The specificity characteristic of analytical methods—II

Talonto, Vol.24,pp.611-623. Pergamon Press, 1977. Printed in Great Britain.

THE SPECIFICITY CHARACTERISTIC OF ANALYTICAL METHODS-II ESTIMATION AND EXPER~ENTAL VERIFICATION OF THE NON-SPECIFICITY COEFFICIENTS LEON PSZONICKI and ANNA EUKSZO-BIE&KOWSKA

Institute of Nuclear Research, Department of Analytical Chemistry, Warsaw (Received 13 September 1976. Revised 1 February 1977. Accepted 22 April 1977) Su~ry-A procedure for estimation of non-s~~city coefficients by use of small amounts of standard samples is described. This procedure enables also the simultaneous verification if the specificity of a method with respect to the investigated interferent may be characterized by means of these coefficients. Four various analytical methods were tested and a good agreement of experimentally found values of interferent effects with the values calculated by means of non-specificity coefficients have been stated.

NOTATION USED

ESTIMATION OF NON-SPECIFICITY COEFFICIENTS

a

-intercept of a straight line graph on the ordinate b slope of the straight line --concentration of interferent Cb --concentration of analyte Cd --relative concentration of interferent, relae, tion of cs to c, KP and KA -non-specificity coefficients R --total analytical signal itnalytical signal of analyte Rd -“parallel shift” component of interfering Rb signal -standard deviation for results indicated by sx subscript mean “lack of fit” square S,. mean “within group” square SW -number of standards in series Y t -number of parallel determinations of a standard sample

The non-specificity coefficients must be calculated from the experimental data obtained by analysis of standard samples. The standards must be prepared separately for every interferent tested The most precise way to estimate the coefficients is to prepare a set of calibration curves with varied interferent concentration,

followed

values

every

for

were defined

two

kinds

such

However,

because

if every curve

method

is tested

interferent, samples

cussed

Here

coefficient

of their

of anaiyt&l

we present

of these coefhcients

methods

a procedure

with solution

was dis-

standards

for esti~~on

very strongly

from the analyte signal;

polarogra-

trations

and would usually chosen

difficult

and thus provide

ideas. In addition,

affect the an~~cal

before measurement

effects are particularly tatively,

to

prepare

be separated

the analyte

used for preparation

to

interferent

improve

to characterize

quanti-

the results obtained

requires

the

for which

to make q = 5

and the total

are easily pres-

617

of

simple, and it may not be used

choice

of

the

samples

procedure,

interferent allows

however.

concen-

simplification

According

I,’ the analytical

to the signal

of

analytical

signal in the presence

(1) of one

is R = kdcd -I- k,c,

ented graphically.

even

the preparation

Rd = k,,c,,

interferent

at

confidence

is

such large

a good test of the proposed

of

of interferer-it.

be lower than 3, testing

one

(4) and (5) Part

of the analytical because

standard

is very time-consuming

in the standard

equations

q2

practice.

of the testing investigated

they were specially

of

and

is relatively

Appropriate

phy and coulometry. results

concentrations

for even

samples,

in analytical

verification

as spectrophotometry,

All the ~terferents

and

q

curve in absence

or 6. Such a procedure

application

and its experimental

for such methods

on q standards

against

the value of q cannot

9 standards,

of

is based

plus the standards calibration

the

range

is un~no~c,

limits of the results it is recommended

and the possibility

to the characterization

a method

it is necessary

the normal

least

of non-specificity

whether

interest.

of the method

I,’

of the coefficient shows

over the concentration

Because

In Part

which

values are constant

the

~TRODU~ON

by calculation

curve,

+ kacdG.

(2)

618

LEONPSZONICKIand ANNALUKSZO-BIE&KOWSKA

The ratio of R to Rd is k + 2% kd

k A, k,

R,=l+

(3)

where c, is the relative concentration of interferent (c, = %/Cd). Taking into account the definitions of K, and KA we obtain R, = 1 + Kpc, + KAc,,.

