Question

Complete the following table by calculating the missing entries. In each case indicate whether the solution is acidic or basic. $$ \begin{array}{llll} \hline\left[\mathrm{H}^{+}\right] & \mathrm{OH}^{-} \text {(aq) } && \mathrm{pH} & &\text { pOH }&& \text { Acidic or basic? } \\ \hline 7.5 \times 10^{-3} \mathrm{M} & & & & \\ & 3.6 \times 10^{-10} \mathrm{M} & & \\ & && 8.25 & & \\ & & &&& 5.70 & \end{array} $$

Short answer

} \\ \hline 7.5 \times 10^{-3} \mathrm{M} & 1.33 \times 10^{-12} \mathrm{M} && 2.12 && 11.88 && \text { Acidic } \\ & 3.6 \times 10^{-10} \mathrm{M} && 4.56 && 9.44 && \text { Basic } \\ & && 8.25 && 5.75 && \text { Basic } \\ & & &&& 5.70 && 8.30 && \text { Basic } \\ \end{array} $$

Step-by-step solution

Calculate missing ion concentrations

Use the relationship \(K_w = \left[\mathrm{H}^{+}\right]\times \left[\mathrm{OH}^{-}\right]\) to find the missing ion concentrations. For the first row, we have \(\left[\mathrm{H}^{+}\right] = 7.5 \times 10^{-3} \mathrm{M}\). Calculate the concentration of \(\left[\mathrm{OH}^{-}\right]\) using: \(K_w = \left[\mathrm{H}^{+}\right]\times \left[\mathrm{OH}^{-}\right]\) \(\implies \left[\mathrm{OH}^{-}\right] = \frac{K_w}{\left[\mathrm{H}^{+}\right]} = \frac{1.0 \times 10^{-14}}{7.5 \times 10^{-3}} = 1.33 \times 10^{-12}\,\mathrm{M}\) For the second row, we have \(\left[\mathrm{OH}^{-}\right] =3.6 \times 10^{-10} \mathrm{M}\). Calculate the concentration of \(\left[\mathrm{H}^{+}\right]\) using: \(K_w = \left[\mathrm{H}^{+}\right]\times \left[\mathrm{OH}^{-}\right]\) \(\implies \left[\mathrm{H}^{+}\right] = \frac{K_w}{\left[\mathrm{OH}^{-}\right]} = \frac{1.0 \times 10^{-14}}{3.6 \times 10^{-10}} = 2.78 \times 10^{-5}\,\mathrm{M}\)

Calculate missing pH and pOH values

Use the relationships pH = \(-\log{(\left[\mathrm{H}^{+}\right])}\) and pOH = \(-\log{(\left[\mathrm{OH}^{-}\right])}\) to find the missing pH and pOH values. For the first row, calculate pH as: pH = \(-\log{(7.5 \times 10^{-3})} = 2.12\) For the second row, calculate pOH as: pOH = \(-\log{(3.6 \times 10^{-10})} = 9.44\) For the third row, we have pH = 8.25. Calculate pOH using the relationship pH + pOH = 14: pOH = 14 - pH = 14 - 8.25 = 5.75 For the fourth row, we have pOH = 5.70. Calculate pH using the relationship pH + pOH = 14: pH = 14 - pOH = 14 - 5.70 = 8.30

Identify whether the solutions are acidic or basic

Determine whether the solution is acidic or basic based on the calculated pH values. For the first row: pH = 2.12 < 7, so the solution is acidic. For the second row: pH = 9.44 > 7, so the solution is basic. For the third row: pH = 8.25 > 7, so the solution is basic. For the fourth row: pH = 8.30 > 7, so the solution is basic. The completed table is as follows: $$ \begin{array}{llll} \hline\left[\mathrm{H}^{+}\right] & \mathrm{OH}^{-} \text {(aq) } && \mathrm{pH} && \text { pOH } && \text { Acidic or basic? } \\ \hline 7.5 \times 10^{-3} \mathrm{M} & 1.33 \times 10^{-12} \mathrm{M} && 2.12 && 11.88 && \text { Acidic } \\ & 3.6 \times 10^{-10} \mathrm{M} && 4.56 && 9.44 && \text { Basic } \\ & && 8.25 && 5.75 && \text { Basic } \\ & & &&& 5.70 && 8.30 && \text { Basic } \\ \end{array} $$

