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eISSN: 2379-6367

Pharmacy & Pharmacology International Journal

Research Article Volume 6 Issue 4

Optimization of intersecting straight lines methods for the evaluation of acidity constants of single equilibria from spectrophotometric data

Julia Martin,1 Agustin G Asuero,2 M Angeles Herrador,2 Gustavo Gonzalez3

1Department of Analytical Chemistry, Escuela Politecnica Superior, University of Seville, Spain
2Department of Analytical Chemistry, Faculty of Pharmacy, University of Seville, Spain
3Department of Analytical Chemistry, Faculty of Chemistry, University of Seville, Spain

Correspondence: Julia Martin, Department of Analytical Chemistry, Escuela Politecnica Superior, University of Seville, C/ Virgen de Africa, 7, E-41011 Seville, Spain, Tel 34-9-5455-6250

Received: July 12, 2018 | Published: August 30, 2018

Citation: Martín J, Asuero AG, Herrador MA, et al. Optimization of intersecting straight lines methods for the evaluation of acidity constants of single equilibria from spectrophotometric data. Pharm Pharmacol Int J. 2018;6(4):333-338. DOI: 10.15406/ppij.2018.06.00197

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Abstract

The acid dissociation constant (Ka) is among the most frequently used physicochemical parameters, and its determination is of interest to many research fields. A number of graphical methods has been proposed for the spectrophotometric evaluation of acidity constants of single equilibrium HR=H+R (charges omitted for the sake of generality), in those cases in which the limit absorbance’s of the pure species HR or R, A1 and A0, respectively, are unknowns. The raw values of absorbance’s versus pH data are transformed in linear functions, which allow to evaluate the unknown parameter Ka and A0/or A1 by a graphical approach or by the application of the least squares method. Among the various methods proposed the double reciprocal, Agren-Sommer and Nash methods yield to a family of intersecting straight lines when experimental measurements of absorbance and pH are made at different wavelengths. The purpose of this paper is to critically examine these methods on the basis of the requirements of single linear regression.

Keywords: intersecting straight lines methods, acidity constants, spectrophotometric data

Introduction

Among the physico-chemical properties of molecules, the acidity constants are of vital importance both in the analysis of drugs as well as in the interpretation of their mechanism of action.1–4 The solution of many galenical problems requires the knowledge of the acidity constants of compounds4 of pharmaceutical interest. Many compounds of biological interest have acidity constants, which lie close to each other. Their absorption, further transport and effect in the living organism are affected by the ratio of concentration of protonated and non-protonated forms in various media. Thus, the knowledge of acidity constants is worthy. The evaluation of acidity constants of organic reagents is also of great value to plan analytical experiments,5 e.g., the acidity constants can be employed in the design of titration procedures6 and to examine the likelihood of separation of mixtures of compounds by extraction.

Although black-box computer software’s are easily available nowadays, the spectrophotometric determination of equilibrium constants7 of single equilibria, such as acidity constants (Ka) of monoprotic acids by graphical plots seems to be valuable, due to the transparency of the linearized methods applied. Spectrophotometry is the choosen method when pKa values are either enough high or low and/or solubility problems8 appear. The correct evaluation of pKa of substances of therapeutic interest has vital importance in drug analysis and for the interpretation of their mechanisms of action. A number of graphical methods has been proposed9 for the spectrophotometric evaluation of acidity constants, HR=H+R (charges omitted for the sake of generality) in those cases in which the limit absorbance of the pure species HR, A1, or R, A0, are unknown. Among them, the double reciprocal or Maroni-Calmon,10 Agren-Sommer11–12 and Nash13 methods, lead to a family of intersecting straight lines when experimental measurements of absorbance and pH are made at varying wavelengths.

In these classical spectrophotometric methods the experimental absorbance pH curves are linearized. For the straight lines (Y=a0+a1 X) obtained in each case by the least squares method, we may easily evaluate the unknown parameters, that is, the acidity constant Ka, and the unknown absorbance, A0 or A1, from the slope, a1, and the intercept, a0, of the corresponding lines. The expressions applied in the available methods as well as the calculations of Ka values are shown in Table 1. If the A-pH measurements are made at different wavelengths a set of intersecting straight lines is obtained, given the coordinates of the cut-off points included (summarized) in Table 2. The purpose of this paper is to shown which of the three methods is more appropriate for linear regression analysis.

Method

Known absorbance limit

Expression

X

Y

Ka

Agren-Sommer

A0

Y= 1 A 1 + K a A 1 X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIXaaaleqaaaaajugibiabgUcaRKqbaoaalaaakeaaju gibiaadUeajuaGdaWgaaWcbaqcLbmacaWGHbaaleqaaaGcbaqcLbsa caWGbbqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaaqcLbsacaWGyb aaaa@4778@

A A 0 A[ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGbbGaeyOeI0IaamyqaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaa keaajugibiaadgeajuaGdaWadaGcbaqcLbsacaWGibqcfa4aaWbaaS qabeaajugWaiabgUcaRaaaaOGaay5waiaaw2faaaaaaaa@41B3@

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b a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGIbaakeaajugibiaadggaaaaaaa@36D9@

