Problem 81 - Chemistry The Science in Context

Question

The following compounds have been detected in space. Which of them contains the greatest percentage of carbon by mass? Do any two of the following compounds have the same empirical formula? a. naphthalene, \(\mathrm{C}_{10} \mathrm{H}_{8}\) b. chrysene, \(\mathbf{C}_{18} \mathrm{H}_{12}\) c. pentacene, \(\mathrm{C}_{22} \mathrm{H}_{14}\) d. pyrene, \(\mathrm{C}_{16} \mathrm{H}_{10}\)

Step-by-Step Solution

Answer

Answer: Pyrene contains the greatest percentage of carbon by mass (approximately 95.05%), and none of the given compounds have the same empirical formula.

Step 1: 1. Calculating molar mass of all given compounds

For each compound, we need to calculate the molar mass, which is the sum of the atomic masses of all the elements in the compound. We know that the atomic mass of carbon (C) is approximately 12 g/mol, and that of hydrogen (H) is approximately 1 g/mol. For each compound: a. naphthalene, \(C_{10}H_{8}\): Molar mass = (10 × 12) + (8 × 1) = 120 + 8 = 128 g/mol b. chrysene, \(C_{18}H_{12}\): Molar mass = (18 × 12) + (12 × 1) = 216 + 12 = 228 g/mol c. pentacene, \(C_{22}H_{14}\): Molar mass = (22 × 12) + (14 × 1) = 264 + 14 = 278 g/mol d. pyrene, \(C_{16}H_{10}\): Molar mass = (16 × 12) + (10 × 1) = 192 + 10 = 202 g/mol

Step 2: 2. Calculating carbon mass percentage in each compound

To determine the ratio of carbon by mass in each compound, we'll divide the total mass contributed by carbon in the compound by the molar mass of the compound, and multiply the result by 100 to get the percentage. a. naphthalene, \(C_{10}H_{8}\): Carbon mass percentage = (120 / 128) × 100 ≈ 93.75% b. chrysene, \(C_{18}H_{12}\): Carbon mass percentage = (216 / 228) × 100 ≈ 94.74% c. pentacene, \(C_{22}H_{14}\): Carbon mass percentage = (264 / 278) × 100 ≈ 94.96% d. pyrene, \(C_{16}H_{10}\): Carbon mass percentage = (192 / 202) × 100 ≈ 95.05%

Step 3: 3. Identifying the compound with the highest carbon mass percentage

Comparing the carbon mass percentages calculated in the step above, we find that pyrene (d) has the highest percentage of carbon with approximately 95.05% carbon by mass.

Step 4: 4. Determining whether any two compounds share the same empirical formula

To find the empirical formula for each compound, we need to determine the simplest whole number ratio between carbon and hydrogen atoms in each compound. a. naphthalene, \(C_{10}H_{8}\): Ratio = \(C: \frac{10}{2} \cdot H = C_{5}H_{4}\) b. chrysene, \(C_{18}H_{12}\): Ratio = \(C: \frac{18}{6} \cdot H = C_{3}H_{2}\) c. pentacene, \(C_{22}H_{14}\): Ratio = \(C: \frac{22}{2} \cdot H = C_{11}H_{7}\) d. pyrene, \(C_{16}H_{10}\): Ratio = \(C: \frac{16}{2} \cdot H = C_{8}H_{5}\) Comparing these empirical formulas, it appears that none of these compounds have the same empirical formula. In conclusion, pyrene contains the greatest percentage of carbon by mass, and none of the given compounds have the same empirical formula.

Key Concepts

Molar Mass Calculation

Understanding how to calculate the molar mass of a compound is an essential skill in chemistry. The molar mass is the sum of the atomic masses of all the atoms in a compound. We express it in grams per mole (g/mol). For example, to determine the molar mass of naphthalene

• Count the number of each type of atom in the chemical formula: naphthalene is composed of 10 carbon atoms and 8 hydrogen atoms.
• Use the atomic masses: carbon has an atomic mass of approximately 12 g/mol, and hydrogen has an atomic mass of about 1 g/mol.
• Multiply the number of each atom by its atomic mass: for naphthalene, this is (10 × 12) + (8 × 1) = 120 + 8 = 128 g/mol.

This calculation is repeated similarly for chrysene, pentacene, and pyrene. Knowing the molar mass helps in further understanding the composition and reaction activities of compounds.

Empirical Formula

The empirical formula of a chemical compound gives the simplest whole number ratio of atoms of each element in the compound. It differs from the molecular formula, which shows the exact number of each atom in a compound. To find the empirical formula:

• Look at the molecular formula: for instance, chrysene is composed of 18 carbon and 12 hydrogen atoms (\(C_{18}H_{12}\).
• Simplify the ratio of these numbers by dividing by the greatest common factor: in this case, the factor is 6, giving an empirical formula of \(C_{3}H_{2}\).

Analyzing the empirical formula of compounds allows us to identify similarities and differences among them. For the given examples in the exercise, none of the compounds share the same empirical formula, indicating distinct elementary compositions in their simplest forms.

Carbon Mass Percentage

The carbon mass percentage in a compound tells you how much of that compound's mass is made up of carbon atoms. It's calculated using the formula:\[\text{Carbon mass percentage} = \left(\frac{\text{Carbon mass in one mole of compound}}{\text{Molar mass of compound}}\right) \times 100\%\]To illustrate:

• For pyrene \(C_{16}H_{10}\), the carbon mass is \(16 \times 12 = 192\) g.
• The compound's molar mass is 202 g/mol.
• Thus, the carbon mass percentage is \((192 / 202) \times 100\% \approx 95.05\%\).

Among the compounds discussed, pyrene has the highest carbon mass percentage. This calculation is essential for understanding the composition of chemical substances, especially when comparing similar compounds or conducting analytical work.