Question

Calculate the \(\left[\mathrm{OH}^{-}\right]\) in each of the following solutions, and indicate whether the solution is acidic or basic. a. \(\left[\mathrm{H}^{+}\right]=1.34 \times 10^{-2} \mathrm{M}\) b. \(\left[\mathrm{H}^{+}\right]=6.99 \times 10^{-7} \mathrm{M}\) c. \(\left[\mathrm{H}^{+}\right]=4.01 \times 10^{-9} \mathrm{M}\) d. \(\left[\mathrm{H}^{+}\right]=4.02 \times 10^{-13} \mathrm{M}\)

Short answer

a. \(\left[\mathrm{OH}^-\right] = 7.46 \times 10^{-13} \mathrm{M}\) (acidic) b. \(\left[\mathrm{OH}^-\right] = 1.43 \times 10^{-8} \mathrm{M}\) (acidic) c. \(\left[\mathrm{OH}^-\right] = 2.49 \times 10^{-6} \mathrm{M}\) (basic) d. \(\left[\mathrm{OH}^-\right] = 2.49 \times 10^{-2} \mathrm{M}\) (basic)

Step-by-step solution

Recall the ion product of water, \(K_w\)

The ion product of water is given by the equation: \[K_w = \left[\mathrm{H}^+\right]\left[\mathrm{OH}^-\right]\] At 25°C, the value of \(K_w = 1.0 \times 10^{-14}\).

Find the concentration of \(\mathrm{OH}^-\) ions

Using the given concentration of \(\mathrm{H}^+\) ions and the ion product of water, solve for \(\left[\mathrm{OH}^-\right]\): a. For \(\left[\mathrm{H}^+\right]=1.34 \times 10^{-2} \mathrm{M}\): \[\left[\mathrm{OH}^-\right] = \frac{K_w}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{1.34 \times 10^{-2}} = 7.46 \times 10^{-13} \mathrm{M}\] b. For \(\left[\mathrm{H}^+\right]=6.99 \times 10^{-7} \mathrm{M}\): \[\left[\mathrm{OH}^-\right] = \frac{K_w}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{6.99 \times 10^{-7}} = 1.43 \times 10^{-8} \mathrm{M}\] c. For \(\left[\mathrm{H}^+\right]=4.01 \times 10^{-9} \mathrm{M}\): \[\left[\mathrm{OH}^-\right] = \frac{K_w}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{4.01 \times 10^{-9}} = 2.49 \times 10^{-6} \mathrm{M}\] d. For \(\left[\mathrm{H}^+\right]=4.02 \times 10^{-13} \mathrm{M}\): \[\left[\mathrm{OH}^-\right] = \frac{K_w}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{4.02 \times 10^{-13}} = 2.49 \times 10^{-2} \mathrm{M}\]

Determine whether the solution is acidic or basic

Compare the concentrations of \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) ions: a. \(\left[\mathrm{OH}^-\right] < \left[\mathrm{H}^+\right]\): The solution is acidic. b. \(\left[\mathrm{OH}^-\right] < \left[\mathrm{H}^+\right]\): The solution is acidic. c. \(\left[\mathrm{OH}^-\right] > \left[\mathrm{H}^+\right]\): The solution is basic. d. \(\left[\mathrm{OH}^-\right] > \left[\mathrm{H}^+\right]\): The solution is basic. In conclusion, the solutions in parts a and b are acidic, while the solutions in parts c and d are basic.

Key concepts

Acidic and Basic Solutions

An understanding of acidic and basic solutions is fundamental in chemistry, as it helps to identify the nature of a substance in an aqueous solution. Acids are substances that increase the concentration of hydrogen ions \(\mathrm{H}^+\) in a solution, while bases increase the concentration of hydroxide ions \(\mathrm{OH}^-\). The behavior of these ions in water is essential for a variety of biological and chemical processes.

An acidic solution has a higher concentration of \(\mathrm{H}^+\) ions than that of \(\mathrm{OH}^-\) ions. In contrast, a basic or alkaline solution has a greater concentration of \(\mathrm{OH}^-\) ions compared to the \(\mathrm{H}^+\) ions. When these ion concentrations are equal, the solution is considered neutral. These characteristics are not just random; they follow from the ion product of water, which establishes a relationship between \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) ion concentrations.

In the given exercise, we calculated the \(\mathrm{OH}^-\) ion concentration for various solutions and identified whether the solution was acidic or basic by comparing the concentrations of \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) ions. It's crucial to remember that this comparison directly influences the solution's classification as acidic or basic.

Ion Product of Water

The ion product of water \(K_w\) is a constant value that represents the product of the molar concentrations of \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) ions in water. At standard conditions, which is typically at 25°C (298 K), this equilibrium constant \(K_w\) is known to be \(1.0 \times 10^{-14} \text{M}^2\). This number is pivotal for understanding the balance between hydrogen and hydroxide ions in aqueous solutions and is crucial in many calculations in chemistry, particularly when determining the pH or pOH of a solution.

The ion product of water serves as a reference point and implies that, in pure water or any neutral solution, the concentrations of \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) are always equal to \(1.0 \times 10^{-7} \text{M}\). When \(\mathrm{H}^+\) concentration increases, making the solution acidic, the \(\mathrm{OH}^-\) concentration correspondingly decreases, and vice versa for basic solutions. This inverse relationship allows for the calculation of one ion's concentration when the other is known, as demonstrated in the step-by-step solution of the provided exercise. Knowing \(K_w\) and the concentration of either \(\mathrm{H}^+\) or \(\mathrm{OH}^-\), you can easily calculate the missing counterpart.

pH Calculation

The pH scale is a measure of the acidity or basicity of an aqueous solution. It is defined as the negative logarithm (base 10) of the \(\mathrm{H}^+\) ion concentration. Calculating the pH gives a convenient way of expressing the acidity of a solution on a scale from 0 to 14, where a pH less than 7 indicates an acidic solution, a pH greater than 7 indicates a basic solution, and a pH of exactly 7 means the solution is neutral, indicative of pure water.

The pH calculation is straightforward when the \(\mathrm{H}^+\) concentration is known: \[pH = -\log [\mathrm{H}^+]\]. Similarly, the pOH can be calculated using the concentration of \(\mathrm{OH}^-\) ions and is used to express the basicity of a solution: \[pOH = -\log [\mathrm{OH}^-]\]. Since \(pH + pOH = 14\) on the pH scale, if you know one value, you can easily determine the other. This relationship is a direct consequence of the ion product of water.

In the exercise provided, although the pH is not directly calculated, understanding the transformation from \(\mathrm{H}^+\) to \(\mathrm{OH}^-\) concentration using \(K_w\) is a stepping stone towards mastering pH calculation. It is important for students to recognize these related concepts to build a comprehensive knowledge of acid-base chemistry.