Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. Many solutions have a \(\mathrm{pH}\) between 1 and 14. Find the corresponding range of \(\left[\mathrm{H}^{+}\right]\).

Step-by-Step Solution

Answer

The range of \([\text{H}^+]\) is \(10^{-14}\) to \(10^{-1}\) moles per liter.

Step 1: Identify the given pH range

We are given that the pH values range from 1 to 14. This means \( 1 \leq \text{pH} \leq 14 \).

Step 2: Understand the pH formula

The formula to calculate pH is \( \text{pH} = -\log [\text{H}^+] \). Our goal is to find the range for \([\text{H}^+]\) based on the given pH range.

Step 3: Figure out the range for hydrogen ion concentration

Convert the pH range into the hydrogen ion concentration \([\text{H}^+]\). Start with the equation \(-\log [\text{H}^+] = 1\) and \(-\log [\text{H}^+] = 14\).

Step 4: Solve for hydrogen ion concentration for pH = 1

Use the equation for pH = 1: \(-\log [\text{H}^+] = 1\). This implies \([\text{H}^+] = 10^{-1}\) or \([\text{H}^+] = 0.1\) moles per liter.

Step 5: Solve for hydrogen ion concentration for pH = 14

Use the equation for pH = 14: \(-\log [\text{H}^+] = 14\). This implies \([\text{H}^+] = 10^{-14}\) moles per liter.

Step 6: Combine the results to find the range

Combine the results from Step 4 and Step 5 to state the range for \([\text{H}^+]\). Thus, \([\text{H}^+]\) is between \(10^{-14}\) and \(10^{-1}\) moles per liter.

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a way to measure the amount of hydrogen ions, written as \([ ext{H}^+]\) in a solution. These ions are crucial because they determine the acidity or basicity of the solution. When in water, substances can release hydrogen ions, affecting how acidic or basic the solution is.

• High concentrations of hydrogen ions mean the solution is more acidic.
• Low concentrations of hydrogen ions indicate a more basic solution.

The hydrogen ion concentration is measured in moles per liter (mol/L), which expresses the number of hydrogen ions present in one liter of solution. This concentration is a key factor in defining the pH of the solution. Understanding this relationship is essential in various scientific fields, including chemistry and biology.

Acidity

Acidity is all about how many hydrogen ions a solution has compared to other substances. It is a way to measure how 'sour' or 'sharp' a solution feels because more hydrogen ions mean more acidity.

• Acidic solutions have higher hydrogen ion concentrations.
• Typical examples include lemon juice and vinegar.

Acidity is connected to the concept of pH, where lower pH values indicate higher acidity. Bringing these ideas together can help us determine whether a substance is acidic, basic, or neutral. The science behind acidity is crucial for understanding activities ranging from digestion to environmental studies. Consistently monitoring and adjusting the acidity in various industries ensures product safety and effectiveness.

Basicity

Basicity, like acidity, describes a solution, but it focuses on how 'alkaline' or 'soapy' it feels primarily due to lower hydrogen ion concentrations. The fewer hydrogen ions, the more basic the solution.

• Basic solutions have lower hydrogen ion concentrations.
• Examples include baking soda and household bleach.

In terms of pH, solutions with high pH values signify high basicity. Understanding basicity is equally important in many areas, such as health care and environmental management. A balance between acidity and basicity in organisms and the environment is integral to maintaining health and sustainability.

Logarithmic Scale

A logarithmic scale is a way of displaying numbers that grow exponentially. The pH scale is one such scale, where each whole number change represents a tenfold change in hydrogen ion concentration.

• When the pH changes by 1, the hydrogen ion concentration changes by a factor of 10.
• This means that a pH of 3 is ten times more acidic than a pH of 4.

This scale allows us to easily compare very large or small quantities in a more manageable format. By using the log function in the pH equation, \( ext{pH} = - ext{log} [ ext{H}^+]\), we can quickly understand just how acidic or basic a solution is without dealing with unwieldy numbers. This makes the logarithmic scale incredibly useful in scientific calculations and comparisons.