ANALYST, AUGUST 1985, VOL. 110 1001

Quantitative Study of a Selectivity Index Applicable to Spectrophotometric Methods of Analysis

Vicente Peris Martinez, Jose Vicente Gimeno Adelantado, Agustin Pastor Garcia and Francisco Bosch Reig Department of Analytical Chemistry, Faculty of Chemistry, University of Valencia, Burjasot (Valencia), Spain

A selectivity index has been developed that quantifies the selectivity of various spectrophotometric methods of analysis. This has been produced by considering the interaction between the analyte and interferent and developing a n algorithm to calculate the value of the index in each instance. The expression obtained has been applied to practical examples. Keywords: Selectivity; spectrophotometric determinations

In an analytical method, several aspects can be distinguished: the analytical manipulations, the apparatus, the external conditions, the experimental procedures, the calibration and the evaluation, which together establish the amount of information necessary to solve a given problem. The dis- tinguishing characteristics of an analytical procedure can be divided into two types: a functional group and a statistical group.1 To the functional group belong the sensitivity, selectivity, specificity and accuracy, and to the statistical group the precision, detection power and limits of precision.

The concepts of selectivity and specificity are widely known2 and used, and have been the subject of dissent and mathematical argument by various workers aiming to quantify the characteristics peculiar to a particular analytical method. Such terms have frequently been applied by many analysts with imprecise and even contradictory definitions.3 Belcher4 and Belcher and Betteridges suggested characterising the selectivity for reagents and reactions in terms of the qualitative aspects only by means of a parameter named the “selectivity index.” The test proposed by Maurice and Buijs6 also gives qualitative information for establishing the total interferences in an analytical procedure, and that by Wilson3 calculate the effects of the interfering species in an individual manner. Kaiser, who related selectivity to the possibility of the independent determination of all the constituents of a sample ,7 was the first to propose mathematical expressions that can provide information on the selectivity and specificity of an analytical method in order to determine various coexisting chemical species, but the excessive theoretical formulations led to difficulties in putting it into practice.3.8 Later, Fugiwara et aZ.9 used a modified equation to express quantitatively the selectivity in analytical procedures involving HAS and AES, including in the expression the concentration as a variable.

Anastassiadis and Common10 defined the specificity as the degree to which the mean result measured by a method is due to the substance to be determined and not to other substances that may be present. Swietoslawskall defined the specificity of a method in relation to the influence of the presence of an interferent on the accuracy of the determination under conditions such that all the interferents and their concentra- tions remain constant.

Pszonickil2 and Pszonicki and Lukszo-Bienkowskal3 sug- gested “non-specificity coefficients,” which express quanti- tatively the magnitude and sense of the interference in the determination of an analyte. Gottschalkl4 defined the concept of selectivity as the extent of non-interference, the degree of selectivity being expressed by the threshold amount of an interferent that will produce a detectable effect on the main reaction. Inczedyl5 expressed the selectivity in terms of the degree to which a component can be determined in the

presence of other components that behave in a similar manner, suggesting an equation that includes the percentage degree of selectivity based on the additional signal caused by the interfering components.

IUPAC recommended,g for the quantifiable expression of selectivity, the combination of the term with another parameter, the quantitative characterisation of the selectivity of a procedure being expressed in terms such as selectivity constant, selectivity index, selectivity factor or selectivity coefficient. The word “constant” should only be.used for a universal constant or a property of a material and “index” for numerical values arising from ratios, and therefore dimension- less; the expressions “number” and “factor” are satisfactory alternatives to “index. ’’ The term “coefficient” should be used for numerical values included in equations of proportionality.

In this paper, we propose the quantification of the selectivity of an analytical procedure involving a spectro- photometric determination through a “selectivity index,” determined starting from the equations for the absorption of radiation, including the concentrations of the analyte and of the interferent, in addition to the interaction effects. An algorithm was developed for obtaining such an index for each different type of analysis.

Theoretical Considerations: Definition and Meaning of the Selectivity Index

All spectrophotometric determinations are based on the relationship between the concentration of a sample chemical species in solution and the absorbance that the instrument measures. For a concentration of analyte (a) c, and a 1-cm cell, it would be

A, = E,C,

If there are interfering substances (b) that absorb at the wavelength of measurement, the reading will depend on all the relevant substances and therefore one can express the relationship in a general form:

where c b is the concentration of the interfering species. Developing the calibration in equation (1) as a Taylor series: A = E,C, + &blCbl + &b2Cb2 + . . . + kclCaCbl + kc2C&b2 + . . .