(4)

When a series of standards is prepared so that c, is constant, then R, is a function of cb only. This function R&J, described by equation (4), enables us to calculate the K-values and to test their constancy by means of 4 standards. When R&J is linear, then K, and KA are constant hnd they are described by the parameters of its regression line: a-l KP = -

(5)

c, KA = b

(6)

The standard deviations and confidence limits of the

K-coefficients may be calculated from the standard deviations of a and b: %P = s&,

and

sKA= s,.

(7)

In the calculation of the regression parameters and verification of the linearity it is important that the variances of all measurements should be homogeneous. In chemical analysis the variance of the measurements usually increases with increasing concentration and this requirement is not fulfilled. It is therefore necessary to use weighting factors in all the statistical calculations and hence the dispersion of the measurements at various concentrations ought to be estimated. The variances of the R,-values can be calculated from the variances of corresponding values of R,, and R, taking into account the law of propagation of errors. The variations in parallel determinations of Rd and R are not correlated, so the following simplified formula may be used:

Calculation scheme 1

1. Calculate the values of R,, . . . Rrq as the ratios of values of R (Table 1, column 8) to corresponding values of Rd (column 3). 2. Calculate the variances si, from the corresponding variances si and s&. 3. Calculate the parameters a and b of regression line RJc,,) by the least-squares method, using the weighting factors, and then calculate the variances s,’ and s:.

R2

2 2 AL++2 SR, Rd’

R;

The general schedule of measurements shown in Table 1 is sufficient for complete calibration of an analytical method. The variances of Rd and R as well as of Rb must be estimated experimentally on the basis of t parallel determinations on standard samples. The minimum value of t may be taken as 3, but if it does not lead to too much extra work this number ought to be higher, e.g., 5 or 6. In practice, an interferent rarely affects the dispersion of results significantly and the magnitude of the variance is dependent only on the magnitude of the analytical signal. Because small errors in estimation of si or s& values do not have a significant influence on the values of the calculated coefficients the variances of R and Rb may be taken as equal to the variance of Rd obtained for a standard that emits a corresponding magnitude of signal. The variance of R need be determined experimentally only for the range of concentration where R significantly exceeds Rd, but such cases happen rather rarely in practical analysis. When t parallel determinations are done, then the values given in Table 1 represent the mean values and their variances. When the variances for R or Rb are taken as the same as for Rd, then the values of R and Rb can be based on single determinations and the variances of single determinations must also then be used. For all interferents for which Rr(c,,) is linear, the values of the measurements tabulated in columns 6 and 7, 11 and 12 etc. are redundant and can be omitted. The calculations are carried out according to the following scheme.

Rd

Table 1. Schematic table of measurements Analyte Standard No.

$, 3

4

Interferent A c,, = R Cb Rb (Cb= C,,.Cd) (Cd= 0) %b (c, = const.) 5

6

7

8

Interferent B cr, = $

Cb

Rb

s&

R

s;

9

10

11

12

13

14

1

2

1 2

cd, Cd*

cO,‘cd,

C,,‘Cdl

‘k’cdz

cr,‘cdz

4

‘d,

cr.., ’ cd.

&a’

etc.

cd,

Note. The values of cd,, cdl . . cd,,and c, are dependent only on the concentration ranges of interest. The data given in columns 6 and 7, 11 and 12 etc. are necessary only when R, = Rr(cb) is not linear.

Specificity characteristics of analytical methods-11 4. Using the weighting factors, calculate the “within-group” sum of squares Of results [i.e., C~i - yi)‘] and the “lack-of-fit” sum of squares, which is the difference between the residual sum of squares about the regression and the “wi~n-group” sum of squares. 5. Calculate the mean “lack-of-fit” square (St) and mean “within-group” square (S,) and apply the F-test to verify the hypothesis of linearity of R,(c,,):~ F = S,_/S,.

