text:
\section{Introduction}

An acid is a chemical substance that dissolves in water producing hydronium ion, $\ce{H3O+}$, and the acid conjugate base. Weak acids dissolve partially in water reaching chemical equilibrium. In the state of equilibrium, the concentration of the chemical species is constant. The equilibrium concentrations of the chemical species are given by the equations of chemical equilibrium and the constraints of mass balance and electrical neutrality. The aqueous dissociation of a weak acid involves four chemical species and four mathematical relations between these species. An aqueous weak acid buffer solution is formed by a weak acid and one of its soluble salts. The equilibrium of a weak acid buffer involves four chemical substances and four equations. In principle 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 difficult to obtain exact mathematical expressions for the equilibrium 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 $\ce{H3O+}$ \cite{Denbigh1981,Burgot2012,Skoog2022}. The equilibrium concentration of the hydronium 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 it due to its complexity \cite{Ladd1980,McNaught1986}. The raw use of the cubic formula requires more than 30 arithmetic operations which makes its use impossible for manual calculation \cite{Bellova2018,Randall2002}.

A quadratic equation is obtained for the hydronium ion if the auto-ionization of water is ignored. Under this approximation, the equilibrium concentration of the hydronium is obtained simply by solving a quadratic equation. The use of the quadratic formula require seven arithmetic operations, which allows its use for manual calculation. The approximation of ignoring water auto-ionization displays a relative error greater than 10\% for weak acids with $K_\mathrm{a}<10^{-6}\,\mathrm{M}$, and at low analytical concentrations of the acid or the salt of the acid, $C_\mathrm{a,s}<10^{-6}\,\mathrm{M}$. Although the concentrations used in the general chemistry laboratory are generally above $10^{-2}\,\mathrm{M}$, there may be situations in chemical analysis where very low concentrations are handled \cite{Denbigh1981,Bellova2018,Randall2002,Skoog2022}.

General purpose computer algebra systems (CAS) allow the algebraic manipulation of equations and inequalities \cite{Albrecht2013}. The use of CAS in mathematical software, with user friendly graphical interfaces, has extended the use of CAS to all fields of knowledge. Examples of free and non-free CAS are Maple \cite{maple}, Wolfram Mathematica \cite{Mathematica}, Maxima \cite{maxima}, and SageMath \cite{sagemath}. 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 require 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 are employed to obtain closed-form 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 \cite{Denbigh1981,Levie2001, Burgot2012,Skoog2022}. The cubic equations for the hydronium ion are deeply analyzed, their roots are obtained by the use of the ancient Cardano's formula for the associated depressed cubic equations \cite{Dickson1914,Dickson1922}. The direct result of Cardano's formula are the cube roots of two complex numbers. The polar representation of these cube roots allows to obtain only three real roots. The use of Descartes rule of signs \cite{Curtiss1918}, and Vieta's formulas \cite{Dickson1914,Dickson1922} allows to demonstrate that only one root is positive. The titration curves of using a strong base to neutralize solutions of weak acids and buffers are analyzed by using the same analytical methods. 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 $\ce{H3O+}$ is presented. This equation gives the pH for any of the four systems studied in this work in terms of an effective acid constant $k$ and an effective acid concentration $c$.

\section{Theory and Methods}\label{sec:theory}

\subsection{Weak acids}
The aqueous dissociation equilibrium of a weak acid $\ce{HB}$ is given by the chemical equations
\begin{align}
    \ce{HB + H2O &<--> H3O+ + B-},\label{ce:HA_diss}\\
    \ce{2 H2O &<--> H3O+ + OH-}.\label{ce:H2O_auto}
\end{align}
Relevant chemical species are $\ce{H3O+}$, $\ce{OH-}$, $\ce{HB}$, and $\ce{B-}$ with molar concentrations  $\ce{[H3O+]}$, $\ce{[OH^-]}$, $\ce{[HB]}$, and $\ce{[B^-]}$ \cite{Denbigh1981,Burgot2012,Skoog2022}.

A low concentration solution of the acid $\ce{HB}$ is prepared in water at analytical concentration $C_\mathrm{a}$. Once the system reaches chemical equilibrium, the concentrations of the chemical species are given by four physical conditions: the weak acid dissociation constant, $K_\mathrm{a}$, the water auto-ionization constant, $K_\mathrm{w}$, the electric neutrality, and the mass balance,
\begin{align}
    K_\mathrm{a}&=\frac{\ce{[H3O+]}\ce{[B^-]}}{\ce{[HB]}},\label{eq:Ka}\\
    K_\mathrm{w}&=\ce{[H3O^+]}\ce{[OH^-]},\label{eq:Kw}\\
    \ce{[H3O+]}&=\ce{[OH^-]}+\ce{[B^-]},\label{eq:charge_balance}\\
    C_\mathrm{a}&=\ce{[B^-]}+\ce{[HB]},\label{eq:matter_balance}
\end{align}
respectively. Acid constants $K_\mathrm{a}$ have units of concentration and their value range typically between $10^{-10}$ and $10^{-1}$ $\mathrm{M}$. It is convenient to define the dimensionless variables and constants ${x=\ce{[H3O^+]}/\sqrt{K_\mathrm{w}}}$, ${y=\ce{[OH^-]}/\sqrt{K_\mathrm{w}}}$, ${w=\ce{[HB]}/\sqrt{K_\mathrm{w}}}$, ${z=\ce{[B^-]}/\sqrt{K_\mathrm{w}}}$, ${c_{\mathrm{a}}=C_\mathrm{a}/\sqrt{K_\mathrm{w}}}$,  ${k_{\mathrm{a}}=K_\mathrm{a}/\sqrt{K_\mathrm{w}}}$, and ${k_\mathrm{w}=1}$. In terms of these dimensionless variables and constants, equations \eqref{eq:Ka}--\eqref{eq:matter_balance} are replaced by
\begin{align}
    k_\mathrm{a}&=\frac{xz}{w},\label{eq:ka}\\
    k_\mathrm{w}&=xy=1,\label{eq:kw}\\
    x&=y+z,\label{eq:cb}\\
    c_\mathrm{a}&=w+z.\label{eq:c}
\end{align}
The combined use of equations \eqref{eq:kw}$-$\eqref{eq:c} gives the equations ${z=x-\frac{1}{x}}$ and ${w=c_\mathrm{a}+\frac{1}{x}-x}$. The use of these equations and the physical constraints on the concentrations $x$, $w$ and $z$, gives, after some algebraic manipulation, the mathematical conditions
\begin{align}
    x&>0,\\
    x^2&>1, \\
    -x^2+c_\mathrm{a} x+1&>0.
\end{align}
Algebraic manipulation of these inequalities leads to the range of the dimensionless concentration $x$
\begin{equation}\label{eq:concentration_x_weak_acid}
    1<x<\tfrac{1}{2}\left(c_\mathrm{a}+\sqrt{c_\mathrm{a}^2+4}\right).
\end{equation}
The left inequality gives the condition of acid pH, $[\ce{H3O+}]>10^{-7}\,\textrm{M}$, the right inequality gives the strong acid dissociation limit.

The combined use of equations \eqref{eq:kw}$-$\eqref{eq:c} in equation \eqref{eq:ka} gives
\begin{equation}\label{eq:ka_equation}
    k_\mathrm{a}=\frac{x\left(x-\tfrac{1}{x}\right)}{c_\mathrm{a}-\left(x-\tfrac{1}{x}\right)}.
\end{equation}
Algebraic manipulation on this equation gives the cubic equation $P_\mathrm{a}=0$, with the cubic polynomial
\begin{equation}\label{eq:polynomial_P}
    P_\mathrm{a}=x^3+k_\mathrm{a}x^2-\left(1+c_\mathrm{a}k_\mathrm{a}\right)x-k_\mathrm{a}.
\end{equation}
The coefficients of $P_\mathrm{a}$ are given by the 4-tuple
\begin{equation}\label{eq:coefficients_Pa}
\begin{split}
    \coef[P_\mathrm{a}]&=\left(a_3,a_2,a_1,a_0\right)\\
    &=\left(1,k_\mathrm{a},-\left(1+c_\mathrm{a}k_\mathrm{a}\right),-k_\mathrm{a}\right).
\end{split}
\end{equation}
The signs of the coefficients $ \coef[P_\mathrm{a}]$ are given by
\begin{equation}\label{eq:signs_coeff_Pa}
\begin{split}
    \sgn[P_\mathrm{a}]&=\left(\sgn a_3,\sgn a_2,\sgn a_1, \sgn a_0\right)\\
    &=\left(+,+,-,-\right).
\end{split}
\end{equation}
Descartes' rule of signs says 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 \cite{Curtiss1918}. The use of Descartes's rule on the sequence \eqref{eq:signs_coeff_Pa} indicates that only one root is positive. The full characterization of the roots of $P_\mathrm{a}=0$ is given by the discriminant of the polynomial \eqref{eq:polynomial_P}, $\Delta[P_\mathrm{a}]$ \cite{Irving2004,Dickson1914,Dickson1922},
\begin{equation}\label{eq:discriminant_x}
    \Delta[P_\mathrm{a}]=4\left(c_\mathrm{a}^3k_\mathrm{a}^3+\left(k_\mathrm{a}^2-1\right)^2+c_\mathrm{a} k_\mathrm{a}\left(5k_\mathrm{a}^2+3\right)\right)+c_\mathrm{a}^2k_\mathrm{a}^2\left(k_\mathrm{a}^2+12\right).
\end{equation}
Clearly $\Delta[P_\mathrm{a}]>0$, hence the cubic equation $P_\mathrm{a}=0$ must have three distinct real roots $\rho_i$, with $i=1,2,3$ and $\rho_1<\rho_2<\rho_3$. Since $x=\ce{[H3O^+]}/\sqrt{K_\mathrm{w}}$ is related to a molar concentration, only one of the roots $\rho_i$ is expected to be a positive number. Although Descartes' rule has already shown that there must be only one positive root, a further analysis is necessary to find the local extremes of $P_\mathrm{a}(x)$. These extremes are given by the solutions of $P'_\mathrm{a}(x)=0$. The derivative of $P_\mathrm{a}(x)$ is given by
\begin{equation}
    P'_\mathrm{a}(x)=3x^2+2k_\mathrm{a} x-\left(1+c_\mathrm{a}k_\mathrm{a}\right),
\end{equation}
which has discriminant $\Delta[P_\mathrm{a}']=4\left(k_\mathrm{a}^ 2+3c_\mathrm{a}k_\mathrm{a}+3\right)$.
The roots of $P_\mathrm{a}'(x)=0$ are
\begin{equation}
    x_{1,2}=\tfrac{1}{3}\left(-k_\mathrm{a}\mp\tfrac{1}{2}\sqrt{\Delta[P_\mathrm{a}']}\right).
\end{equation}
Since $\Delta[P_\mathrm{a}']>0$, it is easily seen that $x_1<0$. Algebraic manipulation on the inequality $x_2>0$ leads to $1+c_\mathrm{a}k_\mathrm{a}>0$, corroborating that $x_2>0$. Since $P_\mathrm{a}(x)$ is a cubic polynomial and the coefficient of the cubic term is positive, it must be given that $\lim_{x\to\pm\infty} P_\mathrm{a}(x)=\pm\infty$, therefore $x_1$ must be a local maximum and $x_2$ a local minimum, with $P_\mathrm{a}(x_1)>0$, and $P_\mathrm{a}(x_2)<0$. The three roots of $P_\mathrm{a}=0$ then obey the inequalities $\rho_1<x_1<\rho_2<x_2<\rho_3$. These inequalities give $\rho_1<0$, and $\rho_3>0$. The use of Vieta's theorem allows to determine the sign of $\rho_2$ \cite{vanderWaerden1991,Virberg2003}, these formulas for $P_\mathrm{a}(x)=0$ are \cite{Dickson1914,Dickson1922}
\begin{align}
    \rho_1+\rho_2+\rho_3&=-k_\mathrm{a},\\
    \rho_1\rho_2+\rho_1\rho_3+\rho_2\rho_3&=-\left(1+c_\mathrm{a}k_\mathrm{a}\right),\\
    \rho_1\rho_2\rho_3&=k_\mathrm{a}.
\end{align}
The use of the last of the formulas gives $\rho_2<0$, therefore only $\rho_3$ is positive.

