1. Introduction

The relationship between the absorbance (A) and the composition of any given solution of a diprotic acid having concentration C_R and mixed acidity constants

(1)

and

(2)

corresponding to the dissociation equilibria

(3)

where the charges are omitted for the sake of simplicity,

and

(4)

respectively, is given by [1, 2]

(5)

(6)

where A₀, A₁ and A₂ are the limit absorbance of R, HR and H₂R species, respectively, and f₂, f₁ and f₀ the molar fractions

(7)

(8)

(9)

(path length one-centimetre); ε_j is the molar absorptivity of the species H_jR, and C_R the analytical (total) concentration of the acid.

At low pH values one would expect the formation of H₂R and HR only and in consequence Eqn. (6) is reduced to

(10)

At high enough pH only the species HR and R are found in solution from a practical point of view, and from Eqn. (6) we have instead

(11)

Expressions (10) and (11) have the general form

(12)

corresponding to the single equilibria

(13)

and then we may evaluate the acidity constants K_an as

(14)

by measuring absorbances at a given wavelength and pH, once the values of A_N+1-n and A_N-n are known. Taking decadic logarithms on Eqn. (14) and rearranging terms we get

(15)

By plotting the left hand of Eqn. (15) against pH we get a straight line (least squares method) with slope unity which cuts the x-axis at pH=pK_a. Eqns. (12) or (14) may also be arranged into several forms [3] suitable for linear plotting.

The aim of this work is to find out in which experimental conditions a diprotic acid H₂R may be treated as a monoprotic one, HR, in order to determining their acidity constants spectrophotometrically.

A bilogarithmic method for the spectrophotometric evaluation of acidity constants of two-step overlapping equilibria has been devised on this basis. The method has been checked with synthetic theoretical data. Finally, another non-approximate bilogarithmic isosbestic point method has been devised and applied to A-pH (literature) data for benzidine, vanillyl ammonium ion and isonicotinic acid, being also the A-pH data obtained in this work for isonicotinic acid the subject of study and application.

2. Spectrophotometric Evaluation of Acidity Constants of Diprotic Acids by Monoprotic Acid Methods

From Eqn. (6) we get

(16)

which may be rearranged into

(17)

Taking decadic logarithms on Eqn. (17) we get

(18)

which has the form

(19)

where

(20)

A representation of the left hand of Eqn. (18) against the pH gives a straight line (least squares method) of slope -1 which cut the x-axis at a value equals to pK_a2.

Similarly, Eqn. (16) on rearrangement gives

(21)

Taking decadic logarithms on Eqn. (21) we get

(22)

where

(23)

A representation of the left hand of Eqn. (22) against the pH gives a straight line (least squares method) of slope 1 which cut the x-axis at a value equals to pK_a1.

At high enough pH values it is expected that only the two RH and R species will be present in solution; so, from a practical point of view, since The equilibrium given by Eqn. (4) is thus obtained, and Eqn. (19) becomes

(24)

At low enough pH values it is expected that only the two H₂R and HR species will be present in solution; so, from a practical point of view, since The equilibrium given by Eqn. (3) is thus obtained, and Eqn. (22) becomes

(25)

Both expressions, Eqns. (24) and (25) have the form of Eqn. (14) and are applicable when the limit absorbance of the intermediate specie A₁ is known, which can be systematically varied up to minimize a given parameter, such as the standard deviation of the regression line, for example.

3. Spectrophotometric Evaluation of Acidity Constants of Diprotic Acids from Absorbance Measurements at the Isosbestic Point Wavelength

If measurements are made at the wavelength in which A₁ = A₀ = A_i, then from Eqn. (16) we get

(26)

which on rearrangement gives

(27)

Taking into account that

(28)

where

(29)

(30)

(31)

being in this case

(32)

we get

(33)

Then, from Eqns. (33) and (28), after taking decadic logarithms we get

(34)

A representation of the left hand of Eqn. (34) against the second term of the right hand gives a straight line (least squares method) of slope unity which cut the x-axis at a value equals to ΔpK_a.