6. If F > Ftab the function R&J is linear. Calculate the values of the K, and K:, coefficients and their standard deviations. An example of calculation according to scheme 1 of K-values for spectrophotometric determination of uranium with respect to thorium as interferent is given in Appendix I. When an analytical method does not fulfll all the simplifying assumptions made in deducing the K-coefficients, then the function Rr(cb) is not linear and at least one of the K-coefficients is not constant. If the interferent signal R, is responsible for the deviation from the linearity and this effect is not large, the specificity of the method may be characterized by means of K-coefficients expressed as linear functions of c,. In such a case the &-value must be calculated from the functions Rd(cd) and R$.z,,), the second of which describes the component of the interfering signal that is responsible for the parallel‘shift of the calibration curve and is inde~ndent of the concentration of anaiyte (see Part I).’ For this purpose it is necessary to prepare an additional series of standards (see columns 6, 11 etc. in Table 1) that contain the corresponding concentrations of interferent (cb = c,-cd)but no analyte. The total number of standards necessary for testing the method with respect to such an interferent is 2q. The K-coefficients must then be calculated according to scheme 2. Calculation scheme 2

Steps l-5 are in scheme 1. 6. When F < Ftab calculate the parameters of the regression lines for R&cd) and R&T,,), using the weighting factors, verify the linearity by means of the F-test and check whether RJO) = 0 and R;(O) = 0 by means of the t-test. Rd(0) and R;(O) may be accepted as equal to zero when the corresponding values of lai/s, are less than tfab. Note. When Rd(cd) or R&J is linear and does not pass through the origin then the analytical procedure is incorrect, e.g., it is necessary to use a blank correo tion. 7~. When Rd(cd) and R;I(c,,) are linear and pass through the origin Kp must be constant. Calculate its value as the relation of the slope of R&J to that 0f R&i). 7b. When R&J is not linear, calculate the local values of k, and then the local values of Kp as the ratio of k, to kd, find the parameters of the regression line of K&c,,) and verify its linearity. If it is

619

linear then KP = a + bc,. Its confidence limit may be estimated only locally for individual c,-values: KP f Cf+bQ.*f0.95@-2). Note. If R&J is not linear or K&J is not linear and not constant, the specificity of the method relative to the interferent tested cannot be characterized by means of the K-coefficients. 8. Using the constant value of Kp or local values of Kp, calculate the local values of KA and their variances. 9. When Kr # const., test whether KA = const. by means of the t-test. Note. K, may be constant despite R&J not being linear, when only the Rb-signal is responsible for this deviation from linearity. 10. If KA Z const. calculate the parameters of the regression line K&z,) and verify its linearity. When it is linear KA may be expressed as KA = a + bcb and its confidence limit may be estimated only locally (see 76). Note, If KA Z const. and KA(c,,) is not linear, the specificity of the method relative to the interferent tested cannot be characterized by means of the K-coefficients. An example of calculation (according to scheme 2) of the K-coefficients for coulometric determination of uranium with molybdenum as interferent is given in Appendix II. Sometimes the coefficient KA ought to be expressed as a linear function of the analyte concentration; then all the operations given in step 10 must be carried out for K&J. The expression of the K-coefficients as linear functions is necessary when the influence of the interferent on the analytical signal is approximated better by a second-order function than a linear function and it usually concerns the coefficient KA that describes the inter-effect between the analyte and the interferent. The application of this procedure widens the possibilities of the approach. It is particularly useful when the K-coefficients are used for mathematical correction of the analytical results obtained in the presence of certain interferents, some of which do not fultil the ~mplif~g assump~ons well. The K-coefficients can also be expressed by means of functions other than linear, but this complicates the problem and finds application only for special cases. The procedure described allows the complete testing of an analytical method for all interferents of interest by use of relatively few standards, and also whether the simplifying assumptions are satisfied. However, it does not permit verification of the assumption that the activity of individual interferents is independent of their mutual proportions in the sample. One way of verifying this is to compare the result for a standard mixture containing known concentrations of the analyte and all interferents of interest with the result (corrected by means of the K-coeficients) for a sample containing the same concentration of pure analyte. The complete statistical evaluation of the data used

