Calculating the pH of a buffer solution[edit | edit source]
![{\displaystyle {\mbox{pH}}=-\log _{10}{\left(K_{a}{\frac {\left[{\mbox{acid}}\right]}{\left[{\mbox{salt}}\right]}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5550ba9989eb85090779711f2f10e5eaa05af2fb)
For any equilibrium
the equilibrium constant, K, is defined as
![{\displaystyle K={\frac {[{\mbox{C}}]^{c}[{\mbox{D}}]^{d}}{[{\mbox{A}}]^{a}[{\mbox{B}}]^{b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b495992a71b709753fac6ae1a81256d2d75890e)
Therefore, for the dissociation equilibrium of any acid
the acid dissociation constant, Ka, is defined as
![{\displaystyle K_{a}={\frac {[{\mbox{H}}^{+}{\mbox{(aq)}}][{\mbox{A}}^{-}{\mbox{(aq)}}]}{[{\mbox{HA}}{\mbox{(aq)}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3683dd50c0cfa026599d8cc321f852060f6fd5)
This equation can be rearranged to make [H+(aq)] the subject:
![{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{HA}}{\mbox{(aq)}}]}{[{\mbox{A}}^{-}{\mbox{(aq)}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5789474b85496e2e43f0ac0a90a50bb4897e861)
Two assumptions are required:
1 Every A− ion comes from the salt
- Although this is not quite true, it is a close enough that the pH value we get from the final equation is very close to that found experimentally. It allows us to assume that
![{\displaystyle \left[{\mbox{HA}}{\mbox{(aq)}}\right]=\left[{\mbox{acid}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0f0573d12144b71ef41b66ff7d29113d02e091)
2 Every HA molecule remains undissociated
- Again, despite being slightly inaccurate, this assumption creates the following useful equation
![{\displaystyle \left[{\mbox{A}}^{-}{\mbox{(aq)}}\right]=\left[{\mbox{salt}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1759f5b5f33c0f70e97977224986923b1f8ac85)
The equations in assumptions 1 and 2 allow us to replace [A−(aq)] with [salt] and [HA(aq)] with [acid] as follows.
The effect of assumption 1 is that
becomes
![{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{HA}}{\mbox{(aq)}}]}{[{\mbox{salt}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0d961b708a18258d841deb5efdf497a779398c)
The effect of assumption 2 is that
becomes
![{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{acid}}]}{[{\mbox{salt}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f66fbb6cbc5d7a71e14fb8b05582843bc5fa51bc)
By definition,
![{\displaystyle {\mbox{pH}}=-\log _{10}{\left(\left[{\mbox{H}}^{+}{\mbox{(aq)}}\right]\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f380a2b0ac542c30cd9747420c0ccac2c572b97)
so
