Problem 150 - Introductory Chemistry Atoms First

Question

What is the \(\mathrm{pH}\) of a solution whose \(\mathrm{OH}^{-}\) concentration is \(10^{-14} \mathrm{M} ?\) Is the solution acidic or basic?

Step-by-Step Solution

Answer

The pH of the solution with an OH⁻ concentration of \(10^{-14}\) M is 0. The solution is acidic.

Step 1: Given concentration of OH⁻ ions

The given concentration of OH⁻ ions in the solution is \(10^{-14}\) M (Molar).

Step 2: Use the ionic product of water

The ionic product of water, \(K_w\), relates the concentration of both H₃O⁺ and OH⁻ ions in the solution. According to the ionic product of water: \[K_w = [\mathrm{H}^+] [\mathrm{OH}^-]\] Where \(K_w\) is the ionic product of water (equals to \(1 \times 10^{-14}\)), \([\mathrm{H}^{+}]\) is the concentration of H⁺ ions and \([\mathrm{OH}^{-}]\) is the concentration of OH⁻ ions.

Step 3: Calculate the concentration of H⁺ ions

Since we know the concentration of OH⁻ ions and the value of \(K_w\), we can now calculate the concentration of H⁺ ions using the ionic product equation: \[1 \times 10^{-14} = [\mathrm{H}^+] (10^{-14})\] \[[\mathrm{H}^+] = \frac{1 \times 10^{-14}}{10^{-14}}\]

Step 4: Find the concentration of H⁺ ions

By solving for the concentration of H⁺ ions, we get: \[[\mathrm{H}^+] = \frac{1 \times 10^{-14}}{10^{-14}} = 1\]

Step 5: Calculate the pH value

Now, we can calculate the pH value of the solution: \[\mathrm{pH} = -\log_{10}[\mathrm{H}^+]\] \[\mathrm{pH} = -\log_{10}(1)\]

Step 6: Find the pH value

By solving for the pH value, we get: \[\mathrm{pH} = -\log_{10}(1) = 0\]

Step 7: Determine if the solution is acidic or basic

A solution with a pH value less than 7 is acidic, pH equal to 7 is neutral, and pH greater than 7 is basic. Since the pH value found is 0, the solution is acidic. To summarize, the pH of the solution is 0, and the solution is acidic.

Key Concepts

Ionic Product of Water

The ionic product of water, often represented as \(K_w\), is a fundamental concept in chemistry that describes the relationship between the concentrations of hydrogen ions \([\mathrm{H}^+]\) and hydroxide ions \([\mathrm{OH}^-]\) in water. This constant value is crucial for understanding the balance between acids and bases in a solution.
In pure water at 25°C, the ionic product is:

• \(K_w = [\mathrm{H}^+][\mathrm{OH}^-] = 1 \times 10^{-14}\)

This small number indicates that in pure water, the concentrations of hydrogen and hydroxide ions are both very low, each being \(1 \times 10^{-7} \mathrm{M}\).

The relationship is also fundamental when we want to calculate the concentration of one ion if we know the concentration of the other. For instance, if we know \([\mathrm{OH}^-]\), we can find \([\mathrm{H}^+]\) by rearranging the formula to:

• \([\mathrm{H}^+] = \frac{K_w}{[\mathrm{OH}^-]}\)

Acidic Solution

An acidic solution is one wherein the concentration of hydrogen ions \([\mathrm{H}^+]\) is greater than that of hydroxide ions \([\mathrm{OH}^-]\). This imbalance results in a \(\text{pH}\) value of less than 7, indicating acidity.

Acids are classified by their capability to donate hydrogen ions in aqueous solutions, which increases \([\mathrm{H}^+]\) concentration, thereby lowering the solution's \(\text{pH}\). For example, a solution with \( ext{pH} = 0\) indicates a high concentration of \([\mathrm{H}^+]\), making the solution strongly acidic.

Important points to remember about acidic solutions:

• \(\text{pH} < 7\)
• Higher \([\mathrm{H}^+]\) than \([\mathrm{OH}^-]\)
• Common examples include lemon juice and vinegar

pH Calculation

The \(\text{pH}\) is a measure of the acidity or basicity of a solution. It is calculated as the negative logarithm (base 10) of the hydrogen ion concentration:

• \(\text{pH} = -\log_{10}[\mathrm{H}^+]\)

This formula comes in handy for transforming a wide range of hydrogen ion concentrations into a more manageable scale. For example, in the original problem, given the \([\mathrm{OH}^-] = 10^{-14} \, M\), first we find \([\mathrm{H}^+]\) using the ionic product of water, which in turn allows us to calculate the \(\text{pH}\):

• \(\text{pH} = -\log_{10}(1) = 0\)

It indicates a very acidic environment.

Remember:

• \(\text{pH}\) measures how acidic or basic a solution is
• Values range from 0 to 14
• Lower \(\text{pH}\) means higher acidity