LEON PSZONKKI

620

and

ANNA LUKSZO-BI~~KOWSKA

for calibration of an analytical method according to schemes 1 and 2 is rather arduous and requires the use of a computer. However, particularly when the results are read from a graphical calibration curve, a much simpler procedure can often be used, that gives sufficiently correct results. The estimation of the parameters of all the test-functions as well as the verification of their linearity may be done graphically and the K-coefficients these parameters.

calculated.

simply on the basis of

EXPERIMENTAL

trodes. Equipment for controlled-potential coulometry based on the Tacussel PRT-500 Lc potentiostat with IG-38 integrator. Minicomputer. Methods

used

1. Spectrophotometric

uranium(W) determination with Arsenazo-III according to Savvin (without extraction).3 2. Spectrophotometric manganese determination with formaldoxime according to Marczenko.4 3. Polarographic determination of uranium(V1) in 2N sulphuric acid medium according to Kolthoff and Harris.’ 4. Controlled-potential coulometric determination of uranium(W) in 5N sulphuric acid medium according to Farrar et a1.6

Apparatus

Standard samples

Unicam SP 500 spectrophotometer. Radelkis OH-102 polarograph with dropping mercury and calomel elec-

The standard samples used for testing the methods above were prepared according to Table 1.

Table 2. K, and KA values for spectrophotometric determination of uranium(W) with Arsenazo III (uranium concentration range %201ugl 10 ml)

Interferent

Interf. concn. range, w/l0 ml

Th Ce Gd

&20 620 Cl5

0.95 1.62 1.64

-1.16 x 1O-2 -7.30 x 10-J -7.15 x 10-a

Table 3. KP and KA values for spectrophotometric determination of manganese with formaldoxime (manganese concentration range O-25 pg/ 1Oml)

Interferent

Interf. concn. range w/10 ml

Ni Fe co MO

O-20 O-40 O-300 0-1000

1.30 0.43 0.053 0

0 0 -3.00 x 1o-4 3.70 x 1o-5

Table 4. KP and KA values for polarographic determination uranium(V1) (uranium concentration range O-1000 pg/lO ml)

Interferent

Interf. concn. range w/l0 ml

KP

V cu MO

O-400 @200 O-250

0 2.336

0.357

of

(J;l)-,’ 5.58 x 1O-4 -5.02 x 1O-4 8.01 x lo-’

Table 5. K1’ and KA values for coulometric determination of uranium(W) (uranium concentration range &4 mg/sample) Interf. concn. range Interferent

mglsample

V MO Fe cu Ce

W.5 0.5 O-l.25 k2.0 W3.6

KA

KP

2.242 5.366 2.010 3.607 0.669

@g/sample)-

0.506 3.069-5.433 cb 0 0 0

621

Specificity characteristics of analytical methods-II

,5. UtlOThtSGdtSCe

Mnt2ONi

d

1.C

,’

UtlOGd

ut5ce Ut5Th U

Fig. 1. Comparison of experimentally determined (points) and calculated (dashed lines) interference effects in the spectrophotometric determination or uranium 1-Uranium; 2+individual interferents; 5-mixture of interferents. The numbers by the symbols of interferents indicate the concentrations in pgg/lO ml.

pg

Mn/lOml

Fig. 2. Comparison of experimentally determined (points) and calculated (dashed line) interference effects in spectrophotometric determination of manganese. l-Manganese; 2-5-individual interferents; 6-mixture of interferents. The numbers by the symbols of interferents indicate the concentrations in ng/lO ml.

