Closed-form expressions for monoprotic weak acid aqueous solutions

A general closed-form expression for [H3O+]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\textrm{H}_{3}\textrm{O}^+]}$$\end{document} of solutions of a weak acid and a weak acid buffer, as well as their titration with a strong base, has been obtained and mathematically analyzed with the aid of computer algebra systems. This expression is used to evaluate, without the use of numerical approximations, the precision and accuracy of different approximations commonly employed in general chemistry and chemical analysis courses. The closed-form expression for [H3O+]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{[\textrm{H}_{3}\textrm{O}^+]}}$$\end{document} of a buffer solution is used to obtain an analytical expression for the pH stability when a strong base is added. Finally, it is shown that the [H3O+]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\textrm{H}_{3}\textrm{O}^+]}$$\end{document} expressions for all the systems under study can be obtained from a general closed-form expression in terms of an effective weak acid constant and an effective acid concentration. The formulas found in this work do not display numerical rounding errors and do not require the use of numerical or graphical methods for their evaluation. Graphical TOC: A closed-form expression for the weak acid and weak acid buffer solutions and their titrations with a strong base Graphical TOC: A closed-form expression for the weak acid and weak acid buffer solutions and their titrations with a strong base


Introduction
Weak acids are important physical systems of interest in chemistry, biochemistry, chemical engineering, and physiology. A weak acid is a substance that undergone partial protolysis in water to produce undissociated acid molecules, hydronium ions, H 3 O + , and the acid conjugate base. An aqueous weak acid buffer solution is formed by a weak acid and one, or several, of its soluble salts. The equilibrium of a weak acid, or a weak acid buffer, involves four chemical substances and four equations: the dissociation of the acid, the autoionization of water, and the balances of charge and matter. In principle it is possible to obtain the equilibrium concentrations of all the chemical species for the weak acid dissociation or the buffer solution by solving these systems of equations. In practice, these systems of equations involve nonlinear terms making it difficult to obtain exact mathematical expressions for the concentrations. The algebraic manipulation of the systems of equations for the weak acid dissociation or the buffer solution give cubic equations for the concentration of H 3 O + [1][2][3]. The equilibrium concentration of H 3 O + is obtained by finding the roots of these cubic equations. Although there exists a cubic formula that gives the explicit roots of a cubic equation, it is not practical to use because of its complexity [4][5][6]. The crude use of the cubic formula requires more than 30 arithmetic operations, making its use for hand calculation impractical [6][7][8]. Macleod and Barling [6,9,10] have already reported mathematical 1 3 9 Page 2 of 9 formulas for the pH of weak acid solutions. Although Macleod's [9] expressions are exact, they show numerical rounding errors, so the Newton-Raphson numerical method is used to calculate the pH. Numerical difficulties with the pH formulas have been observed also in titration models of weak acids [11]. In this work exact simplified formulas for the pH of a weak acid and its buffer are shown and evaluated without numerical rounding errors.
A quadratic equation is obtained for the hydronium ion if the autoionization of water is ignored. Under this approximation, the equilibrium concentration of the H 3 O + is obtained simply by solving a quadratic equation. The approximation of ignoring water autoionization displays a relative error greater than 10% for weak acids with K a < 10 −6 , and at low analytical concentrations of the acid or the salt of the acid, C a,s < 10 −6 M , with 1 M = 1 mol L −1 . Although the concentrations used in the general chemistry laboratory are generally above 10 −2 M , there may be situations in chemical analysis where very low concentrations are handled [1,3,7,8].
General purpose computer algebra systems (CAS) allow the algebraic manipulation of equations and inequalities [12]. The use of CAS in mathematical software, with userfriendly graphical interfaces, has extended the use of CAS to all fields of knowledge. Examples of free and non-free CAS are Maple [13], Wolfram Mathematica [14], Maxima [15], and SageMath [16]. Today mathematical software with CAS capability is easily found in research and academic laboratories and classrooms. Complex problems that were considered beyond the scope of the researchers or students can be solved today on the computer by using CAS mathematical software. The mathematical solution of the aqueous dissociation of weak acids or buffer solutions requires functions available in mathematical software. These functions are mainly (i) the simplification and algebraic manipulations of large equations, and inequalities, and (ii) the use of complex analysis, and graphical representation of functions, inequalities, and implicit regions. In this work, Wolfram Mathematica is employed to obtain simplified closedform expressions for the equilibrium concentrations of the chemical species of four aqueous systems: the weak acid dissociation and its titration by a strong base, the buffer solution and its titration by a strong base [1][2][3]17]. The cubic equations for the hydronium ion are analyzed, their roots are obtained by the use of the classical Cardano's method for the associated depressed cubic equations [18,19]. The direct results of Cardano's method are the cube roots of two complex numbers. The polar representation of these cube roots allows one to obtain only three real roots. These roots are simplified and reduced to closed-form expressions that are simple enough to allow exact pH calculations with only a handheld scientific calculator. The use of Descartes' rule of signs [18][19][20] allows one to demonstrate that only one root is positive. The pH stability is measured as the change of the pH as the strong base is added. Finally, a general equation for the concentration of H 3 O + is presented. This general equation gives the pH for any of the four systems studied in this work in terms of an effective acid constant and an effective acid concentration.

