The aqueous dissociation equilibrium of a weak acid \(\hbox \) is given by the chemical equations

$$\begin \hbox + \hbox _2\hbox &\leftrightarrows \hbox _\hbox ^ + \hbox ^, \end$$

(1)

$$\begin 2 \hbox _\hbox &\leftrightarrows \hbox _\hbox ^ + \hbox ^. \end$$

(2)

These equilibria are altered by the addition of a salt \(\hbox \) or the addition of a strong base, e.g., \(\hbox \). The salt and the strong base dissociate completely in water to produce the ions \(^}\), \(^}\), and \(^}\), through the reactions

$$\begin \hbox &\limits ^}} \hbox ^ + \hbox ^,\end$$

(3)

$$\begin \hbox &\limits ^}} \hbox ^ + \hbox ^. \end$$

(4)

The addition of the salt forms a buffer solution meanwhile the addition of the strong base neutralizes the acidity of the solution. Relevant chemical species are \(_\textrm^+}\), \(^}\), \(\hbox \), \(^}\) , and \(^}\), with molar concentrations \([\textrm_\textrm^+]\), \(^]}\), \([\hbox ]\), \(^]}\) , and \(^]}\) respectively [1,2,3].

A solution of the acid \(\hbox \), the salt \(\hbox \), and the strong base \(\hbox \), with concentrations \(C_\textrm\), \(C_\textrm\), and \(C_\textrm\) , respectively, reaches chemical equilibrium. This equilibrium is quantitatively given by five equations [3]. These are the dissociation of the acid \(K_\textrm\), the autoionization of water \(K_\textrm\), the electric neutrality, the matter balance for the acid, and the matter balance for the strong base and salt added:

$$\begin K_\textrm&=\frac_\textrm^+\right] }}}\frac^]}}}\left( \frac]}}\right) ^,\end$$

(5)

$$\begin K_\textrm&=\frac_\hbox ^+\right] }} }\frac^]}}},\end$$

(6)

$$\begin }_}^+\right] } + \left[ \hbox ^\right]&=^]}+^-]},\end$$

(7)

$$\begin C_\textrm+C_\textrm&=[\hbox ] + ^-]},\end$$

(8)

$$\begin C_\textrm + C_\textrm&= ^+]}. \end$$

(9)

with \(C^ =1\,\textrm\). Acid constants \(K_\textrm\) are dimensionless, with values ranging typically between \(10^\) and \(10^\). It is mathematically convenient to define the variables \(_\hbox ^+]}/(C^ \sqrt})},\) \(^-]}/(C^ \sqrt})},\) \(]/(C^ \sqrt})},\) \(^-]}/(C^ \sqrt})},\) \(^+]/(C^ \sqrt})}}\) and the dimensionless constants \(}=C_\textrm/\sqrt}}\), \(}=C_\textrm/\sqrt}}\), \(}=C_\textrm/\sqrt}}\), \(}=K_\textrm/\sqrt}}\), and \(=1}\). In terms of these dimensionless variables and constants, Eqs. (5)–(9) are replaced by

$$\begin k_\textrm&=\frac,\end$$

(10)

$$\begin x+}_\textrm&=\frac+z_1,\end$$

(11)

$$\begin }_\textrm&=z_0+z_1, \end$$

(12)

with effective base and acid concentrations \(}_\textrm=c_\textrm+c_\textrm\), and \(}_\textrm=c_\textrm+c_\textrm\). These effective concentrations are constrained to \(}_\textrm\ge 0\) and \(}_\textrm\ge 0\).

The combined use of Eqs. (10) and (12) in Eq. (11) produces after some algebra

$$\begin x^2+}_\textrm x-1=\frac}_\textrm k_\textrmx}}, \end$$

(13)

which can be conveniently rewritten as

$$\begin \left( x-\sigma _1\right) \left( x-\sigma _2\right) \left( x+k_\textrm\right) =}_\textrmk_\textrmx, \end$$

(14)

where \(\sigma _=\tfrac\left( -}_\textrm\pm \sqrt}_\textrm^2+4}\right)\), with \(\sigma _1\ge 1\), \(\sigma _2\le -1\), \(\sigma _1 \sigma _2=-1\), and \(\sigma _1+\sigma _2=-}_\textrm\).

