Problem 43 - Chemistry Matter and Change

Question

A sample of sodium sulfite \(\left(\mathrm{Na}_{2} \mathrm{SO}_{3}\right)\) has a mass of 2.25 \(\mathrm{g}\) \begin{equation}\begin{array}{l}{\text { a. How many } \mathrm{Na}^{+} \text { ions are present? }} \\ {\text { b. How many } \mathrm{SO}_{3}^{2-} \text { ions are present? }} \\ {\text { c. What is the mass in grams of one formula unit of } \mathrm{Na}_{2} \mathrm{SO}_{3} ?}\end{array}\end{equation}

Step-by-Step Solution

Answer

a. 2.15 × 10^22 Na⁺ ions b. 1.07 × 10^22 SO₃²⁻ ions c. 2.09 × 10⁻²² g/unit

Step 1: Calculate Molar Mass of Sodium Sulfite

First, find the molar mass of sodium sulfite (\(\mathrm{Na}_2 \mathrm{SO}_3\)) by summing the molar masses of its components. Sodium (\(\mathrm{Na}\)) has a molar mass of approximately 22.99 g/mol, sulfur (\(\mathrm{S}\)) is approximately 32.07 g/mol, and oxygen (\(\mathrm{O}\)) is approximately 16.00 g/mol. Thus, the molar mass of \(\mathrm{Na}_2 \mathrm{SO}_3\) is \(2 \times 22.99 + 32.07 + 3 \times 16.00 = 126.05\, \text{g/mol}\).

Step 2: Calculate Moles of Sodium Sulfite

Using the mass of the sample and the molar mass calculated in Step 1, find the number of moles of sodium sulfite. The formula is \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\). Therefore, the moles of sodium sulfite is \(\frac{2.25\, \text{g}}{126.05\, \text{g/mol}} = 0.01785\, \text{mol}\).

Step 3: Calculate Number of Sodium Ions

Since each formula unit of \(\mathrm{Na}_2 \mathrm{SO}_3\) contains 2 \(\mathrm{Na}^+\) ions, multiply the number of moles of sodium sulfite by Avogadro's number \(6.022 \times 10^{23}\) and by 2 to find the number of sodium ions. Thus, \(0.01785\, \text{mol} \times 6.022 \times 10^{23}\, \text{ions/mol} \times 2 = 2.15 \times 10^{22}\, \mathrm{Na}^+\, \text{ions}\).

Step 4: Calculate Number of Sulfite Ions

Each \(\mathrm{Na}_2 \mathrm{SO}_3\) contains one \(\mathrm{SO}_3^{2-}\) ion. So, multiply the moles of sodium sulfite by Avogadro's number to find the number of sulfite ions. Thus, \(0.01785\, \text{mol} \times 6.022 \times 10^{23}\, \text{ions/mol} = 1.07 \times 10^{22}\, \mathrm{SO}_3^{2-}\, \text{ions}\).

Step 5: Calculate Mass of One Formula Unit

To find the mass of one formula unit of \(\mathrm{Na}_2 \mathrm{SO}_3\), divide the molar mass by Avogadro's number. Thus, \(\frac{126.05\, \text{g/mol}}{6.022 \times 10^{23}\, \text{units/mol}} = 2.09 \times 10^{-22}\, \text{g/unit}\).

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and is crucial for converting between mass and moles in stoichiometric calculations. To find the molar mass of a compound, we sum the atomic masses of all atoms present in its chemical formula.
For example, the compound sodium sulfite, \( \mathrm{Na}_2 \mathrm{SO}_3 \), contains:

• 2 sodium (\( \mathrm{Na} \)) atoms, each with a molar mass of approximately 22.99 g/mol,
• 1 sulfur (\( \mathrm{S} \)) atom with a molar mass of about 32.07 g/mol, and
• 3 oxygen (\( \mathrm{O} \)) atoms, each with a molar mass of approximately 16.00 g/mol.

Hence, the molar mass of \( \mathrm{Na}_2 \mathrm{SO}_3 \) is calculated as: \[2 \times 22.99 + 32.07 + 3 \times 16.00 = 126.05\, \text{g/mol}\].This value is essential when determining the number of moles or converting grams to moles.

Avogadro's Number

Avogadro's number is a key concept in chemistry that provides a link between the microscopic world of atoms and the macroscopic world we can measure. It defines the number of atoms, ions, or molecules in one mole of a substance, which is approximately \( 6.022 \times 10^{23} \).
This enormous number is used to convert between the number of entities (like atoms or molecules) and the amount of substance in moles.
For example, in our sodium sulfite problem, understanding Avogadro's number helps calculate the number of sodium ions and sulfite ions present in a given mass. Specifically:

• Each mole of sodium sulfite contains \( 2 \times 6.022 \times 10^{23} \) sodium ions because there are two sodium ions per formula unit of \( \mathrm{Na}_2 \mathrm{SO}_3 \).
• Sulfite ions are calculated as \( 1 \times 6.022 \times 10^{23} \), as each unit contains one sulfite ion.

Understanding Avogadro's number is essential for connecting the mass of a chemical sample to the number of particles present.

Sodium Sulfite Calculations

Calculating various quantities in a substance like sodium sulfite involves multiple steps including finding the molar mass and using Avogadro's number. Here's a brief walkthrough of the process:
1. **Determine the Molar Mass**: We've already found that the molar mass of \( \mathrm{Na}_2 \mathrm{SO}_3 \) is 126.05 g/mol.
2. **Calculate Moles from Mass**: Given 2.25 g of \( \mathrm{Na}_2 \mathrm{SO}_3 \), calculate moles using the formula:\[\text{moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{2.25\, \text{g}}{126.05\, \text{g/mol}} \approx 0.01785\, \text{mol}\]3. **Determine Ions Present**: Utilizing Avogadro's number:

• For sodium ions, multiply moles by 2 and Avogadro's number.
• For sulfite ions, multiply moles by Avogadro's number.

These calculations help in comprehending how many individual atoms or ions are present in a sample, given its mass.

Chemical Formula Calculations

Calculating the components of a chemical formula involves analyzing the numerical ratios and types of atoms or ions involved. When given the formula unit of a compound, you can determine how many ions or atoms it contains and their respective masses.
For \( \mathrm{Na}_2\mathrm{SO}_3 \):

• The chemical formula tells us there are 2 sodium ions, 1 sulfur atom, and 3 oxygen atoms per formula unit.
• This arrangement leads to a 1-to-1 correspondence between formula units and certain ions (e.g., one sulfite ion per unit).

In the original exercise, the mass of a single formula unit can be found by dividing the compound's molar mass by Avogadro's number, giving:\[ \frac{126.05\, \text{g/mol}}{6.022 \times 10^{23} } \approx 2.09 \times 10^{-22} \text{g/unit}\].
This calculation illustrates the ability to navigate between macroscopic measurements and microscopic quantities through Chem formula evaluations.