An explicit formula for $\rho_3$, in terms of the constants, is obtained by solving the associated depressed cubic equation, \emph{i.e.}, a cubic equation with no quadratic term \cite{vanderWaerden1991,Virberg2003}. The change of variable $x=\bar{x}-\frac{k_\mathrm{a}}{3}$ allows to write the cubic equation $P_\mathrm{a}=0$ as the depressed cubic equation
\begin{equation}\label{eq:depressed_cubic}
      \bar{x}^3+p\bar{x}+q=0,
\end{equation}
with real coefficients
\begin{align}
    p&=-\frac{k_\mathrm{a}^2}{3}-c_\mathrm{a} k_\mathrm{a}-1,\\
    q&=\frac{2k_\mathrm{a}^3}{27}+\frac{c_\mathrm{a}k_\mathrm{a}^2}{3}-\frac{2k_\mathrm{a}}{3},
\end{align}
discriminant $-4p^3-27q^2$ equal to $\Delta[P_\mathrm{a}]$, and  ${p=-\Delta[P_\mathrm{a}']/12}$.

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

Vieta's substitution, $\bar{x}=y-\frac{p}{3y}$, on the depressed cubic equation \eqref{eq:depressed_cubic} gives
\begin{equation}\label{eq:vieta_depressed_cubic}
    y^3-\frac{p^3}{27 y^3}+q=0.
\end{equation}
The denominator of the second term on the right hand side is simplified by multiplying equation \eqref{eq:vieta_depressed_cubic} by $y^3$, obtaining
\begin{equation}\label{eq:vieta_6}
    y^6+q y^3-\frac{p^3}{27}=0,
\end{equation}
which is equivalent to the quadratic equation $\xi^2+q\xi-\tfrac{p^3}{27}=0$, in the variable $\xi=y^3$, with roots
\begin{equation}
\begin{split}\label{eq:roots_vieta_6}
        \xi_{1,2}&=-\frac{q}{2}\pm\sqrt{\frac{27q^2+4p^3}{108}}\\
        &=-\frac{q}{2}\pm \frac{i}{2}\sqrt{\frac{\Delta[P_\mathrm{a}]}{27}}.
\end{split}
\end{equation}
Since $\Delta[P_\mathrm{a}]>0$, the roots $\xi_{1,2}$ are complex conjugates, $\xi_1=\xi_2^*$. The definition $\zeta=\xi_1$ allows to use $\zeta$ and $\zeta^*$ as the roots of \eqref{eq:vieta_6}. It is convenient to use the polar representation of $\zeta$, ${\zeta=\|\zeta\|e^{i\theta}}$ with
\begin{equation}\label{eq:z_modulus}
\begin{split}
    \|\zeta\|&=\frac{1}{2}\sqrt{q^2+\frac{\Delta[P_\mathrm{a}]}{27}}\\
    &=\left(\tfrac{1}{6}\sqrt{\Delta[P_\mathrm{a}']}\right)^3,\\
\end{split}
\end{equation}
and
\begin{equation}\label{eq:theta_def}
    \theta=\arctan{\left(-\frac{q}{2},\frac{\sqrt{\Delta[P_\mathrm{a}]}}{6\sqrt{3}}\right)},
\end{equation}
with $\theta\in(0,\pi)$ as the angle between $\zeta$ and the positive real axis on the Argand plane. The angle $\theta$ is related to the trigonometric solution obtained by Nickalls for the roots of the cubic equation \cite{Nickalls1993}.

The roots of the depressed cubic equation are obtained from $\alpha=\sqrt[3]{\zeta}$ and $\beta=\sqrt[3]{\zeta^*}$. The cubic roots $\alpha$ and $\beta$ have three values each, $\alpha_n$ and $\beta_n$,
\begin{align}
    \alpha_n&=\sqrt[3]{\|\zeta\|}\exp{\left(i\left(\frac{\theta}{3}+\frac{2n\pi}{3}\right)\right)},\\
    \beta_n&=\sqrt[3]{\|\zeta\|}\exp{\left(i\left(-\frac{\theta}{3}+\frac{2n\pi}{3}\right)\right)},
\end{align}
with $n=0,1,2$.

Since the cubic equation \eqref{eq:depressed_cubic} has three real roots, the addition of two cubic roots $\alpha_n+\beta_m$ must give a real number. This is possible only if $\operatorname{Im}(\alpha_n)=-\operatorname{Im}(\beta_m)$.
There are only three possible combinations that fulfill this requirement:
\begin{align}
    \alpha_0+\beta_0&=2\sqrt[3]{\|\zeta\|}\cos{\left(\frac{\theta}{3}\right)},\\
    \alpha_2+\beta_1&=-2\sqrt[3]{\|\zeta\|}\cos{\left(\frac{\theta+\pi}{3}\right)},\\
    \alpha_1+\beta_2&=-2\sqrt[3]{\|\zeta\|}\sin{\left(\frac{\theta+2\pi}{3}\right)}.
\end{align}
The roots of the depressed cubic equation \eqref{eq:depressed_cubic} are
\begin{align}
    \bar{r}_1&=\alpha_1+\beta_2,\label{eq:rbar_1}\\
    \bar{r}_2&=\alpha_2+\beta_1,\label{eq:rbar_2}\\
    \bar{r}_3&=\alpha_0+\beta_0,\label{eq:rbar_3}
\end{align}
with $\bar{r}_1<\bar{r}_2<\bar{r}_3$.

The roots of $P_\mathrm{a}=0$ are given by
\begin{align}
    \rho_1=\bar{r}_1-\tfrac{k_\mathrm{a}}{3},\label{eq:rho_1}\\
    \rho_2=\bar{r}_2-\tfrac{k_\mathrm{a}}{3},\label{eq:rho_2}\\
    \rho_3=\bar{r}_3-\tfrac{k_\mathrm{a}}{3}.\label{eq:rho_3}
\end{align}
By using Vieta's formulas it was shown that $\rho_3$ is the only positive root. The positive root $\rho=\rho_3$ is explicitly given by
\begin{equation}\label{eq:rho3_q_negative}
\begin{split}
    \rho&=2\sqrt[3]{\|\zeta\|}\cos{\left(\theta/3\right)}-\tfrac{k_\mathrm{a}}{3}\\
    &=\tfrac{2}{3}\sqrt{k_\mathrm{a}^2+3c_\mathrm{a}k_\mathrm{a}+3}\cos{\left(\theta/3\right)}-\tfrac{k_\mathrm{a}}{3}.
    \end{split}
\end{equation}
The weak acid and low concentration limits give $\lim_{c_\mathrm{a}\to 0}{\rho}=1$, and  $\lim_{k_\mathrm{a}\to 0}{\rho}=1$. The strong acid limit gives
\begin{equation}
\lim_{k_\mathrm{a}\to\infty}{\rho}=\tfrac{1}{2}\left(c_\mathrm{a}+\sqrt{4+c_\mathrm{a}^2}\right).
\end{equation}

The concentration $\ce{[H3O+]}$ is given by $\ce{[H3O+]}=\sqrt{K_\mathrm{w}}\rho$, and the pH is
\begin{equation}\label{eq:pH_weak_acid}
\begin{split}
    \mathrm{pH}&=-\log_{10}{\frac{\ce{[H3O+]}}{C^\circ}}\\
    &=7-\log_{10}{\rho},
    \end{split}
\end{equation}
with $C^{\circ}=1\,\mathrm{M}$. The low concentration and weak acid pH limits are given by $\lim_{c_\mathrm{a}\to 0}{\mathrm{pH}}=7$, and  ${\lim_{k_\mathrm{a}\to 0}{\mathrm{pH}}=7}$. The strong acid limit can be used to show that the pH can be negative for $c_\mathrm{a} > 10^{7}$, $C_\mathrm{a}>1\,\mathrm{M}$. Equation \eqref{eq:pH_weak_acid} is useful to calculate the pH of a weak acid at low concentrations (${1-10\,\mu}$M). College books of chemical analysis \cite{Skoog2022} resort to several approximations to calculate the pH of a weak acid in order to avoid the numerical difficulties of solving the cubic equation $P_\mathrm{a}=0$. First, assuming that ${\sqrt{K_\mathrm{w}}\ll\ce{[H3O+]}}$, \emph{i.e.} ${1\ll x}$, in equation \eqref{eq:ka_equation}, it is obtained a quadratic equation for $x$ with solution ${x_\mathrm{approx}^{(1)} =\tfrac{1}{2}\left(-k_\mathrm{a}+\sqrt{k_\mathrm{a}^2+4c_\mathrm{a}k_\mathrm{a}}\right)}$. Further, by using ${1\ll x\ll c_\mathrm{a}}$ in equation \eqref{eq:ka_equation}, it is obtained second (less exact) approximation, ${x_\mathrm{approx}^{(2)}=\sqrt{c_\mathrm{a}k_\mathrm{a}}}$. The relative error of the approximate concentrations $x_\mathrm{approx}^{(i)}$, is given by
\begin{align}
    E^{(i)}=\frac{x_\mathrm{approx}^{(i)}-\rho}{\rho},
\end{align}
with $i=1,2$. It can be shown that the errors $E^{(i)}$ have extreme values
\begin{align}
    \lim_{c_\mathrm{a}\to 0}E^{(i)}&=-1,\\
    \lim_{c_\mathrm{a}\to\infty}E^{(i)}&=0,
\end{align}
for $i=1,2$.

Figure \ref{fig:x_approximations}(a) displays the region of the $C_\mathrm{a}$-$K_\mathrm{a}$ plane for which $\left|E^{(1)}\right|>0.05$, figure \ref{fig:x_approximations}(b) shows the region of the $C_\mathrm{a}$-$K_\mathrm{a}$ plane for which $\left|E^{(2)}\right|>0.05$. It is clear that the error of the first approximation, $x_\mathrm{approx}^{(1)}$, is very small even at $\mu\mathrm{M}$ concentrations, this approximation fails for $C_\mathrm{a}<1\,\mu\mathrm{M}$. On the other hand, the approximation $x_\mathrm{approx}^{(2)}$ works well for small acid dissociation constants, $K_\mathrm{a}<10^{-6}\,\mathrm{M}$, and large values of the acid concentration, $C_\mathrm{a}>2\,m\mathrm{M}$.
\begin{figure}[htb]
    \centering
    \includegraphics[scale=0.5]{errors.pdf}
    \caption{The rule of 5\%. In blue,  regions with relative error (a) $\left|E^{(1)}\right|>0.05$, and (b) $\left|E^{(2)}\right|>0.05$. The white color indicate the region on the $C_\mathrm{a}$-$K_\mathrm{a}$ plane where (a) $x_{\mathrm{approx}}^{(1)}$,  or (b) $x_{\mathrm{approx}}^{(2)}$, are good approximations.}
    \label{fig:x_approximations}
\end{figure}

The combined use of equations \eqref{eq:kw} and \eqref{eq:cb} give the equilibrium concentration for $z$
\begin{equation}\label{eq:z_of_rho}
    z=\frac{\rho^2-1}{\rho},
\end{equation}
which can be used to obtain the degree of dissociation of the acid $\mathcal{D}_\mathrm{a}=z/c_\mathrm{a}$ \cite{Skoog2022}, which can be rearranged to obtain $c_\mathrm{a}$ as function of $\rho$ and $\mathcal{D}_\mathrm{a}$
\begin{equation}\label{eq:degree_of_dissociation_1}
    c_\mathrm{a}=\frac{\rho^2-1}{\rho\mathcal{D}_\mathrm{a}}.
\end{equation}

