Question

Derive the Henderson-Hasselbalch equation and mention its significance.

Answer 1

The Henderson-Hasselbalch equation was independently developed by the Swedish physiologist K.A. Hasselbalch and the American biological chemist L.J. Henderson to determine the pH of the bicarbonate buffer system of the blood.
The Henderson-Hasselbalch equation establishes a relationship between the pH of the solution containing a mixture of two components to the acid dissociation constant, ${\text{p}}{{\text{K}}_{\text{a}}}$ and the concentration of involved chemical species.

Complete step by step answer:
We will now try to derive the Henderson-Hasselbalch equation in a step-by-step manner to understand it in a broader perspective:
Let us consider a weak acid HA, having the following ionic equilibrium:
${\text{HA}} \rightleftharpoons {{\text{H}}^{\text{ + }}}{\text{ + }}{{\text{A}}^{\text{ - }}}$ ……………………1.)
According the above reaction the acid dissociation constant of HA will be,
${{\text{K}}_{\text{a}}}{\text{ = }}{{{\text{[}}{{\text{H}}^{\text{ + }}}{\text{][}}{{\text{A}}^{\text{ - }}}{\text{]}}}}{{{\text{[HA]}}}}$ ……………………2.)
Rearranging the above equation, we get,
${\text{[}}{{\text{H}}^{\text{ + }}}{\text{] = }}{{\text{K}}_{\text{a}}}{\text{ \times }}{{{\text{[HA]}}}}{{{\text{[}}{{\text{A}}^{\text{ - }}}{\text{]}}}}$ …………………..3.)
Taking negative logarithm of the equation 3.),
${\text{ - log[}}{{\text{H}}^{\text{ + }}}{\text{] = - log}}{{\text{K}}_{\text{a}}}{\text{ - log}} {{{\text{[HA]}}}}{{{\text{[}}{{\text{A}}^{\text{ - }}}{\text{]}}}}$
Since pH= -log ${\text{[}}{{\text{H}}^{\text{ + }}}{\text{]}}$, the above equation can now be written as,
${\text{pH = p}}{{\text{K}}_{\text{a}}}{\text{ - log}} {{{\text{[HA]}}}}{{{\text{[}}{{\text{A}}^{\text{ + }}}{\text{]}}}}$ Or
${\text{pH = p}}{{\text{K}}_{\text{a}}}{\text{ + log}} {{{\text{[}}{{\text{A}}^{\text{ - }}}{\text{]}}}}{{{\text{[HA]}}}}$ ………………………4.)
In the above equation ${\text{[}}{{\text{A}}^{\text{ - }}}{\text{]}}$ is a proton acceptor and [HA] is a proton donor so it can be written in the form of,
${\text{pH = p}}{{\text{K}}_{\text{a}}}{\text{ + log}} {{{\text{(protonacceptor)}}}}{{{\text{(protondonar)}}}}$ …………….5.)
The equation 4.) And 5.) Is known as Henderson-Hasselbalch equation.
Significance of this equation:
The Henderson-Hasselbalch equation can be used to calculate the pH of a buffer solution using the initial concentration of the weak acid and the salt and ${{\text{K}}_{\text{a}}}$.
And if the pH of the solution is given together with other information of initial concentration of weak acid its salt it can be used to calculate the acid dissociation constant of the weak acid.
The equation can also be used to determine the amount of acid and the conjugate base (it's salt) required to prepare a buffer of a particular pH.
This equation considered the backbone of acid-base physiology.

Note: A buffer solution can be defined as a mixture of a weak acid, its conjugate base or a base and its conjugate acid (in a particular ratio for the desired pH). The pH of a buffer solution changes very little when a small amount of strong acid or base is added to it. Buffer solutions are of two types:
Acidic Buffer: It has a pH of less than 7 and made up of weak acid and its salt (often a sodium salt). E.g., a mixture of ethanoic acid and sodium ethanoate in an aqueous medium
Alkaline buffer: It has a pH of more than 7 and made up of a weak base and its salt. For example, a mixture of ammonium hydroxide and ammonium chloride.