Question

The hydronium ion concentration in an aqueous solution of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) is \(2.0 \times 10^{-4} \mathrm{M}\) at \(25^{\circ} \mathrm{C}\). The hydroxide ion concentration in the solution is (a) 0 (b) \(2.0 \times 10^{-4} \mathrm{M}\) (c) \(5 \times 10^{3} \mathrm{M}\) (d) \(5 \times 10^{-11} \mathrm{M}\)

Short answer

The hydroxide ion concentration in the solution is \(5 \times 10^{-11} \mathrm{M}\).

Step-by-step solution

Understand the Relationship Between Hydronium and Hydroxide Ions

At 25 degrees Celsius, the ion-product constant for water \(K_w\) is always \(1.0 \times 10^{-14}\). According to the formula \(K_w = [\text{H}_3\text{O}^+] \times [\text{OH}^-]\), where \[\text{H}_3\text{O}^+\] is the concentration of hydronium ions and \[\text{OH}^-\] is the concentration of hydroxide ions.

Calculate Hydroxide Ion Concentration Using \(K_w\)

Rearrange the formula to solve for the hydroxide ion concentration: \[\text{OH}^-\] = \(\frac{K_w}{[\text{H}_3\text{O}^+]}\). Substitute the known values: \[\text{OH}^-\] = \(\frac{1.0 \times 10^{-14}}{2.0 \times 10^{-4}}\).

Perform the Division to Find the Hydroxide Ion Concentration

Divide the given values to obtain the hydroxide ion concentration: \(\text{OH}^-\) = \(\frac{1.0 \times 10^{-14}}{2.0 \times 10^{-4}}\) = \(5.0 \times 10^{-11} \text{M}\).

Key concepts

Ion-Product Constant for Water

Understanding the ion-product constant for water, denoted as \(K_w\), is crucial when studying acid-base chemistry. This constant represents the equilibrium condition for the self-ionization of water, a process in which water molecules dissociate into hydronium ions \(\text{H}_3\text{O}^+\) and hydroxide ions \(\text{OH}^-\). At a temperature of 25 degrees Celsius, \(K_w\) has a value of \(1.0 \times 10^{-14}\). This property of water is fundamental because it establishes a fixed relationship between the concentration of hydronium ions and hydroxide ions in any aqueous solution.

When working with \(K_w\), it's important to remember that this constant applies to all aqueous solutions, regardless of whether they are acidic, basic, or neutral. The equilibrium suggests that an increase in the concentration of hydronium ions will correspond to a decrease in the concentration of hydroxide ions, and vice versa, keeping the product of their concentrations at this constant value.

Hydronium Ion Concentration

Hydronium ion concentration in a solution is a measure of its acidity. It refers to the amount of hydronium ions present, typically expressed in moles per liter (Molarity, M). In acid-base reactions, the hydrogen ions \(\text{H}^+\) are often depicted as hydronium ions \(\text{H}_3\text{O}^+\) because hydrogen ions tend to associate with water molecules in the solution. The more hydronium ions in a solution, the more acidic the solution is perceived to be.

When calculating the pH of a solution—a measure of its acidity or basicity—the hydronium ion concentration is the key value. The pH is inversely related to the concentration of hydronium ions: a high hydronium ion concentration means a lower pH and a more acidic solution.

Chemical Equilibrium

Chemical equilibrium is a state in which the rates of the forward and reverse reactions in a chemical process are equal, leading to no net change in the concentration of the reactants and products over time. This does not mean the reactions have stopped, but rather that they are proceeding at the same rate in both directions. The concept of equilibrium is essential in acid-base reactions, including the dissociation of water into hydronium and hydroxide ions.

At equilibrium, the concentrations of these ions are stable, and their product is equal to the ion-product constant for water, \(K_w\). Understanding equilibrium concepts allows us to predict the behavior of reactants and products under various conditions and is fundamental in calculating ion concentrations.

Acid-Base Chemistry

Acid-base chemistry is a central theme in chemistry that deals with the substances that donate and accept protons. An acid is a substance that donates hydrogen ions in solution, while a base is a substance that accepts them. The strength of an acid or a base is determined by its ability to donate or accept hydrogen ions, respectively.

In water, a strong acid will almost completely dissociate into its ions, leading to a high concentration of hydronium ions. Conversely, a strong base will dissociate to produce a high concentration of hydroxide ions. The pH scale is used to quantify the acid or base strength of a solution, with values ranging from 0 to 14. A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic solution. The ion-product constant for water \(K_w\) is also an integral part of acid-base chemistry, as it relates the hydronium and hydroxide ion concentrations in any aqueous solution.