The combined use of equations \eqref{eq:ka}--\eqref{eq:c}, the substitution $x=\rho$, and the use of the definition of $\mathcal{D}_\mathrm{a}$, give after some algebra
\begin{equation}\label{eq:ka_in_terms_of_da_and_rho}
    k_\mathrm{a}=\frac{\rho\mathcal{D}_\mathrm{a}}{1-\mathcal{D}_\mathrm{a}}.
\end{equation}

The use of $\rho=10^{7-\mathrm{pH}}$ in equations \eqref{eq:degree_of_dissociation_1} and \eqref{eq:ka_in_terms_of_da_and_rho} give the functions $c_\mathrm{a}(\mathrm{pH},\mathcal{D}_\mathrm{a})$ and $k_\mathrm{a}(\mathrm{pH},\mathcal{D}_\mathrm{a})$. Figure \ref{fig:ca_ka_pH_da} displays curves of constant $C_\mathrm{a}$ and $K_\mathrm{a}$ as functions of the pH and the degree of dissociation, $\mathcal{D}_\mathrm{a}$. The curves for $K_\mathrm{a}=\sqrt{K_\mathrm{w}}k_\mathrm{a}$ constant have sigmoid shape, increasing monotonically with the pH and reaching the highest value of $\mathcal{D}_\mathrm{a}$ at $\mathrm{pH}=7$. The stronger acids reach almost complete dissociation at $\mathrm{pH}=7$, meanwhile the weakest acids reach a dissociation of $\mathcal{D}_\mathrm{a}\approx 0.1$ at $\mathrm{pH}=7$.

\begin{figure}[htb]
    \centering
    \includegraphics[scale=0.7]{pH_diss.pdf}
    \caption{Constant $C_\mathrm{a}$ (dashed) and $K_\mathrm{a}$ (solid) curves as functions of $\mathrm{pH}$ and $\mathcal{D}_\mathrm{a}$. The dashed curves go from left to right, from the highest concentration $C_\mathrm{a}=10\,\mathrm{M}$ on the left (green), to lowest concentration $C_\mathrm{a}=10^{-9}\,\mathrm{M}$, on the right (blue). The solid curves go from left to right, from the stronger acid $K_\mathrm{a}=1\,\mathrm{M}$ on the left (purple), to the weakest acid $K_\mathrm{a}=10^{-8}\,\mathrm{M}$, on the right (blue).}
    \label{fig:ca_ka_pH_da}
\end{figure}

Equation \eqref{eq:ka_in_terms_of_da_and_rho} can be rearranged to have $\mathcal{D}_\mathrm{a}(\mathrm{pH},k_\mathrm{a})$,
\begin{equation}
    \mathcal{D}_\mathrm{a}=\left(1+\frac{10^{7-\mathrm{pH}}}{k_\mathrm{a}}\right)^{-1},
\end{equation}
The sigmoid function $\mathcal{D}_\mathrm{a}(\mathrm{pH},k_\mathrm{a})$ is bounded and differentiable at all values of $\mathrm{pH}$ and $k_\mathrm{a}$, and monotonically non-decreasing with respect to $\mathrm{pH}$. The derivative of $\mathcal{D}_\mathrm{a}(\mathrm{pH},k_\mathrm{a})$ with respect to pH gives
\begin{align}
    \mathcal{P}_\mathrm{a}(\mathrm{pH},k_\mathrm{a})&=\left(\frac{\partial \mathcal{D}_\mathrm{a}}{\partial \mathrm{pH}}\right)\\
    &=\frac{\ln{10}}{k_\mathrm{a}}\frac{10^{7-\mathrm{pH}}}{\left(1+\frac{10^{7-\mathrm{pH}}}{k_\mathrm{a}}\right)^{2}}.
\end{align}
The maximum of $\mathcal{P}_\mathrm{a}$ is given by the solution of $\left({\partial\mathcal{P}_\mathrm{a}}/{
\partial\mathrm{pH}}\right)=0$, which is given by ${\mathrm{pH}^*=7-\log_{10}{k_\mathrm{a}}}$,  $\mathrm{pH}^*=-\log_{10}{K_\mathrm{a}}$, or $\mathrm{pH}^*=\mathrm{p}K_\mathrm{a}$.

\subsection{Titration by a strong base}

The addition of a strong base, \emph{e.g.} $\ce{NaOH}$, has direct effect on the charge balance \eqref{eq:charge_balance} due to the complete dissociation $\ce{NaOH -> Na+ + OH-}$. In terms of dimensionless concentrations, the equilibrium equations for the mixture of a weak acid and a strong base are \cite{Denbigh1981,Burgot2012,Skoog2022}
\begin{align}
    k_\mathrm{a}&=\frac{xz}{w},\label{eq:ka2}\\
    k_\mathrm{w}&=xy=1,\label{eq:kw2}\\
    x+s&=y+z,\label{eq:cb2}\\
    c_\mathrm{a}&=w+z,\label{eq:c2}\\
    c_\mathrm{b}&=s.\label{eq:cbase2}
\end{align}
In this equations $s=\ce{[Na+]}/\sqrt{K_\mathrm{w}}$, and $c_\mathrm{b}=C_\mathrm{b}/\sqrt{K_\mathrm{w}}$, are the concentrations of sodium ions and the base respectively. Algebraic manipulation of equations \eqref{eq:ka2}--\eqref{eq:cbase2} give the polynomial equation $P_\mathrm{at}(x)=0$, with
\begin{equation}
    P_\mathrm{at}(x)=x^3+\left(k_\mathrm{a}+c_\mathrm{b}\right)x^2-\left(1+k_\mathrm{a}\left(c_\mathrm{a}-c_\mathrm{b}\right)\right)x-k_\mathrm{a}.
\end{equation}
The use of Wolfram Mathematica \cite{Mathematica} allows to prove that the discriminant of $P_\mathrm{at}(x)$ is greater than zero, $\Delta[P_\mathrm{at}]>0$, hence there are three real solutions. It can be proved that only one of these roots is positive by following the same algebraic analysis that led to equation \eqref{eq:rho3_q_negative} from $P_\mathrm{a}(x)=0$. The positive root is given by
\begin{equation}\label{eq:roots_acid_base}
    \rho=2\sqrt[3]{\|\zeta\|}\cos{\left(\frac{\theta}{3}\right)-\frac{k_\mathrm{a}+c_\mathrm{b}}{3}},
\end{equation}
with
\begin{align}
    \sqrt[3]{\|\zeta\|}&=\tfrac{1}{3}\sqrt{\left(k_\mathrm{a}+c_\mathrm{b}\right)^2+3k_\mathrm{a}\left(c_\mathrm{a}-c_\mathrm{b}\right)+3},\label{eq:zeta_2}\\
    \theta&=\arctan{\left(-\frac{q}{2},\frac{\sqrt{\Delta[P_\mathrm{at}]}}{6\sqrt{3}}\right)},\label{eq:theta_def2}\\
    \Delta[P_\mathrm{at}]&=-4p^3-27q^2,\label{eq:disc_def2}\\
    p&=-\tfrac{1}{3}\left(k_\mathrm{a}+c_\mathrm{b}\right)^2-k_\mathrm{a}\left(c_\mathrm{a}-c_\mathrm{b}\right)-1,\label{eq:p_def2}\\
    q&=\frac{2}{27}\left(k_\mathrm{a}+c_\mathrm{b}\right)^3+\frac{k_\mathrm{a}}{3}\left(k_\mathrm{a}+c_\mathrm{b}\right)\left(c_\mathrm{a}-c_\mathrm{b}\right)-\frac{2k_\mathrm{a}}{3}+\frac{c_\mathrm{b}}{3}.\label{eq:q_def_2}
\end{align}
The contours of $\mathrm{pH}=7-\log_{10}{\rho}$ as function of $C_\mathrm{a}=\sqrt{K_\mathrm{w}}c_\mathrm{a}$ and $C_\mathrm{b}=\sqrt{K_\mathrm{w}}c_\mathrm{b}$ for acetic acid, $k_\mathrm{a}=180$, are shown in figure \ref{fig:pH_acid_base}. It is interesting noticing that there is a linear relationship between base and acid concentrations at constant $\mathrm{pH}$. The equivalence point is given by the ${\mathrm{pH}=7}$ line, it has slope one and passes through the origin ${(c_\mathrm{a},c_\mathrm{b})=(0,0)}$. Figure \ref{fig:pH_acid_base} also shows that the $C_\mathrm{b}$-intercept is negative for acidic $\mathrm{pH}$ and positive for basic $\mathrm{pH}$. It is also seen in this figure, that the slope of the lines $C_\mathrm{b}(C_\mathrm{a})$ is less than one for acidic $\mathrm{pH}$.

\begin{figure}[htb]
    \centering
    \includegraphics[scale=0.7]{figure2.pdf}
    \caption{Lines of constant $\mathrm{pH}$ on the $C_\mathrm{a}$-$C_\mathrm{b}$ plane.}
    \label{fig:pH_acid_base}
\end{figure}

The titration experiment consists of the addition of a volume $V_\mathrm{b}$ of a strong base solution, with analytical concentration $c_\mathrm{b}^{0}$, to a volume $V_\mathrm{a}$ of a weak acid solution, with analytical concentration $c_\mathrm{a}^0$. As the strong base is added the pH of the mixture increases. The equivalence point is reached when the moles of the acid and the base are equal, \emph{i.e.}, when the concentration of the acid and the base are equal in the mixture. The relation between the mixture concentrations and the analytical concentrations are given by
\begin{align}
    c_\mathrm{a}&=\frac{c_\mathrm{a}^0V_\mathrm{a}}{V_\mathrm{a}+V_\mathrm{b}},\\
    c_\mathrm{b}&=\frac{c_\mathrm{b}^0V_\mathrm{b}}{V_\mathrm{a}+V_\mathrm{b}}.
\end{align}
These equations can be combined to obtain
\begin{equation}
    c_\mathrm{b}=c_\mathrm{b}^0\left(1-\frac{c_\mathrm{a}}{c_\mathrm{a}^0}\right),
\end{equation}
which is the equation of a straight line with slope $-c_\mathrm{b}^0/c_\mathrm{a}^0$ and intercept $c_\mathrm{b}^0$.

