Problem 62 - 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 \(p H\) Wine, 3.4

Step-by-Step Solution

Answer

The hydronium ion concentration is approximately \(3.98 \times 10^{-4}\) M.

Step 1: Understand the Relationship between pH and Hydronium Ion Concentration

The pH of a solution is a measure of its acidity or alkalinity. The formula to relate pH to the hydronium ion concentration is given by:\[pH = -\log_{10} \left[ \mathrm{H}_3\mathrm{O}^+ \right] \] This means that the hydronium ion concentration, \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \), can be calculated by using the inverse logarithmic function.

Step 2: Apply the Formula to Calculate Hydronium Ion Concentration

Given the pH of wine is 3.4, we apply the formula:\[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] = 10^{-\text{pH}} \]Substituting the given pH value:\[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] = 10^{-3.4} \]

Step 3: Perform the Calculation

Now, calculate the power of ten:\[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] = 10^{-3.4} \approx 3.98 \times 10^{-4} \text{ M}\]This means the hydronium ion concentration in the wine is approximately \(3.98 \times 10^{-4}\) moles per liter (M).

Key Concepts

Hydronium Ion Concentration

The hydronium ion concentration \([ \mathrm{H}_3\mathrm{O}^+ ]\) is a direct measure of the number of hydronium ions present in a solution. Knowing the concentration is important because it tells us how acidic or basic a solution is. For a solution like wine with a given pH value, calculating the hydronium ion concentration helps us understand its acidic strength.
To find this concentration, we use the relationship between pH and \([ \mathrm{H}_3\mathrm{O}^+ ]\): \([ \mathrm{H}_3\mathrm{O}^+ ] = 10^{-\text{pH}}\).
Applying the formula is simple once we have the pH. For wine, with a pH of 3.4, the concentration is calculated as follows: \([ \mathrm{H}_3\mathrm{O}^+ ] = 10^{-3.4} \approx 3.98 \times 10^{-4} \, \text{Molar}\).
This calculation is crucial in many fields such as chemistry and food science because it allows the evaluation of how substances will interact based on their acidity.

Inverse Logarithmic Function

To understand the connection between pH and concentration, we need to utilize the inverse of the logarithmic function. A logarithmic function helps to describe how acidity is measured. The pH scale is logarithmic, meaning each unit change represents a tenfold change in hydrogen ion concentration.
In this context, the formula \[pH = -\log_{10} \left[ \mathrm{H}_3\mathrm{O}^+ \right] \] is rearranged to solve for hydronium ion concentration:

• Start with the pH value.
• Use the inverse, which involves raising 10 to the power of the negative pH, to find the concentration.

This use of the base-10 inverse logs is what enables the translation from pH values into actual molar concentrations of hydronium ions in a solution.
This understanding aids in practical applications, allowing chemists and scientists to predict the behavior of solutions in different pH conditions.

Acid-Base Chemistry

Acid-base chemistry revolves around the behavior of acids and bases in water and their ability to donate or accept protons, often visualized through \[ \mathrm{H}_3\mathrm{O}^+ \] and \[ \mathrm{OH}^- \] ions. pH, a central concept in acid-base chemistry, tells us whether a solution is acidic or basic.
An acidic solution like wine, with a pH of less than 7, has more hydronium ions than hydroxide ions. The pH scale helps categorize solutions as:

• Acidic if pH < 7
• Neutral if pH = 7
• Basic if pH > 7

Understanding the interaction between \[ \mathrm{H}_3\mathrm{O}^+ \] ions and other molecules helps us predict reactions, stability of compounds, and effects in biochemical processes.
Researchers and manufacturers rely on acid-base chemistry to tailor the properties of substances in products like food, medicine, and cleaning agents.

Molecular Concentration Calculation

Molecular concentration is the measurement of how much of a substance is present in a unit volume of solution, usually expressed in moles per liter (M). Calculating concentrations is a fundamental aspect of chemistry.
The calculation involves knowing the initial proton concentration or using measurements like pH to back-calculate other relevant concentrations. With pH, \([ \mathrm{H}_3\mathrm{O}^+ ]\) is specifically calculated using: \[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] = 10^{-\text{pH}} \].
This conversion from pH to concentration allows chemists to:

• Determine the extent of chemical reactions.
• Assess the strength and behavior of acidic or basic environments.

Understanding how to calculate these concentrations aids scientists in devising safe compounds and solutions, ensuring precision in laboratory and industrial contexts.