Expansion of Eq. (13) gives the cubic equation \(P=0\), with

$$\begin P=x^3+c_2 x^2+c_1 x + c_0, \end$$

(15)

and coefficients

$$\begin c_3&= 1,\end$$

(16)

$$\begin c_2&= k_\textrm+}_\textrm,\end$$

(17)

$$\begin c_1&= -1+k_\textrm\left( }_\textrm-}_\textrm\right) ,\end$$

(18)

$$\begin c_0&= -k_\textrm. \end$$

(19)

The 4-tuple of coefficients of P \(\,}}=\left( c_3,c_2,c_1,c_0\right) }\) gives important information about the roots of the equation \(P=0\). The signs of the \(\,}}[P]\) give the 4-tuple

$$\begin \begin \,}}\left( \,}}[P]\right)&=\left(\,}},\,}},\,}},\,}}\right) \\&=\left( +,+,\pm ,-\right) . \end \end$$

(20)

Regardless of the value of \(\,}}\), 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), \(\Delta [P]\) [18, 19, 21]. The case of \(\Delta [P]>0\) indicates that the three roots of \(\) are all real and different. In the case \(\Delta [P]<0\), one of the roots is real and the other two roots are complex, which are a complex conjugate pair. The case \(\Delta [P]=0\) indicates multiple roots. Although the mathematical expression of \(\Delta [P]\) is complicated, the use of the function \(\textsf \) of Wolfram Mathematica shows that \(\Delta [P]>0\) under the assumptions \(}_\textrm>0\), \(}_\textrm\ge 0\), and \(k_\textrm>0\).

In the supporting information it is shown that \(\rho _3\), the positive root of \(P=0\), is given by

$$\begin \rho =2\sqrt[3]\cos/\right) -\frac+}_\textrm}}, \end$$

(21)

with

$$\begin \root 3 \of &=\tfrac\sqrt+}_\textrm\right) ^2+3k_\textrm\left( }_\textrm-}_\textrm\right) +3},\end$$

(22)

$$\begin \theta&=\arctan ,\frac}\right) },\end$$

(23)

$$\begin \Delta [P]&=-4p^3-27q^2,\end$$

(24)

$$\begin p&=-\tfrac\left( k_\textrm+}_\textrm\right) ^2-k_\textrm\left( }_\textrm-}_\textrm\right) -1,\end$$

(25)

$$\begin q&=\tfrac\left( k_\textrm+}_\textrm\right) ^3+\frac}\left( k_\textrm+}_\textrm\right) \left( }_\textrm-}_\textrm\right) +\frac}_\textrm-2k_\textrm}. \end$$

(26)

The angle \(\theta\) given in Eq. (23) is restricted to \(\theta \in (0,\pi )\). The function \(\arctan (x,y)\) used in the definition of \(\theta\) gives the arc tangent of y/x taking into account which quadrant the point (x, y) is in. This angle \(\theta\) is related to the trigonometric solution obtained by Nickalls for the roots of the cubic equation [22].

The general Eq. (21) gives the concentration \(_\textrm^+]}=10^ \rho }\), and an approximation to the pH,

$$\begin \begin \textrm&=-\log __\textrm^+}}}\\&\approx -\log __\textrm^+\right] }} }}\\&=7-\log _, \end \end$$

(27)

with \(a__\textrm^+}}\) as the activity of the \(_\textrm^+}\) ion.

The case of a weak acid is described by using \(}_\textrm=c_\textrm\) and \(}_\textrm=0\) in Eq. (21),

$$\begin \rho =\tfrac\sqrt^2+3c_\textrmk_\textrm+3}\cos -\tfrac}, \end$$

(28)

with

$$\begin \theta =\arctan ,\frac}}\right) }, \end$$

(29)

\(\Delta [P]=-4p^3-27q^2\), and

$$\begin p&=-\frac^2}-c_\textrm k_\textrm-1,\end$$

(30)

$$\begin q&=\frac^3}+\frack_\textrm^2}-\frac}. \end$$

(31)

The coefficient p is evidently negative, meanwhile q can have positive or negative values. Algebraic manipulation on the inequality \(q<0\) gives \(k_\textrm^2+\tfracc_\textrmk_\textrm-9<0\), which can be rearranged to obtain \(c_\textrm<\frac}(k_\textrm^2-9)\). The last inequality implies that negative values of q are obtained with \(k_\textrm<3\), and \(<\frac}\left( 9-k_\textrm^2\right) }\). Very weak acids, e.g., HCN (\(k_\textrm=6.2\times 10^\)) or HOCl (\(k_\textrm=0.4\)), would have negative q at micromolar concentrations.

