Problem 69 - Precalculus : Building Concepts and Connections

Question

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Suppose solution A has a pH of 5 and solution B has a pH of \(9 .\) What is the ratio of the concentration of hydrogen ions in solution \(A\) to the concentration of hydrogen ions in solution B?

Step-by-Step Solution

Answer

The ratio of the concentration of hydrogen ions in solution A to the concentration in solution B is 10,000:1

Step 1: Calculating the concentration of hydrogen ions in solution A

The pH of solution A is given as 5. Remember that pH is defined as the negative log base 10 of the concentration of H+ ions. Therefore, to find the concentration of H+ ions in solution A, we calculate 10 raised to the power of -pH. In this case, it would be \(10^{-5}\).

Step 2: Calculating the concentration of hydrogen ions in solution B

The pH of solution B is given as 9. Similar to step 1, apply the formula to find the concentration of H+ ions in solution B, and in this case it would be \(10^{-9}\).

Step 3: Calculating the ratio

The exercise asks for the ratio of the concentration of H+ ions in solution A to that in solution B. This means dividing the concentration from solution A by the concentration from solution B. Therefore, the ratio would be \(\frac{10^{-5}}{10^{-9}}\), which simplifies to \(10^4\) or 10000:1. This means that solution A has 10,000 times as many H+ ions as solution B.

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a crucial concept in understanding the acidity or alkalinity of a solution. It represents the amount of hydrogen ions (H extsuperscript{+}) present, which directly influences a solution's pH. A high concentration of hydrogen ions indicates an acidic solution, while a low concentration signifies basicity.
To determine the hydrogen ion concentration from a known pH value, we use the formula:

• The concentration \( \text{[H}^+\text{]} \) is calculated as \( 10^{-\text{pH}} \).

This formula stems from the definition of pH itself, where pH is the negative logarithm (base 10) of the hydrogen ion concentration. Understanding this relationship helps explain why even a small change in pH reflects a significant change in hydrogen ion concentration.

• For example, if the pH changes from 5 to 6, the hydrogen ion concentration decreases by a factor of 10.

Remember, the smaller the pH value, the greater the number of hydrogen ions present, making the solution more acidic.

Logarithms

Logarithms are essential mathematical tools that help in compressing large ranges of values into more manageable numbers, making them incredibly useful in disciplines like chemistry. In the context of pH, logarithms allow us to express hydrogen ion concentrations succinctly.
The pH scale uses a logarithmic model for this purpose:

• The formula \( \text{pH} = -\log_{10}(\text{[H}^+\text{]}) \) relies on the base-10 logarithm.

This implies that if the hydrogen ion concentration changes by a factor of ten, the pH changes by just one unit. This scale is particularly advantageous because it simplifies the wide range of hydrogen ion concentrations from strongly acidic to strongly basic solutions, spanning from 1 M to 10^{-14} M in a more digestible range of pH values from 0 to 14.
Understanding logarithms in pH calculations is vital because it reveals how the seemingly small changes in pH numbers can signify substantial shifts in acidity or basicity.

Acid-Base Chemistry

Acid-base chemistry is a fascinating field that explores the interactions between acidic and basic substances. Acidic solutions have higher concentrations of hydrogen ions (H extsuperscript{+}) and lower pH values, indicating they are more 'acidic'. In contrast, basic solutions have fewer hydrogen ions and higher pH values, making them more 'basic' or 'alkaline'.
Let's break down some key points:

• Acids donate hydrogen ions to the solution. As a result, they lower the pH.
• Bases, on the other hand, accept hydrogen ions and increase the pH.

The balance between acids and bases is crucial in many biological and chemical processes, including digestion, respiration, and water treatment. Knowing how to manipulate the concentration of hydrogen ions allows chemists to predict the results of chemical reactions and develop useful compounds for everyday use.