The volume of the acid $V_\mathrm{a}$ is fixed from the beginning of the titration experiment. Figure \ref{fig:titration_acetic_acid} displays titration curves for $V_\mathrm{a}=1\,\mathrm{L}$ of an acetic acid solution, $k_\mathrm{a}=180$, with concentration $c_\mathrm{a}^0=0.1/\sqrt{K_\mathrm{w}}$. The concentration of the base has been chosen such that $nc_\mathrm{b}^0=c_\mathrm{a}^0$ with $n=2,4,6,8,10$. All the titration curves of figure \ref{fig:titration_acetic_acid} are monotonically increasing functions of the base volume $V_\mathrm{b}$, with the same $\mathrm{pH}$ at $V_\mathrm{b}=0$, $\mathrm{pH}=2.37$. It is clear that the more concentrated the base, the faster the $\mathrm{pH}$ grows as function of $V_\mathrm{b}$.
\begin{figure}[htb]
    \centering
    \includegraphics[scale=0.7]{figure3.pdf}
    \caption{Titration curves for $V_\mathrm{a}=1\,\mathrm{L}$ of acetic acid solution, $k_\mathrm{a}=180$, with concentration $c_\mathrm{a}^0=0.1/\sqrt{K_\mathrm{w}}$. The concentration of the base is taken such that $n c_\mathrm{b}^0=c_\mathrm{a}^0$, with $n=2,4,6,8, 10$. The black point on each titration curve indicates the condition $c_\mathrm{a}=c_\mathrm{b}$. }
    \label{fig:titration_acetic_acid}
\end{figure}

The acid-base equivalence point is given when the titration curves reach the neutral $\mathrm{pH}$, $\mathrm{pH}=7$. The equivalence point of the titration curves of figure \ref{fig:titration_acetic_acid} are displayed in table \ref{table:table_1}. The analytical condition $c_\mathrm{a}=c_\mathrm{b}$, is reached at the black points indicated on each titration curve on figure \ref{fig:titration_acetic_acid}. The volume of added base $V_\mathrm{b}$, and the $\mathrm{pH}$ at the condition $c_\mathrm{a}=c_\mathrm{b}$ is displayed in table \ref{table:table_1}.

\begin{table}[htb]
\begin{center}
\begin{tabular}{|c | c | c | c|}
\hline
 $n$ ($nc_\mathrm{b}^0=c_\mathrm{a}^0$) & $V_\mathrm{b}$ ($\mathrm{pH}=7$) & $V_\mathrm{b}$ ($c_\mathrm{a}=c_\mathrm{b}$) & $\mathrm{pH}$ ($c_\mathrm{a}=c_\mathrm{b}$)  \\
 \hline
 2 & 1.989 & 2 & 8.634 \\
 4 & 3.978 & 4 & 8.523\\
 6 & 5.967 & 6 & 8.450\\
 8 & 7.956 & 8 & 8.396 \\
 10 & 9.945 & 10 & 8.352 \\
 \hline
\end{tabular}
\caption{Volume of the base, $V_\mathrm{b}$, at the equivalence point ($\mathrm{pH}=7$), at the equal concentrations condition ($c_\mathrm{a}=c_\mathrm{b}$), and $\mathrm{pH}$ at the concentration condition ($c_\mathrm{a}=c_\mathrm{b}$).}
\label{table:table_1}
\end{center}
\end{table}

\subsection{Buffer solution}

An acid buffer solution is the aqueous dissolution of a weak acid ($\ce{HB}$) and a fully soluble salt of the conjugate base ($\ce{NaB}$). The chemical equations of a buffer equilibrium are \cite{Denbigh1981,Burgot2012,Skoog2022}
\begin{align}
    \ce{HB + H2O &<--> H3O+ + B-},\label{ce:HA_diss_buffer}\\
    \ce{B- + H2O &<--> HB + OH-},\label{ce:B_hydrolysis_buffer}\\
    \ce{NaB &-> Na+ + B-}\\
    \ce{2 H2O &<--> H3O+ + OH-}.\label{ce:H2O_auto_buffer}
\end{align}
Relevant chemical species are $\ce{H3O+}$, $\ce{OH-}$, $\ce{HB}$, $\ce{B-}$, and $\ce{Na+}$, with molar concentrations  $\ce{[H3O+]}$, $\ce{[OH^-]}$, $\ce{[HB]}$, $\ce{[B^-]}$, and $\ce{[Na+]}$.

In an acid buffer solution with acid concentration $C_\mathrm{a}$ and salt concentration $C_\mathrm{s}$, the concentrations  of the chemical species are given by four physical conditions: the weak acid dissociation constant $K_\mathrm{a}$, the water auto-ionization constant $K_\mathrm{w}$, the electric neutrality, and the mass balance,
\begin{align}
    K_\mathrm{a}&=\frac{\ce{[H3O+]}\ce{[B^-]}}{\ce{[HB]}},\label{eq:Ka_buffer}\\
    K_\mathrm{w}&=\ce{[H3O^+]}\ce{[OH^-]},\label{eq:Kw_buffer}\\
    \ce{[H3O+]} + \ce{[Na+]} &=\ce{[OH^-]}+\ce{[B^-]},\label{eq:charge_balance_buffer}\\
    C_\mathrm{a} +  C_\mathrm{s}&=\ce{[B^-]}+\ce{[HB]},\label{eq:matter_balance_buffer}
\end{align}
respectively. Since the salt $\ce{NaB}$ dissociates completely, the concentration of $\ce{Na+}$ is given by $C_\mathrm{s}=\ce{[Na^+]}$. The hydrolysis of the conjugate base, chemical equation \eqref{ce:B_hydrolysis_buffer}, gives the conjugate base hydrolysis constant $K_\mathrm{cb}=K_{\mathrm{w}}/K_{\mathrm{a}}$, which is irrelevant for this analysis.

Acid constants $K_\mathrm{a}$ have units of concentration and their value range typically between $10^{-10}$ and $10^{-1}$ $\mathrm{M}$. It is convenient to define the dimensionless variables and constants ${s=\ce{[Na^+]}/\sqrt{K_\mathrm{w}}}$, ${x=\ce{[H3O^+]}/\sqrt{K_\mathrm{w}}}$, ${y=\ce{[OH^-]}/\sqrt{K_\mathrm{w}}}$, ${w=\ce{[HB]}/\sqrt{K_\mathrm{w}}}$, ${z=\ce{[B^-]}/\sqrt{K_\mathrm{w}}}$, ${c_\mathrm{a}=C_\mathrm{a}/\sqrt{K_\mathrm{w}}}$, ${c_\mathrm{s}=C_\mathrm{s}/\sqrt{K_\mathrm{w}}}$, ${k_a=K_\mathrm{a}/\sqrt{K_\mathrm{w}}}$, and ${k_\mathrm{w}=1}$. In terms of the dimensionless variables and constants, equations \eqref{eq:Ka_buffer}-\eqref{eq:matter_balance_buffer} are replaced by
\begin{align}
    k_\mathrm{a}&=\frac{xz}{w},\label{eq:ka_buffer}\\
    k_\mathrm{w}&=xy=1,\label{eq:kw_buffer}\\
    x+c_\mathrm{s}&=y+z,\label{eq:cb_buffer}\\
    c_\mathrm{a}+c_\mathrm{s}&=w+z.\label{eq:c_buffer}
\end{align}

The addition of a strong base, \emph{e.g.} $\ce{NaOH -> Na+ + OH-}$, affects the charge balance, equation \eqref{eq:cb_buffer}. This addition is useful to study the buffer capacity of the solution. The use of a strong base with analytical concentration ${C_\mathrm{b}=\sqrt{K_\mathrm{w}}c_\mathrm{b}}$, gives the charge balance
\begin{equation}
        x+c_\mathrm{s}+c_\mathrm{b}=y+z.\label{eq:cb_buffer_2}\\
\end{equation}

The combined use of equations \eqref{eq:ka_buffer}-\eqref{eq:cb_buffer_2} gives
\begin{equation}\label{eq:ka_x_buffer}
    k_\mathrm{a}=\frac{x\left(x+c_\mathrm{s}+c_\mathrm{b}-\tfrac{1}{x}\right)}{\tfrac{1}{x}-x+c_\mathrm{a}-c_\mathrm{b}}.
\end{equation}
Since $x>0$, and $k_\mathrm{a}>0$, the numerator and denominator of \eqref{eq:ka_x_buffer} give two quadratic constraints for the concentration $x$,
\begin{align}
    x^2+\left(c_\mathrm{s}+c_\mathrm{b}\right)x-1&>0,\label{eq:quadratic_constraint_buffer_1}\\
    -x^2+\left(c_\mathrm{a}-c_\mathrm{b}\right)x+1&>0,\label{eq:quadratic_constraint_buffer_2}
\end{align}
respectively.

Rearrangement of equation \eqref{eq:ka_x_buffer} leads to the polynomial equation $P_\mathrm{ab}(x)=0$, with
\begin{equation}\label{eq:P_polynomial_buffer}
    P_\mathrm{ab}(x)=x^3+\left(k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}\right)x^2-\left(1+k_\mathrm{a}\left(c_\mathrm{a}-c_\mathrm{b}\right)\right)x-k_\mathrm{a}.
\end{equation}
The dimensionless equation $P_\mathrm{ab}(x)=0$ is equivalent to Charlot's equation when ${c_\mathrm{b}=0}$.

The use of Wolfram Mathematica allows to prove that the discriminant of $P_\mathrm{at}(x)$ is greater than zero, $\Delta[P_\mathrm{ab}]>0$, hence there are three real solutions. It can be proved that only one of these roots is positive by following the same algebraic analysis that led to equation \eqref{eq:rho3_q_negative} from $P_\mathrm{a}(x)=0$. The positive root, ${\rho=\rho(c_\mathrm{a},c_\mathrm{b},c_\mathrm{s})}$, is given by
\begin{equation}\label{eq:roots_acid_buffer}
        \rho=2\sqrt[3]{\|\zeta\|}\cos{\left(\frac{\theta}{3}\right)-\frac{k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}}{3}},
\end{equation}
with
\begin{align}
    \sqrt[3]{\|\zeta\|}&=\tfrac{1}{3}\sqrt{\left(k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}\right)^2+3k_\mathrm{a}(c_\mathrm{a}-c_\mathrm{b})+3},\label{eq:zeta_3}\\
    \theta&=\arctan{\left(-\frac{q}{2},\frac{\sqrt{\Delta[P_\mathrm{ab}]}}{6\sqrt{3}}\right)},\label{eq:theta_def3}\\
    \Delta[P_\mathrm{ab}]&=-4p^3-27q^2,\label{eq:disc_def3}\\
    p&=-\tfrac{1}{3}\left(k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}\right)^2-k_\mathrm{a}\left(c_\mathrm{a}-c_\mathrm{b}\right)-1,\label{eq:p_def3}\\
     q&=\tfrac{2}{27}\left(k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}\right)^3+\frac{k_\mathrm{a}}{3}\left(k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}\right)\left(c_\mathrm{a}-c_\mathrm{b}\right)-\frac{2k_\mathrm{a}}{3}+\frac{c_\mathrm{b}+c_\mathrm{s}}{3}.\label{eq:q_def_3}
\end{align}

\begin{figure}
    \centering
    \includegraphics[scale=0.333]{HH_error.pdf}
    \caption{Abslute error $E^{\mathrm{(pH)}}$, in the $\mathrm{pH}$ calculated by the Herderson-Hasselbach equation for a buffer solution of chlorous acid, $K_\mathrm{a}=1.2\times 10^{-2}\,\mathrm{M}$, as a function of the molar concentrations of the acid $C_\mathrm{a}$ and the salt $C_\mathrm{s}$.}
    \label{fig:HH_approx}
\end{figure}

By applying the logarithm of the acid dissociation constant $K_\mathrm{a}$, equation \eqref{eq:Ka_buffer}, is obtained, after some algebraic manipulation, the Henderson-Hasselbach (HH) equation \cite{Henderson1908,Hasselbalch1917,Po2001}
\begin{equation}\label{eq:HH}
    \mathrm{pH}=\mathrm{p}K_\mathrm{a}+\log_{10}{\frac{\ce{[B^-]}}{\ce{[HB]}}}.
\end{equation}
The right hand side of this equation is a function of $\mathrm{pH}$, therefore the HH equation is not practical for direct calculation of the $\mathrm{pH}$. It is common to use the approximations $\ce{[B^-]}\approx C_\mathrm{s}$ and $\ce{[HB]}\approx C_\mathrm{a}$ to obtain
\begin{equation}\label{eq:HH_approx}
    \mathrm{pH}^{\mathrm{(1)}}\approx\mathrm{p}K_\mathrm{a}+\log_{10}{\frac{C_\mathrm{s}}{C_\mathrm{a}}}.
\end{equation}
Figure \ref{fig:HH_approx} shows the absolute error of the $\mathrm{pH}$, for chlorous acid, calculated by the HH equation with respect to the pH calculated using the exact formula, \eqref{eq:roots_acid_buffer} with $c_\mathrm{b}=0$,
\begin{equation}
    E^{\mathrm{(pH)}}=\mathrm{pH}^{\mathrm{(1)}}-\left(7-\log_{10}{\rho}\right),
\end{equation}
with $\rho$ given by equation \eqref{eq:roots_acid_buffer}. Figure \ref{fig:HH_approx} shows that small errors, $E^\mathrm{(pH)}<0.1$, are obtained for buffer solutions with high concentration of the salt, $C_\mathrm{s}>0.1\,\mathrm{M}$, meanwhile large errors, $E^\mathrm{(pH)}>0.5$ are obtained for buffer solutions with $C_\mathrm{s}<0.01\,\mathrm{M}$.

