7.6.2. Calculating the pH of Polyprotic Acids

Polyprotic acids are acids that possess more than one acidic (ionizable) proton per molecule and can donate these protons in a stepwise manner. Examples include carbonic acid (H2 CO3 , diprotic), sulfuric acid (H2 SO4 , diprotic), and phosphoric acid (H3 PO4 , triprotic).

Polyprotic acids can release multiple protons; for instance, carbonic acid has two protons it can donate. This characteristic allows them to undergo several dissociation reactions, one after the other. Each step of dissociation gives rise to new chemical species and can affect the pH of the solution in different ways based on how many protons have been donated.

Think of polyprotic acids like a person with many balloons in their hands. Each balloon represents a proton. When she releases a balloon, it represents the donation of a proton. Depending on her strength (the concentration of the acid), she may let go of just one or multiple balloons at a time.

Each dissociation step for a polyprotic acid has its own unique acid dissociation constant (Ka).

• First Dissociation: H3 A⇌H++H2 A−(with Ka1)
• Second Dissociation: H2 A−⇌H++HA2−(with Ka2)
• Third Dissociation (if applicable): HA2−⇌H++A3−(with Ka3)

Each time a polyprotic acid donates a proton, it goes through a dissociation step that can be quantified by a specific constant, Ka. The first dissociation constant (Ka1) will usually be larger than the second (Ka2), and so on, because it’s harder to remove protons from species that are already negatively charged. This reflects the increasing stability of the remaining species as protons are lost.

Imagine you have a staircase. The first step is easy to step up, which is like the first dissociation of a polyprotic acid. As you keep stepping up (removing more protons), it gets steeper and harder, just like it gets more difficult to remove more protons as the charge on the species increases.

For most weak polyprotic acids, the first dissociation step contributes almost all of the H+ ions to the solution. The contribution from subsequent dissociation steps is usually negligible and can be ignored for typical pH calculations unless dealing with extremely dilute solutions or specific problems where exact concentrations of intermediate species are required.

In calculations, we often find that the first dissociation of a polyprotic acid has the most significant effect on the pH of the solution. This is because the majority of H+ ions produced during the dissociation process come from this first reaction, making it the most impactful in terms of acidity. The following steps release very few additional protons, making them less important for standard pH calculations.

Consider a playground slide. The first slide down (first proton donation) is steep and fast (lots of change in pH), but as you keep sliding down, each following slide becomes less thrilling or rapid (the second and third protons contribute less to the overall change in pH).

For Strong Polyprotic Acids (e.g., Sulfuric Acid, H2 SO4):

• Sulfuric acid is unique in that its first dissociation is strong: H2 SO4 →H++HSO4−. This means that in a solution of H2 SO4, the concentration of H+ initially produced is equal to the initial concentration of the acid.
• The second dissociation is weak (Ka2 =1.2×10−2): HSO4−⇌ H++SO42−. For accurate calculations, especially for more concentrated solutions, the H+ contributed from the second dissociation must be considered.

In the case of sulfuric acid, the first step of donating a proton is complete—meaning it acts almost as a strong acid would, providing H+ ions equal to its concentration. In practice, however, the second dissociation is much weaker, and for precise pH calculations in concentrated solutions, we take this into account as well by treating the second step like a weak acid equilibrium.

Imagine turning on a faucet full blast (first dissociation). You get a strong and steady flow of water (H+ ions). If you want a little more water (the second dissociation), you’ve got to wait for it to trickle out (the contribution is lesser, more akin to a weak flow).

For Weak Polyprotic Acids (e.g., Carbonic Acid, H2 CO3; Phosphoric Acid, H3 PO4):

• Given that Ka1 >> Ka2 (and Ka3), the pH calculation for a solution of a weak polyprotic acid is predominantly determined by the first dissociation step.

When dealing with weak polyprotic acids, the first dissociation step overwhelmingly determines the solution's pH. Subsequent protons released have a minuscule effect, allowing us to simplify calculations by treating the acid as if it were monoprotic and only using the Ka1 value.

Think of a library where only the first few books (H+ ions) are heavily browsed. After that, the remaining books are rarely picked up, just like how the impact of subsequent protons is negligible on the overall acidity. Thus, it makes sense to focus just on the first books when discussing the library's use.

Titration curves for polyprotic acids are distinct because they exhibit multiple equivalence points, each corresponding to the neutralization of one of the dissociable protons.

• A diprotic acid will show two equivalence points.
• A triprotic acid will show three equivalence points.

In a titration process, polyprotic acids show multiple shifts in pH as each proton is neutralized. These shifts correspond to distinct equivalence points on the titration curve where each of the protons is fully reacted with the base. The areas between these points represent buffer regions where pH changes gradually.

Think of climbing a multi-step staircase, where each step corresponds to a proton being removed. At each landing (equivalence point), you stop to take a breath before continuing to the next step. Similarly, as each proton is neutralized, there’s a noticeable change in the titration behavior, praising the complexity of this process.

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