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pH calculation lectures » pH of buffer - Henderson-Hasselbalch equation
Solutions able to retain constant pH regardless of small amounts of acids or bases added are called buffers. Classical buffer contains solution of weak acid and conjugate base. Small amounts of acids or bases added are absorbed by buffer and the pH changes only slightly. In case of high or low pH just solutions of strong acids or bases are used - for example in case of pH=1 acid concentration is relatively high (0.1 M) and small addition of acid or base doesn't change pH of such solution significantly.
How to calculate pH of buffer solution containing both acid and conjugate base? Dissociation constant definition 1.1 can be rearranged into
15.1or
15.2(note that due to sign change [A-] was moved to nominator).
This is so called Henderson-Hasselbalch equation (or buffer equation). It can be used for pH calculation of solution containing pair of acid and conjugate base - like HA/A-, HA-/A2- or B+/BOH. For solutions of weak bases sometimes it s more convenient to use equation in the form
15.3Both equations are perfectly equivalent and interchangeable.
Henderson-Hasselbalch equation is used mostly to calculate pH of solution created mixing known amount of acid and conjugate base (or neutralizing part of acid with strong base). For example, what is pH of solution prepared mixing reagents so that it contains 0.1 M of acetic acid and 0.05 M NaOH? Half of the acid is neutralized, so concentrations of acid and conjugate base are identical, thus quotient under logarithm is 1, logarithm is 0 and pH=pKa.
This approach - while perfectly justifiable in many cases - is dangerous, as it creates false conviction that the equation can be used this way always. That's not true.
Henderson-Hasselbalch equation is valid when it contains equilibrium concentrations of acid and conjugate base. In case of solutions containing not-so-weak acids (or not-so-weak bases) equilibrium concentrations can be far from concentrations of substances put into solution.
Let's replace acetic acid from our example with something stronger - e.g. dichloroacetic acid, with pKa=1.5. The same reasoning leads to result pH=1.5 - which is wrong. Proper pH value can be calculated from the equation 11.13 or using pH calculator - and it is 1.78. The reason is simple. Dichloroacetic acid is strong enough to dissociate on its own and equilibrium concentration of conjugate base is not 0.05 M (as we expected from the neutralization reaction stoichiometry) but 0.0334 M.
As a rule of thumb you may remember that acids with pKa 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 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:
| Ca (M) | buffer pH found with pH calculator |
|---|---|
| 0.1 | 4.76 |
| 0.01 | 4.76 |
| 10-3 | 4.79 |
| 10-4 | 4.95 |
| 10-5 | 5.47 |
| 10-6 | 6.31 |
| 10-7 | 6.89 |
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 autodissociation. pH of 1 mM solution is close enough to the expected (from pKa) value, more diluted solutions deviate more and more. It is worth of noting here that 1 mM buffer solution has so low capacity, that it has very limited practical use.
Henderson-Hasselbalch equation can be also 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=2.148, pKa2=7.199, pKa3=12.35), but not in case of citric acid (pKa1=3.128, pKa2=4.761, pKa3=6.396). In the latter case to calculate pH you should use full equation 11.16 - or pH calculator.
If you are looking for a way to calculate buffer composition, you can reverse the equation. Using known pH and known pKa you can calculate 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.
Note: if you need program that will help in buffer calculation, you can try our Buffer Maker - the ultimate buffer calculator.