The limit of infinite dilution of \(\theta\) gives

$$\begin \lim _\rightarrow 0}=\arctan \left( 9 -k_\textrm^2\right) ,\sqrt\left| k_\textrm^2-1\right| \right) }. \end$$

(32)

Figure 1 displays this limit as a function of pKa for different weak acids. In this limit of infinite dilution, \(\lim _\rightarrow 0}=1\). The left and right extremes of the \(\theta\) curve shown in Fig. 1 display the infinite dilution of \(\theta\) for the case of \(_\textrm^+}\) and \(_2\textrm}\) , respectively. Weak acids with higher (lower) dissociation constants are located on the left (right) of the figure. The dashed line is given by acids with \(1\le k_\textrm\le 3\). The value \(k_\textrm=1\) gives \(\Delta [P]=0\) and \(q<0\), hence \(\theta =0\) and \(\rho =1\). This case gives one simple root and a double root for \(P=0\). It can be shown that the simple root is \(\rho =1\), and the double root is \(\rho =-1\). The particular case of a triple root is impossible since the triple root condition \(^2=-3(1+c_\textrmk_\textrm)}\) cannot be fulfilled with a real \(k_\textrm\). On the other hand, the value \(k_\textrm=3\) produces \(q=0\) and \(\Delta [P]>0\), hence \(\theta =\pi /2\) and \(\rho =1\).

Angle \(\theta\) at infinite dilution as a function of the pKa. The dashed line gives \(\theta\) for weak acids with \(1\le k_\textrm\le 3\). As the concentration of the acid increases, the value of θ tends to \(\pi /2\)

As the concentration \(c_\textrm\) is increased, the angle \(\theta\) tends to \(\theta =\pi /2\) for all the acids, regardless of its p\(K_\textrm\). The angle \(\theta\) displays a maximum as a function of \(c_\textrm\). This maximum is obtained from the condition \(/}=0\), which can be shown to give \(c_\textrm=/}\).

The concentration \(_\textrm^+]}\) is given by \(_\textrm^+]}=\sqrt}\rho\), and the pH is

$$\begin \begin \textrm&\approx -\log __\textrm^+]}} }}\\&=7-\log _, \end \end$$

(33)

with \(C^}=1\,\textrm\). The low concentration and weak acid pH limits are given by \(\lim _\rightarrow 0}}=7\), and \(\rightarrow 0}}=7}\). Equation (33) is useful to calculate the pH of a weak acid at low concentrations (1–10 μM). We calculate the pH of a diluted (\(C_\textrm=5\,\mu \textrm\)) aqueous solution of acetic acid, \(K_\textrm=1.75\times 10^\), as an example of the use of these equations. The concentration and dissociation constants are given by \(=5.\times 10^/10^=50}\), and \(=1.75\times 10^/10^=175}\). The coefficients p and q are easily calculated by the use of Eq. (30) to obtain \(p=-18959.3\) and \(q=907291.\) From the values of p and q the discriminant \(\Delta [P]\) and \(\theta\) are given by \(\Delta [P]=-4p^3-27q^2\) and Eq. (29): \(\Delta [P]=5.03444\times 10^\), and \(\theta =2.69738\,\)radians. Finally, Eq. (28) gives \(\rho =40.6076\), for which \(\textrm=5.3914\). The use of the function \(\textsf \) of Wolfram Mathematica for \(P=0\) gives \(\rho =40.6076\). 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.

A weak acid titration by a strong base requires one to make \(}_\textrm=c_\textrm\) and \(}_\textrm=c_\textrm\) in Eq. (21),

$$\begin \rho =2\root 3 \of \cos \right) -\frac+c_\textrm}}, \end$$

(34)

with

$$\begin \root 3 \of &=\tfrac\sqrt+c_\textrm\right) ^2+3k_\textrm\left( c_\textrm-c_\textrm\right) +3},\end$$

(35)

$$\begin \theta&=\arctan ,\frac}}\right) },\end$$

(36)

$$\begin \Delta [P]&=-4p^3-27q^2,\end$$

(37)

$$\begin p&=-\tfrac\left( k_\textrm+c_\textrm\right) ^2-k_\textrm\left( c_\textrm-c_\textrm\right) -1,\end$$

