Question

Calculate the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) of each aqueous solution with the following [OH \(^{-}\) ]: a. vinegar, \(1.0 \times 10^{-11} \mathrm{M}\) b. urine, \(2.0 \times 10^{-9} \mathrm{M}\) c. ammonia, \(5.6 \times 10^{-3} \mathrm{M}\) d. \(\mathrm{NaOH}\) solution, \(2.5 \times 10^{-2} \mathrm{M}\)

Short answer

a. \(1.0 \times 10^{-3} \text{M}\); b. \(5.0 \times 10^{-6} \text{M}\); c. \(1.79 \times 10^{-12} \text{M}\); d. \(4.0 \times 10^{-13} \text{M}\).

Step-by-step solution

Understand the Relationship

The relationship between the concentration of hydronium ions \(\text{H}_3\text{O}^+\) and hydroxide ions \(\text{OH}^−\) in water is given by the ion-product constant for water: \[K_w = \text{[H}_3\text{O}^+\text{][OH}^−\text{]} = 1.0 \times 10^{-14} \] at 25°C.

Determine \(\text{[H}_3\text{O}^+]\)

To find the concentration of hydronium ions \(\text{[H}_3\text{O}^+]\), we use the formula \[ \text{[H}_3\text{O}^+] = \frac{K_w}{\text{[OH}^−\text{]}} \]

Calculate \(\text{[H}_3\text{O}^+]\) for Vinegar

For vinegar with \(\text{[OH}^−\text{]} = 1.0 \times 10^{-11} \text{M}\), we have \[ \text{[H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-11}} = 1.0 \times 10^{-3} \text{M} \]

Calculate \(\text{[H}_3\text{O}^+]\) for Urine

For urine with \(\text{[OH}^−\text{]} = 2.0 \times 10^{-9} \text{M}\), we have \[ \text{[H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{2.0 \times 10^{-9}} = 5.0 \times 10^{-6} \text{M} \]

Calculate \(\text{[H}_3\text{O}^+]\) for Ammonia

For ammonia with \(\text{[OH}^−\text{]} = 5.6 \times 10^{-3} \text{M}\), we have \[ \text{[H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{5.6 \times 10^{-3}} = 1.79 \times 10^{-12} \text{M} \]

Calculate \(\text{[H}_3\text{O}^+]\) for \(\text{NaOH}\text{ Solution}\)

For \(\text{NaOH}\text{ solution with }\text{[OH}^−\text{]} = 2.5 \times 10^{-2} \text{M}\), we have \[ \text{[H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{2.5 \times 10^{-2}} = 4.0 \times 10^{-13} \text{M} \]

Key concepts

hydronium ion concentration

In acid-base chemistry, understanding hydronium ion concentration is essential. Hydronium ions \(\text{H}_3\text{O}^+\) form when water molecules accept a proton \(\text{H}^+\). The concentration of these ions determines the acidity of the solution. Hydronium ion concentration is crucial in many chemical and biological processes. Typically, this concentration is measured in molarity (M), which represents moles of \(\text{H}_3\text{O}^+\) per liter of solution. In neutral water at 25°C, the \(\text{H}_3\text{O}^+\) concentration is about \(1.0 \times 10^{-7} \text{M}\). Changes in this concentration can significantly alter the solution’s properties.

pH calculation

pH is a scale used to quantify the acidity or basicity of a solution. It's calculated using the hydronium ion concentration. The formula for pH is: \[ \text{pH} = -\text{log}\text{[H}_3\text{O}^+] \] This logarithmic scale means that a small change in \(\text{H}_3\text{O}^+\) concentration results in a significant change in pH. For example, if \( \text{[H}_3\text{O}^+] = 1.0 \times 10^{-3} \text{M} \), the pH would be 3, indicating a highly acidic solution. On the other hand, if \( \text{[H}_3\text{O}^+] = 1.0 \times 10^{-10} \text{M} \), the pH would be 10, indicating a basic solution.

hydroxide ion concentration

Hydroxide ions \( \text{OH}^- \) are key players in determining a solution's basicity. They form when water molecules lose a proton \( \text{H}^+ \). The concentration of hydroxide ions is also measured in molarity. There's an inverse relationship between hydronium ion and hydroxide ion concentrations, described by the ion-product constant for water \( K_w \). When you know one concentration, you can easily find the other using the formula: \[ \text{[H}_3\text{O}^+] \times \text{[OH}^- \text{]} = K_w \] This relationship ensures that if a solution has a high concentration of \(\text{OH}^−\), it has a low concentration of \(\text{H}_3\text{O}^+\), making it basic.

Kw constant

The ion-product constant for water \[K_w = \text{[H}_3\text{O}^+][\text{OH}^-] \] is a fundamental concept in acid-base chemistry. At 25°C, \( K_w \) is equal to \(1.0 \times 10^{-14} \). This constant helps maintain the balance between hydronium and hydroxide ions in water. If \(\text{H}_3\text{O}^+\) concentration increases, \(\text{OH}^-\) concentration must decrease to keep \(\text{Kw}\) constant, and vice versa. Understanding this helps in predicting the behavior of acids and bases in solutions, making it a cornerstone in studying chemical equilibria.

acid-base chemistry

Acid-base chemistry revolves around the transfer of protons \(\text{H}^+\) between substances. Acids are proton donors, while bases are proton acceptors. This interaction affects the concentration of hydronium and hydroxide ions in a solution. The strength of an acid or base is often described by its dissociation in water. Strong acids/bases dissociate completely, while weak acids/bases only partially dissociate. The pH scale ranging from 0 to 14 helps quantify the acidity or basicity of a solution. Here, 7 is neutral, below 7 is acidic, and above 7 is basic. This scale and these interactions play critical roles in various fields, from industrial processes to biological systems.