The Henderson-Hasselbalch Equation
The Henderson–Hasselbalch equation mathematically connects the measurable pH of a solution, the concentration of hydrogen ions, with the pKa, the negative log of the acid dissociation constant, of the acid. The equation is also useful for estimating the pH of a buffer solution and finding the equilibrium pH in an acid-base reaction. The equation can be derived from the formula of pKa for a weak acid or buffer. The balanced equation for an acid dissociation is:
The acid dissociation constant is:
After taking the log of the entire equation and rearranging it, the result is:
This equation can be rewritten as:
Distributing the negative sign gives the final version of the Henderson-Hasselbalch Equation:
In an alternate application, the equation can be used to determine the amount of acid and conjugate base needed to make a buffer of a certain pH. With a given pH and known pKa, the solution of the Henderson-Hasselbach equation gives the logarithm of a ratio which can be solved by performing the antilogarithm of pH/pKa:
FINDING THE pH OF A SOLUTION EXPERIMENTALLY
Titration is commonly used to determine the pKa of a solution. In this method, the pH of the solution is constantly monitored while a known acid or base (called the titrant) is slowly added. The pH of the unknown solution will stay fairly constant until the moles of titrant added equals the moles of unknown acid or base. When the moles of acid and base are the same, further additions of the titrant will cause a dramatic change in pH until the pH eventually stabilizes. A graph of pH versus added titrant is called a titration curve, and the point at which the pH changes drastically is called the equivalence point.
The equivalence point on the graph is where all of the starting solution (usually an acid) has been neutralized by the titrant (usually a base). It can be calculated precisely by finding the molarity of the titration curve and computing the points of inflection. However, in most cases, simply looking at the graph is usually enough. To calculate the pKa, one must find the volume at the half-equivalence point (that is, where half the amount of titrant has been added to form the next compound).
The titration curve for a polyprotic acid will have more than one equivalence point. As the added base completely removes each proton from the acid, the pH will jump significantly. Figure 1 shows the titration curve for ascorbic acid, which is a polyprotic acid also known as Vitamin C. Graphing the pH versus volume of base added during an acid-base titration shows the successive ionization steps taking place. To find the concentration of a polyprotic acid, the volume of base required to reach the first equivalence point is needed. The half-equivalence points on this graph can also be used to obtain the Ka value of each successive ionization.
LIMITATIONS OF THE EQUATION
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 of 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). Also, the equation does not take into effect the dilution factor of the acid and conjugate base in water. If the proportion of acid to base is 1, then the pH of the solution will be different if the amount of water changes from 1mL to 1L.