Research Article Volume 6 Issue 4
^{1}Department of Analytical Chemistry, Escuela Politecnica Superior, University of Seville, Spain
^{2}Department of Analytical Chemistry, Faculty of Pharmacy, University of Seville, Spain
^{3}Department 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, E41011 Seville, Spain, Tel 34954556250
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):333338. DOI: 10.15406/ppij.2018.06.00197
The acid dissociation constant (K_{a}) 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, A_{1} and A_{0}, respectively, are unknowns. The raw values of absorbance’s versus pH data are transformed in linear functions, which allow to evaluate the unknown parameter K_{a }and A_{0}/or A_{1} by a graphical approach or by the application of the least squares method. Among the various methods proposed the double reciprocal, AgrenSommer 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
Among the physicochemical 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 compounds^{4} 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 nonprotonated 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 procedures^{6} and to examine the likelihood of separation of mixtures of compounds by extraction.
Although blackbox computer software’s are easily available nowadays, the spectrophotometric determination of equilibrium constants^{7} of single equilibria, such as acidity constants (K_{a}) 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 pK_{a} values are either enough high or low and/or solubility problems^{8} appear. The correct evaluation of pK_{a} 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 proposed^{9} 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, A_{1}, or R, A_{0}, are unknown. Among them, the double reciprocal or MaroniCalmon,^{10} AgrenSommer^{11–12 }and Nash^{13} 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=a_{0}+a_{1} X) obtained in each case by the least squares method, we may easily evaluate the unknown parameters, that is, the acidity constant K_{a}, and the unknown absorbance, A_{0} or A_{1}, from the slope, a_{1}, and the intercept, a_{0}, of the corresponding lines. The expressions applied in the available methods as well as the calculations of K_{a} values are shown in Table 1. If the ApH measurements are made at different wavelengths a set of intersecting straight lines is obtained, given the coordinates of the cutoff 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 
AgrenSommer 
A_{0} 
$Y=\frac{1}{{A}_{1}}+\frac{{K}_{a}}{{A}_{1}}X$ 
$\frac{A{A}_{0}}{A\left[{H}^{+}\right]}$ 
$\frac{1}{A}$ 
$\frac{b}{a}$ 
A_{1} 
$Y=\frac{1}{{A}_{0}}+\frac{1}{{A}_{0}{K}_{a}}X$ 
$\frac{A{A}_{1}}{A}\left[{H}^{+}\right]$  $\frac{1}{A}$ 
$\frac{a}{b}$ 

MaroniCalmon 
A_{0} 
$Y=\frac{1}{{A}_{1}{A}_{0}}+\frac{{K}_{a}}{{A}_{1}{A}_{0}}X$ 
$\frac{1}{\left[{H}^{+}\right]}$ 
$\frac{1}{A{A}_{0}}$ 
$\frac{b}{a}$ 
A_{1} 
$Y=\frac{1}{{A}_{0}{A}_{1}}+\frac{1}{\left({A}_{0}{A}_{1}\right){K}_{a}}X$ 
$\left[{H}^{+}\right]$ 
$\frac{1}{A{A}_{1}}$ 
$\frac{a}{b}$ 

Nash 
A_{0} 
$Y=\left\frac{{A}_{0}{A}_{1}}{{A}_{0}}\right\frac{1}{{K}_{a}}X\frac{1}{{K}_{a}}$ 
$\frac{1}{1\raisebox{1ex}{$A$}\!\left/ \!\raisebox{1ex}{${A}_{0}$}\right.}$ 
$\frac{1}{\left[{H}^{+}\right]}$ 
$\frac{1}{a}$ 
A_{1} 
$Y=\left\frac{{A}_{1}{A}_{0}}{{A}_{1}}\right{K}_{a}X{K}_{a}$ 
$\frac{1}{1\raisebox{1ex}{$A$}\!\left/ \!\raisebox{1ex}{${A}_{1}$}\right.}$ 
$\left[{H}^{+}\right]$ 
 $a$ 
Table 1 Expressions used according to various methods and other parameters of interest
Method 
Limit absorbance 
Intersecting point 
AgrenSommer and MaroniCalmon 
A_{0} known 
(1/K_{a}, 0) 
A_{1} known 
(K_{a}, 0) 

