involving low molar mass compounds:8-10)


Kf = [MLn] / ( [M]c · [L]m ) (1)

where [MLn] is the concentration of metal ions linked to the polymer, [M]c is the free metal ion concentration (non-complexed), [L] is the polymer concentration expressed in terms of repeat units, and m is a factor. Yet, there is controversy about the validity of this definition due to that for a polychelatogen solution, the L groups are not uniformly distributed in the system. Thus, it has been postulated that the exponent m may become different from the stoichiometric coefficient n.11-12)

Different techniques have been used to determine formation constants of PMCs. Among the most used are potentiometry and electronic spectroscopy.8-10, 13-16) Recently,17-19) the Liquid-phase polymer-based retention (LPR) technique, which was developed by Geckeler and col.,5) has been incorporated to this group. This technique combines the use of polychelatogens with membrane ultrafiltration, which separates low molar mass species (i.e. free ions in solution) from polymeric compounds (precursor polymers and PMCs). With this technique a high selectivity has been achieved to separate metal ions.20-21)

By the washing method of the LPR technique (see experimental) a retention profile is obtained plotting the retention (R) versus the filtration factor (F). R is defined as the concentration of metal ions inside the ultrafiltration cell in every instant divided by the initial metal ion concentration. F is defined as the ratio between the volume in the filtrate and the volume in the cell. By means of this plot, information is obtained about the affinity between the metal species and the polychelatogen. Variables that show an effect on the retention profile are the same that affects the polymer - metal ion interaction: polymer concentration, polymer-to-metal ion concentration ratio, ionic strength, and pH. When the ultrafiltration experiment is performed at constant ionic strength, a retention profile that tends asymptotically to zero is obtained.17) Applying the mathematical models presented previously,17-19) it is possible to calculate the formation constant of a PMC from the respective retention profile. Therefore, in order to obtain this parameter with the lowest uncertainty, the error in the measurements of all the magnitudes necessary to construct the retention profile must be minimized.

The aim of this paper is to establish the relation between errors in the measurement of the variables involved in the ultrafiltration experiment and the variations that are produced either in the pattern of the retention profile and the formation constant of the PMC. This will allow determining which experimental measurement produces the highest impact on the accuracy of ultrafiltration measurements.



Poly(acrylic acid), PAA, MW = 250,000 and poly(sodium vinylsulfonate), PVS - both from Aldrich - were purified and fractionated by ultrafiltration. For PAA and for PVS the fractions between 100,000 and 1,000,000 Da, and 10,000 and 50,000 Da were used respectively. The salts NaNO3, Mg(NO3)2 and Ni(NO3)2, Merck p.a., were used as received. The solutions were prepared with twice distilled water, whose conductivity was lower than 1mS.


The ultrafiltration equipment has been previously described.21) It has a filtration cell with a membrane with a molar mass cut off (MMCO) of 5,000 Da (Filtron, Pal Gelman), a reservoir for the washing solution, a selector, and a pressure source. The metal ion concentration was determined by atomic absorption spectrometry through a spectrometer UNICAM Solaar 5M Series.

Procedure (Washing Method)

50.0 mL of a solution containing 5.0·10-3 eq/L of polymer, 0.010 or 0.10 M in NaNO3, 1.0·10-4 M in metal ion were placed into the solution cell. The pH was adjusted to 5.0 with dilute HNO3. A washing solution (0.010 M or 0.10 M of NaNO3 in water at pH 5.0) was passed under pressure (3 kPa of N2), from the reservoir through the cell solution. All the experiments were carried out at constant ionic strength. As the in- and out flux are rapidly equaled, the initial volume (50.0 mL) is kept constant during the experiment. Ten fractions of 10 mL were collected and then ten more of 20 mL. Each i fraction was collected in graduated tubes, and the corresponding metal ion concentration was determined ([M]i).


A retention profile is a plot of the fraction of the initial metal ion retained in the ultrafiltration cell (R) versus the filtration factor (F). F is defined as the ratio between the volume in the filtrate and the volume in the cell. To calculate these variables from the experimental measurements, the following expressions are used:


where V0 is the volume in the cell, Vi is the volume of the fraction i, M0 is the initial mole amount of metal ions in the cell, and [M]i is the concentration of metal ions in the fraction i.


