Problem 30 - Chemistry The Central Science In Si Units

Question

Calculate \(\left[\mathrm{OH}^{-}\right]\) for each of the following solutions, and indicate whether the solution is acidic, basic, or neutral: (a) \(\left[\mathrm{H}^{+}\right]=0.00010 \mathrm{M} ;(\mathbf{b})\left[\mathrm{H}^{+}\right]=7.3 \times 10^{-14} \mathrm{M} ;(\mathbf{c})\) a solu- tion in which \(\left[\mathrm{OH}^{-}\right]\) is 100 times greater than \(\left[\mathrm{H}^{+}\right]\).

Step-by-Step Solution

Answer

(a) \(\mathrm{[OH^-]} = 1.0 \times 10^{-10} \mathrm{M}\), acidic; (b) \(\mathrm{[OH^-]} \approx 1.37 \mathrm{M}\), basic; (c) \(\mathrm{[OH^-]} = 1.0 \times 10^{-6} \mathrm{M}\), basic.

Step 1: Relationship between \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) Concentration

The concentrations of hydrogen ions \(\mathrm{H}^{+}\) and hydroxide ions \(\mathrm{OH}^{-}\) in a solution are related by the ion-product constant for water at 25 degrees Celsius, which is \(\mathrm{K_w} = 1.0 \times 10^{-14} \mathrm{M^2}\). The formula is: \[\mathrm{[H^+] \times [OH^-]} = 1.0 \times 10^{-14} M^2 \]

Step 2: Calculate \(\mathrm{[OH^-]}\) for Part (a)

Given \(\mathrm{[H^+]} = 0.00010 \, \mathrm{M}\), calculate \(\mathrm{[OH^-]}\) using the formula: \[\mathrm{[OH^-]} = \frac{1.0 \times 10^{-14}}{0.00010} = 1.0 \times 10^{-10} \, \mathrm{M}\] As \(\mathrm{[H^+]} > \mathrm{[OH^-]}\), the solution is acidic.

Step 3: Calculate \(\mathrm{[OH^-]}\) for Part (b)

Given \(\mathrm{[H^+]} = 7.3 \times 10^{-14} \, \mathrm{M}\), calculate \(\mathrm{[OH^-]}\): \[\mathrm{[OH^-]} = \frac{1.0 \times 10^{-14}}{7.3 \times 10^{-14}} \approx 1.37 \, \mathrm{M}\] As \(\mathrm{[OH^-]} > \mathrm{[H^+]}\), the solution is basic.

Step 4: Calculate \(\mathrm{[OH^-]}\) for Part (c)

For the solution where \(\mathrm{[OH^-]}\) is 100 times greater than \(\mathrm{[H^+]}\), let \(\mathrm{[H^+]} = x\) and \(\mathrm{[OH^-]} = 100x\). Use the formula: \[x \times 100x = 1.0 \times 10^{-14}\] \[100x^2 = 1.0 \times 10^{-14}\] \[x^2 = 1.0 \times 10^{-16}\] \[x = 1.0 \times 10^{-8} \, \mathrm{M}\] Therefore, \(\mathrm{[OH^-]} = 100x = 1.0 \times 10^{-6} \, \mathrm{M}\). As \(\mathrm{[OH^-]} > \mathrm{[H^+]}\), the solution is basic.

Key Concepts

Hydrogen Ion Concentration

The concentration of hydrogen ions (denoted as \([\mathrm{H}^+]\) represents the number of hydrogen ions present in a solution. This value directly influences whether a solution is acidic or basic. A higher concentration of \([\mathrm{H}^+]\) compared to \([\mathrm{OH}^-]\) indicates an acidic solution.

Understanding [\mathrm{H}^+] concentration is key in determining the strength of an acid. When we talk about an acidic solution, it means the [\mathrm{H}^+] concentration is greater than [\mathrm{OH}^-]. This means more hydrogen ions are available in the solution, which in turn lowers the pH value.

In the step-by-step solution, for example part (a), when [\mathrm{H}^+] is given as 0.00010 \, \text{M}, we can determine the corresponding [\mathrm{OH}^-] using the formula [\mathrm{H}^+] \times [\mathrm{OH}^-] = 1.0 \times 10^{-14}. It becomes evident that when [\mathrm{H}^+] is greater, the solution is identified as acidic.

Hydroxide Ion Concentration

The concentration of hydroxide ions, represented by \([\mathrm{OH}^-]\), plays a crucial role in classifying solutions as basic. When \([\mathrm{OH}^-]\) is higher than \([\mathrm{H}^+]\), the solution is basic, meaning it typically has a pH greater than 7.

• A basic (or alkaline) solution has more hydroxide ions than hydrogen ions. This high [\mathrm{OH}^-] concentration attracts protons (H⁺ ions), forming water molecules (H₂O), thus reducing the number of free hydrogen ions.
• The step-by-step solution illustrates this in part (b), where [\mathrm{H}^+] = 7.3 \times 10^{-14} \, \text{M}. Substituting in the ion-product constant equation results in [\mathrm{OH}^-] approximately 1.37 \text{ M}, indicating a basic solution with more hydroxide ions than hydrogen ions.

Understanding the balance between [\mathrm{OH}^-] and [\mathrm{H}^+] allows students to classify the nature of solutions easily.

Ion-Product Constant for Water

The ion-product constant for water, represented as \(\mathrm{K_w}\), is an essential concept in understanding acid-base chemistry. This constant is defined as the product of hydrogen ion concentration and hydroxide ion concentration at a constant temperature — typically 25 degrees Celsius.

\(\mathrm{K_w} = [\mathrm{H}^+] \times [\mathrm{OH}^-] = 1.0 \times 10^{-14} \mathrm{M^2}\)

• \(\mathrm{K_w}\) is crucial because it acts as a benchmark for determining whether a solution is acidic, basic, or neutral. For instance, if the concentrations of both ions are equal, the solution is neutral.
• At 25 degrees Celsius, any deviations from the concentration product of 1.0 \times 10^{-14} indicate whether a solution is more acidic or basic. As demonstrated in the original exercise, knowledge of \(\mathrm{K_w}\) allows practical calculations of \([\mathrm{OH}^-]\) when given \([\mathrm{H}^+]\), and vice versa.
• For part (c) of the original solution, we took this constant into account to find out when [\mathrm{OH}^-] is 100 times higher than [\mathrm{H}^+], showing the direct application of \(\mathrm{K_w}\) in real scenarios.

Therefore, \(\mathrm{K_w}\) is more than just a constant; it’s a tool that helps classify whether we are dealing with acidic, basic, or neutral solutions, making it invaluable for students to understand.