Question

Given that \(K_{w}\) for water is \(10^{-13} M^{2}\) at \(62^{\circ} \mathrm{C}\), compute the sum of \(\mathrm{pOH}\) and \(\mathrm{pH}\) for a neutral aqueous solution at \(62^{\circ} \mathrm{C}\) : (a) \(7.0\) (b) \(13.30\) (c) \(14.0\) (d) \(13.0\)

Short answer

13.0

Step-by-step solution

Understanding the Concept of pH and pOH

The pH is a measure of the hydrogen ion concentration in a solution, while pOH is a measure of the hydroxide ion concentration. In a neutral solution, the concentrations of hydrogen ions and hydroxide ions are equal. For water, this is determined by the ion product constant for water, Kw, which at 62 degrees Celsius is given as 10^-13 M^2.

Define the Relationship Between pH, pOH, and Kw

The relationship is given by Kw = [H+][OH-], where [H+] represents the concentration of hydrogen ions and [OH-] represents the concentration of hydroxide ions. Taking the negative logarithm of both sides gives: pKw = pH + pOH.

Calculate pKw at 62 degrees Celsius

Using the given Kw value, pKw is calculated as: \( pKw = -\log(10^{-13}) = 13 \).

Determine the Sum of pH and pOH

In a neutral solution, pH and pOH are equal, so pH + pOH = pKw. Therefore, the sum is 13.

Key concepts

Ion Product Constant for Water

Understanding the ion product constant for water, often denoted as \(K_w\), is crucial when studying the chemistry of aqueous solutions. It is a unique constant at a given temperature that represents the equilibrium concentration of hydrogen ions (\(H^+\)) and hydroxide ions (\(OH^-\)) in pure water.

For water at any temperature, the relation can be expressed as \(K_w = [H^+][OH^-]\), where \([H^+]\) and \([OH^-]\) are the molar concentrations of hydrogen and hydroxide ions, respectively. At 25°C (298 K), \(K_w\) has a value of approximately \(1 \times 10^{-14} M^2\), which increases with temperature because the dissociation of water molecules is endothermic.

Why is \(K_w\) important? It's a fundamental principle that underpins calculations involving pH, pOH, and the acidity or basicity of a solution. In an exercise where you're given a different temperature – say, 62°C – it's important to note that the value of \(K_w\) changes, affecting the pH and pOH calculations for that particular scenario.

Neutral Aqueous Solution

A neutral aqueous solution is one where the concentrations of hydrogen ions and hydroxide ions are perfectly balanced. In other words, the molar concentration of \(H^+\) is equal to that of \(OH^-\).

In such a scenario, the pH and pOH values also reflect this equilibrium. Recall that pH is a measure of the hydrogen ion concentration and pOH is a measure of the hydroxide ion concentration. For a neutral solution at 25°C, the pH and pOH both equal 7, since \(10^{-7} M\) is the concentration of \(H^+\) and \(OH^-\) in pure water at this temperature. This means that in a neutral solution, the product of the hydrogen ion concentration and the hydroxide ion concentration will still equal \(K_w\), and consequently, the sum of pH and pOH equals 14.

However, it's key to remember that this balance and these numerical values will vary with temperature, as temperature can influence the ion product constant for water, and thereby the corresponding pH and pOH values.

Hydrogen and Hydroxide Ion Concentration

Grasping the concepts of hydrogen ion concentration (\([H^+]\)) and hydroxide ion concentration (\([OH^-]\)) is instrumental in understanding aqueous solutions' acidity or basicity. The concentrations of these ions determine the solution's pH and pOH, respectively, where \(pH = -\text{log}([H^+])\) and \(pOH = -\text{log}([OH^-])\).

In a neutral solution, which is neither acidic nor basic, the concentrations of \(H^+\) and \(OH^-\) are equal. Therefore, their product will always be equal to the ion product constant for water at that particular temperature. When given that the constant \(K_w\) is \(10^{-13} M^2\) at 62°C, it implies that both the hydrogen and hydroxide ion concentrations are \(\text{sqrt}(10^{-13}) M\), or approximately \(10^{-6.5} M\).

Remember, the pH and pOH values depend on these concentrations and are thus interconnected. When the sum of the pH and pOH equals the negative logarithm of \(K_w\), this reflects a neutral solution's state at a given temperature. For a comprehensive understanding, one should always consider temperature's impact on ion concentrations and practice calculations to solidify this concept.