Question

An alkane has a \(\mathrm{C} / \mathrm{H}\) ratio of \(5.1428\) by mass. Its molecular formula is: (a) \(\mathrm{C}_{5} \mathrm{H}_{12}\) (b) \(\mathrm{C}_{6} \mathrm{H}_{14}\) (c) \(\mathrm{C}_{8} \mathrm{H}_{18}\) (d) \(\mathrm{C}_{7} \mathrm{H}_{10}\)

Short answer

The molecular formula with the correct C/H mass ratio is \( \text{C}_6\text{H}_{14} \).

Step-by-step solution

Calculate the Mass Ratio of a Given Formula

To determine which molecular formula corresponds to the given \( \frac{\text{C}}{\text{H}} \) mass ratio, calculate the carbon to hydrogen mass ratio for each option.Start with (a) \( \text{C}_5\text{H}_{12} \):- Molar mass of carbon \( \text{C} = 12 \text{ g/mol} \)- Molar mass of hydrogen \( \text{H} = 1 \text{ g/mol} \)**Mass of C in \( \text{C}_5\text{H}_{12} \):**\( 5 \times 12 = 60 \text{ g} \)**Mass of H in \( \text{C}_5\text{H}_{12} \):**\( 12 \times 1 = 12 \text{ g} \)**Mass Ratio = \( \frac{\text{Mass of C}}{\text{Mass of H}} = \frac{60}{12} = 5 \)**

Verify the Mass Ratio for Other Formulas

To ensure accuracy, calculate the \( \frac{\text{C}}{\text{H}} \) mass ratio for the remaining options and compare them with the ratio 5.1428 provided in the problem.**(b) \( \text{C}_6\text{H}_{14} \):**- Mass of C = \( 6 \times 12 = 72 \text{ g} \)- Mass of H = \( 14 \times 1 = 14 \text{ g} \)Mass Ratio = \( \frac{72}{14} \approx 5.1428 \).**(c) \( \text{C}_8\text{H}_{18} \):**- Mass of C = \( 8 \times 12 = 96 \text{ g} \)- Mass of H = \( 18 \times 1 = 18 \text{ g} \)Mass Ratio = \( \frac{96}{18} \approx 5.3333 \).**(d) \( \text{C}_7\text{H}_{10} \):**- Mass of C = \( 7 \times 12 = 84 \text{ g} \)- Mass of H = \( 10 \times 1 = 10 \text{ g} \)Mass Ratio = \( \frac{84}{10} = 8.4 \).

Compare the Calculated Ratios to the Given Ratio

The only mass ratio that matches the value provided in the problem, 5.1428, is for the molecular formula \( \text{C}_6\text{H}_{14} \). The other options have different ratios: 5 for \( \text{C}_5\text{H}_{12} \), 5.3333 for \( \text{C}_8\text{H}_{18} \), and 8.4 for \( \text{C}_7\text{H}_{10} \).

Key concepts

Understanding Mass Ratio Calculation

In chemistry, calculating the mass ratio is crucial for determining the composition of compounds. The mass ratio is a comparison of the mass of one element to another within the same molecule. To find it, you first calculate the mass of each element in the molecule using the number of atoms and their respective molar masses. Let's say you're working with a molecule of the formula \(\text{C}_5\text{H}_{12}\), the carbon (\(\text{C}\)) contributes to the total mass with \(5 \times 12 = 60\) g (since carbon's molar mass is \(12\) g/mol).Hydrogen (\(\text{H}\)), with a molar mass of \(1\) g/mol, adds \(12 \times 1 = 12\) g to the total. Hence, the mass ratio of carbon to hydrogen can be calculated as \(\frac{60}{12} = 5\). This ratio tells us that for every gram of hydrogen, there are five grams of carbon. Such calculations help in verifying the empirical formula of a compound.

Molar Mass of Carbon

The molar mass of an element is the mass of one mole of its atoms. For carbon, this value is \(12\) g/mol, which means a mole (\(6.022 \times 10^{23}\) atoms) of carbon weighs \(12\) grams. This fixed value is used in various chemical calculations, including mass ratio determination, to figure out how much carbon contributes to the mass of different hydrocarbons like alkanes.In the example of an alkane with the formula \(\text{C}_6\text{H}_{14}\), the total carbon mass is \(6 \times 12 = 72\) g. Recognizing the precise molar mass of carbon helps in ensuring accuracy in computations and provides insight into how carbon atoms dominate the weight of a molecule.

Molar Mass of Hydrogen

Hydrogen, the smallest and lightest element, has a molar mass of \(1\) g/mol. This means that one mole of hydrogen atoms weighs \(1\) gram. It features prominently in organic chemistry calculations due to its presence in hydrocarbons.For an alkane like \(\text{C}_6\text{H}_{14}\), we're dealing with \(14\) hydrogen atoms. Therefore, its added mass is \(14 \times 1 = 14\) g. Knowing the molar mass helps assess the element's contribution to a molecule's overall mass, which is essential when you calculate mass ratios for comparison with empirical data.By coupling hydrogen's molar mass with that of carbon, it becomes possible to verify the mass composition of complex molecules.

Chemical Formula Verification

Verifying a chemical formula involves ensuring the correct elements and atom counts by comparing calculated values with known data, such as mass ratios. When given a mass ratio, it can directly point to the accurate formula of a compound by comparing possibilities.This process was used in our alkane example to confirm the molecular makeup of \(\text{C}_6\text{H}_{14}\), which matched a given \(\frac{\text{C}}{\text{H}}\) mass ratio of \(5.1428\). By contrast, alternatives such as \(\text{C}_5\text{H}_{12}\) and \(\text{C}_8\text{H}_{18}\) produced inconsistent ratios of \(5\) and \(5.3333\) respectively.Chemical formula verification is essential to validate experimental results and ensure laboratory observations align with theoretical predictions. This chemistry fundamental enables the accurate identification of compounds and aids in the synthesis of new ones.