I got confused about this way back when as well, so I will do my best below to straighten things out for you.
The great secret to figure out whether or not you should use K
a or K
b as the equilibrium constant for a reaction is to look at the species being added to water in the balanced chemical equation. It is a little known fact that you can only directly use the equilibrium constant for the species being added to water when you do these calculations (the reason being that the concentration of water was actually multiplied into the value of K
a and K
b as part of its definition which is why you don't include the concentration of water as one of the reactants when you write any of the equilibrium expressions).
----
Here's a few examples showing you how this works.
#1: Acid being added to water on the reactants side.
HX + H
20
H
30
+ + X
-The species being added to water is the conjugate acid, so use K
a as the equilibrium constant:
K
a = [H
30
+][X
-]/[HX]
-----------
#2. Base being added to water on the reactants side.
X
- + H
20
HX + OH
-The base is being added to water here, so use K
b:
K
b = [OH
-][HX]/[X
-]
---------
#3. Base being added to water on the *products* side.
HX + OH
- H
20 + X
-The rule still holds...you can only directly use the K for the species being added to water, which is Kb. But the water is on the wrong side of the equation? No problem, the K for the forward reaction here is just the inverse of Kb:
K = 1/Kb = [X
-]/[OH
-][HX]
----------
If you are given the wrong K value, such as if you were given K
a for example #2, you can always find the K
b that you need to use from:
K
w = (K
a)(K
b)
There, that's all my secrets on this topic