1. 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 [1-9]. The solution of many galenical problems requires the knowledge of the acidity constants of compounds [10] having 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, the knowledge of acidity constants [11-15] being thus of great worth. Evaluation of acidity constants of organic reagents is also of great value in planning analytical work [16, 17], e.g., the acidity constants can be employed in the design of titration procedures [18] and examining the possibility of separation of mixtures of compounds by extraction. The complexing properties of a molecule depends on the number and steric disposition of is donor centres as well as on its acid-base properties [19-21].

The ionization equilibrium of a monobasic acid

(1)

is characterized by the acidity constant

(2)

β₁ is the stability constant of HA, i.e. the constant corresponding to the formation equilibria H + R = HR. The ionic strength and temperature of the solution are assumed to be constant, so that mixed or conditional constants are used in the calculations. Charges are omitted for simplicity.

If A is the measured absorbance (for 1-cm pathlength) of a solution containing a total concentration

(3)

of the acid, then assuming that Beer’s law holds, we have [22, 23]

(4)

(5)

where f₀ and f₁ are the molar fractions of R and HR

(6)

and A₀ and A₁ are the limit absorbances of the species R and HR, respectively, i.e. the absorbances of the pure forms of the reagent R and HR, respectively, which have molar absorptivities ε₀ and ε₁; A₀=ε₀C_R and A₁=ε₁C_R. Eqn. (5) on rearrangement gives

(7)

The slope of the A-pH curve (Figure 1) is given by

(8)

Differentiation of Eqn. (8) with respect to pH leads to

(9)

The condition d²A/d(pH)² will locate the point of inflexion (Figure 1) in the graph of A against pH [23-25]. At this point [H] = K_a, and then by applying Eqn. (5) we have at this point (A’’, pH’’)

(10)

Figure 1. Absorbance-pH curve corresponding to a monoprotic acis HR (blue), and first derivative A-pH curve (red) and second derivative A-pH curve (black)

Note that the value of dA/d(pH) at this point [26] is given by

(11)

where ΔA= A₁-A₀.

The pK_a value of a monoprotic acid can thus be determined by plotting absorbances as a function of pH for a series of solutions having a constant concentration C_R of reagent. The inflexion point of the curve A versus pH, i.e. the value of pH (equal to pH’’) that satisfies the condition (10) coincides exactly with the pK_a value. Although this is strictly true for monoprotic acids, one can wonder if this simple procedure is applicable to diprotic acids. In other words: can a diprotic acid be treated as a monoprotic one? An answer to this question is given in that follows.

2. The Diprotic Acid System

For the dissociation of a diprotic acid H₂R we have the equilibriadescribed by the equationswhere we are specifically neglecting charges for the sake of the generality. The absorbance and the composition of any given solution of a diprotic acid having concentration C_R (Figure 2) is given by [22, 27-30]

(14)

Figure 2. Top: A2= 0.2; A1=0.4; A0=0.8; pKa1=5.5 and varying ΔpKa. Middle: A2=0.4; A1=0.8; A0=0.2; pKa1=5.5 and varying ΔpKa. Bottom: A2=0.8; A1=0.2; A0=0.4; pKa1=5.5 and ΔpKa (ΔpKa: 1; 2; 2,5; 3, 4; and 5)

(15)

where A₀, A₁, and A₂ are the limit absorbances of R, HR and H₂R, respectively, and f₂, f₁ and f₀ the molarity fractions (f_j=[H_jR]/CR and A_j=ε_jC_R). Let A^* and A^** be the absorbance values of two samples which have pH values pH^* and pH^**, respectively, which satisfies the following conditions

Substituting condition (16a) into Eqn. (15), upon rearrangement and collecting the terms containing identical powers of [H^*] one obtains

(17)

Solving this quadratic equation for the concentration of hydrogen ions gives

(18)

In this case the negative root has no physical significance. A more compact version of Eqn. (18) can be obtained by usingand thus

(20)

From Eqn. (20) it is seen that [H^*]=K_a1 only if – a 10^-x is negligible compared with the unity. Effectively when x is large to unite there is a wide range of concentration of [H] over which f₁=[HR]/C_R is very close to 1, as well as the square root included in the expression (20), and then

(21)

A not unexpected result. The value of the ratio K_a1/K_a2=10^x is an important property of a diprotic acid, because this relation can be regarded as the equilibrium constant of the following reaction [31]