Nash 
A_{0} known 
(0, 1/K_{a}) 
A_{1} known 
(0, K_{a}) 
Table 2 Coordinates of intersecting points in when measurements are made at varying wavelengths
The correct use of the least squares method requires^{14–15 }several assumptions when it is applied to linear regression analyses:
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.
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 according^{16} to the random error propagation law
${s}_{z}={\left(\frac{\partial z}{\partial A}\right)}^{2}{s}^{2}{}_{A}+{\left(\frac{\partial z}{\partial pH}\right)}^{2}{s}^{2}{}_{pH}$ (1)
where s_{A}^{2} and s_{pH}^{2} 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 
AgrenSommer 
A_{0} 
$\begin{array}{l}{S}_{Y}^{2}={S}_{A}^{2}/{A}^{4}\\ {S}_{X}^{2}={A}_{0}^{2}{S}_{A}^{2}/\left({A}^{4}{\left[{H}^{+}\right]}^{4}\right)+{\left(A{A}_{0}\right)}^{2}{\mathrm{ln}}^{2}10{S}_{pH}^{2}/\left({A}^{2}{\left[{H}^{+}\right]}^{2}\right)\end{array}$ 
A_{1} 
$\begin{array}{l}{S}_{Y}^{2}={S}_{A}^{2}/{A}^{4}\\ {S}_{X}^{2}={\left[{H}^{+}\right]}^{2}{A}_{1}^{2}{S}_{A}^{2}/{A}^{4}+\left({\left(A{A}_{1}\right)}^{2}/{A}^{2}\right){\mathrm{ln}}^{2}10{\left[{H}^{+}\right]}^{2}{S}_{pH}^{2}\end{array}$ 

MaroniCalmon 
A_{0} 
$\begin{array}{l}{S}_{Y}^{2}={S}_{A}^{2}/{\left(A{A}_{0}\right)}^{4}\\ {S}_{X}^{2}={\mathrm{ln}}^{2}10{S}_{pH}^{2}/{\left[{H}^{+}\right]}^{2}\end{array}$ 
A_{1} 
$\begin{array}{l}{S}_{Y}^{2}={S}_{A}^{2}/{\left(A{A}_{1}\right)}^{2}\\ {S}_{X}^{2}={\left[{H}^{+}\right]}^{2}{\mathrm{ln}}^{2}10{S}_{pH}^{2}\end{array}$ 

Nash 
A_{0} 
$\begin{array}{l}{S}_{Y}^{2}={\mathrm{ln}}^{2}10{S}_{pH}^{2}/{\left[{H}^{+}\right]}^{2}\\ {S}_{X}^{2}={{A}_{0}^{2}{S}_{A}^{2}/\left({A}_{0}A\right)}^{4}\end{array}$ 
A_{1} 
$\begin{array}{l}{S}_{Y}^{2}={\left[{H}^{+}\right]}^{2}{\mathrm{ln}}^{2}10{S}_{pH}^{2}\\ {S}_{X}^{2}={{A}_{1}^{2}{S}_{A}^{2}/\left({A}_{1}A\right)}^{4}\end{array}$ 
Table 3 Propagation of errors in the different variables
If it is assumed that s_{A} and s_{pH} have the values of 0.001 and 0.01, respectively, given the precision of common spectrophotometers and pHmeters, we may evaluate the standard deviation of the variables X and Y, s_{X} and s_{Y}, respectively, by using five series of synthetic absorbance versus pH data, with 21 points uniformly distributed in the pH range pK_{a}±1, applying the Monte Carlo method, detailed e.g. in reference.^{17} The results obtained from this study, expressed as relative standard deviations s_{X}/X and s_{Y}/Y are shown in Table 4.
Method 
Known absorbance limit ${S}_{Y}/Y$ ${S}_{X}/X$ 