Moreover, from Eqn. (33) we get

(35)

The same considerations can be applied supposing that the measurements are realized at the wavelength of the isosbestic point in which A₁=A₂=A_j, in order to evaluate pK_a2. Then,

(36)

which on rearrangement gives

(37)

Taking reciprocals in Eqn. (31), subtracting 1, and then taking decimal logarithms of both sides we get an expression analogous to Eqn. (34)

(38)

As before, a representation of the left hand of Eqn. (38) against the second term of the right hand gives a straight line with unity slope and intercept equals to ΔpK_a.

From Eqn. (37) we get

(39)

It can finally be argued that a diprotic acid with overlapping acidity constants may be treated as a monoprotic acid provided that some approximation is made. The simplifying assumption is that the concentration C_R at low enough and high pH values equals to [H₂R] + [HR] and [HR] + [R], respectively.

4. A-pH Curves of Diprotic Acids: Simulated Data

Simulated data have been generated using Eqn. (6). Figure 1 shows the A-pH curves, corresponding to a family of diprotic acids with pK_a2 equal to 8.000, and pK_a1 variable (from 5.500 to 7.600). The ΔpK_a studied are: 0.250; 0.200; 0.150; 0.100 and 0.600. This last value is the smallest ΔpK_a difference that a diprotic acid can exhibit, as it is deduced from the analysis of their microconstants of acidity [4].

Figure 1. A-pH curves for a diprotic acid H2R with pKa2 = 8.000 and variable pKa1 (indicated in the legend in each case)

When the two pKa’s are far away, the two ionization equilibria can be treated independently. If the successive acidity constants are close (ΔpK_a<4.000) a simultaneous adjustment of the data is necessary [5].

Figure 2 shows the singular straight line (bilogarithmic) complete (diprotic acid model (RU), Eqn. 19) or single (monoprotic acid model, Eqn. 24) version applied to the calculation of pK_a2 (assuming known pK_a1 and the limit absorbance for the intermediate specie, A₁). Regardless of the wavelength measurement, which leads to different values for the limit absorbance A₂, A₁ and A₀ of the species, H₂R, RH and R, respectively, a straight line with slope equal to -1 is obtained. Theoretically, this same line is also obtained in spite of the ΔpK_a value.

Figure 2. The singular straight line (bilogarithmic) complete (diprotic acid model (RU), Eqn. 19) or single (monoprotic acid model, Eqn. 24) version applied to the family of curves of Figure 1 (assuming known pKa1 and the limit absorbance for the intermediate specie, A1)

When the diprotic acid is treated as monoprotic one (Eqn. 24) an accentuated curvature is observed at lower ΔpK_a values due to the occurrence of the H₂R specie involved in the first dissociation equilibrium of the acid.

A graph analogous to Fig. 2, with slope +1.000, is obtained when calculated pK_a1 from Eqn. (22) assuming pK_a2 and A₁ known (see Fig. 3). The cut-off point of the straight line with the x-axis is equal to the pK_a1 value of in each case. If the acid were considered as monoprotic one (Eqn. 25), it would get lines whose curvature is more accentuated when the pH increases and ΔpK_a decreases, due to the occurrence of the R specie involved in the second dissociation equilibrium of the acid.

Figure 3. The singular straight line (bilogarithmic) complete (diprotic acid model (RU), Eqn. 22) or single (monoprotic acid model, Eqn. 25) version applied to the family of curves of Figure 1 (assuming known pKa2 and the limit absorbance for the intermediate specie, A1)

Note that A₂ and A₀ can be obtained at enough acid and basic pH values, respectively. However, the singular straight line (bilogarithmic) complete (diprotic acid model) or single (monoprotic acid model) version requires for its application the knowledge of the limit absorbance of the intermediate specie HR (A₁). To do that, a trial and error method may be applied. For example the best value of A₁ is taken as that which minimizes the standard deviation of the corresponding regression line. The pK_a2 value is obtained from

(40)

The LINEST function in EXCEL [6] is very useful on this respect.

We select for study the system shown in Table 1, absorbances theoretically generated from known pH values by applying Eqn. 6 (rounded to four digits). In order to evaluate pK_a2 by means of Eqn. (24), the pH values were selected in the range [7.800-9.000; N=13] when only the two species HR and R are practically present in solution. The value of the fit parameter assumed is not critical, variations between 0.520 and 0.560 for A₁ lead to pK_a2 values of 7.975 and 8.023, respectively. A pK_a2 = 7.980 is found (Table 1) using A₁ = 0.524. Figure 4 shows the minimization process. Another possibility of trial and error method would be taking as A₁ that whose slope is closest to the unit. In this latter case, the optimal value of A₁ is 0.531 and pK_a2=7.988, in line with the previously obtained. The slight difference from the pK_a2 = 8.000 value comes from the fact that the single method is approximate.