RESULTS AND DISCUSSION The

ficients

possibility of applying the non-specificity coeffor determining the practical specificity

characteristics of the four methods mentioned above was investigated. Each method was tested with respect to the strongest interferents. The KP and KA values obtained experimentally are shown in Tables 2-5. The full data and calculations for an example of each computation scheme are given in Appendices I and II. The comparison of the experimental results with the results corrected by means of the K-coefficients is presented in Figs. 14. The points plotted represent the experimental values of the analytical signal for the analyte in presence of one interferent or a mixture of some interferents. The continuous line represents the normal calibration curve in the absence of interferents and the dashed lines represent the theoretically calculated values, i.e., the calibration curve values corrected by means of the K-coefficients for a defined concentration of one interferent or a mixture of interferents. It is seen from Tables 2-5 that for 15 interferents tested in the four methods the K-coefficients are constant and that only in one case, molybdenum in the coulometric determination of uranium, the coefficient KA must be described by a linear function. The agreement of the calculated and experimental values is excellent for the determinations in presence of single interferents in all the methods investigated. This fact confirms that the K-coefficients may be used successfully as a characteristic of analytical methods and that

the proposed calibration procedure based on a small number of standards gives sulkiently precise results. The worst situation is for the multi-interferent mix-

c+‘4.

Ut250

,3.

ut4oov

/5.

Utloocui .IOOV t 50Mo

,I.

u

,, /’

7’ ,

p’

I

I

I’ ,

,’

Id

’

I’ d’ , /

I’ I’

/, 200

I 400 pg

I’

MO

,o

/

I 600

I 800

U /IO

ml

/

/*.,2.

ut2oOcu

I 1000

Fig. 3. Comparison of experimentally determined (points) and calculated (dashed lines) interference effects in polarographic determination of uranium. l-Uranium; 2&individual interferents; 5-mixture of interferents. Numbers by the symbols of interferents indicate the concentrations in pg/lO ml.

LEONPSZONICKIand ANNALUKSZO-B&KOWSKA

622

ture. In this case the deviations of the calculated results from the experimental are significantly larger than the reproducibility of the methods. However, taking into consideration the very strong interfering signals and the possibility of accumulation of errors made in the estimation of K-values for single interferents, the additivity of interfering signals can be accepted as established. On the basis of these experiments we conclude that the non-specificity coefficients characterize quantitatively the specificity of the analytical procedures investigated.

,6.6. ,’

,’

Ut2.OCu

/’ 1 u+o.5v+o.I

/’

t

>-

REFERENCES 1 L. Pszonicki, Talanta, 1971, 24, 613. 2. 0. W. Davies and P. L. Goldsmith (eds.), Statistical Methods in Research and Production, 4th Ed., pp.

I

199-201. Oliver and Boyd, Edinburgh, 1972. 3. S. B. Savvin, Talanta, 1961, 8, 673. 4. Z. Marczenko, Spectrophotometric Determination of Elements, p. 342. Horwood, Chichester, 1976. 5. I. M. Kolthoff and W. E. Harris, J. Am. Chem. Sot.,

i

1946, 68, 1175.

L

6. L. Farrar, P. Thomason and M. T. Kelly, Anal. Chem., 1958, 30, 1511.

mg U /sample

I. S. Brandt, Statistical and Computational Methods in Data Analysis, North-Holland,

Amsterdam, 1976.

8. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969. APPENDIX I Calculation of K, and KA values for spectrophotometric determination of uranium(V1) in presence of thorium. Note. All statistical calculations presented in Appendices I and II were carried out according to the formulae given by Brandt’ and Bevington.’ Number of standards in series; q = 5. Number of parallel determinations of each standard:

t= 5.

Fig. 4. Comparison of experimentally determined (points) and calculated (dashed lines) interference effects in coulometric determination of uranium. l-Uranium; 2-bindividual interferents; 7-mixture of interferents. Numbers by the symbols of interferents indicate the concentrations in mg per sample.

