Problem 57 - A Complete Resource Book in Chemistry for JEE Main

Question

The empirical formula of a compound is \(\mathrm{CH}_{2}\), One mole of the compound has a mass of \(42 \mathrm{~g}\). Its molecular formula is (a) \(\mathrm{C}_{j} \mathrm{H}_{6}\) (b) \(\mathrm{C}_{3} \mathrm{H}_{\mathrm{s}}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{2}\) (d) \(\mathrm{CH}_{2}\)

Step-by-Step Solution

Answer

Option (b) \(\mathrm{C}_3\mathrm{H}_6\).

Step 1: Determine Molar Mass of Empirical Formula

Calculate the molar mass of the empirical formula \(\mathrm{CH}_2\). The molar mass of carbon (C) is approximately 12 g/mol, and hydrogen (H) is approximately 1 g/mol. Therefore, the molar mass of \(\mathrm{CH}_2\) is \(12 + 2 \times 1 = 14\) g/mol.

Step 2: Determine Number of Empirical Units in Molecular Formula

Divide the given molar mass of the compound (42 g/mol) by the molar mass of the empirical formula (14 g/mol) to find how many empirical units make up the molecular formula. \[ \text{Number of units} = \frac{42}{14} = 3 \].

Step 3: Write the Molecular Formula

If the empirical formula \(\mathrm{CH}_2\) represents one unit, multiplying it by 3 (the number of units found) gives the molecular formula. Thus, the molecular formula is \(\mathrm{C}_3\mathrm{H}_6\).

Step 4: Compare to Given Options

Match the calculated molecular formula \(\mathrm{C}_3\mathrm{H}_6\) with the options provided. It corresponds to option (b) \(\mathrm{C}_3\mathrm{H}_6\).

Key Concepts

Empirical Formula

An empirical formula provides the simplest whole-number ratio of atoms of each element present in a compound. It does not show the exact number of atoms, but rather the simplest ratio. For example, with the empirical formula \( \mathrm{CH}_2 \), it indicates that for every one carbon atom, there are two hydrogen atoms.
This basic formula gives a foundational understanding of the compound's composition but not the total number of atoms that might be present in a related molecular formula. The empirical formula is derived from percent composition data, which reveals the percentage of each element within a compound. It's essential because it serves as a stepping stone to figuring out the molecular formula, especially when the compound’s molar mass is known. To get this simplest form, one sometimes has to convert percentages to moles and then divide by the smallest number of moles to find the ratio.
Understanding the empirical formula is a critical step in chemistry problem-solving, providing a straightforward starting point to deducing more detailed molecular structures.

Molar Mass Calculation

The molar mass of a compound is the sum of the masses of its constituent atoms, calculated using the atomic masses from the periodic table. For the empirical formula \( \mathrm{CH}_2 \), the molar mass is computed by adding the mass of one carbon atom (roughly 12 g/mol) and two hydrogen atoms (each about 1 g/mol). Thus, the molar mass equals approximately 14 g/mol.
Calculating the molar mass is both a simple yet integral part of determining a compound’s molecular formula. Knowing this, along with the molar mass of the whole compound, allows chemists to discern how many empirical units compose the molecular formula.
For instance, if one mole of the compound weighs 42 g and the empirical formula weights 14 g, then using the formula:

• \[\text{Number of units} = \frac{42}{14} = 3\]

the molecular formula comprises three empirical units. This understanding is crucial for progressing into more complex chemistry equations and is vital in ensuring the accuracy of chemical analysis.

Chemistry Problem-Solving

Solving chemistry problems, such as determining a molecular formula, involves a systematic approach using known values and calculations. One begins with the empirical formula and utilizes information about molar masses to piece together the molecular formula. In this case, given data like the empirical formula \( \mathrm{CH}_2 \) and the molar mass of the compound at 42 g/mol directs the calculation path.
Here's the step-by-step reasoning:

• First, ascertain the molar mass of the empirical unit which lays the groundwork for recognizing how many of these units fit into the molecular formula.
• Next, divide the compound's given molar mass by the empirical formula's molar mass to find out the multiplicative factor (in this case, 3).
• Finally, multiply each subscript in the empirical formula by this number to obtain the molecular formula.

Complicated problems like these require clear logic, often with successive problem-solving processes and occasional trial and error. Seasoned chemists lean on their understanding of concepts like stoichiometry and balanced equations in tandem with molar mass calculations to ensure a comprehensive solution.