(22)

(23)

On the other hand, Eqns. (16b) and (15) can be combined to give, once upon rearrangement and collecting the terms in powers of [H]

(24)

The meaningful solution of this quadratic equation is

(25)

The form in which Eqn. (25) is presented is important. In effect, although it is evident from Eqn. (20) that [H^*] tends to K_a1 when x is large, it is not clear from Eqn. (25) that [H^**] tends to K_a2 when x is large. In order to place the limiting process on a sounder basis we will demonstrate that the limit of the ratio is actually the desired quantity. Taking into account expression (19a) we get for [H^**]

(26)

which can conveniently be re-written as follows

(27)

where for the sake of brevity

The limit of the term in parenthesis, f(x), in Eqn. (27) when x is very large to the unity is the unity. Effectively, by applying L’Hôpital’s rule [32, 33] we get

(29)

Rewriting Eqns. (20) and (27) in logarithmic form gives

(30)

(31)

The values of α and β as a function of x for a few systems taken as examples are shown in Table 1, where it is seen that the situation of pK_a values in the pH scale with respect to pH^* and pH^** is dependent on the relative values of the limit absorbances A₀, A₁ and A₂, as well as on the value of x=ΔpK_a. However, except for very close pK_a values the acidity constants are easily experimentally obtained making use of expressions (16a,b).

Table 1. Dependence of the α and β values with ΔpKa

Nevertheless, Eqns. (16a,b) though instructive are not very useful since A₁ must be known. In cases in which equilibria overlap A₁ is usually evaluated together with K_a1 or K_a2 by applying graphical methods of evaluation [34], although in some numerical and graphical method of evaluation K_a1 and K_a2 are simultaneously evaluated whereas A₁ is not. Thus, at first glance the calculations given above are not only somewhat a waste of time, but are also philosophically unattractive. How can be avoided this logical absurdity? Are the expression derived above merely an academic exercise?. A new method reported by Polster [35-39] and based on the measurements of absorbances at two wavelengths (λ and λ^*) allows to evaluate graphically the limit absorbances A₁ and A₁^* for the intermediate specie HR, thus making the matter presented in this work useful both in research as in the teaching of chemical equilibria methods at all levels.

3. Evaluation of the Limit Absorbance A₁ from Absorbance Measurements at Two Wavelengths

In the evaluation of acidity constants of overlapping equilibria approximations of various sorts are frequently made in order to carry out calculations. Working at low pH values where it is only assumed the presence of the species H₂R and HR, i.e. f₀ is close to zero, and from Eqn. (14), we have for measurements at two wavelengths λ and λ^*

(32)

(33)

since in this specific case f₂ + f₁ = 1. Whence it follows that (rearrangement f₁ from Eqns. (32) and (33) and equating the result in both cases)

(34)

From which

(35)

and thus plotting A against A^* for a series of solutions we obtain a straight line having a slope equal to k and an intercept on the ordinate axis equal to A₂ – k A₂^*. In much the same fashion one can obtain in the pH range in which the species HR and R are present that the absorbance at any particular wavelength is given by

(36)

since in this case and then from measurements at two wavelengths λ and λ^* we have

(37)

from which

(38)

A plot of A against A^* for a series of solutions will be a straight line. Its slope will be equal to k’ and the point at which A^*=0 is equal to A₀ – k’ A₀^*. It should be noted that Eqns. (32) and (36) are strictly true only when two species are present in solution. However, when equilibria overlaps there is a range of pH in the neighbourhood of ½ (pK_a1+pK_a2) where the three species R, HR and H₂R are present in solution, a curvature being obtained in both representations with these points.

Equations. (34) and (37) are really particular examples of the more general case, easily derivable from Eqns. (33) and (36)

(39)

previously used by Coleman et al. [40] in the determination of the number of species, n, in solution. When n=2 the plot of the absorbance minus the absorbance of a solution j taken as reference, at λ, against the absorbance minus the absorbance of the reference solution, at λ^*, for a series of solutions, gives a straight line passing through the origin. Working with different pairs of wavelengths a family of straight lines intersecting in the origin of coordinates is obtained.