Key concepts

Acid and Base Identification

Understanding whether a solution is acidic or basic is a fundamental aspect of chemistry that involves measuring its pH levels. An acid solution has a pH less than 7, while a basic (or alkaline) solution has a pH greater than 7. A pH of exactly 7 indicates a neutral solution, like pure water.

To identify an acid or base in the given exercise, you must first know the pH value. If a solution's pH is not directly provided, you may calculate it using the hydrogen ion concentration \( [\mathrm{H}^+] \) and the formula pH = \( -\log([\mathrm{H}^+]) \). When the pH is determined, comparing it to 7 allows you to classify the solution accordingly. As seen in the exercise, any pH lower than 7 indicated an acidic solution, and any pH higher than 7 indicated a basic solution.

Hydrogen Ion Concentration

The hydrogen ion concentration \( [\mathrm{H}^+] \) in a solution is used to determine its acidity. It is measured in moles per liter (M). Larger \( [\mathrm{H}^+] \) values correspond to higher acidities, and thus lower pH values. You can calculate the hydrogen ion concentration if it's unknown by using the relationship between the ion product constant of water (\(K_w\)) and the hydroxide ion concentration (\( [\mathrm{OH}^-] \)).

In the textbook example, when we are given the hydroxide ion concentration, \( [\mathrm{OH}^-] \), we can calculate \( [\mathrm{H}^+] \) by re-arranging the neutral water equation \(K_w = [\mathrm{H}^+] \times [\mathrm{OH}^-]\). This concept is crucial because it allows us to determine the pH values necessary for acid-base identification.

Hydroxide Ion Concentration

Similarly to hydrogen ions, the concentration of hydroxide ions \( [\mathrm{OH}^-] \) can reveal the basicity of a solution. It's also expressed in moles per liter. For stronger bases, this concentration will be greater, leading to a higher pOH and thereby a lower pH value (since pH and pOH are inversely related).

The exercise required calculating the missing hydroxide ion concentrations given \( [\mathrm{H}^+] \) and using the equilibrium constant \(K_w\). Once \( [\mathrm{OH}^-] \) was found, the pOH could be calculated using the formula pOH = \( -\log([\mathrm{OH}^-]) \), which is analogous to how pH is found from \( [\mathrm{H}^+] \).

pH and pOH Relationship

The pH and pOH of a solution are directly related through the ion product constant for water: \(K_w = 1.0 \times 10^{-14} \) at 25°C. For any aqueous solution at this temperature, pH and pOH always add up to 14. To find pOH from pH (or vice versa), you subtract the known value from 14.

In our exercise, once we had either the pH or pOH, we could find its counterpart using this relationship. This is how we established that a pH of 8.25 corresponded to a pOH of 5.75, and a pOH of 5.70 meant the pH was 8.30. Understanding this relationship was key to completing the missing entries in the table.

Chemical Equilibrium in Solutions

When a substance dissociates in water, it reaches a state of chemical equilibrium where the rate of dissociation equals the rate of recombination. At this point, the concentrations of reactants and products remain constant over time. The pH, pOH, and the relationship between hydrogen and hydroxide ions reflect this chemical equilibrium.

For any aqueous solution at equilibrium, the constant \(K_w = [\mathrm{H}^+] \times [\mathrm{OH}^-]\) holds true. This expression was used in the exercise to find missing concentrations. The reactions of acids releasing hydrogen ions and bases releasing hydroxide ions must adhere to this principle of chemical equilibrium, thereby allowing us to perform accurate calculations in our pH and pOH problems.