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The Henderson-Hasselbalch equation is a mathematical formula used to calculate, very quickly and easily, the approximate pH of a buffer, buffer, or pH buffer solution . This equation represents an approximation to the exact solution of the acid-base equilibrium in a solution formed by a conjugate acid-base pair. It therefore exists in two different forms, one for buffer systems formed by a weak acid and a salt of its conjugate base, and another for a weak base and a salt of its conjugate acid.
Henderson-Hasselbalch equation for weak acid/conjugate base buffer system
In the case of a weak acid and its conjugate base, the Henderson-Hasselbalch equation is given by:
where pK a represents the negative base ten logarithm of the acidity constant of the weak acid, C salt is the analytical concentration of the salt, and C acid is the analytical concentration of the acid. By analytical concentrations is meant the initial concentration at which the solution was prepared.
Henderson-Hasselbalch equation for weak base/conjugate acid buffer system
In the case of the buffer system formed by a weak base and a salt of its conjugate acid, the Henderson-Hasselbalch equation is given by:
where pK b , C base and C salt represent, respectively, the base ten logarithm of the basicity constant of the weak base, its analytical concentration, and the analytical concentration of the salt of its conjugate acid.
What is a buffer?
A buffer is a solution formed by a mixture between a weak acid and a weak base. These solutions are capable of buffering the pH changes that would occur in the solution by adding strong acids or bases. This is achieved since the weak acid is capable of neutralizing strong bases, while the weak base is capable of neutralizing acids.
Although any mixture of any weak acid with any weak base can regulate pH in this way, buffers are often prepared using a conjugate acid-base or conjugate base/acid pair, since only one ionic balance which greatly facilitates the calculations.
Derivation of the Henderson-Hasselbalch equation
Next, the derivation of the Henderson-Hasselbalch equation for a weak acid/conjugate base buffer system is presented. The equation for the second case (weak base/conjugate acid) is obtained by replacing the weak acid with the weak base, the protons with hydroxide ions, the conjugate base with the conjugate acid, the acidity constant with the basicity constant, and pH by pOH.
Consider a generic weak acid HA. This acid dissociates according to the following chemical equilibrium:
As we can see in the equation, the conjugate base of the HA acid is the anion A – . The relationship between the equilibrium concentrations of these species is given by the law of mass action, which, in this particular case, is represented by the following mathematical equation:
where all the species in brackets represent the respective molar concentrations in the equilibrium state. Rearranging this equation, we obtain the following expression:
Now, applying base ten logarithm to both sides of the equation and then applying the properties of logarithms, this equation becomes:
where we use the relations log(1/a) = – log(a) and log(ab) = log(a) + log(b). The term on the left is nothing more than the pH, while the first term on the right side of the equation represents the pK a , thus obtaining:
This looks very similar to the Henderson-Hasselbalch equation, but is still not the same, since the concentrations in this equation are equilibrium concentrations of undissociated acid and conjugate base while the final equation shows the concentrations respective analytics.
Now let us consider a sodium or potassium salt of the conjugate base, which we will represent as MA, where M is the metal cation. These salts are strong electrolytes that completely dissociate in water according to the following equation:
As you can see, if we dissolve an analytical concentration of the salt C salt , since it is a strong electrolyte and everything dissociates, that same amount of the anion A – will be produced . This anion is the same one that is present in the equilibrium of the weak acid, so its presence in the salt has the effect of the common ion. This effect can be observed when analyzing the dissociation of the weak acid in the presence of the salt:
The effect of the common ion causes the equilibrium of the acid not to advance towards the products, or to move towards the reactants (remember that it is a weak acid, which implies that it by itself has little tendency to dissociate). Under these conditions, we can assume that the amount of HA that dissociates is very small compared to the initial concentrations of HA and A – . For this reason, we can approximate the equilibrium concentrations of these two species to the analytical concentrations of the acid and salt, that is:
Substituting both approximations into the pH formula, the Henderson-Hasselbalch equation is obtained.
Examples of the use of the Henderson-Hasselbalch equation
Example 1: Determine the pH of a buffer solution containing an equimolar mixture of acetic acid and sodium acetate, knowing that the acidity constant of acetic acid is 1.75.10 -5 .
This system corresponds to a weak acid buffer with a salt of its conjugate base, so in this case, the first form of the Henderson-Hasselbalch equation is used to calculate the pH. The equilibrium in this case is:
We also know that C acid = C salt = C since it is indicated that it is an equimolar mixture, therefore:
Example 2: Determine the pH of a buffer solution containing 0.3 M ammonia and 0.5 M ammonium chloride, knowing that the basicity constant of ammonia is 1.8.10 -5 .
This is the opposite case to the previous one. This buffer corresponds to a weak base with a salt of its conjugate acid whose equilibrium equation is:
Using the second form of the Henderson-Hasselbalch equation, the pOH can be determined and then the pH is calculated:
Limitations of the Henderson-Hasselbalch equation
The Henderson-Hasselbalch equation is a very practical equation and, as seen in the two examples, very easy to use, however, being an approximate equation, it has its limitations. To begin with, this equation only applies when the total concentration of the conjugated acid/base pair is not very low.
If the buffer concentration is close to 10 -6 or 10 -7 , then the ionic balance of the water must be taken into account and the Henderson-Hasselbalch equation is no longer valid.
The other necessary condition is that the degree of dissociation of the acid or protonation of the base is minimal (in order to neglect x in the previous equations). If the concentration of either the acid or the base is much less than that of its conjugate pair or vice versa, then this condition is not met, and the equation is once again invalid.
As a general guideline, the concentrations of the acid or base and its salt should not differ by more than one order of magnitude for the most accurate calculation.
References
Chang, R. (2021). Chemistry (11th ed .). MCGRAW HILL EDUCATION.
Fores-Novales, B., Diez-Fores, P., & Aguilera-Celorrio, L. (2016). Assessment of acid-base balance. Contributions of Stewart’s method. Spanish Journal of Anesthesiology and Resuscitation , 63 (4), 212–219. https://www.elsevier.es
Henderson–Hasselbalch equation . (nd). Khan Academy. https://www.khanacademy.org/science/ap-chemistry-beta/x2eef969c74e0d802:acids-and-bases/x2eef969c74e0d802:buffers/v/hendersonhasselbalch-equation
Henderson-Hasselbalch Equation–MCAT Physical . (nd). Varsity Tutors. https://www.varsitytutors.com/mcat_physical-help/henderson-hasselbalch-equation
Libretexts. (2020, August 24). Henderson–Hasselbach Equation . Chemistry LibreTexts. https://chem.libretexts.org/Ancillary_Materials/Reference/Organic_Chemistry_Glossary/Henderson-Hasselbach_Equation
Skoog, D. (2021). Analytical Chemistry (7th ed .). MCGRAW HILL EDUCATION.
