Problem 55 - 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. A grape has a pH of \(3.5,\) and baking soda has a pH of \(8.0 .\) The hydrogen ion concentration of the grape is how many times that of the baking soda?

Step-by-Step Solution

Answer

The Hydrogen Ion concentration of the grape is \(10^{4.5} \) times that of the baking soda.

Step 1: Convert the pH values to Hydrogen Ion Concentration

Use the formula \( \mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) to calculate the Hydrogen Ion concentration. Rearranging the formula results in \(\left[\mathbf{H}^{+}\right] = 10^{-\mathbf{p H}}\). Thus, by substituting given pH values for grape and baking soda respectively: \(\left[\mathbf{H}^{+}\right]_{grape} = 10^{-3.5} \) \(\left[\mathbf{H}^{+}\right]_{baking~soda} = 10^{-8.0}\)

Step 2: Calculate the ratio

The task is to find how many times the hydrogen ion concentration of the grape is that of baking soda. This can be computed by dividing \(\left[\mathbf{H}^{+}\right]_{grape} \) by \( \left[\mathbf{H}^{+}\right]_{baking~soda}\). This results in the ratio = \( \frac{10^{-3.5}}{10^{-8.0}} = 10^{8 - 3.5} = 10^{4.5}\)

Key Concepts

Hydrogen Ion Concentration

In our daily lives, we often encounter different substances with varying degrees of acidity or alkalinity. This property is closely related to the concept of hydrogen ion concentration. The concentration of hydrogen ions in a solution is represented as \( [\mathbf{H}^{+}] \) and measured in moles per liter (mol/L). Hydrogen ions play a crucial role in determining the pH of a solution, which indicates how acidic or basic it is. A high concentration of \( [\mathbf{H}^{+}] \) means the solution is more acidic, while a lower concentration implies it is more basic or alkaline. In our example, we calculated the hydrogen ion concentrations for grape juice and baking soda using their respective pH values. This step allows us to see the true acidity level beyond just a pH number.

Logarithmic Functions

Logarithmic functions are mathematical functions that help us solve equations involving exponential growth or decay. In the context of pH calculation, the logarithmic function simplifies how we express very small concentrations of hydrogen ions.The acidity model uses the formula \( \mathbf{pH} = -\log [\mathbf{H}^{+}] \), which is derived from logarithmic properties. This formula allows us to go from pH to hydrogen ion concentration by rearranging to \( [\mathbf{H}^{+}] = 10^{-\mathbf{pH}} \). This equation is handy because hydrogen ion concentrations can be extremely small numbers, making them cumbersome to work with directly. The logarithmic scale compresses these values into a more manageable form. Understanding this link between logarithms and pH is crucial for interpreting how acidic or basic a solution truly is.

Acidity Model

The acidity model, given by the formula \( \mathbf{pH} = -\log [\mathbf{H}^{+}] \), is a standard way to express the acidity or basicity of a solution. This model is immensely valuable because it provides a simple way to compare different substances. For example, in the original exercise, by applying the acidity model, we calculated and compared the hydrogen ion concentrations of grape juice and baking soda. Grape juice with a lower pH of 3.5 is much more acidic than baking soda with a pH of 8.0. In fact, we found that grape juice is \( 10^{4.5} \) times more acidic than baking soda. This model underpins a range of scientific fields, from chemistry to environmental science, because it encapsulates complex concentration values within a straightforward logarithmic scale. By grasping the acidity model, students can better understand the relationship between pH and real-world applications.