Question

Which of the following statements about \(57.1 \mathrm{g}\) of octane, \(\mathrm{C}_{8} \mathrm{H}_{18},\) is (are) not true? (a) \(57.1 \mathrm{g}\) is 0.500 mol of octane. (b) The compound is \(84.1 \%\) C by weight. (c) The empirical formula of the compound is \(\mathrm{C}_{4} \mathrm{H}_{3}\) (d) \(57.1 \mathrm{g}\) of octane contains \(28.0 \mathrm{g}\) of hydrogen atoms.

Short answer

Statements (c) and (d) are not true.

Step-by-step solution

Determine Molar Mass of Octane

First, we need to calculate the molar mass of octane, \ \(\text{C}_8\text{H}_{18}\). Using the atomic masses, we have 8 carbon atoms and 18 hydrogen atoms: \(8 \times 12.01 \, \text{g/mol} + 18 \times 1.008 \, \text{g/mol} = 96.08 \, \text{g/mol} + 18.144 \, \text{g/mol} = 114.224 \, \text{g/mol}\).

Verify Statement (a)

To determine the number of moles in 57.1 g of octane, divide the mass by the molar mass: \[\frac{57.1 \, \text{g}}{114.224 \, \text{g/mol}} \approx 0.500 \, \text{mol}.\] Statement (a) is true.

Verify Statement (b)

Calculate the percentage of carbon in octane. The molar mass of carbon in octane is \(8 \times 12.01 = 96.08 \, \text{g/mol}\). The percentage of carbon by weight is \[\frac{96.08}{114.224} \times 100\% \approx 84.17\%.\] Thus, statement (b) is true.

Verify Statement (c)

To find the empirical formula, express the subscripts in the lowest terms. The molecular formula is \(\text{C}_8\text{H}_{18}\), which simplifies to \(\text{C}_4\text{H}_9\) (not \(\text{C}_4\text{H}_3\)). Therefore, statement (c) is not true.

Verify Statement (d)

Calculate the mass of hydrogen in 57.1 g of octane. Since the molar mass of hydrogen in octane is \(18 \times 1.008 = 18.144 \, \text{g/mol}\), the mass of hydrogen can be calculated by its proportion: \[\frac{18.144}{114.224} \times 57.1 \, \text{g} \approx 9.072 \, \text{g}.\] Thus, statement (d) is not true.

Key concepts

Empirical Formula

The empirical formula of a compound provides the simplest whole-number ratio of the elements within it. This means that it reflects the lowest terms of molecules present in a compound. To determine the empirical formula, divide the numbers of moles of each element by the smallest number of moles calculated. This may involve dividing by a common factor if applicable.
For octane, the molecular formula is \(\text{C}_8\text{H}_{18}\). To find the empirical formula, we need to express these subscripts in their simplest terms. By dividing 8 by 2 and 18 by 2, the empirical formula simplifies to \(\text{C}_4\text{H}_9\). This shows that the statement given in the original exercise, \(\text{C}_4\text{H}_3\), was incorrect.
The information from empirical formulas is useful as it gives insight into the relative proportions of each element, helping chemists understand compositional makeup and predict how a compound might react.

Percentage Composition

Percentage composition informs us of the proportion, by mass, of each element within a compound. It is calculated by dividing the total mass of each element in a formula unit by the molar mass of the whole compound, then multiplying by 100 to convert it into a percentage.
For calculating the percentage of carbon in octane \((\text{C}_8\text{H}_{18})\), you take the molar mass of all the carbon atoms in the molecule (8 carbons each with a molar mass of 12.01 g/mol for a total of 96.08 g/mol). Divide this by the molar mass of octane (114.224 g/mol) and then multiply by 100% to get \(\approx 84.17\%\).

• This approach can help chemists determine purity and analyze mixture compositions.

Thus, knowing the percentage composition can validate the composition of a sample or compare it against a theoretical model to check for deviations.

Molecular Formula

The molecular formula specifies the exact number of each type of atom present in a molecule, providing a complete representation of its chemical structure. To determine a molecular formula, one often starts by identifying the empirical formula, then uses the compound's molar mass to determine how many empirical formula units are present in one molecule.
For a compound like octane, we directly use the molecular formula, \(\text{C}_8\text{H}_{18}\), given that the molar mass of octane is approximately 114.224 g/mol. Typically, chemists verify molecular formulas by dividing the compound's molar mass by the empirical formula mass.

• The molecular formula provides a more specific description of a compound compared to the empirical formula, making it crucial for understanding chemical reactions and properties.
• Given the right data, any empirical formula can be expanded to a molecular formula; knowing both helps in identifying compounds and predicting their reactivity.

Understanding a substance's molecular formula is essential in fields such as synthetic chemistry and pharmacology, where precise molecular manipulation is key.