Question

When calculating the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) for a polyprotic acid, the second ionization step can often be neglected. Explain why this statement is valid.

Short answer

The second ionization step can often be neglected because it has a much smaller ionization constant than the first, leading to a minimal increase in \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration.

Step-by-step solution

Understanding Polyprotic Acids

A polyprotic acid is an acid that can donate more than one proton (H+) per molecule in a stepwise manner. The ionization of a polyprotic acid occurs in multiple steps, each with its own ionization constant.

Examining the First Ionization Step

In the first ionization step, the acid donates a proton and forms its conjugate base. This step typically has the highest ionization constant (Ka1), indicating it proceeds to the largest extent compared to subsequent ionization steps.

Considering the Second and Subsequent Ionization Steps

Subsequent ionization steps involve the conjugate base formed in the previous step donating a proton. These steps have much smaller ionization constants (Ka2, Ka3, etc.), meaning they occur to a lesser extent and contribute much less to the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) in solution.

Ignoring the Lesser Contributions

Because the Ka values of the second and subsequent ionizations are significantly smaller, the concentrations they contribute to \(\mathrm{H}_{3} \mathrm{O}^{+}\) are comparatively negligible. Therefore, for most practical calculations, considering only the first ionization is sufficient.

Key concepts

Ionization Constant

The term 'ionization constant', often represented as Ka, quantifies the strength of an acid in solution. It details how completely an acid can donate its protons to water, hence creating hydronium ions \(\mathrm{H}_3\mathrm{O}^+\) and its conjugate base. Stronger acids have higher Ka values, meaning they ionize more completely. Polyprotic acids, those that donate more than one proton, have separate ionization constants for each proton donation step.

For a polyprotic acid, the first ionization constant (Ka1) is typically highest, as the initial proton is the easiest to remove. Following ionization steps have much lower constants (Ka2, Ka3, etc.), reflecting the decreased tendency to ionize further. Why? The successive removal of protons becomes increasingly difficult, as the conjugate base has a reduced affinity for losing another positively charged particle.

Conjugate Base

When an acid donates a proton, it forms what's called a 'conjugate base'. This new species retains the ability to re-accept the proton, which highlights the reversible nature of acid-base reactions. For example, when hydrogen sulfate \(\mathrm{HSO}_4^-\) loses a proton, it becomes sulfate \(\mathrm{SO}_4^{2-}\), its conjugate base.

The strength of the conjugate base is inversely related to the strength of the parent acid. Hence, a strong acid like sulfuric acid gives way to a relatively weak conjugate base, which is less likely to attract a proton back. As the ionization progresses in polyprotic acids, consecutive conjugate bases are formed, each step further decreasing in its tendency to act as an acid.

Proton Donation

In chemistry, the term 'proton donation' describes an acid's capacity to release a proton, which is essentially a hydrogen nucleus consisting of a single positive charge. The proton donation process is what classifies a substance as an acid according to the Brønsted-Lowry theory. For polyprotic acids, this happens in stages: one proton is donated at a time, and each event has varying degrees of likelihood.

The initial proton is the most readily donated, as it usually requires less energy and is less reliant on the surrounding chemical environment. Subsequent protons are released with more difficulty due to the increasingly negative charge of the conjugate base, which naturally repels the positive proton.

Acid Dissociation Constant

The acid dissociation constant, Ka, is related to the ionization constant and it is a specific value that measures the propensity of a particular acid to dissociate in water. This dissociation results in the formation of hydronium ions \(\mathrm{H}_3\mathrm{O}^+\) and a conjugate base. The larger the Ka value, the stronger the acid, and vice versa. For polyprotic acids, each dissociation event has its own Ka value because each proton donation presents different stability factors and energy requirements.

When comparing the sequential Ka values of a polyprotic acid, you'll find that Ka1 > Ka2 > Ka3, etc. This reflects the decreasing acidity and the lesser contribution to the overall \(\mathrm{H}_3\mathrm{O}^+\) concentration in solution as one moves through the ionization steps. This underpins the justification given in the original exercise for generally ignoring the second and subsequent ionizations in practical applications.