When Equilibrium meets Arrhenius

Arrhenius Theory of Acids and Bases was a qualitative theory. The most you could say is the extent to which an acid or a base dissociates. You couldn’t even compare the “acidity” of acids to that of bases, as there was no universal scale!

The quantitative aspect came when people started to apply ideas of equilibrium to the Arrhenius theory of dissociation. I actually spent 4 hours trying to pinpoint exactly who and when did this, and I didn’t find anything! It’s as if the notion of Ka somehow magically appeared and no one knows exactly when did it happen.

The Definition of Ka

Today, when people talk about an acid dissociation, we usually write an equation of the type:

$$\mathrm{HA}+\mathrm{H}{}_2\mathrm{O}\rightleftharpoons \mathrm{A}{}^{-}+\mathrm{H}{}_3\mathrm{O}{}^{+}\text{\hspace{1em}(1)}$$

and then say, the measure of acidity is the acidity constant $K_a$ defined as:

$$K_a=\frac{[\mathrm{H}{}_3\mathrm{O}{}^{+}][\mathrm{A}{}^{-}]}{[\mathrm{HA}]}\text{\hspace{1em}(2)}$$

which looks like an equilibrium constant, but it doesn’t have $[\mathrm{H}{}_2\mathrm{O}]$, and it appears as if you’re suggested to have fun figuring out why. You could rationalize it by noting that quite often you deal with dilute solutions, so at most you start with $1.0\mathrm{M}$ of an acid, and way more often you deal with concentrations like $5\cdot 10^{-4}\mathrm{M}$. If you have a 1L of such solution, you could say that you have around 1 L of water molecules, or 1000 g, or 55.(5) moles, meaning that $[\mathrm{H}{}_2\mathrm{O}{]}_0=55.55\mathrm{M}$. Even if an acid dissociates fully, the water concentration doesn’t change significantly, so we can treat it as almost unchanging: $[\mathrm{H}{}_2\mathrm{O}]\approx [\mathrm{H}{}_2\mathrm{O}{]}_0$, and so even if we write the full equilibrium constant:

$$K=\frac{[\mathrm{H}{}_3\mathrm{O}{}^{+}][\mathrm{A}{}^{-}]}{[\mathrm{HA}][\mathrm{H}{}_2\mathrm{O}]}\text{\hspace{1em}(3)}$$

we might notice that the product $K[\mathrm{H}{}_2\mathrm{O}]$ will be almost a constant, so we might as well define it to be $K_a$.

However, I think this is modern day rationalization and although I couldn’t find proof of it, I think the definition of $K_a$ is a historical artifact. For almost 40 years between Arrhenius and Bronsted-Lowry, the equation (1) was written as:

$$\mathrm{HA}\rightleftharpoons \mathrm{H}{}^{+}+\mathrm{A}{}^{-}\text{\hspace{1em}(4)}$$

It wasn’t until 1920s, when Bronsted and Lowry recognized that naked $\mathrm{H}{}^{+}$ actually doesn’t exist in a solution, and instead of acids undergoing dissociation, what happens instead is the proton transfer reaction with water. More on that in Brønsted-Lowry Acid-Base Theory.

If you look at (4), it’s very natural to think that the equilibrium constant is exactly $K_a$! And the rationalization through the point of unchanging $[\mathrm{H}{}_2\mathrm{O}]$ may be a reason why people used the wrong definition of an equilibrium and didn’t have a lot of issues with it!

Revisiting Water

Even Arrhenius noted that $\mathrm{H}{}_2\mathrm{O}$ is a special molecule in a sense that it behaves both as an acid and as a base:

$$\mathrm{H}{}_2\mathrm{O}\rightleftharpoons \mathrm{H}{}^{+}+\mathrm{OH}{}^{-}\text{\hspace{1em}(5)}$$

You could write a $K_a$ for this reaction:

$$K_a=\frac{[\mathrm{H}{}^{+}][\mathrm{OH}{}^{-}]}{[\mathrm{H}{}_2\mathrm{O}]}\text{\hspace{1em}(6)}$$

and again, I couldn’t find out when and who exactly, but at some point, I presume around the time of Sorensen, people noted (experimentally), that the product of concentrations $[\mathrm{H}{}^{+}][\mathrm{OH}{}^{-}]$ is a constant value. Which, given the form of (6) and given our arguments about the apparent lack of change in $[\mathrm{H}{}_2\mathrm{O}]$ makes sense: nothing stops us from noting that $K_a[\mathrm{H}{}_2\mathrm{O}]$ is also a constant:

$$K_a[\mathrm{H}{}_2\mathrm{O}]=[\mathrm{H}{}^{+}][\mathrm{OH}{}^{-}]\equiv K_w\text{\hspace{1em}(7)}$$

which we define to be $K_w$ (the water constant). Experimentally, we figured out that $K_w$ is precisely $10^{-14}$ (quite spectacular, tbh)

It’s actually surprising that it’s so hard to find who was the first person to observe this, as it’s a really significant turning point. Only when we have equation 7 can we now compare acids to bases: if we know $[\mathrm{OH}{}^{-}]$, we can always find $[\mathrm{H}{}^{+}]$, and so now we seem to have a unified scale for acidity: the concentration of $[\mathrm{H}{}^{+}]$.

Introducing pH

You might almost immediately notice, that the $[\mathrm{H}{}^{+}]$ may have extraordinarily different values. A 0.1 M solution of a strong acid will have $[\mathrm{H}{}^{+}]=0.1\mathrm{M}$, while a 0.1 M solution of a strong base will have $[\mathrm{OH}{}^{-}]=0.1\mathrm{M}$, which, from (7) means that $[\mathrm{H}{}^{+}]=1\cdot 10^{-13}\mathrm{M}$. What a range!

It’s a bit difficult to understand numbers that range over several orders of magnitude, plus it’s inconvenient to say “eight times ten to the power of negative three molar”, and much easier to say that something is equal to 4.8. As a result, we switch to a logarithmic scale. Let’s make a small table:

$[\mathrm{H}{}^{+}]$	$1.00$	$1.00\times 10^{-1}$	$1.00\times 10^{-3}$	$1.00\times 10^{-7}$	$1.00\times 10^{-12}$
$\log [\mathrm{H}{}^{+}]$	0	-1	-3	-7	-12
$-\log [\mathrm{H}{}^{+}]$	0	1	3	7	12

Simply try to verbally convey any number from the first row and then repeat it for the second row. Much simpler. If you’re into these kind of things, we’re mathematically justified in switching to a logarithmic scale because log is a monotonous function. Also, we add a negative sign only because concentrations are smaller than 1 much more often than bigger than 1, so most log values will be negative. We can drop the inconvenience of saying “negative” each time by just taking negative log. Finally, we use base 10 for our logarithms simply for convenience of making estimations: if we have 3 (or 3.3, or 3.1) on log value, it’s fairly straightforward to understand that we’re talking about something bigger than a 1000 (it’s much harder to evaluate any other base raised to an arbitrary power).

Bottom line we define pH as the negative of log (base 10) of $[\mathrm{H}{}^{+}]$.

Introducing pOH

When we talk about bases, a significant metric of their presence is the concentration of $[\mathrm{OH}{}^{-}]$. So we might characterize the amount of OH present by introducing the negative log of it and call it pOH. Just as we did with pH.

Connecting pH and pOH

Let’s do some algebraic manipulations with (7), particularly, take the negative log of it:

$$\begin{array}{rl}[\mathrm{H}{}^{+}][\mathrm{OH}{}^{-}]& =10^{-14}\\ -\log ([\mathrm{H}{}^{+}][\mathrm{OH}{}^{-}])& =-\log 10^{-14}\\ -\log [\mathrm{H}{}^{+}]-\log [\mathrm{OH}{}^{-}]& =-(-14)\\ pH+pOH& =14\end{array}$$

This establishes a pretty convenient way of calculating pH if you know pOH or vice versa.

Introducing pKa

For the exact same reason, instead of working with Ka values, we take the negative log and call it pKa. Whenever you see “p” - that means the negative log of some value.