\subsubsection{Buffer capacity}

The pH stability of an acid buffer solution is measured by adding a volume $V_\mathrm{b}$ of a strong base solution with concentration $c_\mathrm{b}^0$ \cite{Skoog2022}. The addition of this volume changes the concentrations $c_\mathrm{a}$, $c_\mathrm{s}$, and $c_\mathrm{b}$,
\begin{align}
    c_{\mathrm{a}}&=\frac{c_{\mathrm{a}}^0 V_{\mathrm{a}}^0}{V_{\mathrm{ab}}^0+V_{\mathrm{b}}},\label{eq:ca_Vb}\\
    c_{\mathrm{s}}&=\frac{c_\mathrm{s}^0 V_{\mathrm{s}}^0}{V_{\mathrm{ab}}^0+V_{\mathrm{b}}},\label{eq:cs_Vb}\\
    c_{\mathrm{b}}&=\frac{c_\mathrm{b}^0 V_{\mathrm{b}}}{V_{\mathrm{ab}}^0+V_{\mathrm{b}}},\label{eq:cb_Vb}
\end{align}
with $c_\mathrm{a}^0$, $c_\mathrm{s}^0$, and $c_\mathrm{b}^0$, as the concentrations of the acid, salt of acid, and base independent solutions, respectively. The volumes $V_\mathrm{a}^0$, $V_\mathrm{s}^0$, and $V_\mathrm{ab}^0=V_\mathrm{a}^0+V_\mathrm{s}^0$, are the initial volumes of the  acid, salt, and acid buffer solutions respectively.

The pH stability, $S_\mathrm{pH}$, of a buffer solution is given by \begin{equation}\label{eq:pH_stability}
\begin{split}
    S_{\mathrm{pH}}&=\frac{d\left(\mathrm{pH}\right)}{d V_\mathrm{b}}\\
    &=\nabla_{\bm{c}}(\mathrm{pH})\cdot\frac{d\bm{c}}{d V_b},
\end{split}
\end{equation}
with $\bm{c}=\left({c_\mathrm{a}},{c_\mathrm{s}},{c_\mathrm{b}}\right)$ as 3-vector of concentrations, and the gradient of $\rho$, $\nabla_{\bm{c}}\rho$, given by
\begin{equation}
    \nabla_{\bm{c}}\rho=\left(\frac{\partial\rho}{\partial c_\mathrm{a}},\frac{\partial\rho}{\partial c_\mathrm{s}},\frac{\partial\rho}{\partial c_\mathrm{b}}\right).
\end{equation}
The use in equation \eqref{eq:pH_stability} of the pH definition, $\mathrm{pH}=7-{\log_{10}}\rho$, and the concentrations of equations \eqref{eq:ca_Vb}-\eqref{eq:cb_Vb}, gives after some algebra
\begin{equation}
    S_\mathrm{pH}=\frac{1}{{\ln{10}}}\frac{1}{\left(V_\mathrm{ab}^0+V_\mathrm{b}\right)^2}\frac{1}{\rho}{\nabla_{\bm{c}}}\rho\cdot{\bm{n}^0},
\end{equation}
with
\begin{equation}
\bm{n}^0=\begin{pmatrix} c_\mathrm{a}^0V_\mathrm{a}^0, & c_\mathrm{s}^0V_\mathrm{s}^0, & -c_\mathrm{b}^0V_\mathrm{ab}^0\end{pmatrix}.
\end{equation}

The gradient $\nabla_{\bm{c}}\rho$ is calculated with respect to the components of $\bm{c}$ but it must be expressed in terms of the volume of base $V_\mathrm{b}$.

Figure \ref{fig:buffer_stability} displays the $\mathrm{pH}$ stability, $S_\mathrm{pH}$, as function of the $\mathrm{pH}$ of the buffer solution. Since the addition of a strong base increases monotonically the $\mathrm{pH}$ of the solution, the curves of figure \ref{fig:buffer_stability} contain the same information as the titration curves. To simplify the analysis all the concentrations used to prepare or titrate the buffers, $c_\mathrm{a}^0$, $c_\mathrm{s}^0$ and $c_\mathrm{b}^0$, have the same value $c^0$. Panel (a) of figure \ref{fig:buffer_stability} displays $S_\mathrm{pH}/100$ for buffer solutions with concentrations $C^0=10^{-7}\times c^0$ ranging from $10^{-2}$ to $10^{-1}\,\mathrm{M}$ (from blue to red). Panel (b) of figure \ref{fig:buffer_stability} displays $S_\mathrm{pH}$ for buffer solutions with concentrations $C^0=10^{-7}\times c^0$ ranging from $10^{-7}$ to $10^{-5}\,\mathrm{M}$ (from purple to re). In both panels of figure \ref{fig:buffer_stability}, curves with lower (higher) values of $S_\mathrm{pH}$ are for buffer solutions prepared and titrated with solutions of lower (higher) concentrations. The maximum of the curves $S_\mathrm{pH}$ is higher, and located at higher pH, for buffer solutions at higher concentrations $C^0$. The highest concentration buffer of figure \ref{fig:buffer_stability}(a), in red) displays numerical instability for basic $\mathrm{pH}s$.

\begin{figure}
    \centering
    \includegraphics[scale=0.5]{Buffer_stability.pdf}
    \caption{$\mathrm{pH}$ stability $S_\mathrm{pH}$ as a function of the pH for buffer solutions of acetic acid ($k_\mathrm{a}=175$) and sodium acetate titrated by NaOH. To simplify the analysis, the initial concentrations $c_\mathrm{a}^0$, $c_\mathrm{s}^0$, and $c_\mathrm{b}^0$, are equal to $c^0$. Panel (a) displays $S_\mathrm{pH}/100$ for solutions with concentrations $C^0=10^{-7}\times c^0$ from $10^{-5}$ to $10^{-1}\,\mathrm{M}$; The lowest curves (blue color) are at lower concentrations, the highest curves (red color) are for higher concentrations. Panel (b) displays $S_\mathrm{pH}$ for solutions with concentrations $C^0=10^{-7}\times c^0$ from $10^{-7}$ to $10^{-5}\,\mathrm{M}$; The lowest curves (purple color) are for low concentration buffers, the highest curves (red color) for higher concentrations. }
    \label{fig:buffer_stability}
\end{figure}

Figure \ref{fig:buffer_titration} displays the titration curves for the same buffer solutions of figure \ref{fig:buffer_stability}. 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 figure \ref{fig:buffer_titration} have the same equilibrium point at $V_\mathrm{b}=1\,\mathrm{L}$ and $\mathrm{pH}=7$. Buffers of higher concentrations, in red, display the largest changes of $\mathrm{pH}$ as the solution is titrated with strong base.

The use of figures \ref{fig:buffer_stability}(a) and \ref{fig:buffer_titration}(a) indicates that the lower the concentration of the solutions of acid and salt of the acid (blue curves), the lower the change in $\mathrm{pH}$ as the base is added.

\begin{figure}
    \centering
    \includegraphics[scale=0.5]{Buffer_titration.pdf}
    \caption{Buffer titration as function of the volume of base added, $V_\mathrm{b}/\mathrm{L}$. To simplify the analysis, the concentrations of the solutions to prepare, and titrate, the buffer are the same, $C^0=10^{-7}c^0$. Panel (a) displays the $\mathrm{pH}$ for solutions with concentrations $C^0=10^{-7}\times c^0$ from $10^{-5}$ to $10^{-1}\,\mathrm{M}$; The blue blue curves are at lower concentrations, the red curves are for higher concentrations. Panel (b) displays $\mathrm{pH}$ for solutions with concentrations $C^0=10^{-7}\times c^0$ from $10^{-7}$ to $10^{-5}\,\mathrm{M}$; The purple curves are for low concentration buffers, the red curves for higher concentrations. }
    \label{fig:buffer_titration}
\end{figure}

\section{Systematic analysis relating the pH expressions}

The four cubic equations for $x={\ce{[H3O+]}}/{\sqrt{K_\mathrm{w}}}$ of the systems under study can be written as
\begin{equation}\label{eq:general_polynomial}
    x^3+a_2 x^2 +a_1 x+a_0=0.
\end{equation}
The coefficients of the four systems under study are given by the rows of table \ref{tab:coeff_syst}

\begin{table}[hb]
    \centering
    \begin{tabular}{|l|c|c|c|c|}
    \hline
       System & $a_2=k$ & $a_1$ & $a_0$ & $c$ \\
       \hline
       w.a.  & $k_\mathrm{a}$ & $-c_\mathrm{a}k_\mathrm{a}-1$ & $-k_\mathrm{a}$ & $c_\mathrm{a}$ \\
       w.a. + s.b.  & $k_\mathrm{a}+c_\mathrm{b}$ &  $-c_\mathrm{a}k_\mathrm{a}-1+c_\mathrm{b}k_\mathrm{a}$ & $-k_\mathrm{a}$ & $c_\mathrm{a}-c_\mathrm{b}$ \\
       w.a.b. & $k_\mathrm{a}+c_\mathrm{s}$  &  $-c_\mathrm{a}k_\mathrm{a}-1$ & $-k_\mathrm{a}$ & $c_\mathrm{a}$ \\
       w.a.b. + s.b.  & $k_\mathrm{a}+c_\mathrm{b}+c_\mathrm{s}$  &  $-c_\mathrm{a}k_\mathrm{a}-1+c_\mathrm{b}k_\mathrm{a}$ & $-k_\mathrm{a}$ & $c_\mathrm{a}-c_\mathrm{b}$\\
       \hline
    \end{tabular}
    \caption{Coefficients of the polynomials for the systems studied: weak acid (w.a.), weak acid titration by a strong base (w.a. + s.b.), weak acid buffer (w.a.b.) and titration of a weak acid buffer by a strong base (w.a.b. + s.b.). The second and fifth columns give the effective acid constant $k$ and concentration $c$, respectively.}
    \label{tab:coeff_syst}
\end{table}

The concentration $x$ is given by the only positive root of these polynomials,
\begin{equation}\label{eq:general_x}
    x=\tfrac{2}{3}\sqrt{k^2+3ck_a+3}\cos{\left(\theta/3\right)}-\tfrac{k}{3},
\end{equation}
with $c$ and $k$ as the effective concentrations and acid dissociation constants, given by the fifth and second columns of table \ref{tab:coeff_syst}, respectively. The angle $\theta$ is given by the expression
\begin{equation}\label{eq:general_theta}
    \theta=\arctan{\left(-\frac{q}{2},\frac{\sqrt{\Delta}}{6\sqrt{3}}\right)},
\end{equation}
with discriminant $\Delta=-4p^3-27q^2$, and
\begin{align}
    p&=-\tfrac{1}{3}k^2-k_\mathrm{a}c-1,\label{eq:general_p}\\
    q&=\tfrac{2}{27}k^3+\tfrac{k}{3}\left(1+k_\mathrm{a}c\right)-k_\mathrm{a}.\label{eq:general_q}
\end{align}
The values of $k$ and $c$ from table \ref{tab:coeff_syst} show that $k \ge k_\mathrm{a}$ and $c\le c_\mathrm{a}$. Equations \eqref{eq:general_x}--\eqref{eq:general_q} with the data of table \ref{tab:coeff_syst} give the full description of the pH for the systems studied.

