The application of the half-wave potential in polarographic investigations is reviewed. In view of the fact that the methods currently used for computation of the half-wave potential do not give very reliable values and do not allow the estimation of experimental
errors, an attempt is made to apply the least-squares method to the computation of the half-wave potential from the current-voltage data

O. A. Weber

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ARHIV ZA KEMIJU 25 (1953) 231 Treatment of Polarographic Data by the Least-Squares Method I. Estimation of the Half-wave Potential 0. A. Weber Institute for Medicai Research, Yugosiav Academy of Sciences and Arts, Zagreb, Croatia Hecei ved No vember V, 1953 The application of the half-wave potential in polarographic investigations is reviewed. In view of the fact that the methods currently used for computation of the half-wave potential do not give very reliable values and do not allow the estimation of ex-perimental errors, an attempt is made to apply the least-squares method to the computation of the half-wave potential from the current-voltage data. P rior to the introduction of the half-wave potential in polarographic literature1 several attempts were made to characterize electroreducible and electrooxidizable substances by arbitrary points on the current-voltage curv.e. In this connexion the deposition potential2• 3, the reduction potential4• 5 and the characteristic potential obtained by applying Le Blanc's procedure6 should be mentioned. However, none of these arbitrarily selected points could be used for characterization of the processes at the dropping electrode and consequently they had little practical significance. The half-wave potential is best defined using the equation of Heyrowsky and Ilkovic6 : RT i Ede = E11 - - ln--2 nF id-i (1) This function has an inflexion point at i = ~ . The potential corresponding 2 to this point is the half-wave potential. Reversible electrode processes ar~ well characterized by this potential, which is independent from the concentration of the reacting substance, the sensitivity of the recording instrument, the drop time, the rate of flow of mercury, the spacing of the apscissa and the value of the diffusion current provided the concentration of the supporting electro-lyte and the temperature are constant. The influence of the concentration of the supporting electrolyte on the half-wave potential was investigated by Lingane7, who found that increase in concentration of the supporting electrolyt~ makes the half-wave potential more negative. The half-wave potential of a given substance does not depend on the concentration of other oxidizable or reducible substances which might be present in the solution. The temperature coefficient of the half-wave potential of simple metal ions is - 0.7 mV/° C if the reduction at the dropping electrode is reversible8• The same temperature coefficient has been obtained with some organic irreversible systems9 • The half-wave potential has a well defined thermodynamic meaning being in a simple way related to the normal potentiaF· 10. The thermodynamic 232 0 . A . WEBER equation of a reversible polarographic wave becomes identical with the equation of the potentiometric titration, if the volume of the titrating solution is subsituted for the current; and the half-wave potential corresponds to the potential obtained when exactly one half of the equivalent amount of the titrating solution is added. Recently it has been observed that the half-wave potential is influenced by the drop time18 and the rate of flow of mercury19 in such a way that LI E11'J. 0.030 ---- ~--L1 log m - n (m is the rate of flow of mercury in mg sec- 1 ; n is the number of equivalents involved). The knowledge of the half-wave potential is important in studying the formation of complex metal ions. The half-wave potential of complex metal ions is usually shifted towards negative values. The composition of the complex and the complexity constant can be calculated from the magnitude of this shifto. u , 12. The half-wave potential is also used in organic polarography in connexion with :the stydy of redox potentials, of stn,1ctural problems, of Raman- and absorption spectra 14• 15• 16 as well as a criterion of reversibility17. Graphical methods are usually employed for determination of the half-wave potential. Tomes10 has proposed an indirect method for determination of the half-wave potential based on graphical solution of the Heyrowsky-Ilkovic6 equation. Linear interpolation was applied by Meites21 to a series of pairs of measurements of current and potential in the neighbourhood of the half-wave potential, while the symmetry of the polarographic wave was used in a method published by Monnier and Rusconi22• By applying a geometrical method they found the centre of symmetry and calculated the corresponding potential, which was identical with the half-wave potential. Recently, Heyrowsky2s determined the halt-wave potentials from the maximum of the function: !::.I/!::.E = f (E) . The oscilloscope has also been used for this purpose 24- 21. Although the