Problem 81 - Chemistry Matter and Change

Question

How many grams of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must be dissolved into 155 \(\mathrm{g}\) of water to create a solution with a molality of 8.20 \(\mathrm{mol} / \mathrm{kg}\) ?

Step-by-Step Solution

Answer

To create a solution with a molality of 8.20 \(\mathrm{mol}/\mathrm{kg}\), 134.76 grams of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must be dissolved into 155 g of water.

Step 1: Understand the concept of molality

Molality (\(m\)) is a measure of the concentration of a solute (in this case, \(\mathrm{Na}_{2} \mathrm{CO}_{3}\)) in a solution, defined as the number of moles of solute per kilogram of solvent (in this case, water). The formula for molality is: \(m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}}\)

Step 2: Determine the molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\)

To find the molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\), we need to add up the molar masses of all the elements in the compound: Molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\): = (2 × molar mass of Na) + (1 × molar mass of C) + (3 × molar mass of O) = (2 × 22.99 g/mol) + (1 × 12.01 g/mol) + (3 × 16.00 g/mol) = 105.98 g/mol

Step 3: Calculate the mass of solvent in kilograms

In order to use the molality formula, we need the mass of the solvent (water) in kilograms. We are given that the mass of water is 155 g, so we just need to convert this to kg: mass of water (kg) = 155 g × (1 kg / 1000 g) = 0.155 kg

Step 4: Calculate the moles of solute needed for the desired molality

We are given the desired molality as 8.20 \(\mathrm{mol} / \mathrm{kg}\). Using the molality formula, we can find the required moles of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\): moles of solute = molality × mass of solvent (kg) = 8.20 mol/kg × 0.155 kg = 1.2710 mol

Step 5: Calculate the mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) required

Now that we have the moles of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) needed, we can multiply it by its molar mass to find the mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) required: mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) = moles of solute × molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) = 1.2710 mol × 105.98 g/mol = 134.76 g So, 134.76 grams of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must be dissolved into 155 g of water to create a solution with a molality of 8.20 \(\mathrm{mol} / \mathrm{kg}\).

Key Concepts

Molar Mass

Molar mass is key to understanding how much of a substance you need to use in a chemical reaction or solution. It represents the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). For any compound, it is calculated by adding together the atomic masses of all constituent elements. This information is often found on the periodic table.
For example, to find the molar mass of sodium carbonate (\(Na_{2}CO_{3}\)), follow these steps:

• Look at the periodic table for the atomic mass of sodium (Na), carbon (C), and oxygen (O).
• The molar mass of Na is approximately 22.99 g/mol.
• Add these up: \( (2 \times 22.99\ g/mol) + (12.01\ g/mol) + (3 \times 16.00\ g/mol) = 105.98\ g/mol \).

Understanding molar mass is important when calculating how much solute you must use to achieve a certain concentration in a solution.

Solute

In chemistry, a solute is a substance that is dissolved in a solvent to create a solution. The solute can be in a solid, liquid, or gas state, but it is the component in a lesser amount compared to the solvent. It is key in determining the concentration of a solution, which is often expressed in terms of molality, molarity, or percentage.
For the given problem, the solute is sodium carbonate (\(Na_{2}CO_{3}\)), which is dissolved in water to create a specific molality. Remember:

• The solute is the component that determines the solution's behavior and properties.
• To calculate how much solute you need, you use its molar mass and desired molality.

Accurate measurements of the solute are crucial for achieving the desired concentration in any experiment or application.

Solvent

The solvent is the medium in which the solute is dissolved to form a solution. In most cases, the solvent is present in a greater quantity than the solute. It plays a crucial role by allowing the solute to disperse and interact at a molecular level. Water is often termed the "universal solvent" because it dissolves many substances.
For this exercise, water is the solvent.

• The mass of the solvent affects the concentration of the solution.
• In this problem, the mass of water is given as 155 g, or 0.155 kg when converted for calculation purposes.

Understanding the solvent helps in applying the correct techniques for dissolving solutes and creating solutions with precise concentrations.