Question

In any aqueous solution at \(25{ }^{\circ} \mathrm{C}\), the sum of \(\mathrm{pH}\) and \(\mathrm{pOH}\) is 14.0. Explain why this is so.

Short answer

At 25C, the sum of pH and pOH is 14 because it is derived from the self-ionization constant of water \( K_w \) which is \( 1 × 10^{-14} \) at that temperature, and the negative logarithms of \( [H^+] \) and \( [OH^-] \) concentrations define pH and pOH.

Step-by-step solution

Understanding the pH and pOH Relationship

The pH is a measure of the hydrogen ion concentration \( [H^+] \) in solution, and pOH is a measure of the hydroxide ion concentration \( [OH^-] \). In water, \( [H^+] \) and \( [OH^-] \) are related by the self-ionization constant \( K_w \) at a given temperature, which is \( 1 × 10^{-14} \) at \( 25{ }^{\boxed{{\text{}}}}C \). The relationship is described by the equation \( K_w = [H^+][OH^-] \) .

Defining pH and pOH

The pH is defined as \( pH = -\text{log}[H^+] \) and the pOH is defined as \( pOH = -\text{log}[OH^-] \) . By taking the negative logarithm of both sides of the \( K_w \) equation, we can connect these two measures.

Calculating the Sum of pH and pOH

Taking the negative logarithm of \( K_w \) equation yields \( -\text{log}(K_w) = -\text{log}([H^+][OH^-]) \) which simplifies to \( \text{pH} + \text{pOH} = \text{pK}_w \) . Since \( \text{pK}_w = -\text{log}(K_w) = -\text{log}(1 × 10^{-14}) = 14 \) at \( 25{ }^{\boxed{{\text{}}}}C \), the sum of pH and pOH for any aqueous solution at this temperature is always equal to 14.

Key concepts

pH Scale

The pH scale is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It ranges from 0 to 14, with 7 being neutral. A pH less than 7 indicates acidity, while a value greater than 7 indicates alkalinity. The pH value is derived from the hydrogen ion concentration \( [H^+] \) and is calculated using the formula \( pH = -\log[H^+] \). This expression tells us that as the hydrogen ion concentration increases, the pH decreases, making the solution more acidic.

It's essential to appreciate that each whole pH value below 7 is ten times more acidic than the next higher value. For example, a solution with a pH of 3 is ten times more acidic than one with a pH of 4. This is due to the logarithmic nature of the scale, which compresses a wide range of ion concentrations into a more manageable range of numbers.

pOH Scale

In parallel to the pH scale, the pOH scale also ranges from 0 to 14 and is a measure of the hydroxide ion concentration \( [OH^-] \) in an aqueous solution. The pOH is calculated using the formula \( pOH = -\log[OH^-] \). A low pOH means a high concentration of hydroxide ions, indicating a basic or alkaline solution. Conversely, a high pOH indicates an acidic solution with a low concentration of hydroxide ions.

The pOH scale helps us understand the relative strength of basic solutions in the same way that the pH scale does for acidic solutions. For instance, a solution with a pOH of 10 is ten times less basic (or more acidic) than a solution with a pOH of 9 because of the logarithmic scaling.

Self-ionization of Water

Water is a polar molecule that can self-ionize; that is, it can react with itself to form hydronium \( H_3O^+ \) and hydroxide \( OH^- \) ions. This process is also referred to as the self-dissociation or autoionization of water and is expressed by the equilibrium equation \( 2H_2O \leftrightarrow H_3O^+ + OH^- \). The equilibrium constant for this reaction is known as the ion-product constant of water, \( K_w \), and at \(25{ }^\circ\text{C}\), \( K_w \) is \( 1 \times 10^{-14} \).

This constant is critical because it holds at any pH and pOH value in pure water and ensures the balance between \( [H^+] \) and \( [OH^-] \) concentrations, contributing to water’s neutral pH of 7.

Hydrogen Ion Concentration

Understanding hydrogen ion concentration \( [H^+] \) is pivotal for discussing acidity. The molar concentration of hydrogen ions in solution determines how acidic the solution is. The more hydrogen ions present, the lower the pH and the more acidic the solution. Hydrogen ions are often written as \( H^+ \) but are typically found as hydronium ions \( H_3O^+ \) in solution.

Maintaining the correct balance of \( [H^+] \) is crucial for many biological and chemical processes. For instance, human blood has a tightly regulated pH level, usually around 7.4, which is maintained through several buffer systems that manage the hydrogen ion concentration.

Hydroxide Ion Concentration

Hydroxide ion concentration \( [OH^-] \) is the analogous base-centric concept to hydrogen ion concentration. It quantifies the number of hydroxide ions in a solution and influences its basicity or alkalinity. The pOH of a solution gives insight into the hydroxide ion concentration, with lower pOH values implying a higher \( [OH^-] \) and a more alkaline solution.

Just as \( [H^+] \) impacts biological systems, so does \( [OH^-] \) where its balance is necessary for many environmental systems such as ocean alkalinity, which influences marine life and ecological stability.