Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (in moles of hydrogen per liter) of a solution. The \(\mathrm{pH}\) of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor?

Step-by-Step Solution

Answer

The hydrogen ion concentration increases by a factor of 10.

Step 1: Understand the concept of pH

The acidity of a solution is defined by the parameter pH which is derived from the concentration of hydrogen ions \( [\mathrm{H}^{+}] \) present per liter of the solution, according to the equation \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\). A decrease in pH corresponds to an increase in the concentration of hydrogen ions.

Step 2: Identify the change in pH

The phrase 'the pH of a solution is decreased by one unit' refers to a change in pH by one unit, that is, \(Δ\mathbf{p H}=-1\).

Step 3: Compute the change in hydrogen ion concentration

The change in concentration of hydrogen ions can be computed noting that the change in \( \mathrm{pH} \) unit by -1 means the hydrogen ion concentration changes by a factor of 10.This is due to the nature of the logarithm base 10 function that underlies the pH scale. Thus, when the pH value decreases by 1 unit, the hydrogen ion concentration increases by a factor of 10.

Key Concepts

Understanding the Acidity Model

The concept of acidity in chemistry revolves around the presence of hydrogen ions \( \[\mathrm{H}^{+}\] \) in a solution. The more hydrogen ions present, the more acidic the solution is. To quantify this acidity, a convenient scale called pH was established. pH is a logarithmic measure, meaning it uses powers of ten to represent changes in hydrogen ion concentration. The acidity model given by the equation \(\mathbf{pH}=-\log \left[\mathbf{H}^{+}\right]\) provides a direct way to compute the acidity of a solution from the hydrogen ion concentration.

The pH scale ranges from 0 to 14, where 7 represents a neutral solution—like pure water—, values less than 7 indicate an acidic solution, and values greater than 7 indicate a basic or alkaline solution. This scale allows for easy comparison of solution acidity and serves as a fundamental concept in fields like chemistry, biology, and environmental science.

Hydrogen Ion Concentration

Delving into hydrogen ion concentration helps us understand the intricacies behind the acidity of solutions. Hydrogen ions \( \[\mathrm{H}^{+}\] \) are basically protons, and their abundance in a solution determines its acidity. The concentration is usually measured in moles per liter (M), and it's a critical factor in the pH equation.

When the exercise mentions a decrease in pH by one unit, it corresponds to a tenfold increase in hydrogen ion concentration. This is a crucial point for students to comprehend as it directly relates the pH scale with hydrogen ion concentration in a proportional manner; a decrease in pH results in a mathematical increase in the concentration of \( \[\mathrm{H}^{+}\] \) and vice versa.

The Logarithm Base 10 in pH Calculation

• Understanding logarithms is key to grasping the pH scale. The base 10 logarithm is used to manage the wide range of hydrogen ion concentrations in solutions. When the equation \(\mathbf{pH}=-\log \left[\mathbf{H}^{+}\right]\) is applied, it implies that for every unit change in pH, the hydrogen ion concentration changes by a factor of 10.
• For students, visualizing the logarithmic scale can be helpful. Think of a staircase where each step goes up or down by a factor of 10 rather than being equally spaced. In terms of pH, moving down one 'step' (a decrease in pH by 1) means climbing up to a ten times higher concentration of hydrogen ions.

This logarithmic relationship ensures that small changes in pH translate to significant modifications in hydrogen ion concentration, which is essential for understanding chemical reactions, biological processes, and environmental impacts.