Question

How is \(K_{\mathrm{w}}\) defined, and what is its numerical value at \(25^{\circ} \mathrm{C}(298 \mathrm{~K}) ?\)

Short answer

\(K_{\mathrm{w}}\) is the ion-product constant of water, numerically \(1.0 \times 10^{-14}\) at \(25^{\circ} \mathrm{C}\).

Step-by-step solution

Understanding the Ion-Product Constant of Water

The ion-product constant of water, denoted as \(K_{\mathrm{w}}\), represents the equilibrium constant for the self-ionization of water. This involves the dissociation of water molecules into hydrogen ions (\(\mathrm{H}^+\)) and hydroxide ions (\(\mathrm{OH}^-\)). The equation for this process is \[2\, \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{H}_3\mathrm{O}^+ + \mathrm{OH}^-.\] However, for simplicity, it is often represented as \(\mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{H}^+ + \mathrm{OH}^-\).

Representing the Expression for K_w

The expression for the ion-product constant \(K_{\mathrm{w}}\) is based on the concentrations of the ions produced from water. Mathematically, it is given as \[K_{\mathrm = \mathrm{[H}^+] \cdot \mathrm{[OH}^-]}\]. Here, \([\mathrm{H}^+]\) denotes the concentration of hydrogen ions and \([\mathrm{OH}^-]\) denotes the concentration of hydroxide ions in pure water at equilibrium.

Numerical Value at Standard Condition

At standard conditions, specifically at \(25^{\circ} \mathrm{C}\) or \(298 \mathrm{~K}\), the numerical value of \(K_{\mathrm{w}}\) is a constant and is approximately \(1.0 \times 10^{-14}\). This value indicates the product of concentrations \([\mathrm{H}^+][\mathrm{OH}^-]\) in pure water under these conditions.

Key concepts

Ion-Product Constant of Water

The ion-product constant of water, symbolized as \(K_{\mathrm{w}}\), is a critical concept in chemistry. It defines the balance of hydrogen ions (\(\mathrm{H}^+\)) and hydroxide ions (\(\mathrm{OH}^-\)) produced when water undergoes self-ionization. This constant is essential because it helps us understand how water maintains a neutral pH of 7 under standard conditions. The process of self-ionization involves the splitting of water into these ions, and \(K_{\mathrm{w}}\) quantifies this equilibrium.

• \(K_{\mathrm{w}}\) is a measure of water's ion activity.
• It is vital for assessing the acidity or basicity in aqueous solutions.
• At \(25^{\circ} \mathrm{C}\), \(K_{\mathrm{w}}\) is \(1.0 \times 10^{-14}\).

Understanding \(K_{\mathrm{w}}\) is fundamental when studying reactions in water, as it explains why changes in ion concentrations impact pH.

Self-Ionization of Water

Self-ionization is a fascinating natural process that occurs with water. In this process, two water molecules interact to form a hydronium ion (\(\mathrm{H}_3\mathrm{O}^+\)) and a hydroxide ion (\(\mathrm{OH}^-\)). This can also be simplified to one water molecule splitting into a hydrogen ion (\(\mathrm{H}^+\)) and a hydroxide ion (\(\mathrm{OH}^-\)). Even though this occurs infrequently, it is significant. It ensures that even pure water has a very low concentration of these ions, playing a crucial role in maintaining chemical balance.

• Self-ionization is responsible for the formation of ions in pure water.
• It is fundamental to why water can slightly conduct electricity.
• The presence of \(\mathrm{H}_3\mathrm{O}^+\) and \(\mathrm{OH}^-\) facilitates chemical reactions in biological and chemical systems.

This self-ionization establishes the baseline for understanding how solutions increase in acidity or basicity when other substances are dissolved in water.

Equilibrium Constant for Water

The equilibrium constant for water, represented by \(K_{\mathrm{w}}\), embodies the principle of chemical equilibrium within aqueous systems. Specifically, it dictates the precise concentration balance between \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) ions. At \(25^{\circ} \mathrm{C}\), this equilibrium constant remains steady at \(1.0 \times 10^{-14}\), reflecting the product of these ion concentrations at equilibrium. This value highlights water's innate ability to sustain a neutral pH and mediates shifts in balance when acids or bases are introduced.

• \(K_{\mathrm{w}}\) helps predict the direction and extent of chemical reactions in water.
• Affects the pH scale, aiming to maintain a stable equilibrium.
• Necessary for calculating concentrations of ions in different reactions.

The concept of \(K_{\mathrm{w}}\) underpins many biochemical processes, showcasing the importance of water as a medium for life and reactions.