Question

Calculate the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) of each aqueous solution with the following \(\left[\mathrm{OH}^{-}\right]:\) a. baking soda, \(1.0 \times 10^{-6} \mathrm{M}\) b. orange juice, \(5.0 \times 10^{-11} \mathrm{M}\) c. milk, \(2.0 \times 10^{-8} \mathrm{M}\) d. bleach, \(2.1 \times 10^{-3} \mathrm{M}\)

Short answer

a. \[1.0 \times 10^{-8} \text{ M}\ b. \[2.0 \times 10^{-4} \text{ M}\ c. \[5.0 \times 10^{-7} \text{ M}\ d. \[4.8 \times 10^{-12} \text{ M}\

Step-by-step solution

- Identify the relationship between \([\text{H}_3\text{O}^+]\) and \([\text{OH}^-]\)

Use the water dissociation constant: \[K_w = [\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14} \] This relationship holds true for all aqueous solutions at 25°C.

- Solve for \([\text{H}_3\text{O}^+]\) - Case (a) Baking soda

Given \[ [\text{OH}^-] = 1.0 \times 10^{-6} \text{ M} \] Substitute into the formula: \[ [\text{H}_3\text{O}^+] = \frac{K_w}{[\text{OH}^-]} = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-6}} = 1.0 \times 10^{-8} \text{ M} \]

- Solve for \([\text{H}_3\text{O}^+]\) - Case (b) Orange juice

Given \[ [\text{OH}^-] = 5.0 \times 10^{-11} \text{ M} \] Substitute into the formula: \[ [\text{H}_3\text{O}^+] = \frac{K_w}{[\text{OH}^-]} = \frac{1.0 \times 10^{-14}}{5.0 \times 10^{-11}} = 2.0 \times 10^{-4} \text{ M} \]

- Solve for \([\text{H}_3\text{O}^+]\) - Case (c) Milk

Given \[ [\text{OH}^-] = 2.0 \times 10^{-8} \text{ M} \] Substitute into the formula: \[ [\text{H}_3\text{O}^+] = \frac{K_w}{[\text{OH}^-]} = \frac{1.0 \times 10^{-14}}{2.0 \times 10^{-8}} = 5.0 \times 10^{-7} \text{ M} \]

- Solve for \([\text{H}_3\text{O}^+]\) - Case (d) Bleach

Given \[ [\text{OH}^-] = 2.1 \times 10^{-3} \text{ M} \] Substitute into the formula: \[ [\text{H}_3\text{O}^+] = \frac{K_w}{[\text{OH}^-]} = \frac{1.0 \times 10^{-14}}{2.1 \times 10^{-3}} \approx 4.8 \times 10^{-12} \text{ M} \]

Key concepts

water dissociation constant

The water dissociation constant, often represented as \(K_w\), is a crucial concept in chemistry. \(K_w\) refers to the equilibrium constant for the autoionization of water. In pure water, a small number of water molecules naturally dissociate into hydronium ions (\( \text{H}_3\text{O}^+ \)) and hydroxide ions (\( \text{OH}^- \)). The constant expression is given by the equation: \[ K_w = [\text{H}_3\text{O}^+][\text{OH}^-] \].

At 25°C, this constant has a value of \(1.0 \times 10^{-14} \). This value remains the same for any aqueous solution at this temperature. Understanding \(K_w\) helps us determine the relationship between hydronium and hydroxide ions in a solution.

Kw

Further exploring \(K_w\), it is essential to realize it's derived from the concentration of the ions produced during water's autoionization. If a solution is neutral, meaning \( pH = 7 \), then the concentrations of \( \text{H}_3\text{O}^+ \) and \( \text{OH}^- \) are equal, both being \(1.0 \times 10^{-7} \).

However, in acidic solutions, the concentration of \( \text{H}_3\text{O}^+ \) exceeds that of \( \text{OH}^- \). In contrast, alkaline solutions have higher concentrations of \( \text{OH}^- \) compared to \( \text{H}_3\text{O}^+ \).

To find the hydronium ion concentration in any solution, we use the relationship: \[ [\text{H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{[\text{OH}^-]} \]. By substituting the known hydroxide ion concentration into this formula, we can conveniently calculate \( \text{H}_3\text{O}^+ \).

pH and pOH

The pH and pOH scales are measures of a solution's acidity or basicity. pH refers to the 'power of hydrogen' and is calculated as: \[ \text{pH} = -\text{log}[\text{H}_3\text{O}^+] \].

On the other hand, pOH stands for the 'power of hydroxide' and is defined as: \[ \text{pOH} = -\text{log}[\text{OH}^-] \].

Both scales are intertwined and follow the equation: \[ \text{pH} + \text{pOH} = 14 \] at 25°C. This relationship allows us to convert between pH and pOH easily. For instance, if you know the pOH of a solution, you can determine its pH by subtracting the pOH value from 14.

In acidic solutions, pH values range from 0 to 7, while basic solutions have pH values from 7 to 14. Understanding these scales is fundamental in analyzing a solution's properties.

hydronium ion concentration

Hydronium ion concentration, symbolized as [\text{H}_3\text{O}^+], is a pivotal factor in determining a solution's acidity. Given the hydroxide ion concentration ([\text{OH}^-]), you can calculate the hydronium ion concentration using the water dissociation constant (\(K_w\)).

For example, in the exercise provided, various aqueous solutions such as baking soda, orange juice, milk, and bleach were analyzed. By substituting the given hydroxide ion concentrations into the formula \[ [\text{H}_3\text{O}^+] = \frac{K_w}{[\text{OH}^-]} \], we determined their respective hydronium ion concentrations.

Remember, this relationship holds true at 25°C, making it a reliable method for assessing whether a solution is acidic or basic. Grasping this concept simplifies tasks involving pH, pOH, and overall chemical equilibrium.