pH
=
−
log
10
(
K
a
[
acid
]
[
salt
]
)
{\displaystyle {\mbox{pH}}=-\log _{10}{\left(K_{a}{\frac {\left[{\mbox{acid}}\right]}{\left[{\mbox{salt}}\right]}}\right)}}
For any equilibrium
a
A
+
b
B
⇌
c
C
+
d
D
{\displaystyle a{\mbox{A}}+b{\mbox{B}}\rightleftharpoons c{\mbox{C}}+d{\mbox{D}}\,\!}
the equilibrium constant, K , is defined as
K
=
[
C
]
c
[
D
]
d
[
A
]
a
[
B
]
b
{\displaystyle K={\frac {[{\mbox{C}}]^{c}[{\mbox{D}}]^{d}}{[{\mbox{A}}]^{a}[{\mbox{B}}]^{b}}}}
Therefore, for the dissociation equilibrium of any acid
HA
(aq)
⇌
H
+
(aq)
+
A
−
(aq)
{\displaystyle {\mbox{HA}}{\mbox{(aq)}}\rightleftharpoons {\mbox{H}}^{+}{\mbox{(aq)}}+{\mbox{A}}^{-}{\mbox{(aq)}}\,\!}
the acid dissociation constant, K a , is defined as
K
a
=
[
H
+
(aq)
]
[
A
−
(aq)
]
[
HA
(aq)
]
{\displaystyle K_{a}={\frac {[{\mbox{H}}^{+}{\mbox{(aq)}}][{\mbox{A}}^{-}{\mbox{(aq)}}]}{[{\mbox{HA}}{\mbox{(aq)}}]}}}
This equation can be rearranged to make [H+ (aq)] the subject:
[
H
+
(aq)
]
=
K
a
[
HA
(aq)
]
[
A
−
(aq)
]
{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{HA}}{\mbox{(aq)}}]}{[{\mbox{A}}^{-}{\mbox{(aq)}}]}}}
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
[
HA
(aq)
]
=
[
acid
]
{\displaystyle \left[{\mbox{HA}}{\mbox{(aq)}}\right]=\left[{\mbox{acid}}\right]}
2 Every HA molecule remains undissociated
Again, despite being slightly inaccurate, this assumption creates the following useful equation
[
A
−
(aq)
]
=
[
salt
]
{\displaystyle \left[{\mbox{A}}^{-}{\mbox{(aq)}}\right]=\left[{\mbox{salt}}\right]}
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
[
H
+
(aq)
]
=
K
a
[
HA
(aq)
]
[
A
−
(aq)
]
{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{HA}}{\mbox{(aq)}}]}{[{\mbox{A}}^{-}{\mbox{(aq)}}]}}}
becomes
[
H
+
(aq)
]
=
K
a
[
HA
(aq)
]
[
salt
]
{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{HA}}{\mbox{(aq)}}]}{[{\mbox{salt}}]}}}
The effect of assumption 2 is that
[
H
+
(aq)
]
=
K
a
[
HA
(aq)
]
[
salt
]
{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{HA}}{\mbox{(aq)}}]}{[{\mbox{salt}}]}}}
becomes
[
H
+
(aq)
]
=
K
a
[
acid
]
[
salt
]
{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{acid}}]}{[{\mbox{salt}}]}}}
By definition,
pH
=
−
log
10
(
[
H
+
(aq)
]
)
{\displaystyle {\mbox{pH}}=-\log _{10}{\left(\left[{\mbox{H}}^{+}{\mbox{(aq)}}\right]\right)}}
so
pH
=
−
log
10
(
K
a
[
acid
]
[
salt
]
)
{\displaystyle {\mbox{pH}}=-\log _{10}{\left(K_{a}{\frac {\left[{\mbox{acid}}\right]}{\left[{\mbox{salt}}\right]}}\right)}}
![{\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)
![{\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)
![{\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)
![{\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)
![{\displaystyle \left[{\mbox{HA}}{\mbox{(aq)}}\right]=\left[{\mbox{acid}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0f0573d12144b71ef41b66ff7d29113d02e091)
![{\displaystyle \left[{\mbox{A}}^{-}{\mbox{(aq)}}\right]=\left[{\mbox{salt}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1759f5b5f33c0f70e97977224986923b1f8ac85)
![{\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)
![{\displaystyle [{\mbox{H}}^{+}{\mbox{(aq)}}]=K_{a}{\frac {[{\mbox{acid}}]}{[{\mbox{salt}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f66fbb6cbc5d7a71e14fb8b05582843bc5fa51bc)
![{\displaystyle {\mbox{pH}}=-\log _{10}{\left(\left[{\mbox{H}}^{+}{\mbox{(aq)}}\right]\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f380a2b0ac542c30cd9747420c0ccac2c572b97)