AgrenSommer 
A_{0} 
$\left(21.4\right)\cdot {10}^{3}$ 
$0.023$ 

A_{1} 
$\left(2.11.4\right)\cdot {10}^{3}$ 
$\left(0.020.01\right)$ 

MaroniCalmon 
A_{0} 
$\left(42\right)\cdot {10}^{3}$ 
$0.023$ 

A_{1} 
$\left(31\right)\cdot {10}^{3}$ 
$0.023$ 

Nash 
A_{0} 
$0.023$ 
$\left(42\right)\cdot {10}^{3}$ 

A_{1} 
$0.023$ 
$\left(21\right)\cdot {10}^{3}$ 
Table 4 Relative precision of the different regression variables
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 AgrenSommer and MaroniCalmon 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
${s}_{K}{}_{{}_{a}}=\left\frac{\partial {K}_{a}}{\partial {a}_{0}}\right{s}_{{a}_{0}}$ (2)
because the acidity constant coincides with the intercept (A_{0} known) or the reciprocal intercept (A_{1} known) of the corresponding straight line obtained.
By using the classical expressions we obtain instead more complex relationships for the standard deviation of Ka
${s}^{2}{{}_{K}}_{{}_{a}}={\left(\frac{\partial {K}_{a}}{\partial {a}_{0}}\right)}^{2}{s}^{2}{}_{{a}_{0}}+{\left(\frac{\partial {K}_{a}}{\partial {a}_{1}}\right)}^{2}{s}^{2}{}_{{a}_{1}}+2\left(\frac{\partial {K}_{a}}{\partial {a}_{0}}\right)\left(\frac{\partial {K}_{a}}{\partial {a}_{1}}\right)\mathrm{cov}\left({a}_{0},{a}_{1}\right)$ (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 
AgrenSommer 
A_{0} 
$Y=\frac{1}{{A}_{1}}+\frac{{K}_{a}}{{A}_{1}}X$ 
$\frac{1}{A}$ 
$\frac{A{A}_{0}}{A\left[{H}^{+}\right]}$ 
$\frac{1}{a}$ 
A_{1} 
$Y=\frac{1}{{A}_{0}}+\frac{1}{{A}_{0}{K}_{a}}X$ 
$\frac{1}{A}$ 
$\frac{\left(A{A}_{1}\right)\left[{H}^{+}\right]}{A}$ 
$a$ 

MaroniCalmon 
A_{0} 
$Y=\frac{1}{{A}_{1}{A}_{0}}+\frac{{K}_{a}}{{A}_{1}{A}_{0}}X$ 
$\frac{1}{A{A}_{0}}$ 
$\frac{1}{\left[{H}^{+}\right]}$ 
$\frac{1}{a}$ 
A_{1} 
$\begin{array}{l}Y=\frac{1}{{A}_{0}{A}_{1}}+\frac{1}{\left({A}_{0}{A}_{1}\right){K}_{a}}X\\ \end{array}$ 
$\frac{1}{A{A}_{1}}$ 
$\left[{H}^{+}\right]$ 
$a$ 
Table 5 Transformed expressions for the AgrenSommer and MaroniCalmon methods
In this paper a pH meter CRISON model 501pH meter with combined glassAg/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 1cm pathlength 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 (4phenyl3thiosemicarbazone) (MGBPT).^{18–19} Although this reagent has two close pKa corresponding to the equilibria of the type expressed in the following scheme^{20}
applying the Coleman or Polster methods1,21–23 it can be shown that in the range of about 1012pH, the two species corresponding to the first ionization equilibrium are found from a practical point of view.
To obtain the ApH curves of the reagent, solutions of MGBPT concentration equal to 1.08105M are prepared in 25mL 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, NN'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.
Though somewhat unpopular in the computer era, graphical analysis is a very appropriate method to study the acidbase 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 justifiable^{2} to apply the methods already seen. The results obtained are shown in Table 6. Applying to these results the appropriate statistical criteria^{24} we can admit that the average values of pK_{a} 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.
$\lambda (nm)$ 
AgrenSommer Method 
${\Delta}^{**}$ 
MaroniCalmon Method 
${\Delta}^{**}$ 
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 pKa_{1} values at different wavelengths
(*) average value ±standard deviation
(**) $\leftp{K}_{a}classicp{K}_{a}transformed\right$
However, some limitations are inherent to this study:
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 activity^{26–28 }including applications in nuclear medicine.^{28} Recent reviews show the biological importance of thiosemicarbazones as anticancer agents.^{25,29–33}
Among the physicochemical 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 blackbox computer software’s. The exchange of the role of the variables X and Y of the classical AgrenSommer and MaroniCalmon 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 pK_{a} 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.
None.
Authors declare that there is no conflict of interest.
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