By applying the method of least squares it is possible to estimate simultaneously the diffusion current and the half-wave potential with precision which exceeds that of the usual graphical methods. The method described in this paper is particularly suitable for analysing composite current-voltage curves. An example is given of the analysis of the composite curve obtained with a solution containing thallous and lead ions in 0.1 N KC

O. A. Weber

P. K. Karmalkar

V. B. Vouk

English

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ARHIV ZA KEMIJU 27 (1955) 9 Treatment of Polarographic Data by the Method of Least-squares. II. Simultaneous Estimation of the Diffusion Current and the Half-wave Potential" V. B. Vouk, P. K. Karmalkar** and 0. A. Weber Institute for Medical Research, Yugoslav Academy of Sciences and Arts, Zagreb, Croatia, Yugoslavia Received March 3, 1955 By applying the method of least squares it 1s possible to estimate simultaneously the diffusion current and the half-wave potential with precis.ion which exceeds that of the usual graphical methods. The method described in this paper is particularly suitable for analysing composite current-voltage curves. An example is given of the analysis of the composite curve obtained with a solution containing thallous and lead ions in 0.1 N KCL INTRODUCTION In a previous communication from our laboratory1 it has been shown how the method of leas·t squares may be succesfully applied if precise values of half wave potentials are needed. This method is, howev·er, limited to cases where the diffusion current can be estimated with sufficient precision by graphical methods. The method described in this paper enables precise esti-mation of both the half wave potential and the diffusion current, if apprnxi-mate values of these parameters are available. METHOD Suppose we have N pairs of observed values (x, y) and we want to fit the equation y = f (x, A, B) (1) to these points; A and B are constants the approximate values (A0 , B 0 ) of which are known so that A = A 0 +a; B = B 0 + b (2) where a and b are small correction terms. Using the method of least squares2 the correction terms a and b can be calculated by solving the normal equations ~(qi - af.11 - bfn) f A = o ~ (qi -afA -bfn)f;; = o • I.: O. A. W e b er, Arhiv kem. 25 (1953) 231. (3) ** Department of Chemistry, Holkar College, Indore, India ; has been awarded a fellowship by the Federal Executive Council of P. R. Yugoslavia. 10 V. B. VOUK, P . K. KARMALKAR ANDO. A. WEBER where cp , fA, fB are known, and a and b unknown quantities. cp, fA and fB are calculated as follows: cp = y (obs .) - f (x, A 0 , B 0 ); of (x, A 0 , B 0 ) f A = oA fn = of (x, ~0, B 0 ) oB (4) (5) Partial derivatives fA and fB are calculated from eq. (1) and evaluated for each observed value x using A 0 and B0 • N values of fA, fB and cp are thus obtained which aire used in solving the normal equations (3). We shall apply this procedure to the equation of Heyrowsk.y and Ilikovic3• RT Ea e = E , - - ln . . " nF ia-i (6) where Ea,e. is the potential of the illopping electrode at any point on the wave, E . , is the half wave potential, i is the current at any point on the wave and id is the diffusion current; symbols R, T, n and F have their usual meaning. In order to 1simplify the notation we shall write Ea.e = y, i = x, E , 2 = A, id = Band k = RTlnF; 2.303 k = k'. In terms of thiis n<Ytaition equation (6) reads: y = f (x, A, B) = A - k ln [x/ (B 0 - x )] (7) Hence fA = 1; fs = k/(B0 -x) and cp = y (obs)-A 0 + k ln[x/(B 0 -x)] (8) and the normal equations (3) reduce to aN + b ~ f n = 2J cp (9) ex~ f n + b ~ f2n = ~cp f B from which it is easy to calculate a and b, if we compute ~ fn, ~ f2B, L: cp and ~cp fn. Approximate values A 0 and B 0 can always be obtained by graphical methods. ANALYSIS OF CURRENT-VOLTAGE CURVES As an example of the application of this method we give the analysis of current voltaige curves obtained with aqueous 0,1 N KCl solutions containing 4.0 X 10-4 N thallium nitrate (Fig. 1, curve 3), 4.0 X 10-4 N lead acetate (Fig. l, curve 2) and 4.0 X 10-4 N thallium nJtrate + 4.0 X 10-4 N lead acetate .(Fig. 1;, curve 1), respectively. Approximate values for the diffussion current (B0 ) and the half wave potential (A 0 ) were obtained graphically. The value o{ the current (x) cor-responding to a given dropping electrode potential (y) was determined by measuring the distance between the extrapolated residual current and the corresponding point on the wave (see for instap.ce fig. 1, curve 3). This method TREATMENT OF POLAROGRAPHIC DATA 11 was used wi1th curve (2) and (3) and with the first part of the curve (1). :For the second part of the curve (1) the difference was measured between the extrapolated diffusion current and the corresponding rpoint on the cur,.. rent voltage curve, and the value for the current (x) calculated from the dif-ierence B 0 - (B0 - x). (See Fig. 1, curve 1) It should be pointed out that only those parts of the composite curve (Tl +Pb) were used in the analysis which were not d1storted by the interference of Pb and TI-ions, that is, the current values corresponding to a potential not greater than-400 mV (for Pb++) and not less than -440 m V (for TI+) were taken. 111 1 - -JSO -•50 ·SJO ..... 11 1'1 11 1' 11 tll ll l a. ~ 1 - tZ: -J50 ·UO fJ/ ~ +oo -JOO .. ~ F ig . 1. Current voltage curves obtained with solutions of thallous and lead ions in 0.1 N KCI. 1.) 4.0 X 10- • N thallium nitrate + 4.0 X 10- • N lead acetate; 2) 4.0 X 10- • N lead acetate; 3) 4.0 X 10-• N thallium nitrate. As an example of the computation procedure, the analysis of the first part -of curve (1) is given in full: A 0 = -395 m V, B0 = 25 mm (from curve (2) estimated graphically) k' = 29.578 mV (calculated from R, T and n = 2); N = 6 .a) computation of f 0 = A 0 - k ln [x/ (B 0 - x)] x Yobs B 0 -x I x/(B 0 -x) liog f x/(Bo -x)] I fo 1.3 -360 23.7 I 0.0549 -1.26081 -357.71 3.2 -370 21.8 0.1468 -0.83326 -370.35 4.9 -375 20.1 0.2438 - 0.61297 -376.87 5.8 -380 19.2 0.3021 -0.51985 -379.62 10.0 -390 15.0 0.6667 - 0.17612 -389.79 1 5.0 -400 10.0 1.5000 - 0.17609 -400.21 12 V. B . VOUK, P . K. KARMALKAR AND 0. A. WEBER b) Computa:tion of qi, fB = k/(B 0 -:t) and the necessary sums. cp fA qi fs - 360 -357.71 - 2.29 0.542 -370 - 370.35 0.35 0.589 - 375 - 376.87 1.87 0.639 -380 -379.62 - 0.38 0.669 - 390 - 389.79 - 0.31 0.856 - 400 -400.21 0.21 1 285 sums - 0.45 4.580 The following sums a·re calculated directly from table b) by means of a calculating machine: lJ f2B = 3.880528; lJ fo = 0.00423 ; (lJ fo)2 "= 20,9764 c) Determinants necessary for calculating the correction terms a and b (see equation 9) were also obtained by means of the calculating machine from. the values given antler b): lJqi 2J f2n N lJ qi fH N 2J f2g a = -1.77/2.31 = - 0.8; b = 2.09/2.31 = + 0.9 Corrected values A and B are thus: = -1.77 2.09 2.31 A = A 0 +a = - 395.8 mV; B = B 0 + b = 25.9 mm. Curves (2) and (3) and second part of curve (1) were analysed in a similar way. Table I compares the half wave potentials (in mV) and the wave heights (in mm) obtained with solutions containing Pb++ and Tl+ alone with those-calculated f.rom the compooite curve (1) obtained with a solution containing both thallous and lead ions. The agreement is very good. It is obvious that curve (1) could noit have been analysed by the usual graphical methods. In ol'."der to see to what extent the first order corrections a and b can. improve the approximate values A 0 and B 0 , we tried to analyse curve (1} starting with A 0 = -400 mV and B 0 = 28 mm, and we obtained A = A 0 + a = = -395.73 mV and B = B 0 + b = 25.74 mm which is in excellent agreement with the results shown in table I. These examples s:how the usefulness of the descr.ibed method for the analysis of composite current voltage curves exhibiting marked interference of the constituents of the ipolaroigraiphed solution. There are many other instances where this method of analysis might be successfully applied , particularly ill analytical applications of polarography. TREATMENT OF POLAROGRAPHIC DATA 13 EXPERIMENTAL A cambridge Polarograph, Pen Recording Type, and a standard .polarographic <:ell with external saturated calomel electrode were used. Dis'Solved oxygen was <li'Splaced by bubbling purified hydrogen through the solution for 15 minutes prior to the rpolarographic determination . The dropping time was arpproximately 3 seconds. TABLE I I Half-wave I Difference Wave I Difference Measured I from composite cu rve Pb -single curve I c.omposite curve Tl single curve Curve No. 1 2 3 potential mV I mV \ -395.74 11 007 -- 395.67 I l - 464.39 --1-1.0l l - 465.40 I I h eight I O/o O/o mm mm i I 25.8 I I I 0.02 + 0.4 1.6 25.4 I I l 31.5 0.22 -0.1 0.32 31.6 'The ·potential readings on the bridge were checked against a standard Weston €lement. The rate of change of the potential was 18.8 mV/min.; dampin g 4, range •0.5/0; 1 cm on the potential axis of the polarogram thus corresponded to 50 mV ; sensitivity 1/2·0 of the maxim al, i. e. 52.65 X 10- 9 A/mm . Analytical grade chemicals and redistilled water were used for preparing the ·solutions. The solutions to be analysed were prepared by dilution from stock solu--tions containing 0.002 N TlN03, in 0.1 N KCI and 0.002 N Pb(CHaCOOh in 0.1 N KCl; ·0.0020/o of gelatin was added as maximum suppressor. Computations were performed with a F·riden Automatic Calculator . REFERENCE S 1. 0. A. W e b er, Arhiv kem. 25 (1953) 231. 2. See e. g. H . Marg e n au and G. M. Murphy, The Mathematics of Physics and Chemistry, Van Nostrand Co., New York, 1943, p . 500. ::3 . J . Hey Tows k y and D. I 1 k o v i c, Collection Czechoslov. Chem. Commns. 7 (1935) 198. IZVOD Obradivanje polarografskih podataka metodom najmanjih kvadrata II. Simultano odredivanje difuzione struje i poluvalnog potencijala* V . B. Vouk, P. K. Karmalkar** i 0 . A . Web er Primjenom metode najmanjih kv2drata m oze se simultano odrediti difuziona ·s truja i poiuvalni potencijal. Preciznost tog odredivanja premasuje preciznost, koja :se moze postici uobica jenim grafickim m etodama. Opisana m etoda osobito. je pri-kladna za analizu slozenih p olarografskih krivulja. Kao primjer d an a je slozena ::polarografska wrivulja, koja je dobivena anali.zom otopine talijevih i olovnih iona u 0.1 N otopini KCL INSTITUT ZA MEDICINSKA ISTRAZIVANJ A . .JUGOSLAVENSKA AKADEMIJA ZNANOSTI I UMJETNOSTI ZAGREB * I. : 0 . A. W eber, Arhiv kem. 25 (1953) 231. Primlj e no 3. marta 1955 . "*' Depertment of Chemistry, Holkar College, Indore, Indija. Stipendist Saveznog izvrsnog vijeca FNRJ. 

Treatment of Polarographic Data by the Method of Least-squares. II. Simultaneous Estimation of the Diffusion Current and the Half-wave Potential

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