Question

Explain the significance of \(K_{\mathrm{w}}\) in aqueous solutions.

Short answer

The significance of \(K_{\mathrm{w}}\) in aqueous solutions lies in its representation of the ion product constant of water, which is crucial for understanding the behavior of acids and bases in water. The balanced chemical equation for the self-ionization of water is: H₂O(l) + H₂O(l) <=> H₃O⁺(aq) + OH⁻(aq) At 25°C, \(K_{\mathrm{w}} = [H_{3}O^{+}] [OH^{-}] = 1.0 \times 10^{-14}\) M². This constant allows us to calculate pH, pOH, and determine the acidity or basicity of a solution: - Basic solutions have pH > 7, with more hydroxide ions than hydronium ions. - Acidic solutions have pH < 7, with more hydronium ions than hydroxide ions. - Neutral solutions have pH = 7, with equal concentrations of hydronium and hydroxide ions.

Step-by-step solution

1. Introduction to Ion Product Constant of Water (Kw)

The ion product constant of water (\(K_{\mathrm{w}}\)) is the equilibrium constant that describes the self-ionization of water molecules. It is defined as the product of the concentrations of the hydronium ion (H₃O⁺) and the hydroxide ion (OH⁻) in an aqueous solution at a given temperature (\(25^\circ\)C or 298K). The balanced chemical equation for the self-ionization of water is: H₂O(l) + H₂O(l) <=> H₃O⁺(aq) + OH⁻(aq)

2. Value of Kw at 25°C

At 25°C (298K), the ion product constant of water (\(K_{\mathrm{w}}\)) has a value of \(1.0 \times 10^{-14}\). This means that the product of the concentrations of hydronium ions and hydroxide ions in pure water at 25°C is a constant: \(K_{\mathrm{w}} = [H_{3}O^{+}] [OH^{-}] = 1.0 \times 10^{-14}\) M² In pure water, there are equal concentrations of H₃O⁺ and OH⁻ ions, each at a concentration of \(1.0 \times 10^{-7}\) M. So, in pure water at 25°C: \([H_{3}O^{+}] = [OH^{-}] = 1.0 \times 10^{-7}\) M

3. Calculating pH and pOH

The \(pH\) and \(pOH\) are convenient ways to represent the acidity and basicity of a solution, respectively. They are defined as the negative logarithm of the concentrations of the H₃O⁺ and OH⁻ ions: \(pH = -\log[H_{3}O^{+}]\) \(pOH = -\log[OH^{-}]\) We can use the \(K_{\mathrm{w}}\) to find the relationship between pH and pOH: Take the logarithm of both sides of the \(K_{\mathrm{w}}\) equation: \(\log{K_{\mathrm{w}}} = \log{([H_{3}O^{+}][OH^{-}])}\) Apply logarithm rules and the definitions of pH and pOH: \(-\log{1.0 \times 10^{-14}} = (-\log[H_{3}O^{+}]) + (-\log[OH^{-}])\) This simplifies to: \(14 = pH + pOH\) This relationship between pH and pOH holds true for any aqueous solution at 25°C.

4. Significance of Kw in Acidity and Basicity

From the values of \(K_{\mathrm{w}}\), pH, and pOH, we can determine the acidity or basicity of a solution: - If a solution has a pH > 7, it is considered a basic solution. This means \([H_{3}O^{+}] < [OH^{-}]\) and the solution has more hydroxide ions than hydronium ions. - If a solution has a pH < 7, it is considered an acidic solution. This means \([H_{3}O^{+}] > [OH^{-}]\) and the solution has more hydronium ions than hydroxide ions. - If a solution has a pH = 7, it is considered a neutral solution (like pure water). This means \([H_{3}O^{+}] = [OH^{-}]\) and the solution has an equal concentration of hydronium and hydroxide ions. In summary, the significance of \(K_{\mathrm{w}}\) in aqueous solutions is essential to understand the behavior of acids and bases in water, as it allows us to calculate pH, pOH, and determine the acidity or basicity of a solution.

Key concepts

Self-ionization of water

Water has a fascinating ability to react with itself, a process known as self-ionization. This phenomenon occurs when two water molecules interact, leading to the creation of a hydronium ion \( (H_3O^+) \) and a hydroxide ion \( (OH^-) \).
This is represented by the chemical equation:

\[H_2O(l) + H_2O(l) \leftrightarrow H_3O^+(aq) + OH^-(aq)\]
Under normal conditions, especially at 25°C (298 K), this process is in a delicate balance. The extent to which this occurs is quantified by the ion product constant of water, \(K_w\).

• At 25°C, \(K_w\) has a constant value of \(1.0 \times 10^{-14}\) \(\text{M}^2\).
• This constant value is the product of the concentrations of \([H_3O^+]\) and \([OH^-]\) in water.

The self-ionization is a rare event, meaning that the concentration of these ions in pure water is very low: \([H_3O^+]\) and \([OH^-]\) each have a concentration of \(1.0 \times 10^{-7}\) M at 25°C. Nevertheless, this balance is crucial for the chemistry of acids and bases.

pH and pOH calculation

To measure how acidic or basic a solution is, we use pH and pOH scales. These scales are derived from the concentrations of hydronium ions \((H_3O^+)\) and hydroxide ions \((OH^-)\), respectively.
The pH is determined by the formula:\[pH = -\log[H_3O^+]\]Meanwhile, the pOH uses a similar approach:\[pOH = -\log[OH^-]\]
The relationship between pH and pOH in any aqueous solution at 25°C is critically tied to the ion product constant \(K_w\).
By applying logarithmic transformations, the equation becomes:\[pH + pOH = 14\]

• This equation allows you to easily find one value if the other is known.
• It also reflects the balance of the hydronium and hydroxide ions in the solution.

Overall, understanding pH and pOH is essential for evaluating the acidic or basic nature of a solution, providing a clear numerical means to describe it.

Acidic and basic solutions

The concepts of acidity and basicity are pivotal in understanding chemical reactions in solution. A solution's acidity or basicity is determined by comparing the amounts of hydronium and hydroxide ions present.
Here's how you can determine the nature of a solution:

• If the pH is less than 7, the solution is acidic. This signifies that \([H_3O^+] > [OH^-]\).
• If the pH is greater than 7, it means the solution is basic, indicating \([H_3O^+] < [OH^-]\).
• A neutral solution, much like pure water, has a balanced pH of 7 where \([H_3O^+] = [OH^-]\).

Understanding whether a solution is acidic or basic has practical implications, from industrial processes to biological systems.
The significance of \(K_w\) is that it provides the foundation for calculating pH and pOH, thereby helping us classify solutions consistently across various conditions.
Grasping this concept is vital for anyone studying chemistry, as it impacts everything from chemical safety to environmental science.