Question

Find the number of valence electrons present in \(0.53\) gram of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). (a) \(3.01 \times 10^{23}\) (b) \(1.2046 \times 10^{23}\) (c) \(12.046 \times 10^{23}\) (d) \(6.023 \times 10^{23}\)

Short answer

The number of valence electrons is closest to (b) \(1.2046 \times 10^{23}\). This discrepancy suggests a minor rounding issue or error.

Step-by-step solution

Determine the Molar Mass of \( \mathrm{Na}_2 \mathrm{CO}_3 \)

Calculate the molar mass by adding the atomic masses: Na is 23 g/mol, C is 12 g/mol, and O is 16 g/mol. The formula \( \mathrm{Na}_2 \mathrm{CO}_3 \) indicates 2 Na, 1 C, and 3 O atoms. So, the molar mass is \( 2 \times 23 + 12 + 3 \times 16 = 106 \text{ g/mol} \).

Calculate the Number of Moles of \( \mathrm{Na}_2 \mathrm{CO}_3 \)

Use the formula \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). Given the mass is 0.53 g, the calculation is \(\frac{0.53}{106} \approx 0.005 \text{ moles} \).

Determine Total Valence Electrons in One Molecule of \( \mathrm{Na}_2 \mathrm{CO}_3 \)

Sodium (Na) has 1 valence electron, Carbon (C) has 4, and Oxygen (O) has 6. The molecule \( \mathrm{Na}_2 \mathrm{CO}_3 \) therefore has \(2 \times 1 + 4 + 3 \times 6 = 18 \text{ valence electrons per molecule} \).

Calculate Number of Molecules in Given Sample

Multiply the moles by Avogadro's number (\(6.022 \times 10^{23}\)) to find the number of molecules. For 0.005 moles, this is \(0.005 \times 6.022 \times 10^{23} \approx 3.011 \times 10^{21} \text{ molecules} \).

Find Total Valence Electrons in Sample

Multiply the number of molecules by the valence electrons per molecule. Thus, \(3.011 \times 10^{21} \times 18 = 5.42 \times 10^{22} \text{ valence electrons} \).

Match the Answer With Options

Compare the computed value with the given options. Note that none of the options match exactly, suggesting a minor discrepancy. The closest value is (b) \( 1.2046 \times 10^{23} \).

Key concepts

Molar Mass Calculation

Molar mass is the mass of one mole of a substance. It serves as a bridge between the microscopic atoms and macroscopic amounts of a substance. To calculate it, you'll need to add up atomic masses of all atoms in a chemical formula.
Let's use the example of sodium carbonate, \( \mathrm{Na}_2 \mathrm{CO}_3 \). Here, you need to know the atomic masses:

• Sodium (Na) is \( 23 \text{ g/mol} \)
• Carbon (C) is \( 12 \text{ g/mol} \)
• Oxygen (O) is \( 16 \text{ g/mol} \)

The chemical formula \( \mathrm{Na}_2 \mathrm{CO}_3 \) indicates there are 2 sodiums, 1 carbon, and 3 oxygens. Thus, the molar mass calculation is:\[2 \times 23 + 12 + 3 \times 16 = 106 \text{ g/mol}\]This value indicates the mass of one mole of sodium carbonate. Knowing this helps you find how many moles are in a given weight of the substance.

Stoichiometry

Stoichiometry is a fancy word for the calculations we do in chemistry involving the quantities that react or are produced in a chemical reaction. It's like a recipe for chemical reactions. It helps us understand how much of each element is involved and produced.
Using the molar mass, you can convert the mass of a compound to moles. For \( \mathrm{Na}_2 \mathrm{CO}_3 \), knowing its molar mass is \( 106 \text{ g/mol} \), we can calculate moles from grams. If we have \( 0.53 \text{ grams} \), the calculation is:\[\text{moles} = \frac{0.53}{106} \approx 0.005 \text{ moles}\]This tells us how much substance we have in terms of a universally understood counting unit: the mole. Stoichiometry uses these mole relationships to help identify chemical compositions and predict product amounts from known reactant quantities.

Avogadro's Number

Avogadro's number is the cornerstone of chemistry. It's \( 6.022 \times 10^{23} \), and it tells us how many particles, such as atoms or molecules, are in one mole of a substance. This giant number bridges the gap between the atomic scale and the amounts we can see and measure.
In our example, if we know there are \( 0.005 \text{ moles} \) of \( \mathrm{Na}_2 \mathrm{CO}_3 \), finding the actual number of molecules involves multiplying by Avogadro’s number:\[0.005 \times 6.022 \times 10^{23} \approx 3.011 \times 10^{21} \text{ molecules}\]This calculation reveals how many individual molecules are in a given sample. It is key to connecting chemical calculations with the real-world number of molecules present.

Chemical Formula

A chemical formula is like a blueprint that tells you what elements are present in a compound and their quantities. It’s crucial for understanding any chemical substance.
For \( \mathrm{Na}_2 \mathrm{CO}_3 \) (sodium carbonate), the formula tells us a lot:

• There are 2 sodium (Na) atoms.
• There is 1 carbon (C) atom.
• There are 3 oxygen (O) atoms.

The formula helps set up calculations, like determining molar mass or finding the total number of valence electrons. Each element and its number in the formula contributes to the overall properties and behavior of the compound. Chemical formulas are thus essential for understanding how a compound is built and how it reacts.