## Lesson 23. DERIVATIONS OF HENDERSON AND HASSELBALCH EQUATION

Module 9. Buffers

Lesson 23
DERIVATIONS OF HENDERSON AND HASSELBALCH EQUATION

23.1 Introduction

Hendersonâ€“Hasselbalch equation describes the derivation of pH as a measure of acidity using the acid dissociation constant (pKa, in biological and chemical systems. The equation is also useful for estimating the pH of a buffer solution and to ascertain the equilibrium pH in acid-base reactions which will be of use to calculate the isoelectric point pI of proteins.

23.2 Derivation of Henderson-Hasselbalch Equation

As per the theory of Bronsted-Lowry an acid (HA) is capable of donating a proton (H+) and a base (B) is capable of accepting a proton. After the acid (HA) has lost its proton, it is said to exist as conjugate base (A-). Similarly, a protonated base is said to exist as conjugate acid (BH+).

The dissociation of an acid can be described by an equilibrium expression:

Consider the case of acetic acid (CH3COOH) and acetate anion (CH3COO-):
Acetate is the conjugate base of acetic acid. Acetic acid and acetate is a conjugate acid/base pair. We can describe this relationship with equilibrium constant

In this simulation, we will use KA for the acid dissociation constant. Taking the negative log of both sides of the equation gives

This can be rearranged:

By definition, pKa = -lnKa and pH = -ln[H+],

so for acid solutions the equation is

For solutions of a weak base the equation below can be used

Both equations are perfectly equivalent and interchangeable. So this equation can be rearranged to give the Henderson-Hasselbalch equation (Fig. 23.1)

23.3 Limitation of Henderson-Hasselbalch Equation

Henderson-Hasselbalch equation is used mostly to calculate pH of solution which is obtained by mixing known amount of acid and conjugate base (or neutralizing part of acid with strong base). When we prepare a solution by mixing 0.1M of acetic acid and 0.05M NaOH half of the acid is neutralized by the alkali and resulting in the equal concentrations of acid and conjugate base, thus quotient under logarithm is 1, logarithm is 0 and pH=pKa.

However, this approach is justifiable in many cases, it creates false conviction that the equation could be applied in all the situations which is not true

Henderson-Hasselbalch equation is valid only when it contains equilibrium concentrations of acid and conjugate base. In the case of solutions containing not-so-weak acids (or not-so-weak bases) equilibrium concentrations can be far from those predicted by the neutralization stoichiometry.

When we replace acetic acid in the above example with a stronger acids like dichloroacetic acid, with pKa=1.5. Repeating the same reasoning we used earlier we will arrive at pH=1.5, which is wrong. By determining the pH using either an equation or by pH meter we find the pH will be 1.78. It is because dichloroacetic acid is strong enough to dissociate on its own and equilibrium concentrations of acid and conjugate base are not 0.05M (as we expected from the neutralization reaction stoichiometry) but 0.0334 M and 0.0666 M respectively.

As a rule of thumb you may remember that acids with pKa value below 2.5 dissociate too easily and use of Henderson-Hasselbalch equation for pH prediction can give wrong results, especially in case of diluted solutions. For solutions above 10 mM and acids weaker than pKa>=2.5, Henderson-Hasselbalch equation gives results with acceptable error. The same holds good for bases with pKb>=2.5. However, the same equation will work perfectly regardless of the pKa value if you are asked to calculate ratio of acid to conjugate base in the solution with known pH.

Similar problem is present in calculation of pH of diluted buffers. Let's see what happens when you dilute acetic buffer 50/50:

Table 23.1 pH of diluted solutions of acetic buffer 50/50 (ionic strength ignored)Ca (M)

The more diluted the solution is, the more solution pH is dominated not by the presence of acetic acid and its conjugate base, but by the water auto dissociation. pH of 1 mM solution is close enough to the expected (from pKa) value, more diluted solutions deviate more and more. It is worth noting here that 1 mM buffer solution has so low capacity, that it has very limited practical use.

Henderson-Hasselbalch equation can be also be used for pH calculation of polyprotic acids, as long as the consecutive pKa values differ by at least 2 (better 3). Thus it can be safely used in case of phosphoric buffers (pKa1=1.963, pKa2=7.199, pKa3=12.35), but not in case of citric acid (pKa1=3.128, pKa2=4.761, pKa3=5.40). It is necessary to determine the pH of the solution directly rather than depending on the above equation.

Using known pH and known pKa you can calculate the ratio of concentrations of acid and conjugate base, necessary to prepare the buffer. Further calculations depend on the way you want to prepare the buffer.