(38)

$$\begin q&=\frac\left( k_\textrm+c_\textrm\right) ^3+\frac}\left( k_\textrm+c_\textrm\right) \left( c_\textrm-c_\textrm\right) +\frac-2 k_\textrm}. \end$$

(39)

The contours of \(\textrm=7-\log _\) as a function of \(C_\textrm=\sqrt}c_\textrm\) and \(C_\textrm=\sqrt}c_\textrm\) for acetic acid, \(k_\textrm=180\), are shown in Fig. 2. It is interesting to note that there is a linear relationship between base and acid concentrations at constant \(\textrm\); in fact, the \(=7}\) line has a slope of 1 and passes through the origin \(,C_\textrm)=(0,0)}\). Figure 2 also shows that the \(C_\textrm\)-intercept is negative for acidic \(\textrm\) and positive for basic \(\textrm\). It is also seen in this figure that the slope of the lines \(C_\textrm(C_\textrm)\) is less than 1 for acidic \(\textrm\).

Lines of constant \(\textrm\) on the \(C_\textrm\)–\(C_\textrm\) plane for acetic acid and a strong base, at concentrations \(C_\textrm\) and \(C_\textrm\) , respectively

The case of a buffer solution requires one to consider \(}_\textrm=c_\textrm+c_\textrm\) and \(}_\textrm=c_\textrm+c_\textrm\) in Eq. (21),

$$\begin \rho =2\root 3 \of \cos \right) -\frac+c_\textrm+c_\textrm}}, \end$$

(40)

with

$$\begin \root 3 \of &=\tfrac\sqrt+c_\textrm+c_\textrm\right) ^2+3k_\textrm(c_\textrm-c_\textrm)+3},\end$$

(41)

$$\begin \theta&=\arctan ,\frac}\right) },\end$$

(42)

$$\begin \Delta [P]&=-4p^3-27q^2,\end$$

(43)

$$\begin p&=-\tfrac\left( k_\textrm+c_\textrm+c_\textrm\right) ^2-k_\textrm\left( c_\textrm-c_\textrm\right) -1,\end$$

(44)

$$\begin q&=\tfrac\left( k_\textrm+c_\textrm+c_\textrm\right) ^3+\frac}\left( k_\textrm+c_\textrm+c_\textrm\right) \left( c_\textrm-c_\textrm\right) +\frac+c_\textrm-2k_\textrm}. \end$$

(45)

By applying the logarithm of the acid dissociation constant \(K_\textrm\), Eq. (5), we obtain, after some algebraic manipulation, the Henderson–Hasselbalch (HH) equation [23,24,25]

$$\begin \textrm=\textrmK_\textrm+\log _^]}}]}}. \end$$

(46)

The right-hand side of this equation is a function of \(\textrm\) and therefore the HH equation is not practical for direct calculation of the pH. It is common to use the approximations \(^]}\approx C_\textrm\) and \([\hbox ]\approx C_\textrm\) to obtain

$$\begin \textrm^}\approx \textrmK_\textrm+\log _}}}. \end$$

(47)

Absolute error \(E^}\), in the \(\textrm\) calculated by the Herderson-Hasselbalch equation for a buffer solution of chlorous acid, \(K_\textrm=1.2\times 10^\), as a function of the molar concentrations of the acid \(C_\textrm\) and the salt \(C_\textrm\)

Figure 3 shows the absolute error of the \(\textrm\), for chlorous acid, calculated by the HH equation with respect to the pH calculated using the exact formula, Eq. (40) with \(c_\textrm=0\),

$$\begin E^}=\textrm^}-\left( 7-\log _\right) , \end$$

(48)

with \(\rho\) given by Eq. (40). Figure 3 shows that small errors, \(E^\mathrm <0.1\), are obtained for buffer solutions with high concentration of the salt, \(C_\textrm>0.1\,\textrm\), meanwhile large errors, \(E^\mathrm >0.5\) are obtained for buffer solutions with \(C_\textrm<0.01\,\textrm\).