= A, + Abl + Ab2 + . . . + A’1 + A$ + . . . which can be expressed as

n n

n n = A, 4- ,X Abi+ ,Z A’, . . . .

1 = 1 1 = 1

in which there is a term showing the influence of the

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1002 ANALYST, AUGUST 1985, VOL. 110

interferent ( i &biCbi = ii A b i ) and another for the

interferent - analyte interaction 2 kciCaCbi = ,; A’,>.

The former induces a shift in the calibration graph for the analyte and the latter (if it is not constant) changes the slope.

The total absorbance A measured for the chemical system studied will generally be different from A, owing to the influence of the interferents and can become equal to the product of E, and c’, where c’, is a hypothetical concentration of analyte that would correspond to an absorbance equal to the total measured value:

From equations (2) and (3) and removing the sum that represents the contribution of the analyte to the total value, we obtain

i = 1 i = 1

( “ i = 1 1 = 1

* . (3) A = cada . . . . . .

(4)

This expression represents the influence of interferents on the system. Dividing equation (4) by A, or by its equivalent E,c,, we obtain a ratio that gives the contribution of the interferents for unit concentration (or absorbance) of the analyte. This ratio is called the “selectivity index” ( S ) and the expression obtained is

n n

We can finally write n n

A b i + ,Z A ’ i i= 1 r = l

S = . . . . A a

For practical applications we consider it more useful to express S as a function of concentration:

1 “ Ca i = 1 I = 1

n

s= - z &riCbi+ . 2 &iCbi . . . . (7)

where

and

Discussion and Properties The selectivity index S determined by equation (7) can be expressed mathematically by S = f(ci, E ~ , kc,), where ci is the concentration of the species present (analyte and interferent), Ei is the molar absorptivity of such species and kci is a proportionality constant that establishes the influence of the interaction between the species present.

Equation (7) is obtained for a system in which the analyte and interfering substances follow independently the Beer - Lambert law, i.e., the usual condition in spectrophotometric determinations. It considers only the interaction effect of the analyte with each of the interfering substances independently; interactions of the analyte with more than one interfering substance simultaneously, and interactions among the inter- fering substances, are considered to be negligible because they correspond to the later terms in the Taylor series.

The variable kCi must be determined in each instance and for this purpose a general method is proposed that allows the calculation of the suggested values of kCi starting from the experimental data obtained in each individual experiment. This special study is necessary because the influence of such a variable depends directly on the mechanisms that regulate the interactions between the species present in the various spectrophotometric procedures studied in each instance.

Interference can be classified into two types according to IUPAC8: an interference SM given by a substance that produces the signal by a similar mechanism to the analyte, and an interference DM given when the analytical signal originates by a different mechanism than the analyte. One must also consider that the value of k,. for each chemical system depends on the characteristics of the latter and can be a constant (including zero) or be a variable function (linear or not) for determined concentration intervals. The selectivity index S proposed here permits a quantitative judgement of the selectivity concept. One can speak of a “fully selective” procedure (a term used by Kaiserl) when S has a value very near zero (theoretically zero). As S increases the system becomes less selective and for high values of S one cannot really refer to selectivity.

On the other hand, the selectivity index S as defined coincides numerically in value with the relative error in the determination of the analyte and is due, exclusively, to the action of the interferents. If the error caused by each interferent is known, the corresponding value of k,, can be deduced if the concentration and molar absorptivity for application in equation (7) are known.

Algorithm for the Calculation of the Selectivity Index: Practical Examples

To establish the value or the variable function of S, one must determine k,; for each analytical method and chemical system. Equation (2) allows the kCi values to be obtained from

n n

kcfachi = A - -k , EbiCbi) 1 = 1

where the right-hand side expresses the difference between the measured absorbance A and the sum of the readings, independent of the analyte and of the interfering substances, at identical concentrations in which they are present in the sample. The number of independent assays required equals the number of interfering species, using for each assay the analyte with one interferent alone. The following scheme was used:

kclCaCbl = A - kc~CaCb2 = A -

+ Eblcb l )

+ Eb2Cb2)

k, is known and can be substituted in equation (7), allowing the direct calculation of S .

