The pH of a weak acid solution can be calculated approximately using the following formula:
![{\displaystyle {\mbox{pH}}=-\log _{10}{\left({\sqrt {K_{a}\left[{\mbox{acid}}\right]}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60080f496f7742c4d218dcf684a087c9a01d8120)
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)
Two assumptions are required:
1 The concentrations of H+(aq) and A−(aq) are equal, or in symbols:
![{\displaystyle \left[{\mbox{H}}^{+}{\mbox{(aq)}}\right]=\left[{\mbox{A}}^{-}{\mbox{(aq)}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f5f79e3f15c210185cb11004666fbb4a540df)
- The reason this is an approximation is that a very slightly higher concentration of H+(aq) exists in reality, due to the autodissociation of water, H2O(l) ⇌ H+(aq) + A−(aq). We neglect this effect since water produces a far lower concentration of H+(aq) than most weak acids. If you were studying an exceptionally weak acid (you won't at A-level), this assumption might begin to cause big problems.
2 The amount of HA at equilibrium is the same as the amount originally added to the solution.
![{\displaystyle \left[{\mbox{HA}}{\mbox{(aq)}}\right]=\left[{\mbox{acid}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0f0573d12144b71ef41b66ff7d29113d02e091)
- This cannot be quite true, otherwise HA wouldn't be an acid. It is, however, a close numerical approximation to experimental observations of the concentration of HA in most cases.
The effect of assumption 1 is that
becomes
![{\displaystyle K_{a}={\frac {[{\mbox{H}}^{+}{\mbox{(aq)}}]^{2}}{[{\mbox{HA}}{\mbox{(aq)}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78858d0f412b1a27fd4afc281f6c6f92d151394f)
The effect of assumption 2 is that
becomes
![{\displaystyle K_{a}={\frac {[{\mbox{H}}^{+}{\mbox{(aq)}}]^{2}}{[{\mbox{acid}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c6aa5aaf34d35777a70435e5ecd5817e7f1d13)
which can be rearranged to give
![{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]^{2}=K_{a}\left[{\mbox{acid}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5313fb96712d194ce14537adbf0be82c5751dd10)
and therefore
![{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]={\sqrt {K_{a}\left[{\mbox{acid}}\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/074fd212a4aa5233c2ddd3449d72247e8f745e14)
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