A1

Y= 1 A 0 + 1 A 0 K a X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIWaaaleqaaaaajugibiabgUcaRKqbaoaalaaakeaaju gibiaaigdaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaajugWaiaaicda aSqabaqcLbsacaWGlbqcfa4aaSbaaSqaaKqzadGaamyyaaWcbeaaaa qcLbsacaWGybaaaa@48C0@

A A 1 A [ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGbbGaeyOeI0IaamyqaKqbaoaaBaaaleaajugWaiaaigdaaSqabaaa keaajugibiaadgeaaaqcfa4aamWaaOqaaKqzGeGaamisaKqbaoaaCa aaleqabaqcLbmacqGHRaWkaaaakiaawUfacaGLDbaaaaa@41B4@

1 A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeaaaaaaa@368D@

a b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGHbaakeaajugibiaadkgaaaaaaa@36D9@

Maroni-Calmon

A0

Y= 1 A 1 A 0 + K a A 1 A 0 X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0IaamyqaKqbaoaaBaaale aajugWaiaaicdaaSqabaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqz GeGaam4saKqbaoaaBaaaleaajugWaiaadggaaSqabaaakeaajugibi aadgeajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0Ia amyqaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaaaKqzGeGaamiwaa aa@5156@

1 [ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajuaGdaWadaGcbaqcLbsacaWGibqcfa4aaWbaaSqabeaa jugWaiabgUcaRaaaaOGaay5waiaaw2faaaaaaaa@3BF3@

1 A A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeacqGHsislcaWGbbqcfa4aaSbaaSqaaKqz adGaaGimaaWcbeaaaaaaaa@3AED@

b a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGIbaakeaajugibiaadggaaaaaaa@36D9@

A1

Y= 1 A 0 A 1 + 1 ( A 0 A 1 ) K a X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIWaaaleqaaKqzGeGaeyOeI0IaamyqaKqbaoaaBaaale aajugWaiaaigdaaSqabaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqz GeGaaGymaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyqaKqbaoaaBaaale aajugWaiaaicdaaSqabaqcLbsacqGHsislcaWGbbqcfa4aaSbaaSqa aKqzadGaaGymaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaam4saKqbao aaBaaaleaajugWaiaadggaaSqabaaaaKqzGeGaamiwaaaa@54CB@

[ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWadaGcbaqcLbsaca WGibqcfa4aaWbaaSqabeaajugWaiabgUcaRaaaaOGaay5waiaaw2fa aaaa@39F7@

1 A A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeacqGHsislcaWGbbqcfa4aaSbaaSqaaKqz adGaaGymaaWcbeaaaaaaaa@3AEE@

a b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGHbaakeaajugibiaadkgaaaaaaa@36D9@

Nash

A0

Y=| A 0 A 1 A 0 | 1 K a X 1 K a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaabdaGcbaqcfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaa jugWaiaaicdaaSqabaqcLbsacqGHsislcaWGbbqcfa4aaSbaaSqaaK qzadGaaGymaaWcbeaaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaajugW aiaaicdaaSqabaaaaaGccaGLhWUaayjcSdqcfa4aaSaaaOqaaKqzGe GaaGymaaGcbaqcLbsacaWGlbqcfa4aaSbaaSqaaKqzadGaamyyaaWc beaaaaqcLbsacaWGybGaeyOeI0scfa4aaSaaaOqaaKqzGeGaaGymaa GcbaqcLbsacaWGlbqcfa4aaSbaaSqaaKqzadGaamyyaaWcbeaaaaaa aa@55DA@

1 1 A A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaaigdacqGHsisljuaGdaWccaGcbaqcLbsacaWG bbaakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaa aaaaaaaa@3D7A@

1 [ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajuaGdaWadaGcbaqcLbsacaWGibqcfa4aaWbaaSqabeaa jugWaiabgUcaRaaaaOGaay5waiaaw2faaaaaaaa@3BF3@

1 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgkHiTKqbaoaala aakeaajugibiaaigdaaOqaaKqzGeGaamyyaaaaaaa@3829@

A1

  Y=| A 1 A 0 A 1 | K a X K a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaabdaGcbaqcfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaa jugWaiaaigdaaSqabaqcLbsacqGHsislcaWGbbqcfa4aaSbaaSqaaK qzadGaaGimaaWcbeaaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaajugW aiaaigdaaSqabaaaaaGccaGLhWUaayjcSdqcLbsacaWGlbqcfa4aaS baaSqaaKqzadGaamyyaaWcbeaajugibiaadIfacqGHsislcaWGlbqc fa4aaSbaaSqaaKqzadGaamyyaaWcbeaaaaa@5154@

  1 1 A A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaaigdacqGHsisljuaGdaWccaGcbaqcLbsacaWG bbaakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaa aaaaaaaa@3D7B@

  [ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWadaGcbaqcLbsaca WGibqcfa4aaWbaaSqabeaajugWaiabgUcaRaaaaOGaay5waiaaw2fa aaaa@39F7@

 - a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadggaaaa@34B1@

Table 1 Expressions used according to various methods and other parameters of interest

Method

Limit absorbance

Intersecting point

Agren-Sommer and Maroni-Calmon

A0 known

(-1/Ka, 0)

A1 known

(-Ka, 0)

Nash

A0 known

(0, -1/Ka)

A1 known

(0, -Ka)

Table 2 Coordinates of intersecting points in when measurements are made at varying wavelengths

Theory

The correct use of the least squares method requires14–15 several assumptions when it is applied to linear regression analyses:

  1. The measurement of the variable X is assumed error-free.
  2. The Y values obtained (replicates) for the same X value must show a Gaussian distribution.
  3. The standard deviation of the Y values should not change in the range of values covered by the X values (homocedasticity).

