Question

Explain why \(\left[\mathrm{PO}_{4}^{3-}\right]\) in \(1.00 \mathrm{M} \mathrm{H}_{3} \mathrm{PO}_{4}\) is not simply \(\frac{1}{3}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right],\) but much, much less than \(\frac{1}{3}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)

Short answer

The [PO4^3-] is not simply 1/3 [H3O+] because H3PO4 is a polyprotic acid and undergoes stepwise dissociation. Each additional step of dissociation becomes less likely, resulting in a much lower concentration of PO4^3-, the product of the third dissociation step, compared to the H3O+ concentration.

Step-by-step solution

Understanding phosphoric acid and its dissociation

Phosphoric acid, H3PO4, is a polyprotic acid. This means it has more than one hydrogen atom it can lose when it donates a proton. The dissociation of phosphoric acid is a three step process, each step representing the donation of one proton:\n1. H3PO4 -> H2PO4^- + H+2. H2PO4^- -> HPO4^2- + H+3. HPO4^2- -> PO4^3- + H+It should be noted that each step above is an equilibrium reaction and has its own equilibrium constant. Thus, the dissociation process is not complete and all species exist simultaneously in the solution.

Why is [PO4^3-] not simply 1/3 [H3O+] ?

It is tempting to believe that the concentration of PO43- would be one third the concentration of H3O+ since phosphoric acid 'donates' three protons. However, this is not the case. The reason lies in the stepwise dissociation process where each step becomes less likely as the phosphate ion becomes more negatively charged and therefore less likely to donate a proton. This results in a lesser concentration of PO43- compared to H3O+.

Understanding of pH and dissociation steps

This can be further understood by considering pH. For most acids, the second and third dissociations do not greatly contribute to [H3O+]. Therefore, even as [H3O+] might be high, [PO4^3-] (which is product of third dissociation step) is much less. Hence, [PO4^3-] is not simply 1/3 [H3O+], but much, much less than 1/3 [H3O+].

Key concepts

Polyprotic Acids

Polyprotic acids are unique because they can donate more than one proton (H⁺) per molecule. This characteristic significantly affects their dissociation in solution. Each proton is lost in a stepwise manner, which means the dissociation into ions happens in stages. For example, phosphoric acid (H₃PO₄) is a common polyprotic acid. It dissociates in three distinct steps: first into dihydrogen phosphate (H₂PO₄⁻), then into hydrogen phosphate (HPO₄²⁻), and finally into phosphate (PO₄³⁻).

This stepwise dissociation affects the acid's concentration and equilibrium. As each proton is donated, the acid becomes more negatively charged, making it increasingly difficult to lose more protons. That's why, in a solution of 1.00 M H₃PO₄, the [PO₄³⁻] is not simply a third of [H₃O⁺], but much less. The stepwise nature of polyprotic acids is key to understanding their behavior in solutions.

Dissociation Constant

The dissociation constant, often symbolized as Ka, refers to the equilibrium constant for the dissociation of an acid in water. For polyprotic acids, there is a separate dissociation constant for each step of proton donation. In the case of phosphoric acid (H₃PO₄), it has three dissociation constants: Ka₁, Ka₂, and Ka₃, each corresponding to one step of dissociation.

Each subsequent dissociation step typically has a smaller dissociation constant, indicating it becomes harder for the acid to further lose a proton. This is due to the increased negative charge after each proton is lost. For instance, in H₃PO₄, the first dissociation has a relatively high Ka, making it likely to occur, while the third has a much smaller Ka, making the formation of PO₄³⁻ much less favorable. These decreasing dissociation constants are the reason why the concentration of PO₄³⁻ is much less than might be naively expected in solutions of phosphoric acid.

Equilibrium Reactions

Equilibrium reactions are fundamental in understanding how acids dissociate in solution. An equilibrium is reached when the rate of the forward reaction (acid dissociating into ions) equals the rate of the reverse reaction (ions reforming the acid). This concept is crucial for polyprotic acids, where each dissociation step has its own equilibrium.

For phosphoric acid, each proton's dissociation forms an equilibrium that determines the concentration of ions like H₂PO₄⁻, HPO₄²⁻, and PO₄³⁻ in solution. The forward dissociation reaction is less favored as the molecule becomes more negative, influencing each equilibrium position. As a result, while the concentrations of H₂PO₄⁻ and HPO₄²⁻ might be substantial at equilibrium, the concentration of PO₄³⁻ remains low. Understanding these equilibria helps explain why PO₄³⁻ is not simply a third of the hydronium ion concentration, despite the acid's ability to donate three protons.