Problem 48 - Precalculus with Calculus Previews

Question

Determine the hydrogen-ion concentration \(\left[\mathrm{H}^{+}\right]\) of a solution with the given \(\mathrm{pH}\). $$ 8.1 $$

Step-by-Step Solution

Answer

\([ ext{H}^+]\) is approximately \(7.94 \times 10^{-9}\) mol/L.

Step 1: Understanding the pH Formula

The pH of a solution is defined by the formula: \( ext{pH} = - ext{log}_{10}[ ext{H}^+] \). This formula relates the pH of a solution to the hydrogen-ion concentration, \([ ext{H}^+]\).

Step 2: Rearranging the Formula to Find \([ ext{H}^+]\)

To find the hydrogen-ion concentration \([ ext{H}^+]\) from the pH, we need to rearrange the formula: \([ ext{H}^+] = 10^{- ext{pH}}\).

Step 3: Calculating \([ ext{H}^+]\) for pH 8.1

Given the pH value of 8.1, we apply the rearranged formula: \([ ext{H}^+] = 10^{-8.1}\).

Step 4: Solving for \([ ext{H}^+]\)

Calculate \(10^{-8.1}\) using a calculator to find \([ ext{H}^+]\). The calculated hydrogen-ion concentration is approximately \(7.94 imes 10^{-9}\) mol/L.

Key Concepts

Hydrogen-ion Concentration

Hydrogen-ion concentration, often denoted as \([H^+]\), is a measure of the number of hydrogen ions present in a solution. In the context of acid-base chemistry, this concentration helps us understand the acidity or basicity of a solution. The more hydrogen ions in a solution, the more acidic it is. Conversely, fewer hydrogen ions indicate a more basic (or alkaline) solution.

This concentration is crucial because it directly affects the pH of the solution. A high concentration of hydrogen ions results in a low pH, making the solution acidic. On the other hand, a low concentration results in a high pH, indicating an alkaline solution.

When calculating hydrogen-ion concentration, it’s essential to understand its relationship with pH, which is commonly used to express the acidity of a solution.

Logarithmic Functions

Logarithmic functions are used to deal with data that change over several orders of magnitude. In chemistry, the logarithmic scale of pH is used to express hydrogen-ion concentration. This scale is logarithmic because the concentration of hydrogen ions in solutions can vary over a wide range.

The pH scale is based on the formula: \(\text{pH} = -\log_{10}[H^+]\). This equation shows that pH is the negative logarithm (base 10) of the hydrogen-ion concentration. Because of the logarithmic nature, a small change in pH represents a significant change in hydrogen-ion concentration.

• A decrease in pH by 1 unit means a 10-fold increase in hydrogen-ion concentration.
• An increase in pH by 1 unit means a 10-fold decrease in hydrogen-ion concentration.

This property makes the logarithmic scale essential in comparing the acidity or alkalinity levels in different solutions effectively.

Acid-base Chemistry

Acid-base chemistry is a vital branch of chemistry that deals with the qualitative and quantitative aspects of acids and bases in solutions. Acids are substances that increase the hydrogen-ion concentration when dissolved in water. Bases, on the other hand, decrease it.

Understanding acid-base chemistry involves grasping how hydrogen-ion concentrations influence the properties and behaviors of solutions. The pH scale, which ranges from 0 to 14, is central to this study:

• pH less than 7 signifies an acidic solution where hydrogen-ion concentration is relatively high.
• pH of 7 is neutral, typical for pure water at 25°C.
• pH greater than 7 indicates a basic solution with lower hydrogen-ion concentration.

In practical terms, acid-base chemistry helps us predict how substances like acids and bases will react when mixed. For example, adding a base to an acidic solution will decrease the solution's hydrogen-ion concentration and increase the pH, making the solution less acidic.