From this, specific cases can be defined that can be applied to real samples. Case I

If z kciC,Cbi = 0, then there is no appreciable influence of the n

i = 1

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ANALYST, AUGUST 1985, VOL. 110 1003

interaction between the analyte and the interferent on the measured signal. Therefore, the absorbance measured will be the sum of that due to the analyte and that due to the interfering species and the expression that gives the value of S remains as

which is a linear function when S is plotted against llc, (for a fixed concentration of interferent) [Fig. l ( a ) , line 11. When the analyte concentration is fixed and the interferent concen- tration Varies one obtains a linear function of S versus cb that will also pass through the origin [Fig. l(b), line 11. If more than one interferent is present, a family of straight lines with different slopes passing through the origin, for a fixed concentration of interferent, was obtained [Fig. l(a), lines 2-41. For a fixed analyte concentration and with one interfer- ent concentration varying and the others maintained constant, parallel lines were obtained [Fig. l(b), lines 2-41.

The simultaneous determination of Cr(V1) in the presence of Mn(VI1) (or vice versa) by spectrophotometry16 is an example that verifies the above hypothesis, as the absorbance of the mixture coincides with the sum of the individual absorbances, which implies that the term that includes the influence of interactions between the species is zero.

Case I1

If j z kcic,cbi = constant, the value of kcj must depend on the concentration of interferent and analyte. To obtain kci, if there is only one interferent a series of experiments must be performed in which the absorbance of the system is measured, keeping the analyte concentration constant and altering the concentration of the interferent. It was found that the term

X k .c c is independent of the concentration of the analyte.

fl

fl

i = 1 ci a bi

1 lca-m

/ 3 / /

/ /

Fig. 1. Variation of S when the interaction term is zero. (a ) Graphs of S versus l/c, (with c b . constant); and ( b ) graphs of S versus cb (with l/c, constant). Graphs f-4, in the presence of one, two, three and four interferents, respectively

Table 1. Calculation of k,. values for the Cr(V1) - PGR V(V) system. Conditions: pH, 4 (acetic'acid - sodium acetate buffer); ionic strength, 0.1 M; temperature, 25 "C; [PGR], 6 x M; h, 490 nm; fixed time, 5 min

V(V), C r ( W 9

p.p.m. p.p.m. A , Ah A -A'; k,; 0.81 0.10 0.700 0.803 0.651 0.852 10.52 0.81 0.01 0.700 0.817 0.667 0.850 104.94 0.20 0.10 0.796 0.803 0.748 0.851 42.55

n

I = 1 Determination of the value of the term , 2 kcic,cbi

From the reported simultaneous determination of Cr(V1) and V(V) using the oxidation of Pyrogallol Red (PGR)" and from the experimental values given in Table 1, it was found that the difference between the sum of the absorbances when each species is alone and when both species are present, is a constant (-0.852), which corresponds to the term kclc,ch in equation (2), giving k,, = -0.852/c,cb. Therefore, a graph of k,, versus the inverse of the concentration of either the analyte or interferent (when the other is kept constant) is linear [Fig. 2(a)].

Calculation of S

When it is known how k,. changes, we can determine Kci according to equation (8). Replacing the corresponding expression in equation (7), we can study analytically or graphically the variation of S with the concentrations that appear in the expression [Fig. 2(b)].

Case I11 If kciCaCbi is not constant, one must first evaluate the c o k h t value or the function representing the variation of kci. The practical example used will be the same chemical system as discussed above, but under different conditions. Table 2 shows the experimental data obtained for the system V(V) - PGR - Cr(V1) in which for a fixed concentration of vanadium (0.81 p.p.m.) and a variable concentration of chromium, the measured absorbance is not equal to the sum of the absorbances of each species when it is alone. The term -A'i that appears in Table 2, represents the inter-species influence in equation (2). The values of kCi obtained from the ratio of this difference in calculated absorbances and the product of the concentrations of analyte and interferent is a hyperbolic

n

I I 1

l/C, - Fig. 2. Variation of k,. and S against the inverse of concentration of analyte or interferent when the interaction term is constant. (a) Graph of k,. versus l/c,; (b ) S versus l/c,. In both instances c, > cb. 1 , c, = c, with'c, constant; 2, c, = cb with c, constant

Table 2. Calculation of k,. values for the spectrophotometric determination of V(V) with' PGR in the presence of Cr(V1) as interferent. Conditions: pH, 4 (acetic acid - sodium acetate buffer ; ionic strength, 0.1 M; temperature, 25 "C; [PGR], 4,X 10-5 M; [V(V)], 0.81 p.p.m.; A, 490 nm; A, , 0.406; fixed time, 5 min

chi , P*P*m* Cr(V1) A hi A -A'; - k c ,

0.01 0.530 0.403 0.533 81 0.05 0.524 0.395 0.535 16.2 0.1 0.518 0.378 0.546 8.1 0.2 0.505 0.350 0.561 4.05 0.4 0.479 0.297 0.588 2.03 0.6 0.452 0.244 0.614 1.35 0.8 0.426 0.203 0.629 1.01

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1004 ANALYST, AUGUST 1985, VOL. 110

9 2- I

6

3

60

f 40

20

0.2 0.4 0.6 0.8 Cbr P.P.m.