\section{Conclusions}

A systematic algebraic analysis has been applied on the ideal aqueous chemical equilibrium of three acid-base systems of academic and practical interest. The weak acid dissociation, the titration of a weak acid solution by a strong base, the acid buffer solution and its titration by a strong base, have been algebraically analyzed without approximations. Simple analytical expression for the the roots of the cubic equations for the $\ce{[H3O+]}$ are obtained. It has been shown that all the systems under study have one real positive root and two negative roots, no complex roots were obtained. The existence of only one positive root, with physical meaning, was proved in several ways: by analyzing the discriminant of the cubic polynomials, by using Descates' rule of signs on the polynomial and its derivative, and by analyzing Vieta's formulas. The cubic equations for the systems of interest were solved by the method of the depressed cubic equation of Vieta and Cardano. It has been found that, for all the systems under study, $\ce{[H3O+]}$ is the subtraction of two terms; The first term is the product of a trigonometric function and the square-root of a quadratic form of the acid dissociation constant, $k_\mathrm{a}$, and the concentrations, $c_\mathrm{a}$, $c_\mathrm{s}$, and $c_\mathrm{b}$; The second term is simply one third of the sum of the acid constant and the concentrations. The  analytical forms of $\ce{[H3O+]}$ for the four systems analyzed are easily related by simple mathematical substitutions.

In addition to the expression for $\ce{[H3O+]}$, another analytical expression for the degree of dissociation of a weak acid $\mathcal{D}_\mathrm{a}$, the $pH$ titration curves for weak acids and buffer solutions, and the stability of the pH $S_\mathrm{pH}$ have been obtained and studied. Exact titration curves allow to predict accurately the neutralization of weak acids, and buffer solutions, by using strong bases. The expression obtained for the pH stability $S_\mathrm{pH}$ allows to quantify precisely the  pH stability of a buffer solution when a strong base is added.

The exact $\mathrm{pH}$ has been compared against the approximate expression commonly used for calculating the $\mathrm{pH}$. The error of these approximations is analyzed in detail. It was found that the known rule of 5\% is based on assuming that ${\sqrt{K_\mathrm{w}}\ll\ce{[H3O+]}}$. It has been shown that the Henderson-Hasselbach (HH) equation exhibits large error for buffer solutions with salt concentrations  $C_\mathrm{s}>0.01\,\mathrm{M}$.

\section*{Disclosure statement}

No potential conflict of interest was reported by the authors.

\section*{Funding}

This project has been fully financed by the internal research grants of the University.

redpajama_set_name: RedPajamaArXiv


text:
Защитные волокна — один из элементов защиты банкнот и другой защищённой продукции от подделки. Представляет собой волокна, внедрённые в банкнотную бумагу на стадии её изготовления.

Волокна обычно хлопковые или текстильные. Они тонкие и достаточно длинные (от 4 до 6—8 мм). Располагаются волокна в толще бумажного листа, как правило в местах, свободных от печати, для удобства их проверки. Они могут быть как цветными, легко различимыми невооружённым глазом, так и бесцветными, светящимися разными цветами под ультрафиолетовым излучением. Волокна могут иметь особый профиль, видимый только под микроскопом.

Волокна являются достаточно хорошим защитным признаком, так как они препятствуют использованию фальшивомонетчиками обычной бумаги, в которой таких волокон нет. Кроме того, стоимость этого защитного элемента сравнительно низкая. При подделке банкнот волокна, как правило, дорисовываются (допечатываются), либо вдавливаются в бумагу. Допечатанные волокна могут состоять из цветных точек и являться результатом неумышленной имитации при копировании изображения. При впрессовывании волокон в бумагу появляется рельеф, ощутимый на ощупь.

Российский Гознак разработал два новых вида защитных волокон: «Профилированные» и «Зона». «Профилированные» имеют профиль в виде различных геометрических фигур, они предназначены для экспертной оценки подлинности. Волокна «Зона» могут служить как для экспертной, так и для публичной оценки подлинности. Они используются в российских рублях начиная с модификации 2004 года. Волокна «Зона» представляют собой цепочки цилиндров, диаметр которых меняется ступенчато, причём каждая такая «ступень» может быть как бесцветной, так и иметь определённый цвет. В настоящее время в российских рублях применяются красные, светло-зелёные, двухцветные и серые защитные волокна.

Литература

 Ионов В. М. Технологии обработки денежной наличности: бизнес-энциклопедия. — Московский финансово-промышленный университет «Синергия»; ЦИПСиР, 2012.

Защита денег от подделки

redpajama_set_name: RedPajamaWikipedia


The VG-065-00-00 series has a lightweight Delrin® construction that provides maximum chemical resistance, while also allowing for maximum flexibility and easy mounting. Built with no moving parts, these Gast vacuum generators require minimal maintenance and operate very quietly. In addition, their innovative Gast multi-venturi design uses less compressed air, yet delivers substantially more vacuum flow than single-venturi units – making them the ideal choice for applications where a regular pump may not be preferred.
This unit includes a sound attenuator that results in low noise levels in the 60-65 db range.


Berthold City "Moment of Truth" 7" second pressing on gold!
Formed in 2017 by Andrew Kline -- guitarist for Strife and World Be Free and WAR Records owner - who here commands the mic, BERTHOLD CITY's lineup is rounded out by guitarists Dennis McDonald (Internal Affairs) and Devin Vale (Final Fight), drummer Adam Galindo (One Choice), and bassist John Eightclip (Allegiance). The straight edge band released a three-song demo last July and have been keeping busy playing shows with such bands as Terror, Judge, Decline, Regulate, Dare, and Fall Silent, among others.
Written in the musical and lyrical spirit of the seminal Judge 7", Kline wrote all the songs and recorded them with Galindo. BERTHOLD CITY's upcoming Moment Of Truth 7" was tracked by Aaron Jamili at Wormhole Studios and mastered by Don Fury, the NYHC legend who has worked with everyone from Gorilla Biscuits to Agnostic Front to Youth of Today. The 7" will include three songs from the demo, plus three additional cuts tracked during the same session.


Parul University has organised the Indian Summer School for the period of 01 Month from 4th to 28th February 2019 in which 02 students from Wroclaw University of Science & Technology, Poland had participated along with the students of Parul University.
In this one-month exchange program; the participants pursued short courses on the topics: – Indian Culture & Cross Cultural Communication. Students were taken to visit some local attractions like Champaner Fort and Laxmi Vilas Palace along with Swaminarayan Temple (Poicha), Goraj Muni Seva Ashram, Askhardham Temple and Gandhi Ashram, Ahmedabad. They also participated in Cultural sessions and activities such as Rangoli, Mehndi, Indian Costumes, Indian Dance, Celebration of Indian Festivals and Indian Cuisine.
Parul University has organized the Indian Summer School for the period of two weeks from 4th to 16th February 2019 in which 02 students along with 01 professor from Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, Russia had participated along with 05 students from Parul University.
In this two weeks exchange program; the participants pursued short courses on the topic "Internet of Things" along with practical sessions on LED blinking, Soil Sensor/ Bluetooth using Arduino with C Programming , flashing micro python in Node MCU . Participants were taken for visits to some local attractions like Laxmi Vilas Palace and Swaminarayan Temple (Poicha). They also participated in Cultural sessions and activities such as Rangoli, Mehndi, Indian Costumes, Indian Dance, Celebration of Indian Festivals.
Parul University has organised the Indian Summer School for the period of two weeks from 4th to 16th February 2019 in which 12 students along with 04 professors from Bern University of Applied Sciences, Switzerland; 02 students along with 01 professor from HES-So Fribourg University, Switzerland; and 01 professor from Instituto Superior Técnico- Universidade de Lisboa, Portugal had participated.
In this two weeks exchange program; the participants pursued short courses on the topics:- Rural Development , Corporate Social Responsibility Introduction to NGOs and Social Problems in India & Switzerland . Participants were taken for visit to the NGOs- Goraj Muni Seva Ashram, Xavier's Green School and local villages along with some local attractions like Laxmi Vilas Palace and Swaminarayan Temple (Poicha).
11 Professors and 18 students from PU's Partner Universities in Switzerland, Russia, Czech Republic, Portugal and Lithuania are currently at PU to attend the flagship programme of Indian Summer School. They would undergo classes in Social Work, Computer Science and Indian Culture.
Two students from Bern University of Applied Sciences, Switzerland arrived at Parul University campus on 5th August, 2018. They were welcomed in Indian tradition by the IRC team, and Program coordinators, Parul University.
The students were taught both theoretical and practical aspects of their internship curriculum subjects by the professors of Parul University and health care personnel of Parul Sevashram Hospital in all areas of health care which includes medicine, surgery, obstetrics, gynecology, ophthalmology, ENT, Pediatrics, and Emergency & ICU.
During their clinical hours, students were also allowed to have hands on practice. The schedule also covered practical and experiential session to enhance their learning on Indian Culture.
Students were also taken to visit different health care facilities to gain in-depth knowledge of their working system and staffing system including multi specialty corporate hospital, Trust Hospital, Hospital managed by NGO and Primary Health Center.


text:
#include "phonenumbers/regex_based_matcher.h"

#include <string>

#include "phonenumbers/base/memory/scoped_ptr.h"
#include "phonenumbers/phonemetadata.pb.h"
#include "phonenumbers/regexp_adapter.h"
#include "phonenumbers/regexp_cache.h"
#include "phonenumbers/regexp_factory.h"

namespace i18n {
namespace phonenumbers {

// Same implementations of AbstractRegExpFactory and RegExpCache in
// PhoneNumberUtil (copy from phonenumberutil.cc).
RegexBasedMatcher::RegexBasedMatcher()
    : regexp_factory_(new RegExpFactory()),
      regexp_cache_(new RegExpCache(*regexp_factory_, 128)) {}

RegexBasedMatcher::~RegexBasedMatcher() {}

bool RegexBasedMatcher::MatchNationalNumber(
    const string& number,
    const PhoneNumberDesc& number_desc,
    bool allow_prefix_match) const {
  const string& national_number_pattern = number_desc.national_number_pattern();
  // We don't want to consider it a prefix match when matching non-empty input
  // against an empty pattern.
  if (national_number_pattern.empty()) {
    return false;
  }
  return Match(number, national_number_pattern, allow_prefix_match);
}

bool RegexBasedMatcher::Match(
    const string& number,
    const string& number_pattern,
    bool allow_prefix_match) const {
  const RegExp& regexp(regexp_cache_->GetRegExp(number_pattern));

  if (regexp.FullMatch(number)) {
    return true;
  }
  const scoped_ptr<RegExpInput> normalized_number_input(
      regexp_factory_->CreateInput(number));
  return regexp.Consume(normalized_number_input.get())
      ? allow_prefix_match
      : false;
}

