Problem 61 - 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 \(p H\). Soda pop, 2.7

Step-by-Step Solution

Answer

The hydronium ion concentration is approximately \(5.0 \times 10^{-3}\, \text{M}\).

Step 1: Understanding the Formula

The concentration of hydronium ions \([\mathrm{H}_3\mathrm{O}^+]\) can be calculated from the pH using the formula \( pH = -\log_{10}([\mathrm{H}_3\mathrm{O}^+]) \). To find \([\mathrm{H}_3\mathrm{O}^+]\), we need to solve this equation for \([\mathrm{H}_3\mathrm{O}^+]\).

Step 2: Rearrange the Formula

Rearranging the formula \( pH = -\log_{10}([\mathrm{H}_3\mathrm{O}^+]) \) gives us \([\mathrm{H}_3\mathrm{O}^+] = 10^{-pH}\). This formula allows us to calculate the concentration directly from the given pH value.

Step 3: Substitute the Given pH into the Formula

Substitute the given pH value of 2.7 into the formula: \([\mathrm{H}_3\mathrm{O}^+] = 10^{-2.7}\).

Step 4: Calculate the Hydronium Ion Concentration

Using a calculator, compute \([\mathrm{H}_3\mathrm{O}^+] = 10^{-2.7} \), which results in approximately \[5.0 \times 10^{-3}\, \text{M}\].

Key Concepts

pH

The term "pH" is often heard in chemistry, but what does it truly mean? pH is a measure of how acidic or basic a solution is. This numeric scale ranges from 0 to 14, where:

• Values below 7 indicate an acidic solution.
• Values above 7 indicate a basic solution.
• A pH of exactly 7 represents a neutral solution, like pure water.

To further simplify, think of pH as a shorthand to express the concentration of hydrogen ions (u[H^+]u) in a solution. In more scientific terms, pH is defined as the negative base 10 logarithm of the u[H^+]u concentration: \[ pH = -\log_{10}([H^+]) \]This formula helps us understand and quantify how acidic or basic a solution is by simply observing the pH value.

Acidic Solutions

Acidic solutions are fascinating and quite common in everyday life. An acidic solution, by definition, has a pH less than 7. This means there is a higher concentration of hydrogen ions compared to a neutral solution. Some common examples include:

• Lemon juice with a pH around 2.
• Vinegar typically having a pH of about 2.5.
• Soda pop, which ranges from pH 2 to 4 depending on the brand.

These substances are characterized by their sour taste and ability to turn blue litmus paper red. By understanding the pH and the concentration of hydrogen ions, scientists can determine how reactive or corrosive an acid may be, which is crucial for applications in chemistry and industry.

Logarithmic Functions

Logarithmic functions might sound complex, but they play an essential role in chemistry, especially when dealing with pH calculations. Here's how they work:

• The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.
• In chemistry, we often use base 10 logarithms.

For example, if \( log_{10}(x) = y \), it means that\(10^y = x \). This relationship is crucial when calculating the hydronium ion concentration from a pH value, as seen in the formula:\([u H_3O^+] = 10^{-u pH} \). Understanding logarithmic functions allows chemists to convert complicated multiplicative relationships into more straightforward additive (or subtractive) ones, simplifying calculations.

Chemistry Calculations

Mastering chemistry calculations means understanding how to translate problems into mathematical statements that can be solved. Calculating the hydronium ion concentration from the pH is an excellent example of chemistry calculations in action:

• The first step is recognizing the relevant formula: \[ pH = -\log_{10}([H_3O^+]) \]
• Next, rearrange this to solve for the hydronium ion concentration: \([H_3O^+] = 10^{-pH} \)
• By substituting the given pH value into the formula, perform the calculation using a calculator.
• In our example, with a pH of 2.7, you would compute\([H_3O^+] = 10^{-2.7} \), resulting in approximately \(5.0 \times 10^{-3}\, ext{M}.\)

Chemistry calculations like these involve understanding, rearranging formulas, and accurately substituting values to find solutions. This approach ensures precise and efficient problem-solving in chemistry.