# How do pH and pKa relate?

May 5, 2018

The $\text{p} H$ of a solution is directly related to the $\text{p"K_"a}$ of a solution via the Henderson-Hasselbach equation,

$\text{p"H = "p"K_"a} + \log \left(\frac{\left[{A}^{-}\right]}{\left[H A\right]}\right)$

Let's do an example:

What is the $\text{p} H$ of a $\text{1-L}$ solution of $0.12 \text{M}$ of $N {H}_{4} C l$ to which $\text{1 L}$ of $0.03 \text{M}$ of $N a O H$ was added ($\text{p"K_"a}$ of $N {H}_{4}^{+}$ is ${9.25}^{\left[1\right]}$)?

Consider the equilibrium,

$N {H}_{4}^{+} + O {H}^{-} r i g h t \le f t h a r p \infty n s {H}_{2} O + N {H}_{3}$

It is safe to assume that the hydroxide ion will consume one equivalent ammonium's protons, leaving $0.09 \text{mol}$ $N {H}_{4}^{+}$ ions and $0.03 \text{mol}$ $N {H}_{3}$.

Since the total volume cancels out in the ratio of concentrations, we can translate these concentrations into mols and proceed.

To be sure, the hydroxide is treated as a strong base, and the ammonium as a weak acid.

Hence,

"p"H = "p"K_"a" + log(([NH_3]_"eq")/([NH_4^+]_"eq"))

= 9.25 + log("0.03 mols"/("0.12 mols" - "0.03 mols")) approx 8.77

: Nelson, D. L., Cox, M. M., & Lehninger, A. L. (2017). Lehninger Principles of Biochemistry. New York, NY: W.H. Freeman and Company.