Henderson-Hasselbalch equation

In chemistry, the Henderson-Hasselbalch (frequently misspelled Henderson-Hasselbach) equation describes the derivation of pH as a measure of acidity (using pKa, the acid dissociation constant) in biological and chemical systems. The equation is also useful for estimating the pH of a buffer solution and finding the equilibrium pH in acid-base reactions (it is widely used to calculate isoelectric point of the proteins).

Two equivalent forms of the equation are


 * $$\textrm{pH} = \textrm{pK}_{a}+ \log_{10} \frac{[\textrm{A}^-]}{[\textrm{HA}]}$$

and


 * $$pH = pK_{a}+\log_{10} \left ( \frac{[\mathrm{base}]}{[\mathrm{acid}]} \right ).$$

Here, $$pK_{a}$$ is $$-\log_{10} (K_{a})$$ where $$K_{a}$$ is the acid dissociation constant, that is:


 * $$pK_{a} = - \log_{10}(K_{a}) = - \log_{10} \left ( \frac{[\mbox{H}_{3}\mbox{O}^+][\mbox{A}^-]}{[\mbox{HA}]} \right )$$ for the non-specific Brønsted acid-base reaction: $$\mbox{HA} + \mbox{H}_{2}\mbox{O} \rightleftharpoons \mbox{A}^- + \mbox{H}_{3}\mbox{O}^+$$

In these equations, $$\mbox{A}^-$$ denotes the ionic form of the relevant acid. Bracketed quantities such as [base] and [acid] denote the molar concentration of the quantity enclosed.

In analogy to the above equations, the following equation is valid:


 * $$\textrm{pOH} = \textrm{pK}_{b}+ \log_{10} \frac{[\textrm{BH}^+]}{[\textrm{B}]}$$

Where $$\mbox{B}^+$$ denotes the salt of the corresponding base B.

History
Lawrence Joseph Henderson wrote an equation, in 1908, describing the use of carbonic acid as a buffer solution. Karl Albert Hasselbalch later re-expressed that formula in logarithmic terms, resulting in the Henderson-Hasselbalch equation. Hasselbalch was using the formula to study metabolic acidosis, which results from carbonic acid in the blood.

Limitations
There are some significant approximations implicit in the Henderson-Hasselbalch equation. The most significant is the assumption that the concentration of the acid and its conjugate base at equilibrium will remain the same as the formal concentration. This neglects the dissociation of the acid and the hydrolysis of the base. The dissociation of water itself is neglected as well. These approximations will fail when dealing with relatively strong acids or bases (pKa more than a couple units away from 7), dilute or very concentrated solutions (less than 1 mM or greater than 1M), or heavily skewed acid/base ratios (more than 100 to 1).