General equation for the concentration of [H 3 O + ]
The aqueous dissociation equilibrium of a weak acid HB is given by the chemical equations A solution of the acid HB , the salt NaB , and the strong base NaOH , with concentrations C a , C s , and C b , respectively, reaches chemical equilibrium. This equilibrium is quantitatively given by five equations [3]. These are the dissociation of the acid K a , the autoionization of water K w , the electric neutrality, the matter balance for the acid, and the matter balance for the strong base and salt added: with C • = 1 M . Acid constants K a are dimensionless, with values ranging typically between 10 −10 and 10 −1 . It is mathematically convenient to define the varia b l e s In terms of these dimensionless variables and constants, Eqs. The combined use of Eqs. (10) and (12) in Eq. (11) produces after some algebra which can be conveniently rewritten as Regardless of the value of sgn c 1 , this 4-tuple shows only one change of sign, from positive to negative. Descartes' rule of signs states that the number of positive roots of a polynomial is, at most, equal to the number of sign changes of its ordered list of coefficients [20]. The use of Descartes's rule indicates that P = 0 has only one positive root, no matter what system is dealt with-an acid solution, a buffer solution, or the neutralization of any of these solutions.
The full characterization of the roots of P = 0 is given by the discriminant of the polynomial (15), Δ[P] [18,19,21]. The case of Δ[P] > 0 indicates that the three roots of P = 0 are all real and different. In the case Δ[P] < 0 , one of the roots is real and the other two roots are complex, which are a complex conjugate pair. The case Δ[P] = 0 indicates multiple roots. Although the mathematical expression of Δ[P] is complicated, the use of the function of Wolfram Mathematica shows that Δ[P] > 0 under the assumptions c a > 0 , c b ≥ 0 , and k a > 0.
In the supporting information it is shown that 3 , the positive root of P = 0 , is given by

3
9 Page 4 of 9 with a H 3 O + as the activity of the H 3 O + ion.

Weak acid
The case of a weak acid is described by using c a = c a and The coefficient p is evidently negative, meanwhile q can have positive or negative values. Algebraic manipulation on the inequality q < 0 gives k 2 a + 9 2 c a k a − 9 < 0 , which can be rearranged to obtain c a < 2 9k a (k 2 a − 9) . The last inequality implies that negative values of q are obtained with k a < 3 , and c a < 2 9k a 9 − k 2 a . Very weak acids, e.g., HCN ( k a = 6.2 × 10 −3 ) or HOCl ( k a = 0.4 ), would have negative q at micromolar concentrations.
The limit of infinite dilution of gives (32) lim fulfilled with a real k a . On the other hand, the value k a = 3 produces q = 0 and Δ[P] > 0 , hence = ∕2 and = 1.
As the concentration c a is increased, the angle tends to = ∕2 for all the acids, regardless of its p K a . The angle displays a maximum as a function of c a . This maximum is obtained from the condition ∕ c a = 0 , which can be shown to give c a = 8∕k a .
The Mathematica for P = 0 gives = 40.6076 . Although the numerical solution is identical to the analytical solution, the former has a small imaginary part that must be removed to calculate the pH. The contours of pH = 7 − log 10 as a function of C a = √ K w c a and C b = √ K w c b for acetic acid, k a = 180 , are shown in Fig. 2. It is interesting to note that there is a linear relationship between base and acid concentrations at constant pH ; in fact, the pH = 7 line has a slope of 1 and passes through the origin (C a , C b ) = (0, 0) . Figure 2 also shows that the C b -intercept is negative for acidic pH and positive for basic pH . It is also seen in this figure that the slope of the lines C b (C a ) is less than 1 for acidic pH.

