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\title{04 - Example 01 - Ideal Solution}
\author{}
\date{}
\maketitle
\section{Example 01 - Ideal Solution} % (fold)
Ideal solution:
Example, 7 components, 3 reactions: \\
\begin{tabular}{llcll|l}
\hline
$H_2SO_3$ & + $H_2O$ & $\rightleftharpoons$ & $H_3O(+)$ & + $HSO_3(-)$ & $pKa_1=1.856985200$ \\
$HSO_3(-)$ & + $H_2O$ & $\rightleftharpoons$ & $H_3O(+)$ & + $SO_3(2-)$ & $pKa_2=7.171984936$ \\
$H_2O$ & + $H_2O$ & $\rightleftharpoons$ & $H_3O(+)$ & + $OH(-)$ & $pKa_3=13.99567863$\\
\hline
\end{tabular}
\\
(1) $Kc_1 \times Ce_{H_2O} = 10^{-1.856985200} = \frac{Ce_{H_3O(+)} \times Ce_{HSO_3(-)}}{Ce_{H_2SO_3}}$
\\
(2) $Kc_2 \times Ce_{H_2O} = 10^{-7.171984936} = \frac{Ce_{H_3O(+)} \times Ce_{SO_3(2-)}}{Ce_{HSO_3(-)}}$
\\
(3) $Kc_3 \times Ce_{H_2O}^2 = 10^{-13.99567863} = Ce_{H_3O(+)} \times Ce_{OH(-)}$\\
\\
\begin{tabular}{ll}
\hline
$C_{0,H2SO3}$ & = 0 mol/L\\
$C_{0,HSO3(-)}$ & = 5.10E+00 mol/L \\
$C_{0,SO3(2-)}$ & = 0 mol/L \\
$C_{0,H3O(+)}$ & = 1.00E-07 mol/L \\
$C_{0,OH(-)}$ & = 1.01E-07 mol/L \\
$C_{0,H2O}$ & = 5.54E+01 mol/L \\
$C_{0,Na(+)}$ & = 5.10E+00 mol/L \\
\hline
\end{tabular}
\\
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(4) 0 &= -$C_{H2SO3}$  &  &  &  &  &  &  &  & -$\xi_1$ & & \\
(5) $-5.10\frac{mol}{L}$ &=  & -$C_{HSO3(-)}$  &  &  &  &  &  &  & +$\xi_1$ & -$\xi_2$ &  \\
(6) 0 &=  &  &  -$C_{SO3(2-)}$  &  &  &  &  &  &  & +$\xi_2$ &  \\
(7) $-1.00 \times 10^{-7} \frac{mol}{L}$ &=  &  &  & -$C_{H3O(+)}$  &   &  &  &  & +$\xi_1$ & +$\xi_2$ & +$\xi_3$  \\
(8) $-1.01 \times 10^{-7} \frac{mol}{L}$  &= &  &  &  & -$C_{OH(-)}$  &  &  &  &  & & +$\xi_3$ \\
(9) $-55.4 \frac{mol}{L}$  &= &  &  &  &  & -$C_{H2O}$  &  &  & -$\xi_1$ & -$\xi_2$ & -2$\xi_3$ \\
(10) $-5.10\frac{mol}{L}$ &= &  &  &  &  &  &  & -$C_{Na(+)}$ &  &  & \\
\end{tabular}
\\
Non-linear system to solve in matrix form:
\\
\[
\left(
\begin{tabular}{cccccccccc}
-1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 0 & +1 & -1 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & +1 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & +1 & +1 & +1 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & +1 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & $\frac{Ceq_{H_3O(+)}}{Ceq_{H_2SO_3}}$ &
& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & $\frac{Ceq_{SO_3(2-)}}{Ceq_{HSO_3(-)}}$ &
0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & $C_{H_3O(+)}$ & 0 & 0 & 0 & 0 & 0 & 0
\end{tabular}
\right)
\times
\left(
\begin{tabular}{c}
$Ceq_{H2SO3}$ \\
$Ceq_{HSO3(-)}$ \\
$Ceq_{SO3(2-)}$ \\
$Ceq_{H3O(+)}$ \\
$Ceq_{OH(-)}$ \\
$Ceq_{H2O}$ \\
$Ceq_{Na(+)}$ \\
$\xi eq_1$ \\
$\xi eq_2$ \\
$\xi eq_3$ \\
\end{tabular}
\right)
=
\left(
\begin{tabular}{c}
0 mol/L\\
5.10E+00 mol/L \\
0 mol/L \\
1.00E-07 mol/L \\
1.01E-07 mol/L \\
5.54E+01 mol/L \\
5.10E+00 mol/L \\
$10^{-1.856985200}$ \\
$10^{-7.171984936}$ \\
$10^{-13.99567863}$ \\
\end{tabular}
\right)
\]
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