Problem 48 - Chemistry The Central Science In Si Units

Question

Calculate the concentration of an aqueous solution of \(\mathrm{Ca}(\mathrm{OH})_{2}\) that has a pH of 10.05 .

Step-by-Step Solution

Answer

The concentration of the \( \mathrm{Ca(OH)_2} \) solution is approximately \( 5.6 \times 10^{-5} \) M.

Step 1: Understand pH and pOH Relationship

The pH of a solution is a measure of the hydrogen ion concentration. The relationship between pH and pOH is given by the equation: \[ \text{pH} + \text{pOH} = 14 \]. Given that the pH is 10.05, we can use this relationship to calculate the pOH.

Step 2: Calculate pOH

Subtract the given pH from 14 to find the pOH: \[ \text{pOH} = 14 - 10.05 = 3.95 \].

Step 3: Calculate Hydroxide Ion Concentration

The pOH is related to the hydroxide ion concentration \([\text{OH}^-]\) by the equation: \[ \text{pOH} = -\log[\text{OH}^-] \]. Solve for \([\text{OH}^-]\) using the formula: \[ [\text{OH}^-] = 10^{-3.95} \].

Step 4: Solve for [OH⁻]

Calculate \( [\text{OH}^-] = 10^{-3.95} \approx 1.12 \times 10^{-4} \) M.

Step 5: Relate [OH⁻] to [Ca(OH)₂] Concentration

Since \( \mathrm{Ca(OH)_2} \) dissociates into one \( \mathrm{Ca^{2+}} \) ion and two \( \mathrm{OH^{-}} \) ions, the concentration of \( \mathrm{Ca(OH)_2} \) is half of the \( \mathrm{OH^{-}} \) concentration. Therefore, \[ [\mathrm{Ca(OH)_2}] = \frac{1.12 \times 10^{-4}}{2} = 5.6 \times 10^{-5} \text{ M} \].

Key Concepts

pH and pOH relationship

In chemistry, understanding the relationship between pH and pOH is crucial to determine the nature of an aqueous solution, whether it is acidic or basic.
The sum of the pH and pOH of any aqueous solution at 25°C is always equal to 14. This makes it easy to find one if you know the other:

• pH measures the concentration of hydrogen ions \( [\text{H}^+] \) in the solution.
• pOH, on the other hand, is related to hydroxide ions \( [\text{OH}^-] \).

If you know the pH of a solution, like a pH of 10.05 in this case, you can quickly find the pOH by subtracting the pH from 14: \( \text{pOH} = 14 - \text{pH} \). This formula is derived from the constant \( \text{pH} + \text{pOH} = 14 \). This relationship helps us understand the acidity or basicity of the solution.

Hydroxide ion concentration

Once you have determined the pOH of a solution, finding the concentration of hydroxide ions is a straightforward process.

• The relationship is expressed through the formula: \( \text{pOH} = -\log[\text{OH}^-] \).
• This can be rearranged to solve for \( [\text{OH}^-] \): \( [\text{OH}^-] = 10^{-\text{pOH}} \).

For instance, with a pOH of 3.95, the hydroxide ion concentration \( [\text{OH}^-] \) would be \( 10^{-3.95} \), which approximately equals \( 1.12 \times 10^{-4} \) M. This concentration tells us how many hydroxide ions are present per liter of solution, giving insight into the solution's basic properties.

Dissociation of Ca(OH)₂

Calcium hydroxide \( \text{Ca(OH)}_2 \) dissolves in water and dissociates into its component ions. This dissociation is crucial for understanding its effects on a solution's pH and pOH.

• Each formula unit of \( \text{Ca(OH)}_2 \) produces one \( \text{Ca}^{2+} \) ion and two \( \text{OH}^- \) ions when it disassociates in an aqueous solution.
• This means that the concentration of \( \text{OH}^- \) ions is double that of the \( \text{Ca(OH)}_2 \) solution's concentration.

If you know the hydroxide ion concentration from a previous calculation, you can easily determine the concentration of \( \text{Ca(OH)}_2 \). Simply divide the \( [\text{OH}^-] \) concentration by 2 to find the concentration of \( \text{Ca(OH)}_2 \): \( [\text{Ca(OH)}_2] = \frac{[\text{OH}^-]}{2} \).

Chemistry calculations

Chemistry calculations can seem daunting at first, but by breaking them down into manageable steps, they become much easier. These calculations often involve applying formulas and understanding chemical relationships. Here's a simple approach to handle these calculations:

• Identify the givens - Start by identifying what you know, such as pH values or concentrations.
• Apply known relationships - Use key formulas, like \( \text{pH} + \text{pOH} = 14 \), which relate hydrogen and hydroxide ion concentrations.
• Solve step-by-step - Break the problem into smaller steps, calculating one component at a time (e.g., pOH, \( [\text{OH}^-] \), then \( [\text{Ca(OH)}_2] \)).

This method of solving chemistry problems by following a logical sequence will enhance understanding and ensure accuracy. Remember that understanding the chemistry behind the formulas is just as important as performing the calculations.