Problem 53 - 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 { Compute }\left[\mathrm{H}^{+}\right] \text {for a solution in which } \mathrm{pH}=5.8 $$

Step-by-Step Solution

Answer

\([H^+] = 1.58 \times 10^{-6}\) moles of hydrogen per liter.

Step 1: Understanding Logarithmic Relations

The pH of a solution can be calculated by \( pH = -log[H^+]\). Therefore, to get the concentration of hydrogen ions [H+] from a known pH, you would rearrange the formula to \([H^+] = 10^{-pH}\).

Step 2: Substitution

Substitute the given value of pH=5.8 into the equation to find [H+]. This will look like \([H^+] = 10^{-5.8}\).

Step 3: Simplification

Simplify \([H^+] = 10^{-5.8}\) using a calculator to get the final answer. Thus, the Hydrogen ion concentration of the solution can be found.

Key Concepts

Hydrogen Ion Concentration

When we talk about acidity in chemical solutions, the concept of hydrogen ion concentration \[ \left[ H^+ \right] \] is central. It's a measure of the number of hydrogen ions present in a liter of solution. This concentration is expressed in terms of moles per liter, which tells us how acidic or basic a solution is.
Understanding the concentration involves looking at how the ions interact in the solution. More hydrogen ions mean a more acidic solution, while fewer ions indicate a more basic solution. To get a bit more technical:

• Higher \[ \left[ H^+ \right] \] means lower pH value.
• It directly impacts the reactivity and chemical behavior of the solution.

In practice, knowing the hydrogen ion concentration helps in fields like chemistry, biology, and environmental science, where the acidity or basicity of a solution is crucial for reactions and processes.

Logarithmic Equations

Logarithmic equations are essential when dealing with pH and acidity. The pH formula uses a logarithmic scale: \[ pH = -\log \left[ H^+ \right] \]. But why logarithms? The answer lies in the range of hydrogen ion concentrations in water.The concentration can vary from one molar in very acidic solutions to one picomolar in very basic ones. That’s a range spanning 12 orders of magnitude! Logarithms help simplify and express these numbers meaningfully and concisely.
The equation well illustrates why this is helpful:

• Transforming multiplication into addition/subtraction.
• Scaling large ranges into compact, manageable numbers.

The logarithmic scale conveniently handles the vast differences in hydrogen ion concentration, making it easier for scientists and students to analyze acidity levels.

pH Scale

The pH scale is a tool that helps us understand the acidity or basicity of a solution. It's a logarithmic scale ranging from 0 to 14, where:

• A pH less than 7 indicates an acidic solution.
• A pH of exactly 7 is neutral, like pure water.
• A pH greater than 7 signifies a basic (or alkaline) solution.

This scale is very useful for quickly assessing a liquid's nature by just looking at its pH value.
Practically, the pH scale tells us the concentration of hydrogen ions. With each step on the pH scale representing a tenfold difference in \[ \left[ H^+ \right] \], understanding this scale helps in grasping why small pH changes can significantly impact the chemical properties and reactions in solutions.
Mastering the pH scale is crucial for various applications, from testing water quality to agricultural impact, and even in healthcare settings where body pH needs to be monitored.