}  // namespace phonenumbers
}  // namespace i18n

redpajama_set_name: RedPajamaGithub


text:
Caballo Press of Ann Arbor
Receive press releases from Caballo Press of Ann Arbor: By Email PDF Version RSS Feeds:
Caballo Press Author Thomas Foley to Visit Ireland
Caballo Press of Ann Arbor author Thomas Foley will be visiting Ireland from December 13 to January 7. He will be promoting his recently released book, The Simple Game: An Irish Jockey's Memoir, and his role in Walt Disney Studios' major motion picture Secretariat.
Nenagh, Ireland, December 03, 2010 --(PR.com)-- Caballo Press of Ann Arbor author, as well as professional flat and jump jockey Thomas Foley, will be visiting Ireland from December 13 to January 7. He will be in the country promoting his recently released book, The Simple Game: An Irish Jockey's Memoir and his role in Walt Disney Studios' feature film Secretariat. Foley plays the part of Jimmy Gaffney in the movie. Caballo Press officials announced that plans call for Foley to visit a number of bookstores as well as the central public libraries located in Nenagh, Galway, and Dublin, as well as a stopover at Race Academy and Centre of Education (RACE) in Kildare. Foley graduated from the academy in 1997 and shortly thereafter immigrated to the United States. At each stop, he will be reading from and discussing his memoir, and signing copies for those in attendance. A complete list of Foley's itinerary while in Ireland can be found on the publisher's website at www.caballopress.com.
In making the announcement, Caballo Press' CEO and publisher, Rudolph Alvarado, noted that Foley's memoir has been praised in the United States and Ireland for the author's willingness to share the painful details of his life story as he struggled to recapture the passion for horses and horse racing that he enjoyed as a boy in Ireland and came to despise as a man in America.
Writing in the Irish Independent, Kevin Corbett noted that, "From wide-eyed apprentice, to big fish in the small pond of US jump racing, to burnt-out wreck teetering on physical and mental ledges, Thomas Foley reflects on a life lived fast and often heedless in his memoir, A Simple Game.... In it he frankly recounts everything, from the emotional turmoil that sent him down the path to bulimia to the tragicomic circumstances of his confrontation of the problem."
Foley was born in Limerick, Ireland, and brought up in Nenagh and Lorrha in North Tipperary. After completing his training as a jockey at RACE in the Curragh, he moved to America where in his first full year as a professional jockey, he was the leading money-winning apprentice in steeplechase racing and eventually the leading money winner over jumps in 2003. However, the price paid to reach the top of his profession included the loss of his family, his self-respect and his love of the game and the horses that made it all possible.
However, since the release of his memoir by award-winning publisher Caballo Press, and with the aid of his partner Kristin Close, Foley has renewed his passion for horses and and the sport that continues to define his life. He has turned to training thoroughbreds to run on the flat and over jumps. Most recently, he assisted in preparing the Irish-bred Percussionist for the Colonial Cup in the United States.
As far as his acting career, Foley is currently being considered for a number of projects being produced by major motion picture studios in America and he is developing a number of projects for television. Discussions are also underway to turn his memoir into a major motion picture. Most recently, he was invited by Walt Disney Studios Ireland to be a part of the premiere of Secretariat in Newbridge, Kildare. The event was attended by top Disney Ireland officials, as well as by notable members of Ireland's horse racing community. The movie is currently showing nationwide in Ireland.
In addressing his success off the racetrack, Foley stated, "This has certainly been a year to remember and to cap it all off by coming home is something special. I look forward to discussing and reading from my book, as well as attending various screenings of Secretariat."
Foley's The Simple Game: An Irish Jockey's Memoir can be ordered in Ireland and the United Kingdom through O'Mahonys Booksellers. Library orders originating in Ireland and the United Kingdom can be made by contacting Frances Tracey at frances.tracey@omahonys.ie. Special orders can be placed through O'Mahonys Booksellers by contacting Dawn McCarthy at 061-418-155 and special orders placed internationally can also be placed through Ms. McCarthy by dialing the proper international code followed by the number 353-61-418-155. To learn more about The Simple Game: An Irish Jockey's Memoir by Thomas Foley, visit the publisher's website at www.caballopress.com.
Yvette Robertson
www.caballopress.com
Excerpt from The Simple Game: An Irish Jockey's Memoir
Read an excerpt from the opening of Thomas Foley's The Simple Game: An Irish Jockey's Memoir.
Filename: FoleyMemoirSample.pdf
Click here to view the list of recent Press Releases from Caballo Press of Ann Arbor

