Charlot equation

The Charlot equation, named after Gaston Charlot, is used in analytical chemistry to relate the hydrogen ion concentration, and therefore the pH, with the formal analytical concentration of an acid and its conjugate base. It can be used for computing the pH of buffer solutions when the approximations of the Henderson–Hasselbalch equation break down. The Henderson–Hasselbalch equation assumes that the autoionization of water is negligible and that the dissociation or hydrolysis of the acid and the base in solution are negligible (in other words, that the formal concentration is the same as the equilibrium concentration).

For an acid-base equilibrium such as HA ⇌ H⁺ + A⁻, the Charlot equation may be written as

${\displaystyle \mathrm {[H^{+}]} =K_{a}{\frac {c_{a}-\Delta }{c_{b}+\Delta }}}$

where [H⁺] is the equilibrium concentration of H⁺, K_a is the acid dissociation constant, C_a and C_b are the analytical concentrations of the acid and its conjugate base, respectively, and Δ = [H⁺] − [OH⁻]. The equation can be solved for [H⁺] by using the autoionization constant for water, K_w, to introduce [OH⁻] = K_w/[H⁺]. This results in the following cubic equation for [H⁺], which can be solved either numerically or analytically:

${\displaystyle \mathrm {[H^{+}]^{3}} +(K_{a}+C_{b})\mathrm {[H^{+}]^{2}} -(K_{w}+K_{a}C_{a})\mathrm {[H^{+}]} -K_{a}K_{w}=0}$