Problem 5 - Chemistry and Chemical Reactivity

Question

What mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must you add to \(125 \mathrm{g}\) of water to prepare \(0.200 \mathrm{m} \mathrm{Na}_{2} \mathrm{CO}_{3} ?\) What is the mole fraction of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) in the resulting solution?

Step-by-Step Solution

Answer

2.65 g of \( \mathrm{Na}_{2} \mathrm{CO}_{3} \) is needed, and its mole fraction is approximately 0.0036.

Step 1: Understand Molality

Molality ( extit{m}) is defined as the number of moles of solute per kilogram of solvent. Here, we need to calculate the mass of \( \mathrm{Na}_{2} \mathrm{CO}_{3} \) to achieve a molality of \( 0.200 \).

Step 2: Convert Solvent Mass to Kilograms

Convert the mass of water from grams to kilograms:\[125 \text{ g} = 0.125 \text{ kg}\]

Step 3: Calculate Moles of Na2CO3

Using the formula for molality:\[m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}}\]So,\[0.200 = \frac{\text{moles of } \mathrm{Na}_{2} \mathrm{CO}_{3}}{0.125}\]Solve for moles of \( \mathrm{Na}_{2} \mathrm{CO}_{3} \):\[\text{Moles of } \mathrm{Na}_{2} \mathrm{CO}_{3} = 0.200 \times 0.125 = 0.025\]

Step 4: Calculate Mass of Na2CO3

The molar mass of \( \mathrm{Na}_{2} \mathrm{CO}_{3} \) is:\[\text{(2 \times 23) + 12 + (3 \times 16) = 106 \text{ g/mol}}\]Multiply by the number of moles to find the mass:\[0.025 \times 106 = 2.65 \text{ g}\]

Step 5: Calculate Mole Fraction of Na2CO3

First, calculate moles of water:\[\text{Moles of water} = \frac{125}{18} = 6.94 \text{ moles}\]Calculate the mole fraction:\[X_{\mathrm{Na}_{2} \mathrm{CO}_{3}} = \frac{0.025}{0.025 + 6.94} = \frac{0.025}{6.965} \approx 0.0036\]

Key Concepts

Solute

The term 'solute' refers to the substance that is dissolved in a solution. In this particular exercise, sodium carbonate (\(\mathrm{Na}_{2} \mathrm{CO}_{3}\)) is the solute. When a solute is added to a solvent, it creates a mixture, which we call a solution.

A solute's role is to allow the solution to achieve particular chemical properties or behaviors. In different contexts, solutes can vary from gases to solids, like sodium carbonate in this exercise. Understanding the role of a solute is important in calculating molality, mole fraction, and various other solution properties.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of the number of moles of that component to the total number of moles of all components in the mixture.

In this exercise, to find the mole fraction of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\), we first determine the moles of the solute and the solvent. You calculate it by taking the moles of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) divided by the sum of the moles of both \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) and water. This results in:

• \(X_{\mathrm{Na}_{2} \mathrm{CO}_{3}} = \frac{0.025}{0.025 + 6.94} \approx 0.0036\)

Mole fraction is useful because it does not change with temperature and pressure, unlike other concentration units.

Molar Mass

Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). It is crucial in converting between grams of a substance and moles, which allows us to perform various calculations.

The molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) is calculated by adding the atomic masses of all the atoms in the formula:

• 2 sodium (Na) atoms: \(2 \times 23\)
• 1 carbon (C) atom: \(12\)
• 3 oxygen (O) atoms: \(3 \times 16\)

When added together, the molar mass for \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) is \(106 \text{ g/mol}\). This value is crucial when calculating the mass of the solute needed for creating a solution with a specific molality.

Moles Calculation

Calculating moles is essential for understanding many concepts in chemistry, as moles provide a bridge between the atomic world and the macroscopic world we experience.

To calculate the moles of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) needed to achieve a specific molality in this exercise, we use the definition of molality. Molality (extit{m}) is defined as follows:

• \(m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}}\)

For this problem, we rearranged the definition to solve for moles:

• \(\text{Moles of } \mathrm{Na}_{2} \mathrm{CO}_{3} = 0.200 \times 0.125 = 0.025\)

By understanding how to perform these calculations, you can determine the amount of solute needed in various chemical reactions and solutions.