Buffer solution
The case of a buffer solution requires one to consider c a = c a + c s and By applying the logarithm of the acid dissociation constant K a , Eq. (5), we obtain, after some algebraic manipulation, the Henderson-Hasselbalch (HH) equation [23][24][25] The right-hand side of this equation is a function of pH and therefore the HH equation is not practical for direct calculation of the pH. It is common to use the approximations [B − ] ≈ C s and [HB] ≈ C a to obtain Figure 3 shows the absolute error of the pH , for chlorous acid, calculated by the HH equation with respect to the pH calculated using the exact formula, Eq. (40) with c b = 0, (46) pH = pK a + log 10 [ (47) pH (1) ≈ pK a + log 10 C s C a .

Buffer capacity
The pH stability of an acid buffer solution is measured by adding a volume V b of a strong base solution with concentration c 0 b [3]. The addition of this volume changes the concentrations c a , c s , and c b , with c 0 a , c 0 s , and c 0 b , as the concentrations of the acid, salt of acid, and base independent solutions, respectively. The The gradient ∇ c is calculated with respect to the components of c but it must be expressed in terms of the volume of base V b . Figure 4 displays the pH stability, S pH , as a function of the pH of the buffer solution. Since the addition of a strong base increases monotonically the pH of the solution, the curves of Fig. 4 contain the same information as the titration curves. To simplify the analysis all the concentrations used to prepare or titrate the buffers, c 0 a , c 0 s and c 0 b , have the same value c 0 . Panel (a) of Fig. 4 displays S pH ∕100 for buffer solutions with concentrations C 0 = 10 −7 × c 0 ranging from 10 −2 to 10 −1 M (from blue to red). Panel (b) of Fig. 4 displays S pH for buffer solutions with concentrations C 0 = 10 −7 × c 0 ranging from 10 −7 to 10 −5 M (from purple to red). In both panels of Fig. 4, curves with lower (higher) values of S pH are for buffer solutions prepared and titrated with solutions of lower (higher) concentrations. The maximum of the curves S pH is higher, and located at higher pH, for buffer solutions at higher concentrations C 0 . The highest concentration buffer of Fig. 4a (in red) displays numerical instability for basic pH. Figure 5 displays the titration curves for the same buffer solutions as in Fig. 4. To simplify the analysis, the concentrations of the solutions to prepare, and titrate, the buffer are the same, C 0 = 10 −7 c 0 . All the titration curves of panels (a) and (b) of Fig. 5 intercept at V b = 1 L and pH = 7 . Buffers of higher concentrations, in red, display the largest changes of pH as the solution is titrated with (52) (54) S pH = 1 ln 10  Fig. 5(a) that the steepest change in the pH is given for pH ≈ 9 , as shown in Fig. 4a. The use of Figs. 4a and 5a indicates that the lower the concentration of the solutions of acid and salt of the acid (blue curves), the lower the change in pH as the base is added.

Systematic analysis relating the pH expressions
The four cubic equations for x = [H 3 O + ]∕ √ K w of the systems under study can be written as  The coefficients of the four systems under study are given by the rows of Table 1.
The concentration x is given by the only positive root of these polynomials, with c and k as effective concentrations and acid dissociation constants, given by the fifth and second columns of Table 1, respectively. The angle is given by the expression with discriminant Δ = −4p 3 − 27q 2 > 0 , and The values of k and c from Table 1 show that k ≥ k a and c ≤ c a . Equations (57)-(60) with the data of Table 1 give the full description of the pH for the systems studied. otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.