Question

Autoionization of Water and pH Calculate \(\left[\mathrm{OH}^{-}\right]\) in each aqueous solution at \(25^{\circ} \mathrm{C},\) and classify the solution as acidic or basic. $$\begin{array}{l}{\text { a. }\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.2 \times 10^{-8} \mathrm{M}} \\ {\text { b. }\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=8.5 \times 10^{-5} \mathrm{M}} \\ {\text { c. }\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=3.5 \times 10^{-2} \mathrm{M}}\end{array}$$

Short answer

a) \([OH^-]= 8.33 \times 10^{-7} M\), basic; b) \([OH^-]= 1.18 \times 10^{-10} M\), acidic; c) \([OH^-]= 2.86 \times 10^{-13} M\), acidic.

Step-by-step solution

Applying the Water Ionization Constant (Kw)

The autoionization of water can be represented by the equation: \[H_2O(l) \leftrightarrow H^+(aq) + OH^-(aq)\]. At 25°C, the ionization constant of water (Kw) is \(1.0 \times 10^{-14}\). This value is the product of the concentrations of the hydrogen ions \([H^+]\) and the hydroxide ions \([OH^-]\): \(Kw = [H^+][OH^-]\). To find \([OH^-]\), we can rearrange this equation to \([OH^-] = \frac{Kw}{[H^+]}\).

Calculating \[OH^-\rbrack in Solution a

For solution a, where \([H_3O^+] = 1.2 \times 10^{-8} M\), we can use \([OH^-] = \frac{Kw}{[H^+]}\) to find the hydroxide ion concentration. Plugging the numbers in, we get: \[ [OH^-] = \frac{1.0 \times 10^{-14}}{1.2 \times 10^{-8}} = 8.33 \times 10^{-7} M \]. Since \([OH^-]\) is higher than \(1.0 \times 10^{-7} M\), the solution is basic.

Calculating \[OH^-\rbrack in Solution b

For solution b, with \([H_3O^+] = 8.5 \times 10^{-5} M\), we find \([OH^-]\) as follows: \[ [OH^-] = \frac{1.0 \times 10^{-14}}{8.5 \times 10^{-5}} = 1.18 \times 10^{-10} M \]. In this case, \([OH^-]\) is less than \(1.0 \times 10^{-7} M\), so the solution is acidic.

Calculating \[OH^-\rbrack in Solution c

For solution c, where \([H_3O^+] = 3.5 \times 10^{-2} M\), we calculate \([OH^-]\) as: \[ [OH^-] = \frac{1.0 \times 10^{-14}}{3.5 \times 10^{-2}} = 2.86 \times 10^{-13} M \]. This concentration of \([OH^-]\) is also less than \(1.0 \times 10^{-7} M\), indicating that the solution is acidic.

Key concepts

pH calculation

Understanding the pH of a solution is critical for determining its acidity or basicity. The pH is a logarithmic scale that measures the hydrogen ion concentration ([H+] ) in a solution, calculated using the formula: pH = - log ([H+]). A lower pH value indicates a more acidic solution, while a higher pH means the solution is more basic. When we have the concentration of hydronium ions ([H3O+] ), which are often interchangeable with hydrogen ions in calculations, we can directly apply it to find the pH. If the concentration of hydronium ions in a solution decreases, inversely the pH increases, indicating a shift towards basicity. Remember that at 25°C, a neutral solution has a pH of 7; any value below this is acidic, and any value above it is basic.
For example, in step 2 of the provided solution where the concentration of hydronium ions ([H3O+] ) is 1.2 x 10^-8 M, the pH would be calculated as pH = -log (1.2 x 10^-8).

acidic and basic solutions

Acidic and basic are two extremes that describe chemical properties of solutions. Acidic solutions have a higher concentration of hydrogen ions (H+) and a pH below 7. In contrast, basic solutions contain a higher concentration of hydroxide ions (OH^-) and have a pH above 7. When the concentration of hydrogen ions exceeds that of hydroxide ions, the solution becomes acidic, and inversely, when the hydroxide ion concentration is greater, the solution turns basic. The calculations in the textbook solution illuminate this concept by comparing the resulting [OH^-] concentrations to the neutral water concentration, which is 1.0 x 10^-7 M at 25°C. Solutions with a lower hydronium ion concentration than this neutral marker are basic, as seen in step 2, while higher concentrations, as in steps 3 and 4, indicate acidic solutions.

water ionization constant

Every chemical reaction has an equilibrium constant, and for the autoionization of water, this is the water ionization constant, denoted as Kw. At 25°C, Kw is constant and equal to 1.0 x 10^-14. It is calculated based on the product of the molar concentrations [H+] and [OH^-] at equilibrium: Kw = [H+][OH^-]. This equilibrium lies far to the left, meaning that under standard conditions, water exists mostly as H2O molecules with very few hydrogen or hydroxide ions. The equilibrium can shift if additional acids or bases are added, affecting the concentration of H+ and OH^- ions, and thereby the pH. Understanding Kw is essential for calculating the hydroxide ion concentration from a known hydrogen ion concentration and vice versa.

hydroxide ion concentration

The concentration of hydroxide ions in a solution is key to classifying the solution's acidity or basicity. This concentration can be readily determined if we know the concentration of hydrogen ions and the water ionization constant (Kw). The relationship [OH^-] = Kw / [H+] allows us to solve for the hydroxide ion concentration when hydrogen ion concentration is provided, as done in the exercise solutions. With the concentration of OH^-, we can assess the solution's nature; concentrations above 1.0 x 10^-7 M indicate basic solutions, and those below indicate acidic solutions. Often, discovering the hydroxide ion concentration is an intermediate step for finding pH or pOH values, which are more commonly used to describe the acidity or basicity of a solution.