If the various expressions proposed in Table 1 are examined in this regard, it can be accepted in the first instance that conditions b) and c) are met, but not a). Given the different nature of the mathematical function that relates the variables of regression X and Y with the experimental values of A and pH in each method, condition a) should be checked through the propagation of errors made in the mean values of A and pH over the various pairs of data X, Y object of the regression.

Error analysis

Let z be a function of the variables A and pH, z=f(A, pH). The errors involved in the measurement of A and pH will be propagated through Z according16 to the random error propagation law

s z = ( z A ) 2 s 2 A + ( z pH ) 2 s 2 pH MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaamOEaaWcbeaajugibiabg2da9Kqbaoaa bmaakeaajuaGdaWcaaGcbaqcLbsacqGHciITcaWG6baakeaajugibi abgkGi2kaadgeaaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqz adGaaGOmaaaajugibiaadohajuaGdaahaaWcbeqaaKqzadGaaGOmaa aajuaGdaWgaaWcbaqcLbmacaWGbbaaleqaaKqzGeGaey4kaSscfa4a aeWaaOqaaKqbaoaalaaakeaajugibiabgkGi2kaadQhaaOqaaKqzGe GaeyOaIyRaamiCaiaadIeaaaaakiaawIcacaGLPaaajuaGdaahaaWc beqaaKqzadGaaGOmaaaajugibiaadohajuaGdaahaaWcbeqaaKqzad GaaGOmaaaajuaGdaWgaaWcbaqcLbmacaWGWbGaamisaaWcbeaaaaa@6399@                             (1)

where sA2 and spH2 are the variances (squared standard deviations) of the absorbance and pH, respectively. Measurements of A and pH are independent, and then cov(A, pH)=0. The application of the Eqn. (1) to the expression of the three methods tested is compiled in Table 3.

Method

Known absorbance limit

Variance of variables

Agren-Sommer

A0

S Y 2 = S A 2 / A 4 S X 2 = A 0 2 S A 2 / ( A 4 [ H + ] 4 ) + ( A A 0 ) 2 ln 2 10 S pH 2 / ( A 2 [ H + ] 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaam4uaKqbao aaDaaaleaajugWaiaadMfaaSqaaKqzadGaaGOmaaaajugibiabg2da 9KqbaoaalyaakeaajugibiaadofajuaGdaqhaaWcbaqcLbmacaWGbb aaleaajugWaiaaikdaaaaakeaajugibiaadgeajuaGdaahaaWcbeqa aKqzadGaaGinaaaaaaaakeaajugibiaadofajuaGdaqhaaWcbaqcLb macaWGybaaleaajugWaiaaikdaaaqcLbsacqGH9aqpjuaGdaWcgaGc baqcLbsacaWGbbqcfa4aa0baaSqaaKqzadGaaGimaaWcbaqcLbmaca aIYaaaaKqzGeGaam4uaKqbaoaaDaaaleaajugWaiaadgeaaSqaaKqz adGaaGOmaaaaaOqaaKqbaoaabmaakeaajugibiaadgeajuaGdaahaa WcbeqaaKqzadGaaGinaaaajuaGdaWadaGcbaqcLbsacaWGibqcfa4a aWbaaSqabeaajugWaiabgUcaRaaaaOGaay5waiaaw2faaKqbaoaaCa aaleqabaqcLbmacaaI0aaaaaGccaGLOaGaayzkaaaaaKqzGeGaey4k aSscfa4aaSGbaOqaaKqbaoaabmaakeaajugibiaadgeacqGHsislca WGbbqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaOGaayjkaiaawMca aKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaciiBaiaac6gaju aGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaaigdacaaIWaGaam4u aKqbaoaaDaaaleaajugibiaadchacaWGibaaleaajugWaiaaikdaaa aakeaajuaGdaqadaGcbaqcLbsacaWGbbqcfa4aaWbaaSqabeaajugW aiaaikdaaaqcfa4aamWaaOqaaKqzGeGaamisaKqbaoaaCaaaleqaba qcLbmacqGHRaWkaaaakiaawUfacaGLDbaajuaGdaahaaWcbeqaaKqz adGaaGOmaaaaaOGaayjkaiaawMcaaaaaaaaa@92DC@

