Problem 79 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

ACIDITY (pH) OF A SOLUTION The acidity of a solution is measured by its \(\mathrm{pH}\) value, which is defined by \(\mathrm{pH}=-\log _{10}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\), where \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) is the hydronium ion concentration (moles/liter) of the solution. On average, milk has a pH value that is three times the \(\mathrm{pH}\) value of a lime, which in tum has half the \(\mathrm{pH}\) value of an orange. If the average \(\mathrm{pH}\) of an orange is \(3.2\), what is the average hydronium ion concentration of a lime?

Step-by-Step Solution

Answer

The average hydronium ion concentration of a lime is approximately 0.0251 moles/liter.

Step 1: Understand the pH Formula

The pH of a solution is defined as \(\text{{pH}} = -\text{{log}}_{10} \big[\text{{H}}_3\text{{O}}^+\big]\). We need to find \( \big[\text{{H}}_3\text{{O}}^+\big] \) for the lime.

Step 2: Relate pH values of fruits

We know the average \( \text{{pH}} \) of an orange is \(3.2\), and the pH value of lime is half of the pH of an orange. So, the \( \text{{pH}} \) of lime is \(3.2 / 2 = 1.6\)

Step 3: Use the pH value to find hydronium ion concentration

The lime has a \( \text{{pH}} \) of \(1.6\). We can use the formula \( \text{{pH}} = -\text{{log}}_{10} \big[\text{{H}}_3\text{{O}}^+\big] \) to find \( \big[\text{{H}}_3\text{{O}}^+\big] \).\(1.6 = -\text{{log}}_{10} \big[\text{{H}}_3\text{{O}}^+\big] \) implies \( -1.6 = \text{{log}}_{10} \big[\text{{H}}_3\text{{O}}^+\big] \)

Step 4: Convert log value to hydronium ion concentration

To solve for \( \big[\text{{H}}_3\text{{O}}^+\big] \, we convert from logarithmic form: \big[\text{{H}}_3\text{{O}}^+\big] = 10^{-1.6}\). Using a calculator, \big[\text{{H}}_3\text{{O}}^+\big] \approx 0.0251 \text{{ moles/liter}}

Key Concepts

Hydronium Ion Concentration

The hydronium ion concentration \(\big[\text{{H}}_3\text{{O}}^+\big]\) represents the amount of hydronium ions in a solution, measured in moles per liter (M). It is crucial in determining a solution's acidity. Hydronium ions \(\text{{H}}_3\text{{O}}^+\) form when acids dissolve in water, donating protons to water molecules. Calculate it from the pH using the formula: \(\big[\text{{H}}_3\text{{O}}^+\big] = 10^{-\text{{pH}}}\). For the lime example, with a pH of 1.6, the hydronium ion concentration is approximately 0.0251 M.

Logarithmic Functions

Logarithmic functions are involved in calculating pH from hydronium ion concentration. The pH is derived from a negative logarithm base 10 of \(\big[\text{{H}}_3\text{{O}}^+\big]\). In mathematical terms: \(\text{{pH}} = -\text{{log}}_{10} \big[\text{{H}}_3\text{{O}}^+\big]\). Therefore, to find the concentration from pH, rearrange to \(\big[\text{{H}}_3\text{{O}}^+\big] = 10^{-\text{{pH}}}\). This approach simplifies both large and small numbers, making complex calculations manageable.

Acid-Base Chemistry

Acid-base chemistry studies the properties and reactions of acids and bases. Acids typically increase the concentration of hydronium ions \(\text{{H}}_3\text{{O}}^+\) in aqueous solutions. Acidity is described quantitatively by pH, leading to concepts like strong and weak acids. Strong acids (e.g., HCl) dissociate completely in water, significantly increasing \(\big[\text{{H}}_3\text{{O}}^+\big]\). Weak acids (e.g., acetic acid) only partially dissociate. These principles help understand how various substances affect solution pH and vice versa.

Solution Acidity Measurement

The pH scale ranges from 0 to 14, reflecting a solution's acidity or basicity. A pH of 7 is neutral (pure water), below 7 is acidic, and above 7 is basic. Measuring pH provides insight into a solution's chemical properties. The pH of common substances like milk, lime, and orange juice reflects their acidity. The given exercise about fruit pH values exemplifies how measuring pH helps compare acidity levels across different solutions, supporting practical and theoretical understanding of acid-base interactions.