Problem 52 - Precalculus with Limits

Question

Levels In Exercises \(51-56,\) use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. $$ \text { Find the } \mathrm{pH} \text { when }\left[\mathrm{H}^{+}\right]=1.13 \times 10^{-5} $$

Step-by-Step Solution

Answer

The pH of the solution with a hydrogen ion concentration of \(1.13 \times 10^{-5}\) is approximately 4.95.

Step 1: Insert into Formula

The first step requires the provided [\(\mathrm{H}^{+}\)] value to be plugged into the given formula. The \([\mathrm{H}^{+}]\) value is \(1.13 \times 10^{-5}\), so the calculation becomes \(\mathrm{pH} = -\log(1.13 \times 10^{-5})\).

Step 2: Calculate

The next step is to compute the value using a scientific calculator. Care should be taken while entering the value into the calculator. Enter the number as 1.13 EXP -5 instead of 1.13 * 10 ^ -5.

Step 3: Find pH

After evaluating expression, the hydroxide ion concentration (\(pH\)) will be found.

Key Concepts

The Acidity Model

The acidity model is a fundamental concept that helps us measure how acidic or basic a solution is. It uses a scale called the pH scale, which ranges from 0 to 14.
\(\mathrm{pH} = 7\) is considered neutral, like pure water.
Values below 7 are acidic, and values above 7 are basic (or alkaline). This model helps scientists and chemists determine the concentration of hydrogen ions \(\left[\mathrm{H}^{+}\right]\) in a solution, which is crucial for various applications, including biology, chemistry, and environmental science.

• Acidic solutions have a higher concentration of \(\mathrm{H}^{+}\) ions.
• Basic solutions have a lower concentration of \(\mathrm{H}^{+}\) ions.

By using the acidity model, one can predict how a substance will interact with other chemicals, which is essential for safety and environmental reasons.

Understanding Hydrogen Ion Concentration

Hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) refers to the number of hydrogen ions present in one liter of a solution. It's expressed in moles per liter, often using scientific notation.
For instance, a concentration of \(1.13 \times 10^{-5}\) moles/liter indicates a slightly acidic solution.
A higher concentration of hydrogen ions means a more acidic solution, reflected by a lower pH value.
This indicator plays a critical role in many biological and chemical processes.

• It affects enzyme activity, protein stability, and the behavior of ions in a solution.
• Accurately measuring \(\left[\mathrm{H}^{+}\right]\) helps control processes like fermentation and water treatment.

Understanding this concentration helps in predicting the properties and reactions of solutions.

Exploring the Logarithmic Function

The pH calculation involves a logarithmic function, which is essential for transforming large variations in hydrogen ion concentration into a manageable scale.
This function, written as \(\mathrm{pH} = -\log\left[\mathrm{H}^{+}\right]\), compresses vast ranges of data into a more comprehensible form.
Logarithms are the inverse operations of exponentials, enabling us to simplify numbers like \(1.13 \times 10^{-5}\) into everyday figures.

• The minus sign in the formula ensures the pH is a positive number.
• The logarithmic scale is non-linear, meaning each unit change represents a tenfold change in \(\left[\mathrm{H}^{+}\right]\).

This makes it possible to easily compare the acidity of various solutions.

Using a Scientific Calculator

Calculating pH accurately requires a scientific calculator. These devices are designed to handle logarithmic and exponential functions effortlessly.
When working with scientific notation, such as \(1.13 \times 10^{-5}\), it's essential to enter it as \(1.13 \; \text{EXP} \; -5\).
This ensures accuracy and saves time by avoiding common errors associated with manual exponent entry.

• Always double-check the entry to avoid calculation mistakes.
• Using the calculator's log function is crucial for finding pH values.

With practice, you'll find using a scientific calculator straightforward and invaluable for tackling chemistry problems.