redpajama_set_name: RedPajamaCommonCrawl


text:
By Yossi Schwartz, Revolutionary Communist International Tendency, 29 October 2019, www.thecommunists.net
The violence committed by organized, as well as street crime, in the Arab Palestinian citizens of Israel communites, is a very seroius problem. According to a police report published on October 3rd, 2019, there have been more than 70 killings this year, nearly as many as in the past two years combined and more than half of all murder victims nationwide.
There are a few questions to ask: why is there organized crime in the Arab communities? Why have these gangs been allowed to accumulate so many weapons and kill so many? Does the police have an interest in ending organized crime in the Arab communities? And assuming for a moment that the police wants to eliminate organized crime, is it even capable of disarming gangs? How to protect communities against such criminals?
Organized crime has always been an alternative way to accumulate wealth in a poverty stricken society. The poverty rate for Arab families increased to 49.2% in 2017 from 47.1% in 2016. [1] Since very few Palestinian citizens of Israel have been able to become mainstreame capitalists by exploiting workers legally, crime became the answer.
The Arab middle class in Israel is, however, somewhat growing, but so is poverty. Prof. Haidar of the Hebrew University says: "The rate (of the middle class) has grown from 17% [of the Arab population] two decades ago to 27% today. There are still not a lot of wealthy people, only 3% fit that category, and the majority is still in the lower income brackets, but without a doubt it's a growing segment among the young, educated professionals." [2]
Thus while the Arab middle class may have illusions that the police will defend them, it is easy for the organized crime to recruit young poor people.
Until recently, most killing of Arabs by Arabs was ignored by the mass media. "If an Arab kills another Arab, the police don't care, it's only when the weapons make their way out of our communities does anyone pay attention," said Renin Oudeh Shawana, the director of the "Mustaqbal," or "Future," a youth program in Jaljuliya. Her relative was one of the two victims in 2007 of killings in the village following a disagreement between two gangs. After the shootings, police came to look for evidence, but never returned to follow up." [3]
Gun-Free Kitchen Tables Campaign, a coalition of women's organizations working to eliminate gun violence says: "most illegal firearms in Israel are actually registered to the army. According to Public Security Minister Gilad Erdan, 90 percent of illegal firearms in the north of Israel can be traced back to the military. According to Irshaid, there are around 300,000 registered firearms in Israel, and they are mostly owned by law enforcement and Jewish citizens.
In 2013, police estimated around 400,000 unlicensed guns in Israel, and that most of them are owned by Palestinians citizens. The problem is that Israeli authorities treat violence in Arab communities as an internal Arab problem. All it would take is a single gunshot in a Jewish neighborhood of Tel Aviv and police would do everything in their power to find the perpetrator.
The state is directly responsible for the violence. Based on the organization's research, in addition to weapons stolen or illegally purchased from the army, the primary source of weapons in the Arab community in Israel are those smuggled from Jordan, guns that were once legally licensed in Israel, and guns that were manufactured in the occupied territories. [4]
Since most of the weapons in the hands of the gangsters come from the army, we should ask how it is possible that the army does such a poor job of guarding its own weapons. Is it possible that the army allows this theft so Arabs kill Arabs? In addition, is it more than possible that the police collaborate with gangs as it was proven in other countries?
Ayman Odeh, the head of the Joint List, says that the racist government has neglected the Arbs and the police have abandoned the Arab neighborhoods to gangs and criminals. He demands more and effective policing.
The police reject the allegations of indifference and claim that the police are doing everything they can to stem the violence. However, to succeed in disarming the gangs, the police says, local leaders need to do more to cooperate with police and prevent violence.
In other words, the police are blaming the Arab citizens of Israel for the gang violence. Erdan, the Public Security Minister, decided to allocate 620 additional police officers to the Arab communities. A decision was also made to transfer responsibility for handing organized crime among Israeli Arabs to the Lahav 433 Major Crimes Unit.
It is the same unit that a former top police corruption investigator was convicted of failing to report an attempted bribe offered to him by a well-known rabbi. Former police deputy commissioner Menashe Arviv, who headed the national police force's top corruption and fraud investigative body Lahav 433, was convicted of the relatively minor crime of "failure to carry out an official duty." The conviction comes as part of a plea bargain that saw fraud and breach of trust charges excised from the indictment. [5] This is not the first nor the last time that the police are involved in criminal activity.
Ayman Odeh is right when he says that racist policy of the government is responsible for the problem. So what makes him think that under popular pressure, all of a sudden, the police will change its policy and start defending the Arabs?!
In reality calling for more policing of the Arabs citizens of Israel will just open the door to more police violence against Arabs. It is a dangerous reformist illusion to believe that the Zionist Police will eliminate Arab organized crime.
The Israeli police force has a long history of using excessive and even murderous violence against Arab citizens of Israel. To give but a few examples: the 1956 Kafr Qasim massacre, the 1973 "Land Day" massacre and the October 2000 massacre. We ask Odeh: what has changed? Do you believe that demonstrations of 20,000 Arabs will change the nature of the Zionist Police?
This attempt to pressure the Zionist police to defend the Palestinian citizens of Israel is a continuation of the line of recommending that the Blue and White party should form the new government. This party is led by general Gantz who was in charge of massacring 2000 Palestinian in Gaza during "Operation Pillar of Defense" in 2012.
This position reflects the illusion of the Arab middle class that it is possible to be integrated into the Zionist society. Similar illusion Odeh and Tibby shared with the leadership of the Kurds that they can rely on US imperialism and now when they were deserted by Trump they join the army of Assad the butcher and Russian imperialism.
The police is a very important element of the state apparatus. Contrary to the bourgeois propaganda that the state was formed to create a peaceful social order for the citizens, the police is a brutal force created to defend the ruling class against the working class and the oppressed.
Yet reformists, centrists and petit bourgeois nationalists tell the exploited and the oppressed that they can trust the police to protect them from criminal gangs and facist thugs. According to libral and reformist theories the police is in a contradictory position. Under the pressures of the ruling class on one hand and the working class and oppressed on the other, it may act sometimes to protect the ruling class but in other times to protect the oppressed.
To give two examples: Prior to the Nazis taking power in Germany the SPD (German Social Democrats) refused to mobilize the youth movement against the Nazi Huligans because they trusted that the state would defend them. This of course did not happen. Instead they were sent to concentration camps. Prior to the Nazis taking over Austria and Germany, sections of the police considered themselves "socialists". However, when the fascists in Austria were ready to seize power by a coup, the police joined them and attacked the workers. In Germany, they joined the storm troopers.
Following Trump's Immigration and Customs Enforcement (ICE), agents have been dragging undocumented immigrants to deportation centers. Kshama Sawant, a leading member of the right wing centrist CWI and a member of Seattle's city council, at a rally against the deportations on February 17, 2017 stated:
"I'm urging Mayor Murray – if this is a sanctuary city – do not use Seattle police against peaceful protesters. Deploy Seattle police to block ICE from seizing immigrants." [6]
Spreading illusions in the police is the opposite position to Marx, Lenin or Trotsky.
On this question Lenin wrote: "On the one hand, the bourgeois, and particularly the petty-bourgeois, ideologists, compelled under the weight of indisputable historical facts to admit that the state only exists where there are class antagonisms and a class struggle, "correct" Marx in such a way as to make it appear that the state is an organ for the reconciliation of classes. According to Marx, the state could neither have arisen nor maintained itself had it been possible to reconcile classes. From what the petty-bourgeois and philistine professors and publicists say, with quite frequent and benevolent references to Marx, it appears that the state does reconcile classes. According to Marx, the state is an organ of class rule, an organ for the oppression of one class by another; it is the creation of "order", which legalizes and perpetuates this oppression by moderating the conflict between classes. In the opinion of the petty-bourgeois politicians, however, order means the reconciliation of classes, and not the oppression of one class by another; to alleviate the conflict means reconciling classes and not depriving the oppressed classes of definite means and methods of struggle to overthrow the oppressors". [7]
You do not have to be a revolutionary Communist to understand the class nature of the police. You have only to look with open eyes at the role of the police.
In the US police officers kill every year at least 1000 people, especially people of color. According to a 2016 study published in Injury Prevention, the police killed or injured over 55,000 people in 2012; the police-brutality monitoring website Mapping Police Violence reports that over 1,000 people have been killed by police in 2017. Previous studies have found that the police disproportionately use excessive force, search, and kill people of color, particularly black people".
…"In 1983, City University of New York School of Law professor Sidney L. Harring published Policing a Class Society: The Experience of American Cities 1895–1915, which addresses the history of working-class repression by the police. In Policing a Class Society, Harring argues that police have not historically "served and protected" the general population, but rather violently divided the working class and protected the interests of the upper classes. Harring uses Milwaukee and Chicago as case studies to illustrate repressive tactics of the police during the late 19th century and turn of the 20th century.
Upon the release of his book's second edition, Pacific Standard interviewed Harring about the relationship between class, race, and the police; working-class resistance to police repression; and the role of pop culture in signaling messages about the police.
"Looking at the trajectory of policing in working-class communities from the early 1800s until today, does it seem that policing is getting worse for working people, or has its relationship to the community remained steady?
"I think in general it's worse. Partly it's because there are more guns, more technology, and better police databases. It's more technocratic and militaristic. And as far as strikes, if we even have strikes, they're pervasively policed." [8]
The only way to fight organized crime is self-defense by the Palestinians themselves. But for this it is necessary to form a revolutionary Palestinian leadership.
Some Examples of Self Defense
There are examples of workers and oppressed nationalities who came to realize that they cannot rely on the police to defend them, so they organized self defense.
When the Tzar's government in Russia organized pogroms from the end of the 19th century against the Jews, the most known was in Kishinev in 1903, the revolutionary socialist worker movement together with the Bund and Po'ale Zion Left organized self defence against the black hundreds and repealed them.
In 1905 a pogroms was organized against the Jews in Odessa. This time the hooligans faced organized self defense. Just before Easter the National Committee of Jewish Self-Defense urged all Jews to join self-defense brigades and prepare to counter any attack on Jews and their property. Men were told to arm themselves with guns, knives, clubs, and whips, and women were encouraged to prepare solutions of sulfuric acid. Bundists, Bolsheviks, and Mensheviks joined in these efforts
On this Lenin wrote in 1905: "There can be no doubt that the experiment will be further extended. The laurels won by some of the Black Hundreds will give the others no rest until they too will have put their strength to the test. Where there is revolution there is counter-revolution too, and, therefore, Borisoglebsk must also be prepared to experience the organizing skill of the eminent representatives of the Black-Hundred trend. We have reason to expect also in Borisoglebsk pogroms against the Jews, against the workers, and against the intellectuals; therefore, in preparation for proper resistance to the 'illegal measures' which the government has adopted to suppress the revolutionary movement, the Borisoglebsk group is starting a subscription for the organization of armed self-defense, and invites all those whose sympathies do not lie with the government and the Black Hundreds to help in the organization of self-defense groups with money and arms." [9]
Another example of self defense is the civil rights movement in the US. According to the myth the movement was pacifist. However, this is not true, as Charles E. Cobb Jr., who participated in the movement, proved in his book, This Nonviolent Stuff'll Get You Killed: How Guns Made the Civil Rights Movement Possible.
He wrote that the nonviolent organizers in the deep South were often protected by armed farmers and workers. In Louisiana, Mississippi, Alabama and other states run by white supremacists who used terrorism to impose their domination, guns were not an option but a necessity for African Americans. [10]
The black Panther party that was established in 1966 was famous for self defense against police brutality. They grew to a size of 5,000 full time party workers, organized in 45 branches across America. At their peak, they sold 250,000 papers every week. Opinion polls of the day showed the Panthers to have 90% support amongst Blacks in the major cities.
Of course the US capitalist state at the end won because the struggle of the oppressed did not end with a socialist revolution, but they left a legacy. For example the Black Women's Defense League, a Dallas-based organization that teach women to use arms to fight against white racism. The idea of black self- defense is, today, on the rise again.
In South Mexico the native population organized self-defense against organized crime and corrupt police. For example, in March 2013 a force of 1500, armed with rifles entered the Mexican town of Tierra Colorada, which sits on the highway from Mexico City down to the Pacific coast. They seized at gunpoint 12 police officers and a local security official, whom they believed responsible for the murder of their commander. The men were members of a self-defense group, one of a growing number of organizations aiming to restore order to Mexican communities. " [11]
In April this year an association of shopkeepers in Mexico City's historic center has sent a letter to the Attorney General's Office demanding action from authorities against the Unión de Tepito, one of the most powerful criminal groups in the capital. The letter says they have been the victims of extortion, torture, kidnappings and even killings at the hands of the group, El Universal reported.
"They must understand that everything has a limit … [and] don't be surprised by the emergence of self-defense groups … given the slowness and inaction of the authorities in charge of security and law enforcement [in Mexico City]," said the letter signed by around 240 vendors, including some that have closed up shop due to the costs of extortion". [12]
As this article is being written we hear that during the Sukkot holiday not only Arabs but Jews were killed by the gangs. If the police cannot prevent the killing of Jews there is no chance they will stop the killing of the Palestinian citizens of Israel by organized crime.
However, more policing will allow the police to terrorize communities even more. If you want to know how the Zionist police enforce a colonialist racial hierarchy, you can ask the Ethiopian Jews suffering from police brutality. How the police protect Arabs you can ask the Palestinians in the West Bank. According to the human rights organization B'Tselem:
"Violence by settlers (and sometimes by other Israeli civilians) toward Palestinians has long since become part of daily life under occupation in the West Bank. These actions range from blocking roads, throwing stones at cars and houses, raiding villages and farmland, torching fields and olive groves, and damaging crops and property to physical assault, sometimes to the point of hurling Molotov cocktails or using live fire. Over the years, this widespread violence toward Palestinians has resulted in injuries to life and limb, as well as damage to property and land.
Under international law, Israel has a duty to protect Palestinians in the West Bank from this conduct. However, Israeli authorities routinely shirk this responsibility, even when the violent actions can be anticipated. Thousands of testimonies, videos and reports, as well as many years of close monitoring by B'Tselem and other organizations, reveal that Israeli security forces not only allow settlers to harm Palestinians and their property as a matter of course – they often provide the perpetrators escort and back-up". [13]
Army of occupation in oppressed communities
In the Black and Brown communities, the police play the role of a foreign occupation army and practice a form of cruelty and brutality which differs only in degree from the U.S. occupation army in Vietnam or Cambodia.
As a token of the high esteem and affection in which these communities hold the police, they have coined the word "pig" as synonym for cop and this word has passed into the universal language of the oppressed.
Every cent paid to the police comes out of the hides of the workers. Every cent they get is at the expense of welfare, housing, schools, and other facilities and services that are needed by the people and is responsible for reducing crime and domestic violence.
Unlike workers, when police go out on strike [or engage in job negotiations], they are not trying to retrieve money withheld from them for useful work done on behalf of society. Their services are solely and exclusively in the interests of one class of society only: the ruling class.
Clarity on this point is absolutely indispensable. If the police find themselves in a controversy with the ruling class over the amount of money they should get as mercenaries, the workers should treat this as an internal struggle in the camp of the enemy and not confuse it with a struggle of their own class.
Of course, there are exceptional cases where police strikers, in a struggle with the capitalist state, have no alternative but to turn for support to the workers. These cases are rare indeed, such as the Boston police strike of 1919, which Calvin Coolidge, then governor of Massachusetts, broke. In such cases, it is the duty of the workers' leaders to adroitly intervene in the struggle.
In doing so, they must make clear that their intervention is not motivated by any class solidarity with the police (who on the morrow of their victory will again proceed to club the heads of striking workers) but out of motives of working-class expedience — that is, to help the police undermine the capitalist state structure. The longer a police strike lasts, the more it undermines capitalist law and order. In that task, a revolutionary worker should help, while helping even more to build workers' self-defense groups.
The various parasitic elements which constitute the capitalist state are always in conflict with each other on how to divide among themselves the juiciest portions of the city, state and federal treasuries. Like thieves, they are invariably at each other's throats, each seeking a greater share of the loot.
These parasitic elements comprise of the police, detectives, prison officials, executioners, various state and local anti-subversive squads, and the judicial bureaucracies. These are not to be compared to firefighters, sanitation workers or other workers who have been co-opted by the government into the capitalist state apparatus so as to keep their wages in check. These workers perform useful tasks and will continue to do so even in the highest form of socialist society.
Will there be cops when classes are gone?
One way for a Marxist to judge whether a specific social group in the present capitalist state setup is parasitic or really performs socially necessary and useful work is to ask whether such groupings would be needed in a socialist system after the abolition of all class divisions.
Clearly police will not be needed. With the abolition and disappearance of all vestiges of class privilege, the need for a coercive special force, even a workers' militia, becomes superfluous.
However, men and women who work to make a more sanitary social environment and make it free from all sorts of hazards, such as fire, will of course be needed. If even in a socialist society the need for a coercive force such as police continually diminishes as the socialist system develops to a higher and higher form, then all the less do we need police in a capitalist society.
Here its fundamental function is to suppress the working class and in particular use the most brutal violence against the Black, Chicanx and Puerto Rican people.
It is to be noted that the current wave of police insurgency comes after a considerable period when they have been engaged in actual civil war against the Black and Brown communities.
The ruling class has felt itself more and more indebted to the police precisely because of this. Having been highly flattered for their brutal role in the recent period, the police are now demanding extra privileges and remuneration for their stormtrooper role in those communities and on college campuses as well as in the recent strike struggles throughout the whole country.
The police have also become more vociferous in denouncing the so-called lenient judges and demanding that the government "take the handcuffs off the police." This cry is nothing but a fascist demand for the right to unrestricted use of force and violence against the civil population. It is in this context that we must view the police strikes [and job negotiations] as well as the general historical role that they play in the class struggle.
Paris Commune dispelled cops — and crime
That the working class needs no capitalist police to secure and defend them, was never more clearly demonstrated than in the first great proletarian revolution, more than a hundred years ago — during the Paris Commune.
Scarcely had the Paris Commune been established (the first truly working class government had just begun to survey the tasks ahead of it) when the world had its first vision since the dawn of class society of what would happen to the entire capitalist police establishment on the day of the proletarian revolution.
"No more corpses at the morgue, no nocturnal burglaries, scarcely any robberies," says Karl Marx about the Paris Commune in his celebrated book, "The Civil War in France."
"In fact, for the first time since the days of February 1848," he remarks, "the streets of Paris were safe and without police of any kind."
Is there a capitalist government anywhere in the world that can make such a claim even for one day? Is there any large city anywhere in the capitalist world which is free even for a single day of any crime and could do without any police of any kind as was the case with the Paris Commune?
Merely to ask the question is to answer it. To put an end to crime it is first of all necessary to put an end to the thoroughly criminal rule of the bourgeoisie. It is their very existence which breeds not only crime and corruption but virulent racism, imperialist war and genocide.
To infuse the working class with a revolutionary attitude toward the police is at the same time ideological preparation for the overthrow of the capitalist class.
ORGANIZE SELF DEFENSE OF THE PALESTINIANS BY THE PALESTINIANS IN ISRAEL!
NO TRUST IN THE ZIONIST POLICE!
NO SUPPORT FOR NEITHER BLUE NOR WHITE!
[1] Einhorn A. (December 31, 2018). 21.2% Of Israeli Population Lives Below the Poverty Line - New Report. Jerusalem Post.
[2] Arlosoroff M. (June 13, 2017). Arabs Are Israel's New Yuppies. Haaretz.
[3] Rubin S. (April 13, 2017). The Arab Gangsters of Israel. Daily Beast.
[4] Younis R. (December 19, 2018). Israeli authorities treat gun violence in our community as an Arab problem. +972 Magazine.
[5] Times of Israel Staff. February 22, 2018). Former top graft investigator convicted of failing to report bribery attempts. Times of Israel.
[6] Flakin N. (March 2, 2017). Can the Police Stop Deportations? A Debate with Kshama Sawant. Left Voice.
[7] Lenin V.I. (1917). Class Society and the State. The State and Revolution.
[8] King E. (November 27, 2017). Revisiting a History of Police and the Working Class. City Lab
[9] Lenin V.I. (August 29, 1905). The Black Hundreds and the Organisation of an Uprising. Lenin Collected Works, Progress Publishers, 1972, Moscow, Volume 9, pages 200-204.
[10] Miah M. (January/February, 2015). African-American Self-Defense. Solidarity-US.
[11] Asfura-Heim P. and Espach R.H. (July/August 2013). The Rise of Mexico's Self-Defense Forces. Foreign Affairs.
[12] Asmann P. (April 22, 2019.) Will Self-Defense Groups Form in Mexico City?. InSight Crime.
[13] B'Tselem. (November 11, 2017). Settler Violence: Absence of Law Enforcement.


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