Theoretical Expressions

Geckeler and col.17) developed a mathematical model applying the kinetics theory of chemical reactions that occur in partially open systems. This model establishes that the retention profiles at constant ionic strength may be fitted by the following function:

R = exp(-aF) (4)

where a, the fit parameter, is the total retention coefficient, which includes the participation of the polymer as well as the membrane in the retention of the metal ion. The value of a varies in the range 0 < a< 1. Subsequently, Rivas and Moreno-Villoslada,18-19) with a treatment fundamentally thermodynamic, achieve the same results and they established that the a parameter could be decomposed in two factors:

a = km · kp (5)

where km, the membrane sieving coefficient, is given by:

km = [M]f / [M]c (6)

and kp, the retention coefficient of the polymer, is expressed as follows:

kp = (1 + Kass)-1 (7)

with Kass, the association constant, given by the following relation:

Kass = [MLn] / [M]c (8)

In Equation (6) [M]f = S[M]i·Vi / SVi, is the concentration of metal ion in the filtrate.

Combining Equations (1) and (8), an expression that relates the association constant with the formation constant of the PMC is obtained:

Kf = Kass / [L]m (9)

It has been previously demonstrated that, independently of the coordination sphere composition of the complexed metal-ion, m equals 1,22) which is consistent with previous kinetic studies that showed that the formation of the PMCs takes place in only one step. Moreover, from blank experiments (in the absence of the polymer), it has been established that the membrane do not affect the metal ion retention process, that is km = 1. Thus, once the value of retention coefficient a is determined, it is possible to calculate the value of Kf from the following expression:

Kf = (a-1 - 1) / [L] (10)

The relative error involved in the determination of the formation constant of the PMC (eK) depends on the relative error in a (ea). Taking in account that, and neglecting errors in the measurment of [L], from Equation (10) it is obtained that:23)

eK = ea · (1 - a)-1 (11)

Therefore, if one wishes to minimize the error in the formation constant, it is necessary to diminish ea, and moreover, it is important to work in such conditions that the value of a is not close to 1. As can be seen in Figure 1 and deduced from Equation (11), a values close to 1 will increase substantially the eK value in relation to ea. If a maximum error of 5% in the calculation of Kf is accepted, the higher the value of a is, the higher precision is necessary for its determination: if a = 0.8, the maximum permitted value for ea is 1%; if a = 0.4, the maximum permitted value for ea is 3%; if a = 0.1, ea can be as high as 4.5% without eK reaching higher values than the accepted 5%.

Fig. 1. Relation between relative errors for Kf and a, for different values of a

Error Simulation

Now that the relation between eK and ea is known, we face the following questions: (i) how does ea depend on errors produced in the measurements of the variables to determine a?; (ii) for a values close to 1, is it possible to reduce ea so that Kf can be obtained with an acceptable relative error?; (iii) which is the maximum value of a that allows calculating Kf with an acceptable relative error?

To answer these questions, two experiments were performed in two replicates, which confirmed the reproducibility. Their corresponding retention profiles are shown in Figure 2(a). For Experiment 1, performed with PAA and Ni(II) under 0.010 M NaNO3, the retention profile decreases slightly with F. For Experiment 2, performed with PVS and Mg(II) under 0.10 M NaNO3, the retention profile decreases strongly with F due to the higher ionic strength. These retention curves fit very well to the plot of a function given by Equation (4). The semilogarithmic form of these profiles are shown in Figure 2 (b). Deviations from linearity, observed for Experiment 2 from F = 3, were attributed to high relative errors produced in R when this variable takes small values, and therefore the values between F = 0 and F = 3 were used for both experiments and their results compared. These profiles were assumed to be obtained without error and then exact values of a = 0.092 and 0.913 were given respectively for Experiments 1 and 2 by lineal fit of least squares, where the condition that the regression straight line passes through the point (0,0) was included. Then, a was calculated from the following expression:


(a) (b)

Fig. 2. Ultrafiltration experiments: () Experiment 1; () Experiment 2. Retention profiles: (a) in %; (b) in logarithmic representation

Error simulations have been done based on possible errors produced in the measurement of the variables involved in the calculations of F and R (see Equations (2) and (3)). Table 1 summarizes the descriptions of the variables whose error have been simulated, were [M]i is the metal ion concentration in every filtration fraction. Figures 3 and 4 show the changes in the retention profiles of Experiments 1 and 2 respectively, when errors in one of the variables described in Table 1 are considered. The plots at the right correspond to the logarithmic representation of the profiles. From these plots the value of a was also obtained by lineal fit of least squares. These a values are shown in Table 2 and Table 3 along with the corresponding correlation coefficients (R2). It is not possible to include standard deviation values due to that one of the fundamental conditions is not satisfied: errors must be independent (see Equations (2) and (3)). The column headed with the legend Dev. a (%) gives a difference in percentage between the values of a obtained with error and those considered as reference. Because error accumulation, the errors of the first fractions have more influence on the results, so a fortuitous error in V1 and [M]1 has been considered in the variables analyzed.

