Problem 41 - Technical Mathematics with Calculus

Question

Applications. A pH of 7 is considered neutral, while a lower \(\mathrm{pH}\) is acidic and a higher \(\mathrm{pH}\) is alkaline. What is the hydrogen ion concentration at a pH of \(7.0 ?\)

Step-by-Step Solution

Answer

The hydrogen ion concentration at a pH of 7.0 is \(1 \times 10^{-7}\) M.

Step 1: Understanding the pH scale

The pH is a measure of the hydrogen ion concentration in a solution and is calculated using the formula \( pH = -\log[H^+] \), where \( [H^+] \) is the hydrogen ion concentration.

Step 2: Computing the hydrogen ion concentration

Given that the pH is 7.0, we compute the hydrogen ion concentration by reversing the pH formula: \( [H^+] = 10^{-pH} \).

Step 3: Calculation of hydrogen ion concentration

Substitute the given pH value into the equation: \( [H^+] = 10^{-7.0} = 1 \times 10^{-7} \) M, where M stands for molarity, the unit for concentration in moles per liter.

Key Concepts

pH Calculation

The pH scale is an essential concept in chemistry as it pertains to the acidity or alkalinity of solutions. The pH is defined as the negative logarithm of the hydrogen ion concentration, expressed by the formula: \( pH = -\log[H^+] \). To calculate the pH, you simply take the negative log of the molarity of the hydrogen ions (\( [H^+] \) stands for hydrogen ion concentration). For example, with a known pH, calculating the hydrogen ion concentration becomes straightforward. Given a neutral pH of 7.0, the hydrogen ion concentration is found by reversing the pH equation: \( [H^+] = 10^{-pH} = 10^{-7.0} \), which simplifies to \( 1 \times 10^{-7} \) M. It's important to note that a neutral pH does not mean there are no hydrogen ions present but indicates a balance with hydroxide ions, leading to a stable, neutral environment.

Logarithmic Scale in Chemistry

Understanding a logarithmic scale is vital for interpreting pH values, as pH is a logarithmic measure of hydrogen ion concentration. This type of scale represents exponential relationships, showing how a quantity multiplies by a fixed factor for each unit increase. In terms of the pH scale, a one-unit change in pH corresponds to a tenfold change in hydrogen ion concentration. This implies that a solution with a pH of 6 has ten times more hydrogen ions than a solution with a pH of 7. This exponential relationship means that even small changes in pH can signify significant changes in the chemical nature of a solution. Therefore, grasping log scales is essential for a deep understanding of chemical concentrations and reactions.

Molarity and Concentration

Molarity, symbolized as M, is a unit that measures the concentration of a solute within a solution. It is quantified as moles of solute per liter of solution. Molar concentration is integral to numerous chemical calculations, including the determination of pH. For instance, in our pH calculation example, the hydrogen ion concentration is expressed as \( 1 \times 10^{-7} \) M, meaning that there are \( 1 \times 10^{-7} \) moles of hydrogen ions for every liter of solution. Understanding molarity allows students to quantify and prepare solutions with precise chemical properties and is a fundamental aspect of laboratory work in chemistry. It’s the cornerstone for many processes, such as dilution and stoichiometric calculations.