Question

(a) Write a chemical equation that illustrates the auto-ionization of water. (b) Write the expression for the ion-product constant for water \(K_{w}\) . (c) If a solution is described as basic, which of the following is true: (i) \(\left[\mathrm{H}^{+}\right]>\left[\mathrm{OH}^{-}\right],\) (ii) \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{OH}^{-}\right],\) or (iii) \(\left[\mathrm{H}^{+}\right]<\left[\mathrm{OH}^{-}\right] ?\)

Short answer

The short answer to the question is: (a) The chemical equation for the auto-ionization of water is: \[\mathrm{2\,H_2O\, \leftrightarrow\, H_3O^+ + OH^{-}}\] (b) The expression for the ion-product constant for water (\(K_w\)) is: \[K_w = [\mathrm{H^+}][\mathrm{OH^{-}}]\] (c) In a basic solution, the correct relationship between H⁺ and OH⁻ concentrations is (iii): \([\mathrm{H^+}] < [\mathrm{OH^{-}}]\]

Step-by-step solution

Part (a): Auto-Ionization of Water Chemical Equation

The auto-ionization of water is a process in which two water molecules spontaneously react to form a hydronium ion (H₃O⁺) and a hydroxide ion (OH⁻). The chemical equation for the auto-ionization of water is given by: \[\mathrm{2\,H_2O\, \leftrightarrow\, H_3O^+ + OH^{-}}\]

Part (b): Ion-Product Constant for Water \(K_w\) Expression

The ion-product constant for water (\(K_w\)) is the equilibrium constant for the auto-ionization of water, representing the product of the concentrations of hydronium ions (H₃O⁺ or simply H⁺) and hydroxide ions (OH⁻) at a particular temperature (25°C, for instance). The expression for \(K_w\) is given by: \[K_w = [\mathrm{H^+}][\mathrm{OH^{-}}]\] At 25°C, the value of \(K_w\) is approximately \(1.0 \times 10^{-14}\).

Part (c): H⁺ and OH⁻ Concentrations in a Basic Solution

Basic solutions are defined by their pH being higher than 7, meaning that they have a lower concentration of H⁺ (hydronium or hydrogen ions) compared to OH⁻ (hydroxide ions). Hence, the correct relationship between H⁺ and OH⁻ concentrations in a basic solution is: \([\mathrm{H^+}] < [\mathrm{OH^{-}}]\) Therefore, the correct answer is (iii).

Key concepts

Ion-Product Constant for Water

Have you ever wondered why even the purest water has a pH very close to 7? This all comes down to a process known as the auto-ionization of water. Water, H₂O, can act as both an acid and a base. This means that some water molecules donate protons (H⁺) while others accept them, forming hydronium (H₃O⁺) and hydroxide (OH^-) ions.

Now, let's focus on the ion-product constant for water, often symbolized as 'K_w'. This is a special type of equilibrium constant that applies to the auto-ionization of water. In chemical terms, it's the product of the molar concentrations of H⁺ and OH^- ions in pure water. At room temperature, which is often taken as 25°C, this constant has a value of approximately 1.0 x 10^-14. It’s crucial to note that this value changes with temperature but remains constant at a given temperature.

Why is K_w Important?

Understanding K_w is fundamental in chemistry because it ties directly to the concept of pH. It is also used to determine whether a solution is acidic or basic without the need to test the pH. Both students and chemists use this constant to make essential calculations about the composition and properties of aqueous solutions.

Kw Expression

The ion-product constant for water (K_w) isn't just a number; it is an expression that reveals much about the intrinsic properties of water. The expression for K_w is written as:

�Here’s the fascinating part: this value remains remarkably constant in all aqueous solutions, regardless of the presence of acids or bases. This gives us a powerful tool for understanding the balance of H⁺ and OH^- ions.

Taking a glance at the molecular level, we learn that in pure water at equilibrium, the concentration of H⁺ and OH^- ions is equal, which means both are at concentrations of 1.0 x 10^-7 mol/L. These concentrations multiply to give us the ion-product constant for water, K_w.

pH and pOH Relationships

Diving deeper into the world of acidity, bases, and neutrality requires us to understand the pH and pOH scales. These scales offer us a straightforward way to express the acidity or basicity of a solution.

Simply put, the pH is a measure of the hydrogen ion concentration in a solution, while pOH measures the hydroxide ion concentration. The math behind it is pretty elegant – pH is the negative logarithm of the hydrogen ion concentration, and pOH is the negative logarithm of the hydroxide ion concentration.

These scales go hand in hand due to the K_w expression, which has a crucial relationship with pH and pOH: namely,

�pH + pOH = 14 (at 25°C)

This means if one knows either the pH or pOH of a solution, the other can be easily calculated, provided the temperature is known since this relationship is temperature dependent.

When referring to basic solutions, such as in the exercise, a pH greater than 7 means that the pOH is less than 7, confirming that the hydroxide ion concentration is greater than that of the hydrogen ions. Understanding the relationship between pH and pOH not only helps students solve chemistry problems but also allows scientists and industries to maintain the proper conditions for various chemical processes.