In chemistry, the Henderson–Hasselbalch equation describes the derivation of pH as a measure of acidity (using pKa, the negative log of 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 the isoelectric point of proteins).
Contents
The equation is given by:
Here, [HA] is the molar concentration of the undissociated weak acid, [A⁻] is the molar concentration (molarity, M) of this acid's conjugate base and pKa is −log10 Ka where Ka is the acid dissociation constant, that is:
In these equations, A⁻ denotes the ionic form of the relevant acid. Bracketed quantities such as [base] and [acid] denote the molar concentration of the quantity enclosed.
For bases
For the standard base equation:
A second form of the equation expressed in terms of
In analogy to the above equations, the following equation is valid:
where BH+ denotes the conjugate acid of the corresponding base B. Using the properties of these terms at 25 degrees Celsius, one can synthesise an equation for pH of basic solutions in terms of pKa and pH:
Derivation
The Henderson–Hasselbalch equation is derived from the acid dissociation constant equation by the following steps:
Taking the log, to base ten, of both sides gives:
Then, using the properties of logarithms:
Identifying the left-hand side of this equation as -pKa and the
Adding pH and pKa to both sides:
The ratio
History
In 1908, Lawrence Joseph Henderson wrote an equation describing the use of carbonic acid as a buffer solution. Later, Karl Albert Hasselbalch re-expressed that formula in logarithmic terms, resulting in the Henderson–Hasselbalch equation. Hasselbalch was using the formula to study metabolic acidosis.
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 binding of H+ to the base. The dissociation of water and relative water concentration 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). In high buffer dilutions, where the concentration of protons arising from water become equally or more prevalent than the buffer species themselves (at pH 7, this means buffer component concentrations of <10−5 M formally, but practically much higher), the pKa of the "buffer" system will tend towards neutrality.
Conventions
Note that it is conventional for basic compounds to quote a pKa which refers to the pKa of the conjugate acid rather than that of the basic compound itself, likewise for pKb when discussing acids. Thus a low pKa quoted for a basic compound means that the conjugate acid readily dissociates and therefore that the basic compound is likely to be neutral at physiological pH. This can be contrasted with a low pKa for an acid, which means that the acid readily dissociates and the molecule is therefore likely to exist as its conjugate base at physiological pH. To interpret literature pKa and pKb values an inspection of the molecule to determine whether it contains acid or base functionality (or both) is important.
The Henderson-Hasselbalch equation can also be rearranged to calculate the net charge of an amino acid at a certain pH. If the Henderson-Hasselbalch was rewritten in terms of degree of dissociation, it would be written as follows:
If you solve for α in the equation above, the following equation is derived:
This equation above can be used to calculate the degree of dissociation for each ionizable group at the desired pH. For ionizable groups that are able to deprotonate to a charge of -1, multiply the calculated dissociation constant by -1, such as carboxyl or sulfhydryl groups. For ionizable groups that are able to deprotonate to a charge of 0, take the complement of the dissociation constant (1-α) and multiply the constant by +1. The net charge of the amino acid will be the sum of the charges of all of the ionizable groups.
Estimating blood pH
The Henderson–Hasselbalch equation can be applied to relate the pH of blood to constituents of the bicarbonate buffering system:
where:
This is useful in arterial blood gas, but these usually state pCO2, that is, the partial pressure of carbon dioxide, rather than H2CO3. However, these are related by the equation:
where:
Taken together, the following equation can be used to relate the pH of blood to the concentration of bicarbonate and the partial pressure of carbon dioxide:
where: