Arrhenius Theory of Acids and Bases

Remember that Arrhenius theory was developed for inorganic compounds. Every inorganic compound can be categorized into one of the following groups:

• oxides
• acids
• bases
• salts
• other binary compounds

Every inorganic compound can be thought of as consisting of a positive ion (cation) and a negative ion (anion): let’s denote it as CatAn. Or we could say ${\text{Cat}}_x{\text{An}}_y$, meaning that the charges of our ions are: ${\text{Cat}}^{y+}$ and ${\text{An}}^{x-}$. It’s taken that almost any inorganic compound, when dissolved in water (if it can be dissolved), dissociates into ions (corresponding to cations and anions).

fun fact: the idea that compounds dissociate into ions is the subject of the 1903 Nobel Prize in Chemistry given to Svante Arrhenius.

Within Arrhenius theory:

• an acid is a compound for which the cation is $\mathrm{H}{}^{+}$. Plain and simple. Examples: $\mathrm{HCl},\mathrm{HBr},\mathrm{HI},\mathrm{H}{}_2\mathrm{SO}{}_4,\mathrm{HClO}{}_4,\mathrm{HClO}{}_3,\mathrm{HNO}{}_3$ and $\mathrm{H}{}_2\mathrm{SO}{}_3,\mathrm{HNO}{}_2,\mathrm{H}{}_3\mathrm{PO}{}_4,\mathrm{CH}{}_3\mathrm{COOH},\mathrm{H}{}_2\mathrm{CO}{}_3,\mathrm{HClO}{}_2,\mathrm{HClO}$.
• a base is a compound for which the anion is $\mathrm{OH}{}^{-}$. Again, dead simple. Examples: $\mathrm{LiOH},\mathrm{NaOH},\mathrm{KOH},\mathrm{Ba}(\mathrm{OH}){}_2,\mathrm{Ca}(\mathrm{OH}){}_2$ and $\mathrm{NH}{}_4\mathrm{OH},\mathrm{Mg}(\mathrm{OH}){}_2,\mathrm{Al}(\mathrm{OH}){}_3,\mathrm{Fe}(\mathrm{OH}){}_2,\mathrm{Be}(\mathrm{OH}){}_2$.
• a salt is a compound for which the cation is not $\mathrm{H}{}^{+}$ and the anion is not $\mathrm{OH}{}^{-}$. Another way to look at it — salt forms when a base reacts with an acid. examples: $\mathrm{NaCl},\mathrm{K}{}_2\mathrm{SO}{}_4,\mathrm{LiNO}{}_3$.

If the cation is $\mathrm{H}{}^{+}$ and the anion is $\mathrm{OH}{}^{-}$, this is our friend water $\mathrm{H}{}_2\mathrm{O}$.

Arrhenius Acids

Any compound which produces $\mathrm{H}{}^{+}$ upon dissociation is an Arrhenius acid. I find it amusing that the formal definition of an acid by Arrhenius is “the compound which increases the concentration of $\mathrm{H}{}^{+}$ ions in solutions.” The amusing part is the lack of any information about the cause of such effect, the focus is on what happens and what we can observe, not propositions about the structure of molecules themselves. Anyway, let’s take any two acids and write the equations of their dissociation.

$$\mathrm{HCl}\to \mathrm{H}{}^{+}+\mathrm{Cl}{}^{-}\text{\hspace{1em}(1)}$$

and:

$$\mathrm{HNO}{}_2\to \mathrm{H}{}^{+}+\mathrm{NO}{}_2{}^{-}\text{\hspace{1em}(2)}$$

it turns out that some acids dissociate fully and some do not. As a result, if we take an initial concentration of the acid as $c_0=0.1\mathrm{M}$, in the first case, we’ll end up with $[\mathrm{H}{}^{+}]=0.1$ and $[\mathrm{HCl}]=0$. In the second case, we’ll end up with $[\mathrm{H}{}^{+}]<0.1$ and $[\mathrm{HNO}{}_2]>0$. You’ll be surprised, but within Arrhenius theory, we cannot make any further predictions! We can’t say how much of $[\mathrm{H}{}^{+}]$ we will have exactly.

What scientists usually did, at the time, to compare the strengths of two acids was to take solutions with equal concentration, and then measure the electric conductivity of each. The electrical current is conducted by ions, so the more ions we have, the higher the conductivity. As a result, you can measure the concentration of ions by measuring electrical conductivity of the solution.

As a result, you can take a solution of 0.1 M HNO2 and find that the concentration of $\mathrm{H}{}^{+}$ is around 0.008 M. Knowing stoichiometry, you presume there are 0.008 M of $\mathrm{NO}{}_2{}^{-}$ ions. I hope you agree that if you never knew the equilibrium constant, the most intuitive relative metric you could compute using this data is the fraction of acid that is dissociated (we call it $\alpha$): i.e. 0.008M/0.1M which gives you ~8%. It turned out that this number depended on the initial concentration of the acid! (particularly, the $[\mathrm{A}{}^{-}]\propto \sqrt{c_0}$) So if you wanted to calculate $[\mathrm{H}{}^{+}]$ of 0.1 M HNO2, you’d have to look up some reference table and see that the 8% of the acid dissociates, so $[\mathrm{H}{}^{+}]=0.008\mathrm{M}$

Quite inconvenient, but it’s the best we could do by early 1890s. We could, then, empirically observe that the following acids dissociate fully: $\mathrm{HCl},\mathrm{HBr},\mathrm{HI},\mathrm{H}{}_2\mathrm{SO}{}_4,\mathrm{HClO}{}_4,\mathrm{HClO}{}_3,\mathrm{HNO}{}_3$, while the following dissociate only partially: $\mathrm{HF},\mathrm{H}{}_2\mathrm{SO}{}_3,\mathrm{HNO}{}_2,\mathrm{H}{}_3\mathrm{PO}{}_4,\mathrm{CH}{}_3\mathrm{COOH},\mathrm{H}{}_2\mathrm{CO}{}_3,\mathrm{HClO}{}_2,\mathrm{HClO}$. We then define strong acids as those which dissociate fully, while weak acids as those which dissociate partially.

Arrhenius Bases

Just as acids, bases dissociate to form some cation and a $\mathrm{OH}{}^{-}$ anion.

$$\mathrm{NaOH}\to \mathrm{Na}{}^{+}+\mathrm{OH}{}^{-}\text{\hspace{1em}(3)}$$

or

$$\mathrm{NH}{}_4\mathrm{OH}\to \mathrm{NH}{}_4{}^{+}+\mathrm{OH}{}^{-}\text{\hspace{1em}(4)}$$

again, some bases dissociate fully, like $\mathrm{LiOH},\mathrm{NaOH},\mathrm{KOH},\mathrm{Ba}(\mathrm{OH}){}_2,\mathrm{Ca}(\mathrm{OH}){}_2$ and they’re called strong bases.

Some bases are insoluble in water, so they physically can’t dissociate and form ions, and we call them weak: $\mathrm{Mg}(\mathrm{OH}){}_2,\mathrm{Al}(\mathrm{OH}){}_3,\mathrm{Fe}(\mathrm{OH}){}_2,\mathrm{Be}(\mathrm{OH}){}_2$.

There’s one more special base - $\mathrm{NH}{}_4\mathrm{OH}$. Within Arrhenius theory it’s considered weak. Essentially any soluble base is strong except for one base: $\mathrm{NH}{}_4\mathrm{OH}$.

Water

Water is a special molecule. When it dissociates, it forms $\mathrm{H}{}^{+}$ and $\mathrm{OH}{}^{-}$, so under Arrhenius theory it can be considered both as an acid and a base.

$$\mathrm{H}{}_2\mathrm{O}\to \mathrm{H}{}^{+}+\mathrm{OH}{}^{-}$$

Salts

When any Arrhenius acid reacts with an Arrhenius base, you get a salt. You can even write a general formula or a recipe for such reaction:

$$\mathrm{Cat}\mathrm{OH}+\mathrm{H}\mathrm{An}=\mathrm{Cat}\mathrm{An}+\mathrm{H}{}_2\mathrm{O}$$

or, if you want to introduce non-unitary charges on ions:

$$x\mathrm{Cat}(\mathrm{OH}){}_y+y\mathrm{H}{}_x\mathrm{An}=\mathrm{Cat}{}_x\mathrm{An}{}_y+\mathrm{x}\cdot y\mathrm{H}{}_2\mathrm{O}$$

you can replace Cat with $\mathrm{Na}{}^{+},\mathrm{Li}{}^{+},\mathrm{K}{}^{+},\mathrm{NH}{}_4{}^{+},\mathrm{Ba}{}_2{}^{+},\mathrm{Ca}{}_2{}^{+}$ and An with $\mathrm{Cl}{}^{-},\mathrm{Br}{}^{-},\mathrm{SO}{}_4{}^{2-},\mathrm{NO}{}_3{}^{-}$.