Table 1. Description of the simulated errors

Notation Description

M0 ± a %

Error in M0 of a a % by excess (+) and by defect (-)
V1 ± b % Error in V1 of b % by excess (+) and by defect (-)
[M]1 ± c % Error in [M]1 of a c % by excess (+) and by defect (-)
V0 ± d % Error in V0 of d % by excess (+) and by defect (-)
Vi ± e % Error in every Vi of e % by excess (+) and by defect (-)
[M]i ± f % Error in every [M]i of f % by excess (+) and by defect (-)

Fig. 3. Error simulation, by excess () and defect (D), on Experiment 1 (): (a) 5% of error in M0; (b) 30% of error in V1; (c) 10% of error in V0; (e) 20% of error in every Vi; (c) 5% of error in every [M]i. (a'), (b'), (c'), (d'), (e'), (f'), logarithmic representations.

Table 2 shows that, for Experiment 1, high errors in V1 and [M]1 produced very small deviations in a. On the contrary, M0 and V0 are critical variables since errors in their measurement produce important changes on the slope of the regression straight curve and then in a. An error of 5-10% in either M0 or V0 produced deviations of the same percentage in a. In general, in order to clearly appreciate distortions generated on the retention profiles, errors have been over-estimated. It is possible that due to the pressure fitting process at the beginning of the experiment, the cell solution volume varies with respect to the initial volume of 50.0 mL, so it is necessary to measure the volume inside the ultrafiltration cell at the end of the experiment.

Table 2. Results of the lineal fit by least squares for the different simulations of errors considering the Experiment 1.

Simulation a R2 Dev a
Kf · 10-3 a)
Dev Kf

Exp 1 0.0919 0.9984 - 1.98 -
M0 + 5% 0.087 0.9988 5.3 2.10 6.2
M0 - 5% 0.0973 0.9978 5.9 1.86 6.1
M0 + 2% 0.0899 0.9985 2.2 2.02 2.4
M0 - 2% 0.094 0.9981 2.3 1.93 2.5
V1 + 30% 0.0917 0.9980 0.2 1.98 0.2
V1 - 30% 0.0921 0.9987 0.2 1.97 0.2
[M]1 + 10% 0.0927 0.9986 0.9 1.96 1.0
[M]1 - 10% 0.091 0.9980 1.0 2.00 1.1
V0 + 10% 0.1011 0.9984 10.0 1.78 10.0
V0 - 10% 0.0827 0.9984 10.0 2.22 12.3
V0 + 2% 0.0937 0.9984 2.0 1.93 2.1
V0 - 2% 0.09 0.9984 2.1 2.02 2.3
Vi + 20% 0.0939 0.9964 2.2 1.93 2.3
Vi - 20% 0.0900 0.9995 2.1 2.02 2.3
[M]i + 5% 0.0971 0.9978 5.7 1.86 5.9
[M]i - 5% 0.0868 0.9988 5.5 2.10 6.5
[M]i + 2% 0.0939 0.9981 2.2 1.93 2.3
[M]i - 2% 0.0898 0.9985 2.3 2.03 2.6

a) [L] = 5.0 · 10-3 M

Systematic errors produced in the measurements of every Vi and [M]i were also considered. These errors changed the observed value of a but not the lineal pattern of the logarithmic representation, as can be seen in Table 2 and Figures 3(e') and 3(f'). On the other hand, Table 3 shows that, for Experiment 2, in which a took a value close to 1, errors in the variables produced a greater change on the observed values of a and Kf. Besides, an error of 5% in M0 and 10% in every Vi produced a sharp change on the shape of the logarithmic representation for high values of F (see Figure 4 (a') and (e')), and therefore, these simulations were not included in Table 3.

Table 3. Results of the lineal fit by least squares for the different simulations of errors considering the Experiment 2


