Question

Calculate the pH corresponding to each of the hydroxide ion concentrations given below. Tell whether each solution is acidic, basic, or neutral. a. \(\left[\mathrm{OH}^{-}\right]=1.4 \times 10^{-6} \mathrm{M}\) b. \(\left[\mathrm{OH}^{-}\right]=9.35 \times 10^{-9} \mathrm{M}\) c. \(\left[\mathrm{OH}^{-}\right]=2.21 \times 10^{-1} \mathrm{M}\) d. \(\left[\mathrm{OH}^{-}\right]=7.98 \times 10^{-12} \mathrm{M}\)

Short answer

a. pH = 8.15, basic b. pH = 5.97, acidic c. pH = 13.34, basic d. pH = 2.90, acidic

Step-by-step solution

Find the hydrogen ion concentration using the ion-product constant for water

For each of the given hydroxide ion concentrations, we'll use the ion-product constant (Kw) for water to find the hydrogen ion concentration. The Kw is given by: \[K_{w} = [\mathrm{H^{+}}]\times [\mathrm{OH^{-}}] =1\times 10^{-14} \mathrm{M^2}\] So, given a hydroxide ion concentration, we can find the corresponding hydrogen ion concentration by rearranging this equation to be: \[[\mathrm{H^{+}}]=\frac{K_{w}}{[\mathrm{OH^{-}}]}\]

Calculate the pH value for each solution

Now that we can find the hydrogen ion concentration for each solution, we'll use the pH formula to determine the pH value for each solution. The pH formula is given by: \[\mathrm{pH}=-\log [\mathrm{H^{+}}]\]

Determine whether the solution is acidic, basic, or neutral

With the pH value in hand, we'll determine the nature of each solution using the pH scale: - If pH = 7, the solution is neutral. - If pH < 7, the solution is acidic. - If pH > 7, the solution is basic. Now let's do the calculations for each solution: a. \(\left[\mathrm{OH}^{-}\right]=1.4 \times 10^{-6} \mathrm{M}\) - Calculate [H⁺]: \[[\mathrm{H^{+}}]=\frac{1\times 10^{-14}}{1.4 \times 10^{-6}}= 7.14 \times 10^{-9} \mathrm{M}\] - Calculate pH: \[\mathrm{pH}=-\log (7.14 \times 10^{-9})=8.15\] - Since pH > 7, this solution is basic. b. \(\left[\mathrm{OH}^{-}\right]=9.35 \times 10^{-9} \mathrm{M}\) - Calculate [H⁺]: \[[\mathrm{H^{+}}]=\frac{1\times 10^{-14}}{9.35 \times 10^{-9}}=1.07 \times 10^{-6} \mathrm{M}\] - Calculate pH: \[\mathrm{pH}=-\log (1.07 \times 10^{-6})=5.97\] - Since pH < 7, this solution is acidic. c. \(\left[\mathrm{OH}^{-}\right]=2.21 \times 10^{-1} \mathrm{M}\) - Calculate [H⁺]: \[[\mathrm{H^{+}}]=\frac{1\times 10^{-14}}{2.21 \times 10^{-1}}= 4.52 \times 10^{-14} \mathrm{M}\] - Calculate pH: \[\mathrm{pH}=-\log (4.52 \times 10^{-14})=13.34\] - Since pH > 7, this solution is basic. d. \(\left[\mathrm{OH}^{-}\right]=7.98 \times 10^{-12} \mathrm{M}\) - Calculate [H⁺]: \[[\mathrm{H^{+}}]=\frac{1\times 10^{-14}}{7.98 \times 10^{-12}}= 1.25 \times 10^{-3} \mathrm{M}\] - Calculate pH: \[\mathrm{pH}=-\log (1.25 \times 10^{-3})=2.90\] - Since pH < 7, this solution is acidic.

Key concepts

Understanding Hydroxide Ion Concentration

Hydroxide ion concentration (\( \left[ \mathrm{OH}^{-} \right] \)) plays a key role in determining the pH of a solution. It provides insights into whether a solution is acidic or basic. Measured in moles per liter (M), this concentration indicates the amount of hydroxide ions present in the solution. The higher the concentration of hydroxide ions, the more basic the solution becomes.
When we know the hydroxide ion concentration, we can calculate the hydrogen ion concentration using the ion-product constant for water. This connection is vital because it helps us transition from knowing one concentration to finding another, ultimately leading us to the pH.
So, when you have \( \left[ \mathrm{OH}^{-} \right] \), you're halfway to finding out whether the solution is acidic, neutral, or basic.

The Ion-Product Constant for Water (Kw)

The ion-product constant for water, denoted as \( K_{w} \), is a crucial concept in chemistry. It represents the product of the concentrations of hydrogen ions (\( \left[ \mathrm{H^{+}} \right] \)) and hydroxide ions (\( \left[ \mathrm{OH}^{-} \right] \)) in pure water. Mathematically, it is expressed as:\[K_{w} = [\mathrm{H^{+}}] \times [\mathrm{OH^{-}}] = 1 \times 10^{-14} \mathrm{\, M^{2}}\]This constant illustrates that in neutral water, the concentrations of hydrogen ions and hydroxide ions are in balance. However, this balance shifts when acids or bases are introduced into the solution.
By rearranging the equation of \( K_{w} \), you can find the concentration of hydrogen ions when you know the concentration of hydroxide ions, and vice versa. This relationship is fundamental for further calculations leading to pH determination. Understanding \( K_{w} \) helps connect ion concentrations to the broader topic of the acidity or basicity of solutions.

Classifying Solutions: Acidic, Basic, or Neutral

Once you've calculated the hydrogen ion concentration (\( \left[ \mathrm{H^{+}} \right] \)), you can determine the pH, which tells you whether the solution is acidic, basic, or neutral. The pH scale is logarithmic, ranging from 0 to 14:

• pH = 7: The solution is neutral. This is typical for pure water.
• pH < 7: The solution is acidic. More hydrogen ions than hydroxide ions are present.
• pH > 7: The solution is basic. More hydroxide ions than hydrogen ions are present.

With these guidelines, you can quickly assess the nature of any solution once its pH is known. Remember, each unit change in pH represents a tenfold change in hydrogen ion concentration, making pH a powerful tool for understanding chemical equilibria in solutions.
So, knowing how to calculate and interpret pH can aid in solving many real-world chemical problems and applications.