Purpose of a pH Problem

I was doing a multiple choice question as follows.


The pH scale measures the


1. Strength of an acid
2. Concentration of hydrogen ions
3. Strength of hydrogen ions
4. Concentration of an acid


The answer is 2 but why is 1 not sufficient. The way I understand the pH scale is that it measures how acidic or alkaline a substance is. So does it not measure the strength of an acid?




pH is approximated as $\mathrm{-log\ce{[H+]}}$. It specifically has the $\ce{[H+]}$ or concentration of $\ce{H+}$ term, so pH is a direct measure of that.


$\mathrm{p}K\_\mathrm{a}$ is a better measure of the strength of an acid. Some acids are stronger than others, and dissociate more in water. While, yes, more acidic species produce more $\ce{H+}$ ions, thus having lower pH, there isn't a fully direct relationship.


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Suppose we have two different aqueous solutions, one being a strong acid and the other being a weak acid:


$$\ce{HCl(aq)<=>H+(aq) +Cl-(aq)},\quad \pu{K\_a}=10^6$$


$$\ce{HCOOH(aq)<=>H+(aq) +COOH-(aq)},\quad \pu{K\_a}=10^{-3.75}$$


The strength of an acid is a measure of to what extent it is dissociated into its constituent ions. The dissociation fraction $\alpha$ gives us an idea of this, and it is defined as:


$$\alpha=\frac{x}{C\_{\pu{Ao}}}$$


Where $x$ represents the equilibrium concentrations of $\ce{H^+}$ ions or conjugate base $\ce{B-}$ ions, assuming no initial amounts of either are present, and we can make the approximation:


$$\pu{pH}=\pu{-log}\;{C\_{\ce{H^+}}}\implies x=10^{\pu{-pH}}$$


The initial concentration $C\_{\pu{Ao}}$ can also be set in terms of $x$:


$$\pu{K\_a}=\frac{x^2}{C\_{\pu{Ao}}-x}\implies C\_{\pu{Ao}}=\frac{x^2}{\pu{K\_a}}+x$$


Substituting in $\alpha$:


$$\alpha=\frac{10^{\pu{-pH}}}{\frac{10^{\pu{-2pH}}}{\pu{K\_a}}+10^{\pu{-pH}}}=\frac{1}{\frac{10^{\pu{-pH}}}{\pu{K\_a}}+1}$$


We can observe that:


$$\pu{K\_a}\to\infty\implies \alpha\to1$$


$$\pu{K\_a}\to0\implies \alpha\to0$$


If we plot $\alpha$ vs $pH$ for $\ce{HCl}$ and $\ce{HCOOH}$, we get:


[![enter image description here](https://i.stack.imgur.com/pjzFy.png)](https://i.stack.imgur.com/pjzFy.png)


Now, let's suppose someone were to hand you an aqueous solution of an unknown acid with a known $\pu{pH}$ value of 2.


You might be tempted to conclude that since the $\pu{pH}$ is low, the acid must be a strong acid like $\ce{HCl}$ that is completely dissociated ($\alpha=1$); however, as shown in the plot, it could very well be a weak acid like $\ce{HCOOH}$ that is partially dissociated ($\alpha=0.017$).


In other words, $\pu{pH}$ alone is not a reliable criterion for gauging the strength of an acid. If it is coupled with $\pu{K\_a}$ however, it can be.


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