Problem 63 - A Graphical Approach to Precalculus with Limits

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

The hydronium ion concentration is approximately \(1.585 \times 10^{-5} \, M\).

Step 1: Understanding pH and [H3O+] Relationship

The pH of a solution is defined as the negative logarithm (base 10) of the hydronium ion concentration \([\mathrm{H}_{3}\mathrm{O}^{+}]\). The formula is given by: \[pH = -\log([\mathrm{H}_{3}\mathrm{O}^{+}])\] Given \(pH = 4.8\), we'll use this relation to find the hydronium ion concentration.

Step 2: Rearranging the Formula

We need to rearrange the formula \(pH = -\log([\mathrm{H}_{3}\mathrm{O}^{+}])\) to solve for \([\mathrm{H}_{3}\mathrm{O}^{+}]\). By exponentiating both sides with base 10, we get:\[[\mathrm{H}_{3}\mathrm{O}^{+}] = 10^{-pH}\]This rearranges the expression to solve for \([\mathrm{H}_{3}\mathrm{O}^{+}]\).

Step 3: Substituting the Values

Now, substitute the given pH value into the new formula:\[[\mathrm{H}_{3}\mathrm{O}^{+}] = 10^{-4.8}\]This substitution will allow us to calculate the hydronium ion concentration.

Step 4: Calculating the Concentration

Use a calculator to find the value of \(10^{-4.8}\). This will give us the hydronium ion concentration:\[[\mathrm{H}_{3}\mathrm{O}^{+}] = 1.585 imes 10^{-5} \, \text{M}\]This represents the molarity (M) of the hydronium ions in the solution.

Key Concepts

Logarithmic Functions

Logarithmic functions are a crucial aspect of many scientific calculations, such as determining pH in chemistry. The logarithmic function is the inverse operation of exponentiation. When dealing with pH and hydronium ion concentration, we rely on the logarithmic relationship

• The pH of a solution is defined as the negative base-10 logarithm of the hydronium ion concentration.
• The formula used here is:
  \(pH = -\log([\mathrm{H}_{3}\mathrm{O}^{+}])\).

This formula converts the concentration of hydronium ions into a more manageable scale that ranges from 0 to 14, where each unit change represents a tenfold change in concentration. For example, a pH of 4.8 indicates the concentration is ten times more than at pH 5.8. Understanding logarithmic functions allows chemists to easily compare acidity levels between different solutions. It's essential to have a good grasp of exponent rules, since solving for hydronium concentration requires rearranging the equation to find \([\mathrm{H}_{3}\mathrm{O}^{+}]\) by expressing it as \(10^{-pH}\). This capability makes interpreting and manipulating chemical data far more accessible.

Acid-Base Chemistry

Acid-base chemistry is a fundamental area of study within chemistry, focusing on the behavior of acids and bases in solution. The pH scale is an integral part of this topic, measuring how acidic or basic a solution is. Acids generally have a pH less than 7, while bases have a pH greater than 7. Neutral solutions, like pure water, have a pH of 7.
In the context of the given exercise, beer with a pH of 4.8 is considered acidic.

• This is because a lower pH correlates with a higher concentration of hydronium ions \([\mathrm{H}_{3}\mathrm{O}^{+}]\).
• The formula \([\mathrm{H}_{3}\mathrm{O}^{+}] = 10^{-pH}\) helps in calculating this concentration, indicating how acidic the solution is relative to others.

Values provide insights into many properties of the solution, including its potential corrosiveness or its role in chemical reactions. Acid-base chemistry is not just theoretical but impacts fields such as medicine, environmental science, and even the food industry, given the importance of maintaining precise pH levels.

Chemical Concentration Calculations

In chemistry, concentration calculations are essential for determining the amount of a substance dissolved in a particular volume of solution. For acids and bases, this is often expressed in terms of the molarity (M), which is the number of moles of solute per liter of solvent. Using molarity as a measure, we can understand the strength of an acidic or basic solution.
The hydronium ion concentration \([\mathrm{H}_{3}\mathrm{O}^{+}]\) tells us how many hydrogen ions are present in a solution, which is critical for understanding a solution's acidity.

• In the exercise, the calculated \([\mathrm{H}_{3}\mathrm{O}^{+}] = 1.585 \times 10^{-5} \, \text{M} \) implies that there are about \(1.585 \times 10^{-5} \) moles of hydronium ions in every liter of beer.
• These calculations enable chemists to predict behavior in chemical reactions and production processes, ensuring desired outcomes.

Whether ensuring the right processing conditions in industrial applications or examining the taste profile in food sciences, awareness and precision in concentration calculations are vital for credible scientific and commercial results. By comprehensively understanding how to calculate chemical concentrations, chemists can detect, measure, and adjust the properties of solutions to achieve specific goals.