Table 1. Evaluation of the pKa2 of a diprotic acid H2R as monoprotic one

Figure 4. Evaluation of the pKa2 of a diprotic acid H2R as monoprotic one

In a same way, we proceed to calculate pK_a1 according to Eqn. (25) selecting the pH range [5.000 – 6.200; N=13] in this case obtaining analogous good results. Then, a diprotic acid can be treated as monoprotic one without a considerable sacrifice in accuracy.

5. Determination of the Acidity Constants of Diprotic Acids from Absorbance Measurements at the Isosbestic Point Wavelength: Theoretical Model Systems

5.1. Benzidine

Theoretical [A, pH] data (Eqn. 6) are generated (A rounded to four digits) for the model system benzidine, R, a diacid base with parameter values: pK_a1=3.602; pK_a2=4.761; A₂=0.940; A₁=A₀=0.260 (isosbestic point).

The application of the complete bilogarithmic method (Eqn. (19)) to the theoretical data generated over the entire range of pH values leads to the exact value of the second dissociation constant, pK_a2=4.761, assuming a correct value of pK_a1 equal to 3.602 (straight line with slope -1.000). In the same way, the application of the complete bilogarithmic method based on the use of Eqn. (22) leads to the exact value of the first dissociation constant, pK_a1=3.602, assuming known the correct value of pK_a2=4.761 (straight line with slope 1.000).

Additionally, in order to apply the isobestic point bilogarithmic method (Eqn. 34), ten pairs of data [A, pH] were selected in the range of pH from 2.600 to 4.400 (near to pK_a1), from 0.200 to 0.200 units of pH. The minimization criterion leads to a pK_a2 value of 4.763. The bilogarithmic method at the isosbestic point wavelength leads to a pK_a1 value equal to 3.602.

So, it can be concluded that the results obtained are excellent. The value of the selected adjustment parameter is not especially critical on the pK_a1 calculation, i.e., assuming pK_a2 values of 4.700 and 4.800 lead to pK_a1 values equal to 3.619 and 3.592, respectively. If we adopt as the best value of the adjustment parameter those straight line with slope closest to the unit, we get pK_a2=4.761 and pK_a1=3.602, coinciding in both cases with the true values.

5.2. Vanillyl Ammonium Ion and Isonicotinic Acid

Analogous good results are obtained with vanyllil ammonium ion system (pK_a1=9.032; pK_a2=10.460; A₂=0.024; A₁=A₀=0.787) and to isonicotinic acid system (pK_a1=1.820; pK_a2=4.760; A₂=A₁=0.702; A₀=0.407).

6. Determination of the Acidity Constants of Diprotic Acids from Absorbance Measurements at the Isosbestic Point Wavelength: Experimental Systems

Table 2. Benzidine model system: pKa1 evaluation using the bilogarithmic method at the isosbestic point, A1=A0

6.1. Experimental Systems Described in the Literature

The bilogarithmic method has been applied to the literature data for benzidine [7], vanillyl ammonium ion [8], and isonicotinic acid [9]. Results obtained for benzidine are shown in Fig. 5. The pKa values obtained were: Benzidine: pK_a1=3.657 and pK_a2=4.455, ΔpK_a=0.798; Vanillyl ammonium ion: pK_a1=9.044 and pK_a2=10.657, ΔpK_a=1.613; Isonicotinic acid: pK_a1=1.816 and pK_a2=4.751, ΔpK_a=2.935.

Figure 5. Benzidine test system. Top: A-pH data at 245 nm, I=0.01, T=25° C, CR=0.5·10-4 M. Middle: pKa evaluation using the bilogarithmic method at the isosbestic point. Bottom: Reduced residuals (residual divided by sy/x)

These values are of the same order of magnitude as those published in their original papers, and of high quality as can be seen from the low s(y/x) of the regression lines, with the residuals randomly distributed. pK_a are reported with three digits in all cases, even if they are not significant.