It can be easily demonstrated that the point of intersection of the two straight lines is given by

(40)

On the other hand the required unknowns A₁ and A₁^* can be obtained by solving pairs of simultaneous equations derived from Eqns. (35) and (38)

(41)

(42)

which lead to

(43)

(44)

Comparing Eqns. (43) and (44) and Eqn. (40) we see that the coordinates of the point of intersection of both straight lines defined by Eqns. (35) and (38) are given by (A₁^*, A₁). By substituting the expression for k and k’ given by Eqns. (34) and (37), respectively, into Eqn. (40) we also get the coordinates of the intersecting point, but much algebra would be needed to achieve the same results.

Plots of A against A^* are easy to construct, especially with the aid of a spreadsheet. It is a simple matter to evaluate A₁ and A₁^* from such a plot; the extended tangents or limiting slopes of straight lines should intersect at the point (A₁^*, A₁). From an experimental point of view the conditions more favourable for obtain A-pH data is spectrophotometric titration, but fairly precise results can be obtained and good accuracy can be achieved obtaining a number of data closely spaced. However, arithmetic calculation in this method is reduced at a minimum, which undoubtedly constitutes an attractive feature with purposes of teaching in the undergraduate analytical laboratory if comparing with other graphical or numerical methods of evaluation.

Nevertheless, although it is true that at the pH values corresponding to the absorbances A^* and A^** we have (ΔpK_a>2)

(45)

there seems little point in measuring the whole absorbance versus pH graphs merely to determine three points. So, the constants obtained are little efficient in terms of return for effort used [41]. Another way, which is more complicated but which uses the data more efficiently and provides a much more reliable value A₁ and A₁^* is described in the following. An orthogonal regression method should be applied to A versus A^* data because the two axes are affected by errors of similar magnitude.

4. Orthogonal Regression

We can apply a least squares method to the experimental data (A^*, A), but single linear regression is not strictly applicable to fitting the best straight line through data points because both variables A and A^* contain analogous random error of measurement [42-45].

The perpendicular distance from the point (x_i, y_i) to the line whose algebraic representation is

(46)

is given by

(47)

We will minimize the sum of the squares of the distance perpendicular to the least squares line

(48)

Note that there is only to unknown quantities in Q: a₀ and a₁. If Q is to be a minimum the first partial derivatives of Q with respect to a₀ and a₁ must be zero. Then

(49)

and

On the other hand by combing Eqns. (48) and (50b) we have

(51)

where S_XX, S_YY are sums of squares about the mean for two variables (x and y), and S_XY is the corresponding sum of cross-products

(52)

(53)

(54)

and thus

(55)

(56)

(57)

(58)

The meaningful solution of the quadratic Eqn. (57) gives the value of a₁. The sign of a₁ is the same as the sign of S_XY. Once the value of a₁ is known, the value of a₀ is calculated from Eqn. (50b). The (A₁^*, A₁) point is defined by the intersection of the extrapolated linear portion of the two branches obtained (by the least squares method above described) plotting A against A^*, y=a₀+a₁x and y = b₀+b₁x, and then we have

(59)

In those cases in which measurement errors affect both axes in an unequal way, Boccio et al. [46], and McCartin [47, 48] should be consulted, in addition to the references cited at the beginning of this section. Once the limit absorbance, A₁, of the intermediate specie HR, is known the acidity constants can be evaluated as follows.

5. Spectrophotometric Evaluation of Acidity Constants

From Eqn. (15) we get

(60)

which on rearrangement gives

(61)

A plot of the left hand against [H] (A-A₂)/(A-A₁) gives a straight line of slope 1/(K_a2K_a1) and intercept 1/K_a2 from which

(62)

(63)

6. Error Analysis

All analytical measurements are random variables and the information they provide is subject to uncertainty. Changes of interest are usually based on a set of observations and we want to know if the mean and variance of these functions are related to the mean, variance and covariance of the original measurements. The relationship for calculating the variance of a continuous arbitrary function of multiple variables, x₁, x₂ ... x_N, which are normally distributed with variances , is known as the law of random error propagation [49, 50], which is expressed as

(64)

(65)

The law of error propagation applied to the function R = f (a₀, a₁) leads to

(66)

s_a0², s_a1², and cov(a₀, a₁), are the variance of the intercept, the variance of the slope and the covariance between the intercept and the slope of the regression line using the conventional least squares method in this case

(67)

(68)

(69)

(70)

(71)

where s_y/x² is the variance of the regression line, given by

(72)

Note that the s_y/x value can be easily get using linear regression (method of the least squares), in EXCEL, with the function LINEST (but no the covariance between the intercept and the slope).