The pH stability of an acid buffer solution is measured by adding a volume \(V_\textrm\) of a strong base solution with concentration \(c_\textrm^0\) [3]. The addition of this volume changes the concentrations \(c_\textrm\), \(c_\textrm\), and \(c_\textrm\),

$$\begin c_}&=\frac}^0 V_}^0}}^0+V_}},\end$$

(49)

$$\begin c_}&=\frac^0 V_}^0}}^0+V_}},\end$$

(50)

$$\begin c_}&=\frac^0 V_}}}^0+V_}}, \end$$

(51)

with \(c_\textrm^0\), \(c_\textrm^0\), and \(c_\textrm^0\), as the concentrations of the acid, salt of acid, and base independent solutions, respectively. The volumes \(V_\textrm^0\), \(V_\textrm^0\), and \(V_\textrm^0=V_\textrm^0+V_\textrm^0\) are the initial volumes of the acid, salt, and acid buffer solutions, respectively.

The pH stability, \(S_\textrm\), of a buffer solution is given by

$$\begin \begin S_}&=\frac\left( \textrm\right) } V_\textrm}\\&=\nabla _}}(\textrm)\cdot \frac}} V_\textrm}, \end \end$$

(52)

with \(}=\left( },},}\right)\) as 3-vector of concentrations, and the gradient of \(\rho\), \(\nabla _}}\rho\), given by

$$\begin \nabla _}}\rho =\left( \frac},\frac},\frac}\right) . \end$$

(53)

The use in Eq. (52) of the pH definition, \(\textrm=7-}\rho\), and the concentrations of Eqs. (49)–(51), gives after some algebra

$$\begin S_\textrm=\frac}}\frac^0+V_\textrm\right) ^2}\frac}}}\rho \cdot }^0}, \end$$

(54)

with

$$\begin }^0=\begin c_\textrm^0V_\textrm^0,&c_\textrm^0V_\textrm^0,&-c_\textrm^0V_\textrm^0\end. \end$$

(55)

The gradient \(\nabla _}}\rho\) is calculated with respect to the components of \(}\) but it must be expressed in terms of the volume of base \(V_\textrm\).

Figure 4 displays the pH stability, \(S_\textrm\), 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, \(c_\textrm^0\), \(c_\textrm^0\) and \(c_\textrm^0\), have the same value \(c^0\). Panel (a) of Fig. 4 displays \(S_\textrm/100\) for buffer solutions with concentrations \(C^0=10^\times c^0\) ranging from \(10^\) to \(10^\,\textrm\) (from blue to red). Panel (b) of Fig. 4 displays \(S_\textrm\) for buffer solutions with concentrations \(C^0=10^\times c^0\) ranging from \(10^\) to \(10^\,\textrm\) (from purple to red). In both panels of Fig. 4, curves with lower (higher) values of \(S_\textrm\) are for buffer solutions prepared and titrated with solutions of lower (higher) concentrations. The maximum of the curves \(S_\textrm\) 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.

pH stability \(S_\textrm\) as a function of the pH for buffer solutions of acetic acid (\(k_\textrm=175\)) and sodium acetate titrated with NaOH. To simplify the analysis, the initial concentrations \(c_\textrm^0\), \(c_\textrm^0\), and \(c_\textrm^0\), are equal to \(c^0\). Panel (a) displays \(S_\textrm/100\) for solutions with concentrations \(C^0=10^\times c^0\) from \(10^\) to \(10^\,\textrm\). The lowest curves (blue color) are at lower concentrations, the highest curves (red color) are for higher concentrations. Panel (b) displays \(S_\textrm\) for solutions with concentrations \(C^0=10^\times c^0\) from \(10^\) to \(10^\,\textrm\). The lowest curves (purple color) are for low concentration buffers, the highest curves (red color) for higher concentrations

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, \(C^0=10^c^0\). All the titration curves of panels (a) and (b) of Fig. 5 intercept at \(V_\textrm=1\,\textrm\) and \(\textrm=7\). Buffers of higher concentrations, in red, display the largest changes of pH as the solution is titrated with strong base. It is observed in Fig. 5(a) that the steepest change in the pH is given for \(\textrm\approx 9\), 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 \(\textrm\) as the base is added.

Buffer titration as a function of the volume of base added, \(V_\textrm/\textrm\). To simplify the analysis, the concentrations of the solutions to prepare, and titrate, the buffer are the same, \(C^0=10^c^0\). Panel (a) displays the \(\textrm\) for solutions with concentrations \(C^0=10^\times c^0\) from \(10^\) to \(10^\,\textrm\). The blue blue curves are at lower concentrations, the red curves are for higher concentrations. Panel (b) displays \(\textrm\) for solutions with concentrations \(C^0=10^\times c^0\) from \(10^\) to \(10^\,\textrm\). The purple curves are for low concentration buffers, the red curves for higher concentrations