Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Find the \(p H\) for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Limes, \(1.6 \times 10^{-2}\)

Step-by-Step Solution

Answer

The pH of the limes solution is approximately 1.80.

Step 1: Understand the Definition of pH

The pH of a solution is a measure of its acidity or basicity. It is calculated using the formula: \[ pH = -\log_{10}[\text{H}_3\text{O}^+] \] where \([\text{H}_3\text{O}^+]\) is the concentration of hydronium ions in moles per liter (M). The pH scale typically ranges from 0 to 14.

Step 2: Substitute the Given Value

Given the hydronium ion concentration \([\text{H}_3\text{O}^+] = 1.6 \times 10^{-2}] M\), substitute this value into the pH formula:\[ pH = -\log_{10}(1.6 \times 10^{-2}) \]

Step 3: Calculate the Logarithm

Calculate the logarithm of the hydronium ion concentration. The calculation is as follows:\[ \log_{10}(1.6 \times 10^{-2}) = \log_{10}(1.6) + \log_{10}(10^{-2}) \]Using a calculator, we find:\[ \log_{10}(1.6) \approx 0.2041 \] and \[ \log_{10}(10^{-2}) = -2 \].

Step 4: Combine the Logarithm Values

Combine the two logarithm values to find the overall logarithm:\[ \log_{10}(1.6 \times 10^{-2}) = 0.2041 - 2 = -1.7959 \].

Step 5: Calculate the pH

Substitute the combined logarithm value back into the pH formula:\[ pH = -(-1.7959) = 1.7959 \]. This result indicates that the pH of the lime solution is approximately 1.80, which means it is acidic.

Key Concepts

Acidity and Basicity

The concepts of acidity and basicity revolve around the concentration of hydrogen ions in a solution. When we talk about acidity, we refer to solutions with a high concentration of hydronium ions \( \text{H}_3\text{O}^+ \). Conversely, basicity or alkalinity pertains to solutions with a lower concentration of these ions. This scale of measurement is essential for understanding various chemical reactions and processes.

The pH scale is a convenient way to express how acidic or basic a solution is. It ranges from 0 to 14:

• A pH of 7 is considered neutral, like pure water.
• pH values less than 7 indicate an acidic solution.
• pH values greater than 7 signify a basic or alkaline solution.

Understanding acidity and basicity allows us to predict the behavior of substances when mixed or dissolved, which is crucial in fields such as biology, medicine, and environmental science.

Logarithms

The concept of logarithms is pivotal in simplifying complex multiplication and division tasks, especially in the context of pH calculations. A logarithm is essentially the inverse operation of exponentiation, making large numerical values more manageable.

In pH calculations, we use a base-10 logarithm, also known as a common logarithm, denoted as \( \log_{10} \).

• The common logarithm, like in the pH formula \( pH = -\log_{10}[\text{H}_3\text{O}^+] \), helps transform the concentration values into a pH value.
• This transformation is essential because it converts potentially large or small concentration values into a simple numerical scale (0-14).

Using logarithms makes it easier to handle and interpret varying concentrations of hydronium ions, facilitating our understanding of the solution's acidity.

Hydronium Ion Concentration

The hydronium ion concentration \( [\text{H}_3\text{O}^+] \) is a crucial factor in determining a solution's pH. These ions result from the interaction of hydrogen ions (H⁺) with water.

The amount of hydronium ions in a solution directly correlates with its acidity:

• High concentrations of \( [\text{H}_3\text{O}^+] \) result in low pH values, indicating a highly acidic solution.
• Conversely, low concentrations lead to higher pH values, suggesting a more basic solution.

For example, in our exercise, the hydronium ion concentration is given as \( 1.6 \times 10^{-2} \) M, which is a fairly high concentration, resulting in a low pH of around 1.80, confirming its acidic nature. This direct relationship allows scientists to assess and manipulate chemical environments, particularly in acid-base chemistry.