a R2 Dev a
Dev Kf

Exp 2

0.9129 0.9996 - 19.1 -
M0 + 2% 0.8582 0.9969 6.0 33.0 73
M0 - 2% 0.9803 0.9989 7.4 4.02 79
V1 + 20% 1.0253 0.9954 12.3 -4.94 126
V1 - 20% 0.8297 0.9947 9.1 41.1 115
V1 + 5% 0.9639 0.9995 5.6 7.49 61
V1 - 5% 0.8688 0.9976 4.8 30.2 58
[M]1 + 5% 0.9419 0.9998 3.2 13.5 71
[M]1 - 5% 0.8858 0.9990 3.0 81.0 70
V0 + 10% 0.9944 0.9996 8.9 1.13 94
V0 - 10% 0.8314 0.9996 8.9 40.6 112
V0 + 2% 0.9455 0.9996 3.6 11.5 40
V0 - 2% 0.8803 0.9996 3.6 27.2 42
Vi + 5% 1.0134 0.9945 11.0 -2.64 114
Vi - 5% 0.8284 0.9900 9.3 41.4 117
[M]i + 2% 0.9788 0.9990 7.2 4.33 77
[M]i - 2% 0.8572 0.9968 6.1 33.3 75

a) [L] = 5.0 · 10-3 M

Fig. 4. Error simulation, by excess () and defect (D), on Experiment 2 (): (a) 5% of error in M0; (b) 30% of error in V1; (c) 10% of error in V0; (e) 20% of error in every Vi; (c) 5% of error in every [M]i. (a'), (b'), (c'), (d'), (e'), (f'), logarithmic representations.

Then, we observe that in order to obtain Kf with acceptable relative errors, if a takes high values, it must be calculated with higher precision according to Equation (11). However, errors in the measurements of the different magnitudes influence the observed value strongly as a increases.

In order to obtain the maximum value for a that allows obtaining Kf with an acceptable relative error, the relation between ea and a is searched applying statistics laws. a is an aleatory variable (see Equation (12)) that, through Fi and Ri, with i = 1n, is a function of the aleatory independent variables V0, M0, [M]1,...., [M]n, V1,....,Vn. Then, its approximate variance, sa2, is given by the following expression:24)


0000where sx2 represents the variance of the x variable.

Combining Equations (2), (3), (12), and (13), and assuming that the relative errors in Vi and [M]i are independent on i, it is achieved that:


where ex2 represents the relative error in x (given by: ex2 = sx2/x2 ) and:



In Equations (16) and (17), F0 = 0 and R0 = 1. Is clear that f(a), g(a), and h(a) are functions of a, as they depend on Ri (Ri = exp(-aFi), i = 1...n).

From Equation (14) it is directly concluded that the contribution of eV0 in ea is independent of the value that a takes. Results in error simulations concerning Experiments 1 and 2 (see Tables 2 and 3) are consistent with this observation. The functions f(a), g(a), and h(a) were evaluated for Experiments 1 and 2, and the value of ea was calculated applying the following estimated relative errors for the independent variables: 2% for V0, 2% for M0, 2% for every [M]i, and 5% for every Vi. The results are presented in Table 4. It is observed that the values of the functions increase with increasing a and, in consequence, errors in the corresponding variables are more weighted as a increases. In both experiments f(a) takes the highest value, so that eM0 becomes the most important variable influencing ea. On the contrary, h(a) always adopts a low value, so errors in every Vi produce less impact in the measure of ea. Equation (14) was derived considering aleatory errors in the variables, and then, it will not predict the results obtained considering systematic errors in every Vi and [M]i. Nevertheless, the results in Table 4 are qualitatively consistent with those in Tables 2 and 3. Finally, the plot of ea vs a is shown in Figure 5. It has been constructed evaluating Equation (14) from the different values of a corresponding to the Fi given, calculated assuming relative errors in the independent variables listed above. It is also shown the curve eK vs a, obtained by substitution of every (a, ea) in Equation (11). Under these conditions, it can be seen that a = 0.4 is the maximum value of a that yields Kf values with arbitrary eK £ 7%.

Table 4. Evaluation of Equations (14-17) for Experiments 1 and 2

Exp1 Exp2


1.120 3.294


0.320 1.094
h(a) 0.039 0.811
ea 0.031 0.083

Fig. 5: Plot of eK and ea versus a.


The formation constant of the PMC (Kf) can be obtained through the LPR technique. The relative error in Kf (eK) depends on the error in a (ea). Due to the functional relation between these parameters, when a takes high values, its measurement must be performed with higher precision. Meanwhile, errors in the measurements of the different magnitudes that allow the calculation of a influence the observed value strongly as a increases. In order to achieve a measurement of Kf with a relative error lower than 7 %, the experimental a found should not exceed the value 0.4 when relative errors for the independent variables (2% for V0, 2% for M0, 2% for every [M]i, and 5% for every Vi) are considered.



The authors thank to FONDECYT the financial support (Grants No 1030669, No 1020198, and No 2000123). L. N. Sch. thanks to CONICYT the Ph.D fellowship.


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(Received: March 3, 2004 - Accepted: October 8, 2004)


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