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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 solution]]s 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 AH {{eqm}} A<sup>−</sup> + H<sup>+</sup>, the Charlot equation may be written as
:<math>\mathrm{[H^+]} = K_a \frac{C_a - \Delta}{C_b + \Delta}</math>

where [H<sup>+</sup>] is the equilibrium concentration of H<sup>+</sup>, ''K''<sub>a</sub> is the [[acid dissociation constant]], ''C''<sub>a</sub> and ''C''<sub>b</sub> are the analytical concentrations of the acid and its conjugate base, respectively, and Δ = [H<sup>+</sup>] − [OH<sup>−</sup>]. The equation can be solved for [H<sup>+</sup>] by using the [[autoionization constant for water]], ''K''<sub>w</sub>, to introduce [OH<sup>−</sup>] = ''K''<sub>w</sub>/[H<sup>+</sup>]. This results in the following [[cubic equation]] for [H<sup>+</sup>], which can be solved either numerically or analytically:

:<math>\mathrm{[H^+]^3} + (K_a + C_b)\mathrm{[H^+]^2} - (K_w + K_a C_a)\mathrm{[H^+]} - K_a K_w = 0</math>

==Derivation==

Starting from the definition of the [[equilibrium constant]]
:<math>K_a = \mathrm{\frac{[H^+][A^-]}{[HA]}}</math>

one can solve for [H<sup>+</sup>] as follows:
:<math>\mathrm{[H^+] = \mathit{K_a} \frac{[HA]}{[A^-]}}</math>

The main issue is how to determine the equilibrium concentrations [HA] and [A<sup>−</sup>] from the initial, or analytical concentrations ''C''<sub>a</sub> and ''C''<sub>b</sub>. This can be achieved by considering the electroneutrality and mass balance constraints on the system. The first constraint is that the total concentration of cations needs to equal the total concentration of anions, because the system has to be electrically neutral: 

:<math>\mathrm{[M^+] + [H^+] = [A^-] + [OH^-]}</math>

Here M<sup>+</sup> is the [[counterion]] that comes with the conjugate base, [A<sup>−</sup>], that is added to the solution. For example, if HA is [[acetic acid]], A<sup>−</sup> would be [[acetate]], which could be added to the solution in the form of [[sodium acetate]]. In this case, M<sup>+</sup> would be the sodium cation. The equilibrium concentration [M<sup>+</sup>] is constant and equal to the analytical concentration of the base, ''C''<sub>b</sub>. Therefore,

:<math>\mathrm{[A^-] = \mathit{C_b} + [H^+] - [OH^-]} = C_b + \Delta</math>

Because of mass balance, the sum of the equilibrium concentrations of the acid and its conjugate base has to remain equal to the sum of their analytical concentrations. (HA may convert into A<sup>−</sup> and vice versa, but what is lost of HA is gained of A<sup>−</sup>, keeping the sum constant.)

:<math>\mathrm{[HA] + [A^-]} = C_a + C_b</math>

Substituting [A<sup>−</sup>] and solving for [HA]:

:<math>\mathrm{[HA]} = C_a - \Delta</math>

Introducing the equations for [HA] and [A<sup>−</sup>] into the equation for [H<sup>+</sup>] yields the Charlot equation.

==References==
* {{cite journal |title=Utilité de la définition de brönsted des acides et des bases en chimie analytique |author=G. Charlot |journal=Analytica Chimica Acta |volume=1 |year=1947 |pages=59–68 |doi=10.1016/S0003-2670(00)89721-4}}
* {{cite journal | author = de Levie, Robert | journal = [[The Chemical Educator]] | year = 2002 | volume = 7 | pages = 132–135 | doi = 10.1007/s00897020562a | title = The Henderson Approximation and the Mass Action Law of Guldberg and Waage | issue = 3}}

[[Category:Acid-base chemistry]]