Problem 104

Question

Calculate the pH of a solution made by adding \(1.00 \mathrm{~g}\) potassium oxide \(\left(\mathrm{K}_{2} \mathrm{O}\right)\) to enough water to make \(2.00 \mathrm{~L}\) of solution.

Step-by-Step Solution

Verified

Answer

The pH of the solution is 12.03.

1Step 1: Determine the Molar Mass of Potassium Oxide

Calculate the molar mass of potassium oxide (\(\mathrm{K}_2\mathrm{O}\)). Potassium \((\mathrm{K})\) has an atomic mass of 39.10 g/mol and oxygen \((\mathrm{O})\) has an atomic mass of 16.00 g/mol. Therefore: \[ \text{Molar mass of } \mathrm{K}_2\mathrm{O} = 2 \times 39.10 \, \text{g/mol} + 16.00 \, \text{g/mol} = 94.20 \, \text{g/mol} \]

2Step 2: Calculate Moles of Potassium Oxide

Convert the mass of \(\mathrm{K}_2\mathrm{O}\) to moles using its molar mass. Given 1.00 g of \(\mathrm{K}_2\mathrm{O}\): \[ \text{Moles of } \mathrm{K}_2\mathrm{O} = \frac{1.00 \, \text{g}}{94.20 \, \text{g/mol}} = 0.0106 \, \text{mol} \]

3Step 3: Dissolution Reaction of K2O in Water

Determine the reaction that occurs when \(\mathrm{K}_2\mathrm{O}\) dissolves in water. The reaction is: \[ \mathrm{K}_2\mathrm{O} + \mathrm{H}_2\mathrm{O} \rightarrow 2\, \mathrm{KOH} \] This means 1 mole of \(\mathrm{K}_2\mathrm{O}\) produces 2 moles of \(\mathrm{KOH}\).

4Step 4: Calculate Moles and Concentration of KOH

Since \(\mathrm{K}_2\mathrm{O}\) produces 2 moles of \(\mathrm{KOH}\) per mole, calculate moles of \(\mathrm{KOH}\) produced: \[ \text{Moles of } \mathrm{KOH} = 2 \times 0.0106 \, \text{mol} = 0.0212 \, \text{mol} \] Now calculate the concentration in the 2.00 L of solution: \[ \text{Concentration of } \mathrm{KOH} = \frac{0.0212 \, \text{mol}}{2.00 \, \text{L}} = 0.0106 \, \text{M} \]

5Step 5: Calculate pOH from KOH Concentration

Use the concentration of \(\mathrm{KOH}\) to find the pOH. For a strong base like \(\mathrm{KOH}\), pOH is given by: \[ \text{pOH} = -\log[\mathrm{KOH}] = -\log(0.0106) \approx 1.97 \]

6Step 6: Calculate pH from pOH

Use the relation \(\text{pH} + \text{pOH} = 14\) to find the pH: \[ \text{pH} = 14 - \text{pOH} = 14 - 1.97 = 12.03 \]

Key Concepts

Molar massDissolution reactionConcentrationpOH

Molar mass

Understanding molar mass is crucial in chemistry as it allows us to convert between grams and moles. The molar mass of a compound is simply the mass of one mole of that substance. For example, in the compound potassium oxide (fK_2O), we need to find the molar mass to determine how many moles are present in a given mass. Potassium (fK) has an atomic mass of 39.10 g/mol and oxygen (fO) has an atomic mass of 16.00 g/mol. Therefore, the molar mass of potassium oxide is calculated as follows:

2 atoms of potassium: 2 4 39.10 g/mol = 78.20 g/mol
1 atom of oxygen: 16.00 g/mol

This results in a total molar mass of 94.20 g/mol for potassium oxide. Having this molar mass allows us to convert grams of potassium oxide into moles, which is fundamental in determining how it will interact in chemical reactions.

Dissolution reaction

A dissolution reaction describes how a substance interacts with a solvent, usually water, to form a solution. For potassium oxide, when it is dissolved in water, it undergoes a chemical reaction rather than a mere physical dissolution. The reaction for potassium oxide when it dissolves in water is:

fK_2O + fH_2O 4 2 fKOH

This equation shows that one mole of potassium oxide reacts with one mole of water to produce two moles of potassium hydroxide (fKOH), a strong base. Understanding this chemical equation tells us how many moles of product (in this case, fKOH) will be produced from a given amount of reactant (fK_2O). This is critical for determining properties of the solution, like its pH.

Concentration

Concentration is a measure of how much solute is present in a given amount of solvent or solution. It is usually expressed in terms of molarity (M), which is the number of moles of solute per liter of solution. In the context of potassium oxide dissolving in water, we calculated the concentration of potassium hydroxide (KOH) formed.

From the dissolution of potassium oxide, we found 0.0212 moles of fKOH are formed.
The solution volume is 2.00 L, so the concentration is calculated as: f\[ \text{Concentration of } \mathrm{KOH} = \frac{0.0212 \text{ mol}}{2.00 \text{ L}} = 0.0106 \text{ M} \]

This concentration is vital for determining further properties of the solution, such as pOH, which can then be used to find the pH, indicating the acidity or basicity of the solution.

pOH

pOH is a measure used to express the alkalinity of a solution, particularly for solutions containing strong bases like potassium hydroxide (KOH). The pOH can be calculated using the formula:

f\[ \text{pOH} = -\log[\mathrm{KOH}] \]

In our example, with a fKOH concentration of 0.0106 M, the pOH is:f\[ \text{pOH} = -\log(0.0106) \approx 1.97 \]Knowing the pOH is essential, as it allows us to calculate the pH by using the relationship: