Problem 65 - Chemistry An Atoms First Approach

Question

What amount (moles) is represented by each of these samples? a. \(150.0 \mathrm{g} \mathrm{Fe}_{2} \mathrm{O}_{3}\) b. \(10.0 \mathrm{mg} \mathrm{NO}_{2}\) c. \(1.5 \times 10^{16}\) molecules of \(\mathrm{BF}_{3}\)

Step-by-Step Solution

Answer

The amounts in moles for the given samples are: a. \(0.939 \: \mathrm{mol} \: \mathrm{Fe}_{2}\mathrm{O}_{3}\) b. \(2.17 \times 10^{-4} \: \mathrm{mol} \: \mathrm{NO}_{2}\) c. \(2.49 \times 10^{-8} \: \mathrm{mol} \: \mathrm{BF}_{3}\)

Step 1: Determine the molar masses of the substances

First, we need to find the molar masses of Fe₂O₃, NO₂, and BF₃ using the atomic masses of each element. Molar mass of Fe₂O₃: Fe = 55.845 g/mol (2 atoms) O = 16.00 g/mol (3 atoms) Molar mass of Fe₂O₃ = (2 × 55.845) + (3 × 16.00) = 159.69 g/mol Molar mass of NO₂: N = 14.01 g/mol O = 16.00 g/mol (2 atoms) Molar mass of NO₂ = 14.01 + (2 × 16.00) = 46.01 g/mol Molar mass of BF₃: B = 10.81 g/mol F = 19.00 g/mol (3 atoms) Molar mass of BF₃ = 10.81 + (3 × 19.00) = 67.81 g/mol

Step 2: Convert masses to moles and molecules to moles

Now that we have the molar masses, we can use the amounts given for each sample to find the number of moles. We will also need to convert the given mass for NO₂ from milligrams to grams. a. Moles of Fe₂O₃: Moles = (mass) / (molar mass) Moles = (150.0 g) / (159.69 g/mol) = 0.939 mol b. Convert 10.0 mg of NO₂ to grams: 1 g = 1000 mg 10.0 mg = 0.0100 g Moles of NO₂ = (mass) / (molar mass) Moles = (0.0100 g) / (46.01 g/mol) = 2.17 × 10⁻⁴ mol c. Convert the number of molecules of BF₃ to moles: Moles = (number of molecules) / (Avogadro's number) Moles = (1.5 × 10¹⁶ molecules) / (6.022 × 10²³ molecules/mol) = 2.49 × 10⁻⁸ mol

Step 3: Write the final answers

Now we can write the final answers for the amount of moles for each sample: a. 150.0 g of Fe₂O₃ = 0.939 mol b. 10.0 mg of NO₂ = 2.17 × 10⁻⁴ mol c. 1.5 × 10¹⁶ molecules of BF₃ = 2.49 × 10⁻⁸ mol

Key Concepts

Molar Mass

The concept of molar mass is foundational in chemistry and involves determining the mass of one mole of a substance. It is expressed in grams per mole (g/mol). The molar mass of a compound can be found by summing the atomic masses of all its constituent elements, as provided by the periodic table. For example, to find the molar mass of Fe₂O₃, we need to consider:

• the atomic mass of iron (Fe) which is 55.845 g/mol, and it appears twice in the molecule, growing to a mass of \(2 \times 55.845\).
• oxygen (O) has an atomic mass of 16.00 g/mol and appears three times, resulting in \(3 \times 16.00\).

The total molar mass of Fe₂O₃ then becomes \(159.69\ g/mol\.\) This process is repeated similarly for other compounds such as NO₂ and BF₃. Understanding molar mass is crucial for converting between mass and moles, making it a key component of mole calculations.

Chemical Conversions

Chemical conversions are essential for moving between different units of measurement in chemistry. They typically involve converting grams into moles or molecules into moles, and vice versa. One of the most common conversions is from mass to moles, using the formula:\[\text{Moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\]For instance, to find how many moles are in 150.0 g of Fe₂O₃ with a molar mass of 159.69 g/mol, you would divide the mass by the molar mass:\[\text{Moles of } Fe₂O₃ = \frac{150.0}{159.69} \approx 0.939 \text{ mol}\]Similarly, if you have a very small mass like 10.0 mg of NO₂, you would first convert it to grams (0.0100 g) before dividing by its molar mass of 46.01 g/mol.Converting molecules to moles requires knowing Avogadro's number, but we'll address that in more detail in the next section. Mastering these conversions allows chemists to quantify substances effectively, playing a significant role in chemical reactions and stoichiometry.

Avogadro's Number

Avogadro's number is a fundamental constant used to relate quantities at the atomic or molecular scale to more tangible amounts. It is defined as the number of atoms or molecules in one mole of a substance and is approximately \(6.022 \times 10^{23} \text{ molecules/mol}\).This constant is critical when converting molecules to moles. For example, if you have \(1.5 \times 10^{16}\) molecules of BF₃, you can find the number of moles using the equation:\[\text{Moles} = \frac{\text{number of molecules}}{\text{Avogadro's number}}\]For our BF₃ example, it would be: \[\frac{1.5 \times 10^{16}}{6.022 \times 10^{23}} = 2.49 \times 10^{-8} \text{ moles}\]Understanding Avogadro's number is crucial for converting between particles and moles, allowing chemists to approach atomic-scale quantities in the laboratory settings.