6.2. Experimental Data: Isonicotinic Acid

Reagents

Isonicotinic acid (C₆H₅NO₂) M = 123.06 g/mol (Sigma, > 99%); Sodium hydroxide (NaOH) M = 40,01g/mol (Merck, analytical grade); Perchloric acid (HClO₄) M=100.5 g/mol (Merck, 72%); sodium perchlorate (NaClO₄) M=122.5 g/mol (Merck, analytical grade); Water for ACS analysis.

Instruments

Analytical balance (Metler AE200), Granatario (Metler PJ 400), pH-meter Crison GLP 21 (3 decimals), with a combined Ag/AgCl glass electrode. pH-meter is calibrated using pH buffers 4.01, 7.00 and 9.21 (setting the pH meter to 4,010, 7.000 and 9.210, respectively). Ultraviolet-visible molecular absorption spectrophotometer (Shimadzu).

Experimental part

Two point five mL of isonicotinic acid solution (1.750 10^-3 M) is pipetted into a 25 mL volumetric flask containing 2.5 mL of a NaClO₄ solution (1 M). Then varying amounts of HClO₄ and NaOH solutions are added in order to adjust the pH, completing to the mark with distilled water. The absorbance is measured at 266.7 nm (quartz cuvettes, pathlength one-centimetre) versus a blank. Finally the pH of the solution is also measured (passed into a 50 mL beaker).

Details of the experimental data (Fig. 6) follows: A₂ =A₁= 0.7052; A₀=0.4028. [pH, A]: [3.337, 0.6906; 3.962, 0.6793; 4.073, 0.6442; 4.149, 0.6222; 4.230, 0.6277; 4.240, 0.5969; 4.281, 0.6036; 4.383, 0.5756; 4.385, 0.6000; 4.392, 0.5894; 4.429, 0.5817; 4.659, 0.5309; 4.772, 0.5424. The dispersion of the data is greater than that obtained by Evans et al. [9].

Figure 6. Isonicotinic acid experimental system. A-pH data at 266.7 nm, I=0.1 M, T=25ºC, CR=1.675•10-4M

The application of the bilogarithmic method from absorbance measurements at the isosbestic point wavelength (Figure 7, top) leads to pK_a1=1.920 and pK_a2=4.615, ΔpK_a=2.695 ± 0.125, not very distant from those previously published by Evans et al. [9], calculated at a different ionic strength. The results are given with three decimal digits in order to show the precision of measurements achieved even if they are not significant.

Figure 7. Top: Isonicotinic acid pKa evaluation by the bilogarithmic method at the isosbestic point. Bottom: Reduced residuals (residual divided by sy/x) versus x

The residual analysis (Fig. 7, bottom) indicates a random pattern and values of reduced residuals lower to 2, in absolute value, so that no data was eliminated. The non-linear regression method (CSS Statistica) leads to a pK_a2 value of 4.611, although the pK_a1 value by this method is very sensitive to the initial starting value (i.e. Figure 8).

Figure 8. Non-linear regression method applied to the A-pH data obtained in this work for isonicotinic acid (pKa2=4.611)

7. Conclusions

Many molecules having analytical and biological interest [10] behave as diprotic acids. The bilogarithmic procedures (complete or simplified versions) and isobestic point bilogarithmic method applied in this work have been tested with synthetic data and model and experimental (benzidine, vanillyl ammonium ion, isonicotinic acid) systems and found to be satisfactory, yielding results of noticeable precision. The efficiency of the procedure relies upon having a reasonable a priori estimate of pK_a1 or pK_a2. The efficiency of the procedure relies upon having a reasonable a priori estimate of pK_a1 or pK_a2.

In spite of the modern possibility of calculating acidity constants using black-box computer programs, the determination of these constants by linearized plots seems to be more prevalent, probably owing to the transparency of the method used. Proper nonlinear fitting data is often difficult and non-linear least squares are not always free from problems. Occasionally, problems arise because of the choice of the data, initial estimates, convergence or multiple local minima, all typical of nonlinear regression. Therefore, the importance of transformation stems from the various limitations of nonlinear approaches.

ACKNOWLEDGEMENTS

Thanks are due to the University of Seville for the concession of a Grant destined to the publication of the Bachelor´s degree Final Projects of students.