Problem 29 - Chemistry The Central Science In Si Units

Question

Calculate \(\left[\mathrm{H}^{+}\right]\) for each of the following solutions, and indicate whether the solution is acidic, basic, or neutral: (a) \(\left[\mathrm{OH}^{-}\right]=7.3 \times 10^{-10} \mathrm{M}\) (b) \(\left[\mathrm{OH}^{-}\right]=0.015 \mathrm{M}\) (c) \(\mathrm{a}\) solution in which \(\left[\mathrm{H}^{+}\right]\) is 10 times greater than \(\left[\mathrm{OH}^{-}\right]\).

Step-by-Step Solution

Answer

(a) \(1.37 \times 10^{-5} \mathrm{M}\), acidic; (b) \(6.67 \times 10^{-13} \mathrm{M}\), basic; (c) \(3.16 \times 10^{-7} \mathrm{M}\), acidic.

Step 1: Understanding the Relationship

First, recall that the relationship between \ \\(\left[\mathrm{H}^+\right]\) and \(\left[\mathrm{OH}^-\right]\) at 25°C is given by the ion product constant of water: \[\left[\mathrm{H}^+\right] \times \left[\mathrm{OH}^-\right] = 1.0 \times 10^{-14}\] This equation will help us find \(\left[\mathrm{H}^+\right]\) when \(\left[\mathrm{OH}^-\right]\) is known.

Step 2: Calculating [H+] for (a)

Given \(\left[\mathrm{OH}^-\right]=7.3 \times 10^{-10} \mathrm{M}\), substitute in the ion product equation: \[\left[\mathrm{H}^+\right] = \frac{1.0 \times 10^{-14}}{7.3 \times 10^{-10}}\]Calculate \(\left[\mathrm{H}^+\right]\): \[\left[\mathrm{H}^+\right] \approx 1.37 \times 10^{-5} \mathrm{M}\]Since \(\left[\mathrm{H}^+\right] > 1.0 \times 10^{-7}\, \mathrm{M}\), the solution is acidic.

Step 3: Calculating [H+] for (b)

Given \(\left[\mathrm{OH}^-\right]=0.015 \mathrm{M}\), use the ion product equation:\[\left[\mathrm{H}^+\right] = \frac{1.0 \times 10^{-14}}{0.015}\]Calculate \(\left[\mathrm{H}^+\right]\): \[\left[\mathrm{H}^+\right] \approx 6.67 \times 10^{-13} \mathrm{M}\]Since \(\left[\mathrm{H}^+\right] < 1.0 \times 10^{-7}\, \mathrm{M}\), the solution is basic.

Step 4: Analyzing (c)

For the condition \(\left[\mathrm{H}^+\right] = 10 \times \left[\mathrm{OH}^-\right]\), substitute in the ion product equation:\[10 \times \left[\mathrm{OH}^-\right]^2 = 1.0 \times 10^{-14}\]Solve for \(\left[\mathrm{OH}^-\right]\):\[\left[\mathrm{OH}^-\right] = \sqrt{\frac{1.0 \times 10^{-14}}{10}} \approx 3.16 \times 10^{-8} \mathrm{M}\]Now calculate \(\left[\mathrm{H}^+\right]\):\[\left[\mathrm{H}^+\right] = 10 \times 3.16 \times 10^{-8} = 3.16 \times 10^{-7} \mathrm{M}\]As \(\left[\mathrm{H}^+\right] > 1.0 \times 10^{-7}\, \mathrm{M}\), the solution is acidic.

Key Concepts

ion product of water

The ion product of water, often denoted as \(K_{w}\), is a crucial concept in understanding pH calculation. At 25°C, the ion product constant of water is expressed as:\[K_{w} = [\mathrm{H}^+][\mathrm{OH}^-] = 1.0 \times 10^{-14}\] This equilibrium constant reflects the balance between the concentration of hydrogen ions \([\mathrm{H}^+]\) and hydroxide ions \([\mathrm{OH}^-]\) in pure water.

• In pure water, both of these ion concentrations are equal and each measures \(1.0 \times 10^{-7} \ \mathrm{M}\).
• This means that when any substance is added to water, altering these concentrations, understanding their relationship through \(K_{w}\) allows us to predict changes in acidity or basicity.

By manipulating the equation \([\mathrm{H}^+] \times [\mathrm{OH}^-] = 1.0 \times 10^{-14}\), we can calculate unknown concentrations when either hydrogen or hydroxide ion concentration is provided. This relationship is vital when determining the nature of solutions as acidic, basic, or neutral.

acidic and basic solutions

Determining whether a solution is acidic, basic, or neutral is essential in chemistry. We often use the concentration of hydrogen ions \([\mathrm{H}^+]\) to make this classification.

• A solution is considered acidic if \([\mathrm{H}^+] > 1.0 \times 10^{-7} \ \mathrm{M}\). This means the concentration of hydrogen ions is greater than that in pure water.
• A basic solution has \([\mathrm{H}^+] < 1.0 \times 10^{-7} \ \mathrm{M}\), indicating that hydroxide ions are predominant.
• A neutral solution, such as pure water, maintains \([\mathrm{H}^+] = [\mathrm{OH}^-] = 1.0 \times 10^{-7} \ \mathrm{M}\).

The pH scale, which ranges from 0 to 14, is the logarithmic scale used to specify the acidity or basicity of a solution:

Understanding the pH Scale

- Lower pH values (<7) indicate acidity. - A pH of exactly 7 is neutral. - Higher pH values (>7) indicate basicity. By calculating the hydrogen ion concentration, we can identify the solution's nature and use this to predict its behavior in various reactions.

concentration of hydrogen ions

The concentration of hydrogen ions \([\mathrm{H}^+]\) in a solution is a central factor in determining its acidity and is crucial for pH calculation. By using the ion product of water, we can easily find the hydrogen ion concentration if we know the concentration of hydroxide ions \([\mathrm{OH}^-]\). Here's how:

Calculating \([\mathrm{H}^+]\) from \([\mathrm{OH}^-]\)

The ion product equation \([\mathrm{H}^+] \times [\mathrm{OH}^-] = 1.0 \times 10^{-14}\) allows us to solve for \([\mathrm{H}^+]\):

• If \([\mathrm{OH}^-]\) is provided, \([\mathrm{H}^+]\) can be found by rearranging the equation to \([\mathrm{H}^+] = \frac{1.0 \times 10^{-14}}{[\mathrm{OH}^-]}\).
• This simple calculation tells us whether the solution is acidic or basic by comparing \([\mathrm{H}^+]\) to \(1.0 \times 10^{-7} \ \mathrm{M}\).

Understanding the concentration of hydrogen ions is paramount because it not only classifies the solution's nature but also guides us in predicting the solution's reactivity and potential effects in chemical processes.