Problem 63 - A Graphical Approach to College Algebra

Question

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\) Beer, 4.8

Step-by-Step Solution

Answer

\([\mathrm{H}_3\mathrm{O}^+] \approx 1.58 \times 10^{-5} \; \text{M}\).

Step 1: Understanding the pH definition

The pH of a solution is defined as the negative logarithm of the hydronium ion concentration: \( pH = -\log_{10} [\mathrm{H}_3\mathrm{O}^+] \). This means that if we know the pH, we can find the hydronium ion concentration using this relationship.

Step 2: Rearrange the pH formula for hydronium ion concentration

To find the hydronium ion concentration, we rearrange the formula to solve for \([\mathrm{H}_3\mathrm{O}^+]\). The formula becomes: \( [\mathrm{H}_3\mathrm{O}^+] = 10^{-pH} \).

Step 3: Substitute the given pH into the formula

Substitute the given pH value into the formula: \( [\mathrm{H}_3\mathrm{O}^+] = 10^{-4.8} \).

Step 4: Calculate the hydronium ion concentration

Use a calculator to compute \( 10^{-4.8} \). The result is approximately \( 1.58 \times 10^{-5} \; \text{M} \), which represents the concentration of hydronium ions in the beer solution.

Key Concepts

Hydronium Ion Concentration

In the context of acid-base chemistry, hydronium ions (often represented as \( [\mathrm{H}_3\mathrm{O}^+] \)) play a critical role in determining the acidity of a solution. These ions form when water molecules (\( \mathrm{H}_2\mathrm{O} \)) gain a hydrogen ion (proton), resulting in \( \mathrm{H}_3\mathrm{O}^+ \). The concentration of these ions directly influences the pH level of a solution.To find the hydronium ion concentration, we typically first know the pH of the solution. From this value, we can apply the formula: \( [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}} \). You just raise 10 to the negative power of the pH value. This equation derives from the definition of pH itself.

Logarithmic Function

Logarithmic functions are mathematical tools that relate exponents and directly address questions like "What power does the base have to be raised to to get a particular number?" In our context, we deal with logarithms that use base 10.What pH actually tells us is that it is the negative logarithm of the hydronium ion concentration. The equation \( pH = -\log_{10} [\mathrm{H}_3\mathrm{O}^+] \) expresses this relationship. Logarithms simplify calculations related to the exponentially small values of ion concentration in solutions.Understanding logarithms helps us convert between the pH values and the ion concentration smoothly. A core concept is that each change of 1 unit in pH reflects a tenfold change in \( [\mathrm{H}_3\mathrm{O}^+] \). For example, a solution with a pH of 3 has ten times more hydronium ions compared to one with a pH of 4.

Acid-Base Chemistry

The field of acid-base chemistry explores how substances, known as acids or bases, interact to form ions in solution. Acids, like the ones found in everyday substances such as vinegar or lemon juice, contribute hydrogen ions to solutions. When these hydrogen ions join with water molecules, they form hydronium ions \( \mathrm{H}_3\mathrm{O}^+ \).Beer, with a pH of 4.8, is slightly on the acidic side of the pH scale, which ranges from 0 to 14. A lower pH means a higher concentration of hydronium ions, hence more acidity.In acid-base chemistry, the ability to measure and predict changes in ion concentration is vital. Chemists often work with logarithms to easily manage the broad range of ion concentrations encountered in different solutions.Engaging with acid-base reactions can help understand a variety of chemical processes, like those happening in metabolic pathways in our bodies or in industrial chemical reactions. Acid-base titrations are a practical application of these principles, used to determine the concentration of an unknown acid or base in a solution.