A more rigorous theory of $pH$

Mixture of strong base with non-negligible weak base

Exercise 9

    

$100\; mL$ $0.1\; M$ $NaOH$ are mixed with $900\; mL$ of $0.1\; M$ diethylamine a) Find for this mixture 5 equations between molarities at equilibrium: $[Na^+]$, $[H_3O^+]$, $[(C_2H_5)_2NH]$, $[(C_2H_5)_2NH_2^+]$, and $[OH^-]$ (the electroneutrality condition, two conditions of conservation of matter, an equilibrium condition and the ionic product of water) b) Show that the resolution of this system leads to the equation As we are in a strong basic medium, it is best to keep $[OH^-]$ as an argument of f, the values of $[H_3O^+]$ beeing likely too weak for machinel calculation. c) Solve the equation by using →   this

One finds $pH = 12.19$ while the value calculated by by summing the independent actions of the two bases is $pH = 12.29$