A1

S Y 2 = S A 2 / A 4 S X 2 = [ H + ] 2 A 1 2 S A 2 / A 4 +( ( A A 1 ) 2 / A 2 ) ln 2 10 [ H + ] 2 S pH 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaam4uaKqbao aaDaaaleaajugWaiaadMfaaSqaaKqzadGaaGOmaaaajugibiabg2da 9KqbaoaalyaakeaajugibiaadofajuaGdaqhaaWcbaqcLbmacaWGbb aaleaajugWaiaaikdaaaaakeaajugibiaadgeajuaGdaahaaWcbeqa aKqzadGaaGinaaaaaaaakeaajugibiaadofajuaGdaqhaaWcbaqcLb macaWGybaaleaajugWaiaaikdaaaqcLbsacqGH9aqpjuaGdaWcgaGc baqcfa4aamWaaOqaaKqzGeGaamisaKqbaoaaCaaaleqabaqcLbmacq GHRaWkaaaakiaawUfacaGLDbaajuaGdaahaaWcbeqaaKqzadGaaGOm aaaajugibiaadgeajuaGdaqhaaWcbaqcLbmacaaIXaaaleaajugWai aaikdaaaqcLbsacaWGtbqcfa4aa0baaSqaaKqzadGaamyqaaWcbaqc LbmacaaIYaaaaaGcbaqcLbsacaWGbbqcfa4aaWbaaSqabeaajugWai aaisdaaaaaaKqzGeGaey4kaSscfa4aaeWaaOqaaKqbaoaalyaakeaa juaGdaqadaGcbaqcLbsacaWGbbGaeyOeI0IaamyqaKqbaoaaBaaale aajugWaiaaigdaaSqabaaakiaawIcacaGLPaaajuaGdaahaaWcbeqa aKqzadGaaGOmaaaaaOqaaKqzGeGaamyqaKqbaoaaCaaaleqabaqcLb macaaIYaaaaaaaaOGaayjkaiaawMcaaKqzGeGaciiBaiaac6gajuaG daahaaWcbeqaaKqzadGaaGOmaaaajugibiaaigdacaaIWaqcfa4aam WaaOqaaKqzGeGaamisaKqbaoaaCaaaleqabaqcLbmacqGHRaWkaaaa kiaawUfacaGLDbaajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibi aadofajuaGdaqhaaWcbaqcLbmacaWGWbGaamisaaWcbaqcLbmacaaI Yaaaaaaaaa@91DF@

Maroni-Calmon

A0

S Y 2 = S A 2 / ( A A 0 ) 4 S X 2 = ln 2 10 S pH 2 / [ H + ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaam4uaKqbao aaDaaaleaajugWaiaadMfaaSqaaKqzadGaaGOmaaaajugibiabg2da 9KqbaoaalyaakeaajugibiaadofajuaGdaqhaaWcbaqcLbmacaWGbb aaleaajugWaiaaikdaaaaakeaajuaGdaqadaGcbaqcLbsacaWGbbGa eyOeI0IaamyqaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaakiaawI cacaGLPaaajuaGdaahaaWcbeqaaKqzadGaaGinaaaaaaaakeaajugi biaadofajuaGdaqhaaWcbaqcLbmacaWGybaaleaajugWaiaaikdaaa qcLbsacqGH9aqpjuaGdaWcgaGcbaqcLbsaciGGSbGaaiOBaKqbaoaa CaaaleqabaqcLbmacaaIYaaaaKqzGeGaaGymaiaaicdacaWGtbqcfa 4aa0baaSqaaKqzadGaamiCaiaadIeaaSqaaKqzadGaaGOmaaaaaOqa aKqbaoaadmaakeaajugibiaadIeajuaGdaahaaWcbeqaaKqzadGaey 4kaScaaaGccaGLBbGaayzxaaqcfa4aaWbaaSqabeaajugWaiaaikda aaaaaaaaaa@6BB2@

A1

S Y 2 = S A 2 / ( A A 1 ) 2 S X 2 = [ H + ] 2 ln 2 10 S pH 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaam4uaKqbao aaDaaaleaajugWaiaadMfaaSqaaKqzadGaaGOmaaaajugibiabg2da 9KqbaoaalyaakeaajugibiaadofajuaGdaqhaaWcbaqcLbmacaWGbb aaleaajugWaiaaikdaaaaakeaajuaGdaqadaGcbaqcLbsacaWGbbGa eyOeI0IaamyqaKqbaoaaBaaaleaajugWaiaaigdaaSqabaaakiaawI cacaGLPaaajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaaakeaajugi biaadofajuaGdaqhaaWcbaqcLbmacaWGybaaleaajugWaiaaikdaaa qcLbsacqGH9aqpjuaGdaWadaGcbaqcLbsacaWGibqcfa4aaWbaaSqa beaajugWaiabgUcaRaaaaOGaay5waiaaw2faaKqbaoaaCaaaleqaba qcLbmacaaIYaaaaKqzGeGaciiBaiaac6gajuaGdaahaaWcbeqaaKqz adGaaGOmaaaajugibiaaigdacaaIWaGaam4uaKqbaoaaDaaaleaaju gWaiaadchacaWGibaaleaajugWaiaaikdaaaaaaaa@6AF9@

