Problem 101 - Elementary and Intermediate Algebra

Question

Hydrogen lon Concentration. Find the hydrogen ion concentration of a saturated solution of calcium hydroxide whose \(\mathrm{pH}\) is 13.2

Step-by-Step Solution

Answer

The hydrogen ion concentration is approximately \\(6.31 \times 10^{-14} ext{ M}\\).

Step 1: Understanding the Problem

We need to find the hydrogen ion concentration in a solution. The given information is the pH of the solution, which is 13.2. We will use this pH value to calculate the hydrogen ion concentration \([H^+]\).

Step 2: Using the pH Formula

pH is the negative logarithm of the hydrogen ion concentration. The formula to find the hydrogen ion concentration is:\([H^+] = 10^{-pH}\).

Step 3: Substitute the pH Value and Calculate

Substitute the given pH value into the formula: \[ [H^+] = 10^{-13.2} \] Use a calculator to find the result, which gives us the hydrogen ion concentration.

Step 4: Calculation Result

Calculate \(10^{-13.2}\), which results in approximately \(6.31 \times 10^{-14} ext{ M}\). This is the hydrogen ion concentration in the solution.

Key Concepts

pH Value

The term "pH" is a measure of how acidic or basic a solution is, based on its hydrogen ion concentration. Understanding pH is crucial when dealing with solutions like calcium hydroxide. The scale ranges from 0 to 14, with 7 being neutral. Values less than 7 indicate acidity, whereas values greater than 7 indicate basicity or alkalinity.
The pH value is determined through a logarithmic formula:

• A pH of 0 means very high acidity, while a pH of 14 indicates a highly alkaline solution.
• Each unit change represents a tenfold change in hydrogen ion concentration, making it a logarithmic scale.

In our exercise, the pH of 13.2 signifies a strong base, meaning a very low hydrogen ion concentration.

Calcium Hydroxide Solution

Calcium hydroxide is a strong alkaline compound often used in solutions for various industrial and chemical processes.When dissolved in water, calcium hydroxide dissociates to produce calcium and hydroxide ions. This reaction is what gives the solution its basic characteristics.
Key properties of calcium hydroxide solutions:

• The chemical formula is \(Ca(OH)_2\), indicating two hydroxide ions for each calcium ion in solution.
• These solutions are highly basic, often with pH values exceeding 12.
• They have applications in water treatment, construction, and more.

In this exercise, understanding the basic nature of calcium hydroxide helps in predicting the low level of hydrogen ions present due to its high pH.

Acid-Base Equilibrium

The concept of acid-base equilibrium is essential in understanding how acids and bases interact in solution.An acid-base equilibrium involves the balance between hydrogen ions \([H^+]\) and hydroxide ions \([OH^-]\). A strong base like calcium hydroxide shifts this balance significantly due to its ability to donate hydroxide ions.
Important points about acid-base equilibrium:

• The equilibrium constant for the autoionization of water \(Kw\) is \(10^{-14}\) at room temperature.
• To maintain equilibrium, an increase in hydroxide ions leads to a decrease in hydrogen ions.
• Since \(pH + pOH = 14\), a high \(pH\) implies a low \(pOH\), aligning with strong bases.

Thus, in our exercise, we see how a solution's pH and derived hydrogen ion concentration fit the broader acid-base equilibrium concept.

Logarithmic Calculation

Logarithmic calculations are fundamental when working with pH, as the entire scale is logarithmic.Understanding how to maneuver through these calculations enables one to accurately determine ion concentrations from pH values.
Here's how you perform this calculation:

• The formula \([H^+] = 10^{-pH}\) helps find the hydrogen ion concentration from a given pH.
• This formula arises from the definition of pH as the negative logarithm (base 10) of the hydrogen ion concentration: \(pH = -\log [H^+]\).
• Using calculators greatly simplifies computing values like \(10^{-13.2}\), providing the exact hydrogen ion concentration.

In our specific calculus, converting the pH of 13.2 allows calculation of the hydrogen ion concentration accurately, indicating a strong base with a very low \([H^+]\).