methods are numerous, little is known about their precision and reliability. If very small differences of the half-wave potential are involved (for instance when studying the influence of the rate of flow of mercury on the half-wave potential19) it is most important to know the experimental errors and to be able to test the significance of such differences. In such cases the usual procedures do not seem satisfactory28• Therefore, an attempt was made to apply the least-squares method to this problem. ESTIMATION OF THE HALF-WAVE POTENTIAL BY THE LEAST-SQUARES METHOD The purpose of the present paper is to show the method of least-squares can be applied in estimating the half-wave potential from the current-voltage data. By rearranging equation (1) and using logarithms to base 10 we obtain: i 0.4343 nF E 0.4343 nF log-. --. = 112 - Ede. rd-z RT RT (2) ES'DIMATION OF THE HALF-WAVE POTENTIAL n.' setting i lo[1-- = y; id- i _ 0.4343 nF = b and 0.4343 nF = a RT RT E112 ·wt can write equation (2) as y = a + b . Ede "For y = 0, Ed e• = - ~ = E11 b 2 233 (3) By comparing 1/b with its theoretical value 0.0591/n (at 25°C) an estimate of n can also be obtained. Since both y and Ede can be measured, the problem of estimating the half-wave potential reduces to a linear regression analysis of y upon Ede The regression equation Y = a + b ·Ede - -- N 2Ey - L.,£ 2y -with a = y - b E, b = ---=----~ N 2£2 - (2£)2 1 -,,E ) = - ~ de• N is then used for estimating Eae at Y = 0. The standard error of Ede at Y = 0 is easily estimated !from the -e0uation31 : (4) where V is the variance of y about the regression Y = a + bE. For computing V the following expression29 may be used : { ( 2Ey-__!_ 2E 2y) V _ 1 ....:• 2 1 ( ,-. )2 N --- ~Y -- -""Y -N-2 N :sE2 _ __!_ (:sE)2 N N is the number of measured Ede• y pairs. The standard error of 1/b = RT/0.4343 nF can also be estimated from the -equation: 1 S t,b =-Sb b2 where The above procedure has been applied in calculating the half-wave potential -Of - 6 uranium in a supporting electrolyte containing salicylic and sulphuric acid .30 234 0. A. WEBER HEF.KRENCBS 1. J . Hey row sky and D. Il ko·vic, Collection Czechoslov . Chem. Com mns. 1 (1935) HJ8. 2. J. Hey rows k y, Phil. Mag. 45 (1923) 303. 3. M. Shi !k at a, Mem. Coll. Agr. K yoto Imp. Univ ., Ser. No 4 (1927) 1. 4. J . Hey rows k y, Ree. trav. chim. 44 (1925) 488. 5. G. Se me ran o, Gazz. chim. ital. 62 (1932) 518. 6. A. W:in lke l and G. Proske, Ber. 69 (1936) 693. 7. J. J . Ling an e, J . Am. Chem. Soc. 61 (1939) 2099. 8. I. M. Ko 1 t ho ff and J . J. L i n g an e, Polarography. New York 1952. p . 202 . 9. 0 . H. M il 11 er in A. Weis s b e r ge r, Physical Methods of Organic Chemistry , Vol. I, part II. New York 1949. p . 1838. · 10. M. von S tack e lb erg, Z. Elektrochem. 45 (1939) 466. 11. M. von Stackelbe rg and H . von Freyhold, Z. Elektrochem. 46 (UHO) 120. 12. J. J. L i ng a n e, Chem. Revs. 29 (1941) 1. 13. 0 . H. M il 11 er, Ann. N. Y . Acad. Sci. 40 (1940) 91 - see ref. No 9, p. 1829. 14. H . Shi k at a and I. T ac hi, Collection Czechoslov. Chem. Commns. 10 (1938) 369. 15. H. Luc as, J. Am. Chem. Soc. 48 (1926) 1827. 16. A. Winkel and G. P ros ke, Ber. 69 (1936) 1917. 17. 0. H. M il 11 er and J. P. B aumberg e r, Trans. Electroche m. Soc. 71 (1937) 181. 18. H. St re h 1 ow and M. v on St ack e 1 b erg, Z. Elektrochem. 54 (1950) 51. 19. I. Fi 1 i po v i c, Arhiv kem. 22 (1951) 133. 20. J. Tome il, Collection Czechoslov. Chem. Commns. 9 (1937) 12. 21. L . Me i t e s, J. Am. Chem. Soc. 72 (1950) 2293. 22. D. Monnier and Y. Rusconi, Anal. Chim . . Acta 7 (195·2) 567. 23. J. Hey rows k y, Analyst 72 (1947) 229. 24. R.H. Muller, R. L. Garm an, M. E . D roz and J . P et ras, I nd. Eng. Chem., Anal. Ed. 10 (1938) 3·39. 25. L . A. Matheson and N. Ni c ho 1 s, Trans. Electrochem. Soc. 73 (1938) ! 9:3. 26. J. Boetke and H. yan Suchtelen, Philips' tech. Rundschau 4 (1939l 'M3. 27. J. van Cake n .b erg he, Bul. soc. chim. Belges 60 (195'1) 3. 28. I. Fi 1 i p o v i c, private communication. 29. W. J. Youd en, Statistical Methods for Chemists. New York 1951. 30. 0 . .N.. Weber, M. Bra ni c a and V. B. Vouk, Arhiv kem. 25 (1953) 225. 31. See, for instance, 0. L. Davies, Stati stical Methods in Research and P·ro-· duction. London 1949. IZVOD Obrada polar1<>grafskih podataka metodom najmanjih kvadra~a I. lzracunavanje poluvalnog potencijala 0 . A. Weber Dan je pregled fizikalno-kemijskog i praktienog analitickog znacenj a poiu-valnog potencijala u polarografiji. Istaknuta je potreba poznavanja pouzdan;ih vrijednosti za tu velicinu. S obzirom na to da metode, koje se upotreblja!vaju za izracunavanje poluvalnog potencijala ne daju osobito pouzdane vrijednosti i ne dozvoljavaju odredivanje eksperimentalne pogreilke, prikazana je Rrimjena metode najmanjih kvadrata na izracunavanje poluvalnog potencij ala iz eksp erimentalnih podataka. INS'l'I'l'U'l' ZA MEDICINSKA IS'fRAZIVANJA .JUGOSLAVENSKE AKADEMIJE ZNANOS'l'I I UMJK'l'i\'OS'l'f. ZAGREB ' 

Treatment of Polarographic Data by the Least-Squares Method I. Estimation of the Half-wave Potential

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