Question

(a) What are \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right],\left[\mathrm{OH}^{-}\right],\) and \(\mathrm{pOH}\) in a solution with a \(\mathrm{pH}\) of \(8.97 ?\) (b) What are \(\left[\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{J},\left[\mathrm{OH}^{-}\right],\right.\) and \(\mathrm{pH}\) in a solution with a pOH of \(11.27 ?\)

Short answer

(a) \( \text{[H}_3 \text{O}^+] = 10^{-8.97}, \text{[OH}^-] = 10^{-5.03}, \text{pOH} = 5.03 \). (b) \( \text{[H}_3 \text{O}^+] = 10^{-2.73}, \text{[OH}^-] = 10^{-11.27}, \text{pH} = 2.73 \).

Step-by-step solution

Understanding pH and pOH Relationship

Recall that \( \text{pH} + \text{pOH} = 14 \). This relationship will help in finding pOH from pH and vice versa.

Calculate pOH from pH

Given the pH is 8.97, use \( \text{pOH} = 14 - \text{pH} \) to find pOH. Thus, \( \text{pOH} = 14 - 8.97 = 5.03 \).

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

Use the formula \( \text{[H}_3 \text{O}^+] = 10^{-\text{pH}} \) to find the concentration of \(\text{[H}_3 \text{O}^+]\text\). So, \( \text{[H}_3 \text{O}^+} = 10^{-8.97} \).

Calculate \( \text{[OH}^-\text] \)

Use the formula \( \text{[OH}^-] = 10^{-\text{pOH}} \) to find the concentration of hydroxide ions. Thus, \( \text{[OH}^-] = 10^{-5.03} \).

Calculate pH from pOH for Part (b)

Given pOH is 11.27, use \( \text{pH} = 14 - \text{pOH} \). Thus, \( \text{pH} = 14 - 11.27 = 2.73 \).

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

Using the formula \( \text{[H}_3 \text{O}^+] = 10^{-\text{pH}} \), we get \( \text{[H}_3 \text{O}^+} = 10^{-2.73} \).

Calculate \( \text{[OH}^-\text] \) for Part (b)

Using \( \text{[OH}^-] = 10^{-\text{pOH}} \), we get that \( \text{[OH}^-] = 10^{-11.27} \).

Key concepts

Acid-Base Chemistry

In chemistry, understanding acids and bases is crucial as they play a vital role in many chemical reactions. Acids are substances that increase the concentration of hydronium ions \(\text{H}_3 \text{O}^+\) when dissolved in water, while bases increase the concentration of hydroxide ions \((\text{OH}^-)\). The strength of acids and bases is measured using the\ \text{pH} scale, which ranges from 0 to 14.

Substances with a \text{pH} less than 7 are considered acidic, while those with a \text{pH} greater than 7 are basic, and a \text{pH} of 7 is neutral. Knowing how to calculate the \text{pH} and \text{pOH} of a solution allows us to determine its acidity or basicity.

The relationship between \text{pH} and \text{pOH} is given by the formula:
\[ \text{pH} + \text{pOH} = 14 \].

This relationship helps in converting \text{pH} to \text{pOH} and vice versa, which is fundamental in acid-base calculations.

Concentration Calculations

Understanding the concentration of ions in a solution is essential. The concentration of hydronium ions \(\text{[H}_3 \text{O}^+]\) and hydroxide ions \((\text{[OH}^-])\) can be calculated from the \text{pH} and \text{pOH} values using the following formulas:

• \[\text{[H}_3 \text{O}^+] = 10^{-\text{pH}}\]
• \[\text{[OH}^-] = 10^{-\text{pOH}}\]

These formulas help us determine how acidic or basic a solution is beyond just the \text{pH} or \text{pOH} value.

For example, to find the concentration of hydronium ions in a solution with a \text{pH} of 8.97, we use \[\text{[H}_3 \text{O}^+] = 10^{-8.97}\]. Similarly, to find the concentration of hydroxide ions in the same solution, we first need the \text{pOH}, calculated as \[\text{pOH} = 14 - \text{pH} = 5.03\]. Then, \[\text{[OH}^-] = 10^{-5.03}\].

Hydronium and Hydroxide Ions

The hydronium ion \(\text{H}_3 \text{O}^+\) and hydroxide ion \(\text{OH}^-\) are fundamental to understanding acid-base chemistry.

When an acid dissolves in water, it releases protons \(\text{H}^+\), which combine with water molecules to form hydronium ions \(\text{H}_3 \text{O}^+\). For example, hydrochloric acid \(\text{HCl}\) dissociates as follows:
\[ \text{HCl} \rightarrow \text{H}^+ + \text{Cl}^- \rightarrow \text{H}_3 \text{O}^+ + \text{Cl}^- \]

When a base is added to water, it releases hydroxide ions \(\text{OH}^-\). An example is sodium hydroxide \(\text{NaOH}\), which dissociates as follows:
\ \[ \text{NaOH} \rightarrow \text{Na}^+ + \text{OH}^- \].

The concentrations of these ions determine the \text{pH} and \text{pOH} of the solution. By knowing one, you can calculate the other using \[\text{pH} + \text{pOH} = 14\], providing a deeper understanding of the solution's nature.

To illustrate, for a solution with \text{pOH} of 11.27, converting it to \text{pH} gives us \[\text{pH} = 14 - 11.27 = 2.73\]. Using the formulas, we calculate \[\text{[H}_3 \text{O}^+] = 10^{-2.73}\] and \[\text{[OH}^-] = 10^{-11.27}\], which reflect the ion concentrations in solution and indicate its acidic nature.