Animated Solution for Chemistry - Ionic Equilibrium: $p{K}_a$ of a weak acid (HA) and $p{K}_b$ of a weak base (BOH) are 3.2 and 3.4, respectively. The pH of their salt (AB) solution is

Salt Hydrolysis

Imagine you drop a pinch of a special salt, let's call it $AB$, into a beaker of pure water. This isn't your everyday table salt ($NaCl$). This salt is the offspring of a weak acid ($HA$) and a weak base ($BOH$).

Because both of its "parents" are weak, neither the cation $B^{+}$ nor the anion $A^{-}$ is content just floating around. They both want to react with the water molecules. The anion $A^{-}$ tries to snatch a proton to become $HA$, releasing $O{H}^{-}$ ions. Meanwhile, the cation $B^{+}$ tries to grab a hydroxide ion to become $BOH$, releasing $H^{+}$ ions.

This simultaneous reaction of both ions with water is known as salt hydrolysis. The ultimate question is: who wins this tug-of-war? Will the solution end up acidic, basic, or perfectly neutral?

To determine the victor, we don't need to guess; we have a precise mathematical tool. The pH of a solution containing a salt of a weak acid and a weak base is governed by the following elegant formula:

Let's dissect this. The baseline is $7$ (neutral water). The term $\frac{1}{2}p{K}_a$ pulls the pH upwards (towards basic), while the term $-\frac{1}{2}p{K}_b$ pulls it downwards (towards acidic). The final pH is simply a delicate balance between these two opposing forces.

In our specific problem, we are given the "strength" indicators of the parent acid and base: - $p{K}_a=3.2$ - $p{K}_b=3.4$

Let's substitute these values directly into our master equation:

Now, we perform the atomic computations. Half of $3.2$ is $1.6$, and half of $3.4$ is $1.7$.

Combining the decimal terms, $1.6-1.7$ gives us $-0.1$.

The final calculated pH is $6.9$, which is just a hair below $7$. This means the solution is very slightly acidic.

Does this make physical sense? Absolutely! Remember that a smaller $pK$ value indicates a stronger component. Here, $p{K}_a(3.2)<p{K}_b(3.4)$, which means the weak acid is relatively stronger than the weak base. Therefore, the acidic character slightly overpowers the basic character, dragging the pH just below the neutral mark of $7$.