Calculate \(\left[\mathrm{OH}^{-}\right]\)given \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)in each aqueous solution and classify the solution as acidic or basic. (a) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.3 \times 10^{-3} \mathrm{M}\) (b) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=9.1 \times 10^{-12} \mathrm{M}\) (c) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=5.2 \times 10^{-4} \mathrm{M}\) (d) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=6.1 \times 10^{-9} \mathrm{M}\)

Understanding the ion product of water is crucial for students delving into the field of acid-base chemistry. At 25°C, the ion product of water, denoted as \( K_w \), always equals \(1.0 \times 10^{-14} M^2\). This constant represents the equilibrium concentration of hydrogen ions, \( [H_3O^+] \), and hydroxide ions, \( [OH^-] \), in pure water. In mathematical terms, \( K_w = [H_3O^+][OH^-] \).

For example, if the concentration of \( [H_3O^+] \) in a solution increases, the equilibrium shifts to decrease the concentration of \( [OH^-] \) to maintain the constant ion product. Conversely, if the concentration of \( [H_3O^+] \) decreases, the concentration of \( [OH^-] \) increases. This is because the value of \( K_w \) must stay constant, which is a key concept in determining the nature of a solution—whether it is acidic, neutral, or basic.

Calculating the pH and pOH of a solution can help students classify the acidity or basicity of that solution. The pH scale is a measure of the hydrogen ion concentration in a solution, while pOH gives us the hydroxide ion concentration. Both are derived from the molarity (M) of the ions:

\( pH = -\text{log}_{10}[H_3O^+] \), and \( pOH = -\text{log}_{10}[OH^-] \).

The sum of pH and pOH is always 14, reflecting the relationship to the ion product of water, \( K_w \). This sum is derived from the logarithmic relationship with \( K_w \): \( pH + pOH = 14 \). For instance, in a neutral solution at 25°C, where \( [H_3O^+] = [OH^-] \), both pH and pOH equal 7. If the pH is below 7, the solution is acidic, and if it is above 7, the solution is basic. By calculating either pH or pOH, we can deduce the other and, hence, the character of the solution.

The classification of solutions into acidic or basic is an essential part of chemistry education. Acidic solutions have a higher concentrations of hydronium ions (\( [H_3O^+] \)) than hydroxide ions (\( [OH^-] \)), which leads to a pH value of less than 7. On the other hand, basic solutions have lower concentrations of hydronium ions and higher concentrations of hydroxide ions, resulting in a pH value greater than 7. An important point to remember is that the neutrality of a solution is rare and occurs when \( [H_3O^+] \) and \( [OH^-] \) are both equal at a concentration of \(1.0 \times 10^{-7} M\), leading to a neutral pH of 7.

The original exercise provided is a prime example of identifying solution character. In all cases, students must calculate the concentration of hydroxide ions using the ion product of water and the given hydronium ion concentration. Once known, the comparison of these two ion concentrations helps ascertain whether the solution is acidic or basic. This is a foundational skill in chemistry that paves the way for understanding the behavior of acids and bases in various contexts.