Problem 53 - A Complete Resource Book in Chemistry for JEE Main

Question

An alkane has a \(\mathrm{C} / \mathrm{H}\) ratio of \(5.1428\) by mass. Its molecular formula is (a) \(\mathrm{C}_{3} \mathrm{H}_{12}\) (b) \(\mathrm{C}_{6} \mathrm{H}_{14}\) (c) \(\mathrm{C}_{8} \mathrm{H}_{18}\) (d) \(\mathrm{C}_{7} \mathrm{H}_{10}\)

Step-by-Step Solution

Answer

The molecular formula of the alkane is \(\mathrm{C}_8\mathrm{H}_{18}\).

Step 1: Understand the Problem

We need to find the correct molecular formula of an alkane based on its given carbon/hydrogen mass ratio of 5.1428.

Step 2: Calculate Molar Mass for Carbon and Hydrogen

The atomic mass of carbon is approximately 12.01 g/mol, and that of hydrogen is approximately 1.01 g/mol.

Step 3: Calculate Mass Ratio for Each Option

We will calculate the C/H mass ratio for each option: \(\frac{\text{mass of } C}{\text{mass of } H}\). By substituting the number of carbon and hydrogen atoms in each possible molecular formula:

Step 3a: Option A - \(\mathrm{C}_3\mathrm{H}_{12}\)

Calculate mass of carbon: \(3 \times 12.01 = 36.03\) g/mol. Calculate mass of hydrogen: \(12 \times 1.01 = 12.12\) g/mol. Calculate the mass ratio: \(\frac{36.03}{12.12} \approx 2.973\)

Step 3b: Option B - \(\mathrm{C}_6\mathrm{H}_{14}\)

Calculate mass of carbon: \(6 \times 12.01 = 72.06\) g/mol. Calculate mass of hydrogen: \(14 \times 1.01 = 14.14\) g/mol. Calculate the mass ratio: \(\frac{72.06}{14.14} \approx 5.095\)

Step 3c: Option C - \(\mathrm{C}_8\mathrm{H}_{18}\)

Calculate mass of carbon: \(8 \times 12.01 = 96.08\) g/mol. Calculate mass of hydrogen: \(18 \times 1.01 = 18.18\) g/mol. Calculate the mass ratio: \(\frac{96.08}{18.18} \approx 5.287\)

Step 3d: Option D - \(\mathrm{C}_7\mathrm{H}_{10}\)

Calculate mass of carbon: \(7 \times 12.01 = 84.07\) g/mol. Calculate mass of hydrogen: \(10 \times 1.01 = 10.10\) g/mol. Calculate the mass ratio: \(\frac{84.07}{10.10} \approx 8.322\)

Step 4: Compare Calculated Ratios to Given Ratio

We need to match each option’s C/H mass ratio with the given ratio of 5.1428. The closest match indicates the correct formula.

Step 5: Conclusion

The C/H mass ratio for option C \(\mathrm{C}_8\mathrm{H}_{18}\) is 5.287, which is the closest to the given ratio of 5.1428.

Key Concepts

C/H Ratio Calculation

When dealing with alkanes, a key task is to determine the carbon/hydrogen (C/H) mass ratio. This ratio is simply the mass of carbon in the molecule divided by the mass of hydrogen. To solve the problem and find the compound with a given C/H ratio of 5.1428, you will want to calculate this ratio for each of the proposed molecular formulas.
First, calculate the total mass contributed by carbon and hydrogen in each formula. Then, divide the total mass of carbon by the total mass of hydrogen to find the ratio. This quantitative expression helps in comparing different molecular formulas to find the best match with known experimental data.
By performing these calculations, we can identify which formula, among several, aligns most closely with the specified mass ratio, securing a clearer understanding of the compound's composition.

Molar Mass of Elements

Understanding the molar mass of elements is cornerstone in chemistry. Molar mass allows us to convert between the number of atoms or molecules and the mass in grams. For each element, the molar mass is defined as the mass of one mole of its atoms. For example:

• The molar mass of carbon is approximately 12.01 g/mol.
• The molar mass of hydrogen is approximately 1.01 g/mol.

By knowing these values, you can calculate how much mass each element contributes to a molecule. Take the formula \( ext{C}_{6} ext{H}_{14}\):

• The total mass of carbon atoms is calculated as \(6 \times 12.01\) g/mol.
• The total mass of hydrogen atoms is \(14 \times 1.01\) g/mol.

Once you have these individual masses, they can be used to determine the overall mass ratio, aiding in deducing the molecular composition based on empirical data.

Comparing Mass Ratios

When comparing mass ratios, the goal is to find out which molecular formula produces a ratio closest to an experimentally determined value. Each possible formula of an alkane gives a different C/H mass ratio. In our exercise:

• Option A: \(\text{C}_3\text{H}_{12}\) results in a ratio of about 2.973.
• Option B: \(\text{C}_6\text{H}_{14}\) gives a ratio of approximately 5.095.
• Option C: \(\text{C}_8\text{H}_{18}\) leads to a ratio of around 5.287.
• Option D: \(\text{C}_7\text{H}_{10}\) results in a higher ratio of approximately 8.322.

We compare these calculated ratios with the given ratio of 5.1428. The mathematical closeness of a calculated ratio to the given ratio indicates the most probable molecular formula for the alkane. Through these comparisons, you learn not only which molecular structure likely matches an observed chemical compound, but also refine your skills in using empirical data for chemical analysis.