Nash

A0

S Y 2 = ln 2 10 S pH 2 / [ H + ] 2 S X 2 = A 0 2 S A 2 / ( A 0 A ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaam4uaKqbao aaDaaaleaajugWaiaadMfaaSqaaKqzadGaaGOmaaaajugibiabg2da 9KqbaoaalyaakeaajugibiGacYgacaGGUbqcfa4aaWbaaSqabeaaju gWaiaaikdaaaqcLbsacaaIXaGaaGimaiaadofajuaGdaqhaaWcbaqc LbmacaWGWbGaamisaaWcbaqcLbmacaaIYaaaaaGcbaqcfa4aamWaaO qaaKqzGeGaamisaKqbaoaaCaaaleqabaqcLbmacqGHRaWkaaaakiaa wUfacaGLDbaajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaaakeaaju gibiaadofajuaGdaqhaaWcbaqcLbmacaWGybaaleaajugWaiaaikda aaqcLbsacqGH9aqpjuaGdaWcgaGcbaqcLbsacaWGbbqcfa4aa0baaS qaaKqzadGaaGimaaWcbaqcLbmacaaIYaaaaKqzGeGaam4uaKqbaoaa DaaaleaajugWaiaadgeaaSqaaKqzadGaaGOmaaaaaOqaaKqbaoaabm aakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqz GeGaeyOeI0IaamyqaaGccaGLOaGaayzkaaaaaKqbaoaaCaaaleqaba qcLbmacaaI0aaaaaaaaa@722E@

A1

  S Y 2 = [ H + ] 2 ln 2 10 S pH 2 S X 2 = A 1 2 S A 2 / ( A 1 A ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaam4uaKqbao aaDaaaleaajugWaiaadMfaaSqaaKqzadGaaGOmaaaajugibiabg2da 9KqbaoaadmaakeaajugibiaadIeajuaGdaahaaWcbeqaaKqzadGaey 4kaScaaaGccaGLBbGaayzxaaqcfa4aaWbaaSqabeaajugWaiaaikda aaqcLbsaciGGSbGaaiOBaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaK qzGeGaaGymaiaaicdacaWGtbqcfa4aa0baaSqaaKqzadGaamiCaiaa dIeaaSqaaKqzadGaaGOmaaaaaOqaaKqzGeGaam4uaKqbaoaaDaaale aajugWaiaadIfaaSqaaKqzadGaaGOmaaaajugibiabg2da9Kqbaoaa lyaakeaajugibiaadgeajuaGdaqhaaWcbaqcLbmacaaIXaaaleaaju gWaiaaikdaaaqcLbsacaWGtbqcfa4aa0baaSqaaKqzadGaamyqaaWc baqcLbmacaaIYaaaaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyqaKqbao aaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHsislcaWGbbaakiaa wIcacaGLPaaaaaqcfa4aaWbaaSqabeaajugWaiaaisdaaaaaaaa@7178@

Table 3 Propagation of errors in the different variables

If it is assumed that sA and spH have the values of 0.001 and 0.01, respectively, given the precision of common spectrophotometers and pH-meters, we may evaluate the standard deviation of the variables X and Y, sX and sY, respectively, by using five series of synthetic absorbance versus pH data, with 21 points uniformly distributed in the pH range pKa±1, applying the Monte Carlo method, detailed e.g. in reference.17 The results obtained from this study, expressed as relative standard deviations sX/X and sY/Y are shown in Table 4.

Method

Known absorbance limit      S Y /Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcgaGcbaqcLbsaca WGtbqcfa4aaSbaaSqaaKqzadGaamywaaWcbeaaaOqaaKqzGeGaamyw aaaaaaa@3999@                  S X /X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcgaGcbaqcLbsaca WGtbqcfa4aaSbaaSqaaKqzadGaamiwaaWcbeaaaOqaaKqzGeGaamiw aaaaaaa@3997@

Agren-Sommer

A0                                            

                                      ( 21.4 ) 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aIYaGaeyOeI0IaaGymaiaac6cacaaI0aaakiaawIcacaGLPaaajugi biabgwSixlaaigdacaaIWaqcfa4aaWbaaSqabeaajugWaiabgkHiTi aaiodaaaaaaa@41AB@                                   

0.023 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaicdacaGGUaGaaG imaiaaikdacaaIZaaaaa@376A@

A1

                                      ( 2.11.4 ) 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aIYaGaaiOlaiaaigdacqGHsislcaaIXaGaaiOlaiaaisdaaOGaayjk aiaawMcaaKqzGeGaeyyXICTaaGymaiaaicdajuaGdaahaaWcbeqaaK qzadGaeyOeI0IaaG4maaaaaaa@4318@

( 0.020.01 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aIWaGaaiOlaiaaicdacaaIYaGaeyOeI0IaaGimaiaac6cacaaIWaGa aGymaaGccaGLOaGaayzkaaaaaa@3CA6@

Maroni-Calmon

A0

                                      ( 42 ) 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aI0aGaeyOeI0IaaGOmaaGccaGLOaGaayzkaaqcLbsacqGHflY1caaI XaGaaGimaKqbaoaaCaaaleqabaqcLbmacqGHsislcaaIZaaaaaaa@403E@

