Why is the hydronium ion concentration in a solution that is \(0.10 \mathrm{M}\) in \(\mathrm{HCl}\) and \(0.10 \mathrm{M}\) in HCOOH determined by the concentration of HCl?

The term 'acid strength' refers to an acid's ability to donate protons (H+) when it is in an aqueous solution. This characteristic directly influences the concentration of hydronium ions ((H_3O^+)), which is a central concept in understanding an acid's behavior in water. Acid strengths vary widely across different substances, and they are generally categorized into two groups: strong acids and weak acids.

Strong acids, such as hydrochloric acid (HCl), ionize completely in water, releasing all their available protons and thus producing a high concentration of hydronium ions. This complete ionization is why the strength of HCl in a solution can be gauged by its initial concentration. On the other hand, weak acids like formic acid (HCOOH) only partially ionize, resulting in a relatively lower and limited release of protons into the solution.

The strength of an acid can be quantitatively described by its acid dissociation constant (Ka). The Ka value offers insight into the extent of ionization that occurs--the higher the Ka, the stronger the acid, with complete ionization reflected by a significantly large Ka value. Thus, understanding acid strength is crucial for determining the hydronium ion concentration in a mixture of acids.

Acid dissociation is a chemical process where an acid releases protons ((H^+)) into an aqueous solution, which subsequently combine with water molecules to form hydronium ions ((H_3O^+)). The dissociation is reversible and is governed by an equilibrium between the undissociated acid and the ions produced.

The equilibrium position, hence the extent of acid dissociation, is visually represented by the ionization reaction, such as\[\text{HA } \rightleftharpoons \text{ H}^+ + \text{ A}^-\]
where HA is the acid and A- is the conjugate base. The acid dissociation constant (Ka) is an expression of this equilibrium and is calculated using the concentrations of the reactants and products at equilibrium:\[K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\]
For a weak acid, because not all molecules dissociate, the concentration of HA remains significant, and the value of Ka is relatively small. In contrast, a strong acid has a larger Ka, indicating a more extensively shifted equilibrium toward the products, thus releasing more H+ and leading to higher hydronium ion concentration.

The classification of acids into 'strong' and 'weak' is pivotal for predicting the behavior of acids in solution, particularly when determining the hydronium ion concentration. Strong acids, like hydrochloric acid (HCl), dissociate completely in solution, meaning they donate all their protons to water, forming hydronium ions efficiently. Therefore, the concentration of hydronium ions in a solution of a strong acid will equal the original concentration of the acid itself.\[\text{HCl} \rightarrow \text{H}^+ + \text{Cl}^-\]
Conversely, weak acids, such as formic acid (HCOOH), do not ionize fully. They establish an equilibrium between the un-ionized acid and the ions in solution, resulting in fewer hydronium ions compared to a strong acid at the same concentration. Their dissociation can be represented as:\[\text{HCOOH} \rightleftharpoons \text{H}^+ + \text{HCOO}^-\]
Due to this significant difference in dissociation, in a mixture of a strong and weak acid at equivalent concentrations, the strong acid dominates the hydronium ion composition. Consequently, when evaluating a solution that includes both HCl and HCOOH, the primary contributor to the hydronium ion concentration is the fully dissociating strong acid, HCl, and the contribution of HCOOH can often be disregarded for practical purposes.