Relative concentration of interferent: c, = 1. cd and cb in &lOml. R, and R in absorbance units. All R, and R values are the mean values of five measurements and the variances are the variances of these means.

Experimental data Standard No. 1 2 3

4 5

Cd

Rd

s;, x lo6

2 5

0.043 0.107

0.2 0.8

10 15 20

0.214 0.319 0.428

1.8 5.0 5.0

R

s; x lo6

2 5

0.085 0.198

0.8 0.8

10 15 20

0.388 0.568 0.742

3.2 12.8 28.8

cb

Calculated values of the total relative signal R, Standard No:

Ch

R,

$, x 104

1 2 3 4 5

2 5 10 15 20

1.977 1.850 1.813 1.781 1.734

8.6 3.2 2.0 2.8 2.4

Regression parameters of R&J: b = -9.68 x 10m3; S, = 1.621 x lo-*; SW = 8.951 x 1O-4; F = 18.10 > F,.,,(,,,, = 10.13; R, (cb) is linear. SK, = 0.03; SK, = 2.3 x 10-3; to.ss(3j = 3.18; Kp = 0.92 f 0.09 and K, = -0.01 f 0.007 (pg/loml)-1. a = 1.92;

623

Specificity characteristics of analytical methods-II APPENDIX

Relative concentration of interferent: c. = 0.1. cd and c,, in mg/sample. Rdr Rb and R in coulombs. All Rd, Rb and R values are the mean values of five measurements and the variances are the variances of these means.

II

Calculation of KP and K,, values for coulometric determination of uranium(V1) in presence of molybdenum Number of standards in series: q = 5. Number of parallel determinations of each standard:

t= 5. Experimental data Standard No. 1 2 3 4 5

0.2 0.5 1.0 2.0 4.0

R,

& x lo6

Cb

Rb

s&, x lo6

R

s; x lo5

0.142 0.409 0.812 1.621 3.254

1.8 1.8 5.0 7.2 20.0

0.02 0.05 0.10 0.20 0.40

0.086 0.216‘ 0.435 0.868 1.742

1.8 1.8 7.2 9.8 12.8

0.275 0.685 1.455 3.133 6.157

0.32 0.98 2.0 4.6 18.0

.

Calculated values of total relative signal R, Standard No. I

2 3 4 5

0.02 0.05 0.10 0.20 0.40

1.586 1.675 1.792 1.933 1.892

29.5 8.9 5.5 2.8 2.4

Function R&j: a = 1.75; b = 0.445; S, = 1.84 x lo-‘; SW = 8.51 Function R&J: s, = 0.005; b = 0.811; a = 1.41 x 10-J; F = 1500 $ F0.95(1,3)= 10.13. R,(c,) is linear and passes through the origin.

x

10-s; F = 2.16 < Fe.sstf,s) = 10.13; Rt(cb) is not linear. t = 0.28 < t0.s5(aj = 3.18;

SL = 2.10;

S, = 1.4 x 10-d;

Function R&J: t = 1.44< t0,95(3j = 3.18; S, = 0.793; SW = 1.46 x lo-“; a = -9.90 x 10-4; b = 4.36; s, = 6.9 x 10-4; F = 5.4 x 10’ r> F0.95c1,3, = 10.13; R’&)b) is linear and passes through the origin; sK. = 0.047; Kr = 5.37 + 0.15. Local values of K, Standard No. 1

2 3 4 5

cb

0.02 0.05 0.10 0.20 0.40

KA

2.45 2.16 2.55 1.98 0.89

&A

6.4 x 3.2 x 5.2 x 7.2 x 1.6 x

10-s lo-& 1O-5 1O-6 10-e

Function K&J: U = 3.07; b = 5.46; S, = 1.17; S, = 1.19 x lo-+; F = 9800 % FO,,so,,, = 10.13; K&q,) is linear. KA = 3.07 - 5.46 q, (mg/sample)-‘.