0.023 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaicdacaGGUaGaaG imaiaaikdacaaIZaaaaa@376A@

A1

                                      ( 31 ) 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aIZaGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcLbsacqGHflY1caaI XaGaaGimaKqbaoaaCaaaleqabaqcLbmacqGHsislcaaIZaaaaaaa@403C@

0.023 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaicdacaGGUaGaaG imaiaaikdacaaIZaaaaa@376A@

Nash

A0

                                       0.023 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaicdacaGGUaGaaG imaiaaikdacaaIZaaaaa@376A@

( 42 ) 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aI0aGaeyOeI0IaaGOmaaGccaGLOaGaayzkaaqcLbsacqGHflY1caaI XaGaaGimaKqbaoaaCaaaleqabaqcLbmacqGHsislcaaIZaaaaaaa@403E@

A1

                                       0.023 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaicdacaGGUaGaaG imaiaaikdacaaIZaaaaa@376A@

  ( 21 ) 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaqadaGcbaqcLbsaca aIYaGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcLbsacqGHflY1caaI XaGaaGimaKqbaoaaCaaaleqabaqcLbmacqGHsislcaaIZaaaaaaa@403B@

Table 4 Relative precision of the different regression variables

Transformation of Expressions

It can be seen from results included in Table 4 that the method of Nash led to a greater precision in the measurement of the variable X compare to the other two methods. If we are interesting in applying both Agren-Sommer and Maroni-Calmon method in a more rigorous way, the role of the variables X and Y must be interchanged leading to the expressions found in Table 5. An additional advantage of the transformed expressions is that allow a simple calculation of the standard deviation of Ka values

s K a =| K a a 0 | s a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaeaajugWaiaadUeaaKqbagqaamaaBaaabaWaaSbaaeaa jugWaiaadggaaKqbagqaaaqabaqcLbsacqGH9aqpjuaGdaabdaqaam aalaaabaqcLbsacqGHciITcaWGlbqcfa4aaSbaaeaajugWaiaadgga aKqbagqaaaqaaKqzGeGaeyOaIyRaamyyaKqbaoaaBaaabaqcLbmaca aIWaaajuaGbeaaaaaacaGLhWUaayjcSdqcLbsacaWGZbqcfa4aaSba aeaajugWaiaadggajuaGdaWgaaqaaKqzadGaaGimaaqcfayabaaabe aaaaa@5676@                  (2)

because the acidity constant coincides with the intercept (A0 known) or the reciprocal intercept (A1 known) of the corresponding straight line obtained.

By using the classical expressions we obtain instead more complex relationships for the standard deviation of Ka

s 2 K a = ( K a a 0 ) 2 s 2 a 0 + ( K a a 1 ) 2 s 2 a 1 +2( K a a 0 )( K a a 1 )cov( a 0 , a 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaWbaaeqabaqcLbmacaaIYaaaaKqbaoaaBaaabaqcLbmacaWG lbaajuaGbeaadaWgaaqaamaaBaaabaqcLbmacaWGHbaajuaGbeaaae qaaKqzGeGaeyypa0tcfa4aaeWaaeaadaWcaaqaaKqzGeGaeyOaIyRa am4saKqbaoaaBaaabaqcLbmacaWGHbaajuaGbeaaaeaajugibiabgk Gi2kaadggajuaGdaWgaaqaaKqzadGaaGimaaqcfayabaaaaaGaayjk aiaawMcaamaaCaaabeqaaKqzadGaaGOmaaaajugibiaadohajuaGda ahaaqabeaajugWaiaaikdaaaqcfa4aaSbaaeaajugWaiaadggajuaG daWgaaqaaKqzadGaaGimaaqcfayabaaabeaacqGHRaWkdaqadaqaam aalaaabaqcLbsacqGHciITcaWGlbqcfa4aaSbaaeaajugWaiaadgga aKqbagqaaaqaaKqzGeGaeyOaIyRaamyyaKqbaoaaBaaabaqcLbmaca aIXaaajuaGbeaaaaaacaGLOaGaayzkaaWaaWbaaeqabaqcLbmacaaI YaaaaKqzGeGaam4CaKqbaoaaCaaabeqaaKqzadGaaGOmaaaajuaGda WgaaqaaKqzadGaamyyaKqbaoaaBaaabaqcLbmacaaIXaaajuaGbeaa aeqaaiabgUcaRiaaikdadaqadaqaamaalaaabaqcLbsacqGHciITca WGlbqcfa4aaSbaaeaajugWaiaadggaaKqbagqaaaqaaKqzGeGaeyOa IyRaamyyaKqbaoaaBaaabaqcLbmacaaIWaaajuaGbeaaaaaacaGLOa GaayzkaaWaaeWaaeaadaWcaaqaaKqzGeGaeyOaIyRaam4saKqbaoaa BaaabaqcLbmacaWGHbaajuaGbeaaaeaajugibiabgkGi2kaadggaju aGdaWgaaqaaKqzadGaaGymaaqcfayabaaaaaGaayjkaiaawMcaaiGa cogacaGGVbGaaiODamaabmaabaqcLbsacaWGHbqcfa4aaSbaaeaaju gWaiaaicdaaKqbagqaaiaacYcajugibiaadggajuaGdaWgaaqaaKqz adGaaGymaaqcfayabaaacaGLOaGaayzkaaaaaa@A206@                   (3)

However, by using the new expressions proposed with the synthetic data generated, smaller standard deviations values are obtained (approximately half) than by applying the classical equations. To illustrate the above, a practical application has been developed.

