Problem 30 - Chemistry: The Central Science in SI Units, Global Edition

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 solution in which \(\left[\mathrm{OH}^{-}\right]\) is 100 times greater than \(\left[\mathrm{H}^{+}\right]\).

Step-by-Step Solution

Answer

Answer: (a) \([\mathrm{OH}^{-}]\) = \(1.0 \times 10^{-10} \mathrm{M}\), acidic (b) \([\mathrm{OH}^{-}]\) ≈ \(1.37 \times 10^{-1} \mathrm{M}\), basic (c) \([\mathrm{OH}^{-}]\) = \(1.0 \times 10^{-6} \mathrm{M}\), basic

Step 1: Calculate \([\mathrm{OH}^{-}]\)

Using the ion-product constant for water, \(K_w = \left[\mathrm{H}^{+}\right] \cdot \left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-14}\), solve for \([\mathrm{OH}^{-}]\): \[\left[\mathrm{OH}^{-}\right] = \frac{K_w}{\left[\mathrm{H}^{+}\right]} = \frac{1.0 \times 10^{-14}}{0.0001 \mathrm{M}} = 1.0 \times 10^{-10} \mathrm{M}\]

Step 2: Compare \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\)

In this case, \[\left[\mathrm{H}^{+}\right] = 0.0001 \mathrm{M} > \left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-10} \mathrm{M}\]

Step 3: Classify the solution

Since the hydrogen ion concentration is greater than the hydroxide ion concentration, the solution is acidic. (b) \[\left[\mathrm{H}^{+}\right] = 7.3 \times 10^{-14} \mathrm{M}\]

Step 1: Calculate \([\mathrm{OH}^{-}]\)

Using the ion-product constant for water, solve for \([\mathrm{OH}^{-}]\): \[\left[\mathrm{OH}^{-}\right] = \frac{K_w}{\left[\mathrm{H}^{+}\right]} = \frac{1.0 \times 10^{-14}}{7.3 \times 10^{-14} \mathrm{M}} \approx 1.37 \times 10^{-1} \mathrm{M}\]

Step 2: Compare \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\)

In this case, \[\left[\mathrm{H}^{+}\right] = 7.3 \times 10^{-14} \mathrm{M} < \left[\mathrm{OH}^{-}\right] \approx 1.37 \times 10^{-1} \mathrm{M}\]

Step 3: Classify the solution

Since the hydrogen ion concentration is smaller than the hydroxide ion concentration, the solution is basic. (c) \[\left[\mathrm{OH}^{-}\right] = 100\left[\mathrm{H}^{+}\right]\]

Step 1: Calculate \([\mathrm{OH}^{-}]\)

Again, using the ion-product constant for water, solve for \([\mathrm{OH}^{-}]\): \[\left[\mathrm{H}^{+}\right] \cdot \left[\mathrm{OH}^{-}\right] = K_w\] \[\left[\mathrm{H}^{+}\right] \cdot (100\left[\mathrm{H}^{+}\right]) = 1.0 \times 10^{-14}\] \[100\left[\mathrm{H}^{+}\right]^2 = 1.0 \times 10^{-14}\] \[\left[\mathrm{H}^{+}\right]^2 = 1.0 \times 10^{-16}\] \[\left[\mathrm{H}^{+}\right] = 1.0 \times 10^{-8} \mathrm{M}\] Now we can calculate \(\left[\mathrm{OH}^{-}\right]\): \[\left[\mathrm{OH}^{-}\right] = 100\left[\mathrm{H}^{+}\right] = 100(1.0 \times 10^{-8} \mathrm{M}) = 1.0 \times 10^{-6} \mathrm{M}\]

Step 2: Compare \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\)

In this case, \[\left[\mathrm{H}^{+}\right] = 1.0 \times 10^{-8} \mathrm{M} < \left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-6} \mathrm{M}\]

Step 3: Classify the solution

Since the hydrogen ion concentration is smaller than the hydroxide ion concentration, the solution is basic.

Key Concepts

Ion-Product Constant of Water

The ion-product constant of water, often symbolized as \(K_w\), is a crucial concept in understanding acid-base equilibrium. It represents the product of the concentrations of hydrogen ions \([\text{H}^+]\) and hydroxide ions \([\text{OH}^-]\) in water, and it is expressed as:\[K_w = [\text{H}^+] \cdot [\text{OH}^-] = 1.0 \times 10^{-14}\]This constant applies to pure water and aqueous solutions at 25°C. It helps us find the missing ion concentration when one is known. For example, if you're given \([\text{H}^+]\), you can solve for \([\text{OH}^-]\) using this formula. Also, if the temperature changes, \(K_w\) will change, but at room temperature, it remains 1.0 x 10\(^{-14}\).Understanding \(K_w\) is vital, as it lays the foundation for assessing whether solutions are acidic, basic, or neutral based on ion concentrations.

• Neutral solutions have equal \([\text{H}^+]\) and \([\text{OH}^-]\), both \(1.0 \times 10^{-7}~\text{M}\).
• Acidic solutions have a higher \([\text{H}^+]\) than \([\text{OH}^-]\).
• Basic solutions have a higher \([\text{OH}^-]\) than \([\text{H}^+]\).

Acidic and Basic Solutions

Solutions can be categorized as acidic, basic, or neutral based on the concentrations of hydrogen ions \([\text{H}^+]\) and hydroxide ions \([\text{OH}^-]\). Knowing whether a solution is acidic or basic helps understand its chemical behavior. **Acidic Solutions**- These have more hydrogen ions compared to hydroxide ions.- \([\text{H}^+] > [\text{OH}^-]\). - Example: If \([\text{H}^+] = 0.0001~\text{M}\), then \([\text{OH}^-]\) will be smaller, indicating an acidic solution.**Basic Solutions**- These have more hydroxide ions than hydrogen ions.- \([\text{OH}^-] > [\text{H}^+]\).- Example: If \([\text{OH}^-]\) is calculated to be higher than \([\text{H}^+]\) as in a 100 times ratio, the solution is basic.**Neutral Solutions**- These have equal concentrations of \([\text{H}^+]\) and \([\text{OH}^-]\).- At \(25^\circ\text{C}\), \([\text{H}^+] = [\text{OH}^-] = 1.0 \times 10^{-7}~\text{M}\). Recognizing these characteristics aids in determining the acidic or basic nature of a solution by simply comparing ion concentrations.

Hydrogen Ion Concentration

Hydrogen ion concentration, \([\text{H}^+]\), plays a key role in defining a solution's acidity or basicity. It's part of the pH scale, which measures how acidic or basic a substance is. **Understanding pH**- The pH is calculated as \(-\log([\text{H}^+])\).- Lower pH values (below 7) indicate an acidic solution.- Higher pH values (above 7) signify a basic solution.**Importance in Chemistry**- Knowing \([\text{H}^+]\) helps classify solutions and predict their reactions. - Effects like corrosiveness or basicity changes in reactions depend on hydrogen ion concentration. For instance, a solution with \([\text{H}^+] = 7.3 \times 10^{-14}~\text{M}\) is highly basic because the concentration is much lower than the neutral level of \(1.0 \times 10^{-7}~\text{M}\).Using the ion-product constant \(K_w\), these concentrations link directly to \([\text{OH}^-]\), allowing for easy calculations in determining the solution's nature.