Fig. 3. ( a ) Graph of k,, versus ch, ( b )

25 50 75 100 l/cb,(p.p.m.)-’

0.2 0.4 0.6 0.8 Cbr P*P.m.

graph of k,. versus l/cb and ( c ) graph of S versus ch for the deter- mination of V(V) with Pyrogaliol Red in the presence of Cr(V1) as-ihe foreign species. 1, [V(V)l, 0.81 p.p.m.; 2, [V(V)], 1.22 p.p.m.

Table 3. Calculation of k,. values for the spectro hotometric determination of V(V) with’ PGR in the presence o r Cr(V1) as interferent. Conditions: pH, 4 (acetic acid - sodium acetate buffer); ionic strength 0.1 M; temperature, 25 “C; [PGR], 4 x 10-5 M; [V(V)], 1.22 p.p.m. ; A, 490 nm; A,, 0.356; fixed time, 5 min

chi, P-P.m* Cr (VI)

0.05 0.1 0.2 0.4 0.6 0.7 0.8

A bi

0.524 0.518 0.505 0.479 0.452 0.439 0.426

A -A ’; - k C i

0.342 0.538 8.82 0.331 0.543 4.45 0.309 0.552 2.26 0.259 0.576 1.18 0.218 0.590 0.81 0.199 0.596 0.70 0.186 0.596 0.61

function of cb [Fig. 3(a), line 11, as shown by the linearity of the graph of kCi versus 1/cb [Fig. 3(b), line 11. The equation obtained was

k, = 0.1556 -t 0.6563 ( I / C b )

For 1.22 p.p.m. of V(V) (Table 3) the following expression was similarly obtained:

kci = 0.0747 -4- 0.4573 ( 1 / C b )

and the graph of kci versus l/cb is linear [Fig. 3(b), line 21. In this instance the slope and the intercept on the ordinate of the graph of kci versus 1/cb varied inversely with the analyte concentration, c,.

In both instances, for a known kci, Kci was obtained and on substitution of the values in equation (7) the representation of s versus cb was h e a r [Fig. 3(c) , lines 1 and 21, in agreement with the theory.

If c b < c,/2, it is verified that underestimation of the intercept on the ordinate represents a change in kCi of less than l o%, and by means of the general equation (4) we obtain

(slope)ic,i = constant

This simplification takes us into Case I1 above (constant inter-species influence term).

The authors thank Dr. Adela Sevillano for her help in obtaining the cuperimental data.

1. 2.

3. 4. 5. 6.

7. 8.

9.

10.

11. 12. 13.

14. 15. 16.

17.

References Kaiser, H., Spectrochim. Acta, Part B, 1978, 33, 551. Elving, P. J . , and Kienitz, H., in Kolthoff, I. M . , and Elving, P. J . , Editors, “Treatise on Analytical Chemistry,” Second Edition, Part I, Volume 1, Interscience, New York, 1978,

Wilson, A. L., Talanta, 1974, 21, 1109. Belcher, R., Talanta, 1965, 12, 129. Belcher, R., and Betteridge, D., Talanta, 1966, 13, 535. Maurice, M. J., and Buijs, K. , Fresenius 2. Anal. Chem., 1969, 244, 18. Kaiser, H., Fresenius 2. Anal. Chem., 1972,260, 252. den Boef, G., and Hulanicki, A., Pure Appl. Chem., 1983,55, 553. Fugiwara, K., McHard, J . A., Foulk, S. J . , Bayer, S . , and Winefordner, J . D., Can. J . Spectrosc., 1980, 25, 18. Anastassiadis, P. A., and Common, R. H., Anal. Biochem., 1968, 22, 409. Swietoslawska, J., Chem. Anal., 1971, 16, 337. Pszonicki, L., Talanta, 1977, 24, 613. Pszonicki, L., and Lukszo-Bienkowska, A., Talanta, 1977,24, 617. Gottschalk, G., Fresenius Z. Anal. Chem., 1975, 276, 257. Inczedy, J., Talanta, 1982,29, 595. Charlot, G., “Dosages Colorimetriques des ElCments Mini- raux,” Masson, Paris, 1961, pp. 256-257. Sevillano, A., PhD Thesis, University of Valencia, 1983.

pp. 91-92.

Paper A4/430 Received December 7th, I984 Accepted February 18th, I985

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