Since K_a2 is a function of the intercept

(73)

the variance of K_a2 is equal to

(74)

and as pK_a2, is a function of K_a2

(75)

we can calculate the variance of pK_a2 as

(76)

and its standard deviation as

(77)

The first acidity constant is function of the intercept and the slope, so the corresponding expressions are more complicated

(78)

Thus applying the law of random error propagation

(79)

and then, the variance of pK_a1 is equal to

(80)

and as pK_a1, is a function of K_a1, we can calculate the variance of pK_a1 as

(81)

the covariance of measurements can be as important as the variances and both contribute significantly to the total analytical error [51, 52].

7. Evaluation of Acidity Constants in Experimental System: Resorcinol

Resorcinol or 1,3-benzenediol is used in the determination of ascorbic acid in pharmaceuticals and in the synthesis of several organic compounds [53]. However, there are few studies dealing with the molecular structure [54] or the acid–base properties of this compound in solution [10]. Considering that knowledge of the acid dissociation constants (pK_a) becomes essential for the development of new compounds with biological activity [55, 56], in this paper we determine the overlapping pK_a values of resorcinol in water.

In this section, we apply the developed theory to the experimental A-pH data (see Table 2) described in the literature for resorcinol published by Blanco et al. [57]. The A-pH curves corresponding to a resorcinol at 368.3 and 293.5 nm are plotted in Figure 3.

Table 2. A-pH data for the resorcinol system published by Blanco et al. [57]

Figure 3. A-pH data at 268.3 and 193.5 nm, I=0.1, T=25°C, CR=4.76·10-4 M for the resorcinol system

Figure 4 shows the application of the Polster method. The orthogonal regression leads to A₁ values of [0.5353; 0.8271]. After the application of Eqn. (61) the pK_a values were obtained from the slope 1/(K_a2K_a1) and the intercept 1/K_a2. The best value of the limit absorbance for the intermediate specie HR (A₁) was obtained by a trial and error method, i.e. the best value of A₁ is taken as that which minimizes the standard deviation of the corresponding regression line. The value assumed for A₁ was 0.814 (Figure 6).

Figure 4. Evaluation of the limit absorbance of the intermediate specie, HR

Figure 5. Evaluation of the acidity constants of resorcinol by Eqn. 61

Figure 6. s(y/x) as a function of the A1 value assumed

The pK_a values obtained are: pK_a1 = 9.251 ± 0.066 and pK_a2 = 10.879 ± 0.014. These values are of the same order of magnitude as those published in the original work of Blanco et al. [57], or those described in the literature for resorcinol. Values of estimated pK_a are reported with three digits in all cases, even if they are not significant. In addition, the experimental data considered are of high quality as can be seen from the low standard deviation of the regression lines.

8. Conclusions

It can finally be argued that a diprotic acid with overlapping (simultaneous equilibria) acidity constants may be treated as a monoprotic acid provided that some approximation is made. The simplifying assumption is that the concentrations of the species R or H₂R are small compared with the total concentration C_R at enough low and pH values, respectively. Such approximations are often necessary to have an accurate knowledge of the composition of the solution at a certain pH interval.

A new method reported by Polster and based on the measurements of absorbances at two wavelengths (λ and λ^*) allows to evaluate graphically the limit absorbances A₁ and A₁^* for the intermediate species HR of a diprotic acid H₂R, dealing with overlapping (simultaneous) equilibria. Least squares treatment, which takes into account similar errors in both x and y variables, i.e. orthogonal regression, is also included in this work. Few attempts to deal with this problem in the evaluation of equilibrium constants have been made.

Except for very close pK_a values the acidity constants are easily experimentally obtained making use of expressions A_(pKa1)=(A₂+A₁)/2 and A_(pKa2)=(A₁+A₀)/2. However, there seems little point in measuring a whole A-pH curve to determine one point. The constants so obtained are much less efficient in terms of return for effort used.

The theory developed in this work has been successfully applied to the experimental systems described in the bibliography (resorcinol, with ΔpK_a of about 1.7). A detailed analysis of the errors implied is also made, taking into account the strong correlation existing between the slope and intercept of a straight line obtained by the least squares method. The covariance between two variables is so important as the variances and both contribute significantly to the total analytical error.

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.