Method

Known absorbance limit

Expression

X

Y

Ka

Agren-Sommer

A0

Y= 1 A 1 + K a A 1 X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIXaaaleqaaaaajugibiabgUcaRKqbaoaalaaakeaaju gibiaadUeajuaGdaWgaaWcbaqcLbmacaWGHbaaleqaaaGcbaqcLbsa caWGbbqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaaqcLbsacaWGyb aaaa@4778@

1 A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeaaaaaaa@368D@

A A 0 A[ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca WGbbGaeyOeI0IaamyqaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaa keaajugibiaadgeajuaGdaWadaGcbaqcLbsacaWGibqcfa4aaWbaaS qabeaajugWaiabgUcaRaaaaOGaay5waiaaw2faaaaaaaa@41B3@

1 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgkHiTKqbaoaala aakeaajugibiaaigdaaOqaaKqzGeGaamyyaaaaaaa@3829@

A1

Y= 1 A 0 + 1 A 0 K a X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIWaaaleqaaaaajugibiabgUcaRKqbaoaalaaakeaaju gibiaaigdaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaajugWaiaaicda aSqabaqcLbsacaWGlbqcfa4aaSbaaSqaaKqzadGaamyyaaWcbeaaaa qcLbsacaWGybaaaa@48C0@

1 A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeaaaaaaa@368D@

( A A 1 )[ H + ] A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcfa4aae WaaOqaaKqzGeGaamyqaiabgkHiTiaadgeajuaGdaWgaaWcbaqcLbma caaIXaaaleqaaaGccaGLOaGaayzkaaqcfa4aamWaaOqaaKqzGeGaam isaKqbaoaaCaaaleqabaqcLbmacqGHRaWkaaaakiaawUfacaGLDbaa aeaajugibiaadgeaaaaaaa@43D5@

a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgkHiTiaadggaaa a@359E@

Maroni-Calmon

A0

Y= 1 A 1 A 0 + K a A 1 A 0 X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadMfacqGH9aqpju aGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadgeajuaGdaWgaaWc baqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0IaamyqaKqbaoaaBaaale aajugWaiaaicdaaSqabaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqz GeGaam4saKqbaoaaBaaaleaajugWaiaadggaaSqabaaakeaajugibi aadgeajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0Ia amyqaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaaaKqzGeGaamiwaa aa@5156@

1 A A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeacqGHsislcaWGbbqcfa4aaSbaaSqaaKqz adGaaGimaaWcbeaaaaaaaa@3AED@

1 [ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajuaGdaWadaGcbaqcLbsacaWGibqcfa4aaWbaaSqabeaa jugWaiabgUcaRaaaaOGaay5waiaaw2faaaaaaaa@3BF3@

1 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgkHiTKqbaoaala aakeaajugibiaaigdaaOqaaKqzGeGaamyyaaaaaaa@3829@

A1

  Y= 1 A 0 A 1 + 1 ( A 0 A 1 ) K a X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqzGeGaamywaiabg2 da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamyqaKqbaoaa BaaaleaajugWaiaaicdaaSqabaqcLbsacqGHsislcaWGbbqcfa4aaS baaSqaaKqzadGaaGymaaWcbeaaaaqcLbsacqGHRaWkjuaGdaWcaaGc baqcLbsacaaIXaaakeaajuaGdaqadaGcbaqcLbsacaWGbbqcfa4aaS baaSqaaKqzadGaaGimaaWcbeaajugibiabgkHiTiaadgeajuaGdaWg aaWcbaqcLbmacaaIXaaaleqaaaGccaGLOaGaayzkaaqcLbsacaWGlb qcfa4aaSbaaSqaaKqzadGaamyyaaWcbeaaaaqcLbsacaWGybaakeaa aaaa@54DC@

  1 A A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadgeacqGHsislcaWGbbqcfa4aaSbaaSqaaKqz adGaaGymaaWcbeaaaaaaaa@3AEE@

  [ H + ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWadaGcbaqcLbsaca WGibqcfa4aaWbaaSqabeaajugWaiabgUcaRaaaaOGaay5waiaaw2fa aaaa@39F7@

  a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgkHiTiaadggaaa a@359E@

Table 5 Transformed expressions for the Agren-Sommer and Maroni-Calmon methods

Materials and methods

In this paper a pH meter CRISON model 501pH meter with combined glass-Ag/AgCl electrodes (In gold) with a range of use of 0 to 14pH units, has been used in pH measurements. A Spectrophotometer SPECTRONIC 2000 (Bausch & Lomb) provided with a graphic XY recorder and equipped with quartz cells of 1-cm path-length has been used for absorbance measurements.

