Question

Calculate the molar concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) in solutions that have the following \(\mathrm{pOH}\) values. (a) 7.19 (b) 1.26 (c) 10.85 (d) 13.15 (e) 5.24

Short answer

\([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\) for (a) are \(6.46 \times 10^{-8}\) M and \(1.55 \times 10^{-7}\) M, for (b) are \(5.50 \times 10^{-13}\) M and \(1.82 \times 10^{-2}\) M, for (c) are \(1.40 \times 10^{-4}\) M and \(7.14 \times 10^{-11}\) M, for (d) are \(0.71 \times 10^{-1}\) M and \(1.41 \times 10^{-14}\) M, and for (e) are \(5.75 \times 10^{-9}\) M and \(1.74 \times 10^{-6}\) M.

Step-by-step solution

Understand the Relationship between pOH and \(\mathrm{OH}^{-}\) Concentration

The pOH of a solution is related to the concentration of hydroxide ions (\(\mathrm{OH}^{-}\)) by the formula \(\text{pOH} = -\log([\mathrm{OH}^{-}])\). To find the \(\mathrm{OH}^{-}\) concentration, we rearrange the formula to solve for \([\mathrm{OH}^{-}]\) which gives us \([\mathrm{OH}^{-}] = 10^{-\text{pOH}}\).

Calculate the Concentration of \(\mathrm{OH}^{-}\) for Each pOH Value

Using the equation from Step 1, calculate \([\mathrm{OH}^{-}]\) for each pOH value given. For instance, if pOH is 7.19, the hydroxide ion concentration is \([\mathrm{OH}^{-}] = 10^{-7.19}\).

Use the Relationship between \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) Concentrations

In a neutral aqueous solution at 25°C, the product of the concentrations of hydrogen and hydroxide ions is constant: \(\left[ \mathrm{H}^{+} \right] \times \left[ \mathrm{OH}^{-} \right] = 1.0 \times 10^{-14}\) (the ion-product constant for water, \(K_w\)). To find the concentration of \(\mathrm{H}^{+}\), use the equation \(\left[ \mathrm{H}^{+}\right] = \frac{1.0 \times 10^{-14}}{\left[\mathrm{OH}^{-}\right]}\).

Calculate the Concentration of \(\mathrm{H}^{+}\) for Each \(\mathrm{OH}^{-}\) Concentration

Using the calculated \([\mathrm{OH}^{-}]\) from Step 2, apply the equation from Step 3 to find \([\mathrm{H}^{+}]\) for each case. For example, with \([\mathrm{OH}^{-}] = 10^{-7.19}\), calculate \([\mathrm{H}^{+}]\) as \(\frac{1.0 \times 10^{-14}}{10^{-7.19}}\).

Repeat Calculations for All Given pOH Values

Perform the steps 2 and 4 for all the given pOH values to obtain all \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) concentrations.

Compile the Results

After calculating for each case, organize the concentration values of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) according to the given pOH values (a through e).

Key concepts

Hydroxide Ion Concentration

Understanding hydroxide ion concentration is fundamental when studying the chemistry of aqueous solutions. The concentration of hydroxide ions (_OH⁻)) in a solution indicates the solution's basicity, or alkalinity. It is commonly represented by ([OH⁻]) and is measured in moles per liter (Molarity, M). The higher this concentration, the more basic the solution is.

To find the hydroxide ion concentration from the pOH of a solution, we employ the formula ([OH⁻] = 10^−pOH). For instance, if a solution has a pOH of 7.19, we calculate the hydroxide ion concentration as follows: ([OH⁻] = 10^−7.19). It's crucial for students to comprehend how to perform these calculations as they can directly affect the outcomes in experimental settings and inform us about the properties of the solution we're working with.

Hydrogen Ion Concentration

In contrast to hydroxide ions, hydrogen ions (H⁺) determine the acidity of a solution. Their concentration, denoted as [H⁺]), also measured in moles per liter, indicates the solution's acidic level—the higher the concentration, the more acidic the solution.

To calculate the hydrogen ion concentration, especially when the hydroxide ion concentration is already known, we use the ion-product constant for water (K_w) and the formula [H⁺] = K_w / [OH⁻]). Given the K_w value of 1.0 × 10⁻¹⁴ at 25°C, we can rearrange this equation to solve for the hydrogen ion concentration if the hydroxide ion concentration is available from pOH calculations. Mastering this calculation is essential for students who need to balance chemical reactions or determine pH values in various chemical contexts.

Ion-Product Constant for Water

The ion-product constant for water (K_w) is a vital constant in chemistry that quantifies the relationship between hydrogen ions (H⁺) and hydroxide ions (OH⁻) in water. At 25°C, K_w has a value of 1.0 × 10⁻¹⁴. This number represents the product of the concentrations of these two ions in pure water and stays consistent for all aqueous solutions at this temperature.

Understanding K_w allows students to make connections between the acidity and basicity of a solution. It also provides insight into why the multiplication of the hydrogen ion concentration by the hydroxide ion concentration in any aqueous solution at standard temperature always returns this constant value. Comprehension of the ion-product constant is, therefore, indispensable for predicting the behavior of reactions and for solving complex chemistry problems.

pH and pOH Relationship

The pH and pOH of a solution are directly related to one another and inversely related to the concentrations of hydrogen and hydroxide ions, respectively. The pH scale is a measure of how acidic or basic a water-based solution is, with a lower pH indicating a more acidic solution and a higher pH a more basic solution. Similarly, pOH provides a measure for the basicity of a solution. The pH and pURlOH relationship can be summarized with the equation pH + pOH = 14. This is because the sum of the logs of the ion concentrations equals the negative log of the ion-product constant of water (K_w), which is 14 at 25°C.

pH is more commonly used in discussions of acidity, but pOH is equally important for calculating and understanding basic solutions. To find the pH from pOH, simply subtract the pOH from 14, and vice versa. This interrelationship is crucial for laboratory work in chemistry, environmental science, and many industrial processes where controlling the pH level is necessary for the desired outcomes.