Question

Calculate the mass in grams for a single molecule of the following compounds: (a) methane, \(\mathrm{CH}_{4}\) (b) ammonia, \(\mathrm{NH}_{3}\) (c) sulfur trioxide, \(\mathrm{SO}_{3}\) (d) nitrogen dioxide, \(\mathrm{NO}_{2}\)

Short answer

(a) \(2.664 \times 10^{-23}\) g, (b) \(2.829 \times 10^{-23}\) g, (c) \(1.33 \times 10^{-22}\) g, (d) \(7.644 \times 10^{-23}\) g.

Step-by-step solution

Calculate the molar mass of the compound

To calculate the molar mass of a compound, sum the atomic masses of all the atoms in its formula. Atomic masses can be found on the periodic table. For each compound: (a) Methane: \ \( \text{CH}_4 = 1(\text{C}) + 4(\text{H}) = 12.01 + 4(1.008) = 16.042 \, \text{g/mol} \) (b) Ammonia: \ \( \text{NH}_3 = 1(\text{N}) + 3(\text{H}) = 14.01 + 3(1.008) = 17.034 \, \text{g/mol} \) (c) Sulfur trioxide: \ \( \text{SO}_3 = 1(\text{S}) + 3(\text{O}) = 32.07 + 3(16.00) = 80.07 \, \text{g/mol} \) (d) Nitrogen dioxide: \ \( \text{NO}_2 = 1(\text{N}) + 2(\text{O}) = 14.01 + 2(16.00) = 46.01 \, \text{g/mol} \)

Use Avogadro's number to find the mass of one molecule

Avogadro's number, \( 6.022 \times 10^{23} \), is the number of atoms/molecules in one mole. Divide the molar mass by Avogadro's number to find the mass of a single molecule. For each compound:(a) Methane: \ \( \frac{16.042}{6.022 \times 10^{23}} = 2.664 \times 10^{-23} \, \text{g} \) (b) Ammonia: \ \( \frac{17.034}{6.022 \times 10^{23}} = 2.829 \times 10^{-23} \, \text{g} \) (c) Sulfur trioxide: \ \( \frac{80.07}{6.022 \times 10^{23}} = 1.33 \times 10^{-22} \, \text{g} \) (d) Nitrogen dioxide: \ \( \frac{46.01}{6.022 \times 10^{23}} = 7.644 \times 10^{-23} \, \text{g} \)

Key concepts

Avogadro's Number

In chemistry, Avogadro's number is a fundamental constant that helps quantify atoms and molecules. Imagine needing to count the countless tiny entities making up matter; Avogadro's number makes this manageable. It states that one mole of any substance contains exactly \(6.022 \times 10^{23}\) entities, whether they are atoms, molecules, or ions. This allows us to equate macroscopic amounts of substances to their microscopic amounts. For example, when calculating the mass of a single molecule in our initial exercise, understanding Avogadro’s number bridges the gap between the scale we use in real-world measurements and the atomic scale where molecules operate. Without it, conversing about individual molecules in grams would be inconceivably tiny and complicated. This number is the cornerstone of the mole concept, crucial for translating the weight of massive quantities to the weight of single particles when doing chemical calculations.

Molar Mass

Molar mass is a crucial concept when assessing how much a mole of a substance weighs, given in grams per mole (g/mol). Essentially, it's the mass of one mole of a chemical compound or element. When you sum up the atomic masses of all the atoms in a compound's formula, it yields the molar mass.Looking at our example compounds:

• Methane (\(\text{CH}_4\)) has a molar mass of 16.042 g/mol.
• Ammonia (\(\text{NH}_3\)) weighs 17.034 g/mol.
• Sulfur trioxide (\(\text{SO}_3\)) carries a molar mass of 80.07 g/mol.
• Nitrogen dioxide (\(\text{NO}_2\)) counts to 46.01 g/mol.

These calculations become handy when converting between the mass of substances and moles, aiding chemists in working between different scales efficiently. The molar mass helps provide a stepping stone from molecular-level calculations to understandable, tangible quantities.

Atomic Mass Calculation

Calculating atomic mass is key to determining molar masses, which are based on the sum of atomic masses from the periodic table. The atomic mass of each element closely approximates the mass of a single atom and is usually expressed in atomic mass units (amu).To perform an atomic mass calculation, follow these steps:

• Identify each element present in the compound.
• Locate the atomic mass of each element from the periodic table.
• Multiply the atomic mass by the number of those atoms in one molecule.
• Sum these values for the total atomic mass of the molecule.

For instance, methane (\(\text{CH}_4\)) is composed of one carbon atom and four hydrogen atoms:

• Carbon: \(1 \times 12.01 \text{ amu}\)
• Hydrogen: \(4 \times 1.008 \text{ amu}\)
• Total: 16.042 amu (molar mass in g/mol)

By understanding atomic mass, we can accurately calculate not only molar masses but also engage in practical chemistry calculations with real-world relevance.