Problem 54 - 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. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution for which \(\mathrm{pH}=5.8\)

Step-by-Step Solution

Answer

\([H^+] \approx 1.58 \times 10^{-6}\) moles of hydrogen per liter

Step 1: Understand and apply the formula

Given the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\), if we want to find the hydrogen ion concentration \([H^+]\) we need to make \([H^+]\) the subject of the formula. This will give us the formula in terms of \([H^+] = 10^{-\mathbf{pH}}\)

Step 2: Substitute pH value into the formula

Substitute \(\mathbf{5.8}\) for \(\mathbf{pH}\) in the formula, so \([H^+] = 10^{-5.8}\)

Step 3: Calculate the hydrogen ion concentration

Compute \([H^+] = 10^{-5.8}\) by raising 10 to the power of \(-5.8\). Use a calculator to solve this.

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration, denoted as \([H^+]\), is a crucial factor in determining the acidity or basicity of a solution. It measures how many moles of hydrogen ions are present in one liter of solution.
This concentration directly impacts the pH level, providing insights into the solution's chemical nature.

• A high \([H^+]\) indicates an acidic solution, with a pH less than 7.
• A low \([H^+]\) indicates a basic solution, with a pH greater than 7.
• A \([H^+]\) around neutrality means the pH is approximately 7.

Understanding hydrogen ion concentration is essential for fields like biology and chemistry, where the pH influences everything from metabolic reactions to industrial processes.

Logarithmic Function

The acidity model uses a logarithmic function, represented by the formula \( \mathbf{pH} = - \log \left[ \mathbf{H}^{+} \right] \). Logarithms are mathematical operations that help us deal with exponential relationships by turning them into manageable linear forms.
This formula demonstrates that the pH is the negative logarithm of the hydrogen ion concentration. Here's why we use this approach:

• Logarithms condense the wide range of \([H^+]\) values into a smaller, more understandable range of pH values.
• This makes it easier to compare the acidity of solutions since the pH scale is more concise than \([H^+]\).
• Calculating the inverse involves exponentiating the negative pH to find \([H^+]\).

By using logarithms, we simplify chemical and biological analyses, making complex calculations more approachable.

Chemistry Calculations

Performing chemistry calculations, especially involving pH, requires careful understanding of mathematical principles and chemical properties. To calculate the hydrogen ion concentration from a given pH:

• Use the rearranged formula \( [H^+] = 10^{-\text{pH}} \).
• Substitute the pH value into the formula.
• Perform the calculation using exponentiation.

For example, with a pH of 5.8, the calculation becomes \( [H^+] = 10^{-5.8} \). This computation involves raising 10 to the power of \(-5.8\), which may require a calculator for precision.
These calculations are vital in predicting how a solution will react, its effectiveness in chemical processes, and its biological interactions. Mastery of these techniques is fundamental for success in chemistry-related fields.