The two classical methods as well as the new expressions have been applied to the pKa evaluation of the methylglyoxal bis (4-phenyl-3thiosemicarbazone) (MGBPT).18–19 Although this reagent has two close pKa corresponding to the equilibria of the type expressed in the following scheme20

applying the Coleman or Polster methods1,21–23 it can be shown that in the range of about 10-12pH, the two species corresponding to the first ionization equilibrium are found from a practical point of view.

To obtain the A-pH curves of the reagent, solutions of MGBPT concentration equal to 1.0810-5M are prepared in 25-mL volumetric flasks. The appropriate pH is achieved by adding different volumes of KOH or HCl of various concentrations. To ensure the homogeneity of the solutions, N-N'-dimethylformamide  (DMF) is added (the optimum solvent for this reagent is a mixture of DMF and water) so that the samples have 60% V/V of DMF. To fix the ionic strength of the medium in 0.1, 2.5ml of solution of KCl 1M were added. Finally, it is poured with distilled water in 25mL volumetric flasks, and the pH values measured by passing the solutions to 25ml beakers. Absorption spectra are recorded against blanks prepared in the same manner, without reagent. The temperature was about 20±1°C.

Results and discussion

Though somewhat unpopular in the computer era, graphical analysis is a very appropriate method to study the acid-base behaviour of single equilibria from a spectrophotometric point of view. Since we may assume that in the pH range of choice there are only two species in solution, it is justifiable2 to apply the methods already seen. The results obtained are shown in Table 6. Applying to these results the appropriate statistical criteria24 we can admit that the average values of pKa obtained from the classical expressions did not differ from those obtained with the new expressions. Moreover, there are also no significant differences as regards their precisions.

λ(nm) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU7aSjaacIcaca WGUbGaamyBaiaacMcaaaa@38BD@

Agren-Sommer Method

Δ ** MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfs5aeLqbaoaaCa aaleqabaqcLbmacaGGQaGaaiOkaaaaaaa@3876@

Maroni-Calmon Method

Δ ** MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfs5aeLqbaoaaCa aaleqabaqcLbmacaGGQaGaaiOkaaaaaaa@3876@

Nash Method

Classic

Transformed

Classic

Transformed

340

10.903

10.926

0.023

10.961

10.965

0.004

10.951

345

10.909

10.931

0.022

10.963

10.967

0.004

10.939

*

10.906±0.004

10.928±0.003

 

10.962±0.001

10.966±0.001

 

10.945±0.008

Table 6 pKa1 values at different wavelengths

(*) average value ±standard deviation

(**) | p K a classicp K a transformed | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaabdaGcbaqcLbsaca WGWbGaam4saKqbaoaaBaaaleaajugWaiaadggaaSqabaqcLbsacaWG JbGaamiBaiaadggacaWGZbGaam4CaiaadMgacaWGJbGaeyOeI0Iaam iCaiaadUeajuaGdaWgaaWcbaqcLbmacaWGHbaaleqaaKqzGeGaamiD aiaadkhacaWGHbGaamOBaiaadohacaWGMbGaam4BaiaadkhacaWGTb GaamyzaiaadsgaaOGaay5bSlaawIa7aaaa@53B7@

However, some limitations are inherent to this study:

  1. The calculations were only made for values of sA=0.001 and spH=0.01.
  2. It was assumed that absolute values of A are independent of the actual values for the whole range of absorbance found.
  3. If the Gauss law holds for A and pH measurements, this is not necessary so for X and Y of the type used for the calculations in the manuscript; a serious objection, however, the authors does not know whether this is.
  4. The variance analysis did not include the uncertainty of the instruments readings.

Transmittance and concentration are related by means of a logarithmic relationship, in such a way that small errors in transmittance measurement causing large relative absorbance errors. However, the main source of indeterminate error in modern spectrophotometers lies in the measure of the absorbance. However, the procedure devised in this paper has the inherent advantage of its simplicity, which makes it attractive. Bisthiosemicarbazones are clinically relevant for a variety of diseases, e.g. tuberculosis, viral infections, malaria and cancer.25 Copper (II) uncharged lipophilic complexes of bisthiosemicarbazones posses fascinating biological activity26–28 including applications in nuclear medicine.28 Recent reviews show the biological importance of thiosemicarbazones as anticancer agents.25,29–33

Conclusion

Among the physico-chemical properties of molecules, the acidity constants are of vital importance both in the analysis of drugs as well as in the interpretation of their mechanism of action. The solution of many galenical problems requires the knowledge of the acidity constants of compounds having pharmaceutical interest. As it has previously indicated, calculation of acidity constants of monoprotic acids by graphical methods may be advantageously used in spite of the existence of modern black-box computer software’s. The exchange of the role of the variables X and Y of the classical Agren-Sommer and Maroni-Calmon methods resulted in new expressions that fulfil the mathematical conditions required for the regression analysis, and led to more simplified equations for the calculation of the pKa and its individual standard deviation. As far as the results are concerned, there are no significant differences and, therefore, this reformulation may be considered valid. Note that bisthiosemicarbazones are biological relevant as shown at the end of the previous section.

Acknowledgement

None.

Conflict of interest

Authors declare that there is no conflict of interest.

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