Problem 106 - A Complete Resource Book in Chemistry for JEE Main

Question

In an organic compound of molar mass \(108 \mathrm{~g} \mathrm{~mol}^{-1} \mathrm{C}\), \(\mathrm{H}\) and \(\mathrm{N}\) atoms are present in \(9: 1: 3.5\) by weight. The molecular formula of the compound can be (a) \(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{~N}_{2}\) (b) \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{~N}_{2}\) (c) \(\mathrm{C}_{5} \mathrm{H}_{6} \mathrm{~N}_{3}\) (d) \(\mathrm{C}_{4} \mathrm{H}_{12} \mathrm{~N}_{2}\)

Step-by-Step Solution

Answer

The molecular formula of the compound is \( \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{N}_{2} \).

Step 1: Establish Mass Composition

The given mass ratio for C, H, and N is 9:1:3.5. Let's assume the mass of the compound is 108 g (equal to its molar mass). Then the masses are: \( m_C = \frac{9}{13.5}\times108 = 72 \text{ g} \), \( m_H = \frac{1}{13.5}\times108 = 8 \text{ g} \), and \( m_N = \frac{3.5}{13.5}\times108 = 28 \text{ g} \).

Step 2: Calculate Number of Moles of Each Element

Using the atomic masses: \( M_C = 12 \text{ g/mol} \), \( M_H = 1 \text{ g/mol} \), \( M_N = 14 \text{ g/mol} \). \( n_C = \frac{72}{12} = 6 \text{ moles} \), \( n_H = \frac{8}{1} = 8 \text{ moles} \), \( n_N = \frac{28}{14} = 2 \text{ moles} \).

Step 3: Determine Empirical Formula

The empirical formula is based on the smallest whole number ratio of moles, which is 6:8:2. Dividing by the smallest number (2), we get 3:4:1. So, the empirical formula is \( \text{C}_3\text{H}_4\text{N} \).

Step 4: Find the Molecular Formula

The molar mass given is 108 g/mol. Calculate the molar mass of the empirical formula: \( 3 \times 12 + 4 \times 1 + 1 \times 14 = 54 \text{ g/mol} \). Divide the molar mass of the compound by the empirical formula mass: \( \frac{108}{54} = 2 \). Thus, the molecular formula is \( (\text{C}_3\text{H}_4\text{N})_2 = \text{C}_6\text{H}_8\text{N}_2 \).

Step 5: Match with Given Options

Compare the derived molecular formula \( \text{C}_6\text{H}_8\text{N}_2 \) to the given options. It matches choice (b).

Key Concepts

Molar Mass

Molar mass is an essential concept in chemistry that refers to the mass of a given substance divided by the amount of substance, expressed in moles. It is often measured in grams per mole (g/mol). To determine the molar mass, you need to know the sum of the atomic masses of all atoms present in a molecule.
For example, if we are considering a compound with carbon (C), hydrogen (H), and nitrogen (N) atoms, we must sum the atomic masses:

• Carbon: 12 g/mol
• Hydrogen: 1 g/mol
• Nitrogen: 14 g/mol

Let’s say a compound has the formula C_3H_4N; the molar mass will be calculated as: \[3 \times 12 + 4 \times 1 + 1 \times 14 = 54 \text{ g/mol}\] Understanding molar mass allows you to compare it with the given molar mass of a substance to verify or determine the correct molecular formula.

Empirical Formula

The empirical formula of a compound represents the simplest whole-number ratio of atoms of each element present in the compound. It is fundamental in determining a compound's molecular formula, which provides insight into the compound's composition.
To find the empirical formula, you start by calculating the moles of each element present using their respective mass-to-atomic mass ratios, then establish the simplest ratio.

• Convert mass percentages or mass ratios into moles.
• Simplify the mole ratio by dividing each number by the smallest number of moles calculated.
• Write the new numbers as subscripts in the chemical formula.

For example, a compound with 6 moles of C , 8 moles of H , and 2 moles of N converts to an empirical formula of C_3H_4N after simplification. This ratio (3:4:1) is the simplest form.

Elemental Composition

Elemental composition is a term that refers to the relative amount of each different element present in a chemical compound. This information can be expressed as a percentage by weight or as a ratio. Understanding elemental composition is key to the initial steps in determining the compound's empirical formula.
It begins with the known mass ratios or percentage compositions of each element in a compound. For instance, if you are told the elements are present in a 9:1:3.5 weight ratio of C to H to N , that ratio helps establish mass values if a total mass (like 108 g/mol in the exercise) is assumed.
Using these masses, you can calculate the number of moles to further clarify the proportions required for determining formulas. In our problem, assumptions allow a precise breakdown where the mass of C , H , and N are found individualized in grams, with calculations spreading across empirical and molecular formula determination.

Molecular Formula Calculation

Calculating the molecular formula requires further insight beyond just knowing the empirical formula. Once the empirical formula is established, you must factor in the known molar mass of the compound to complete this next step.
Starting with the empirical formula mass, compare this with the molar mass provided for the compound.

• Calculate the molar mass of the empirical formula.
• Divide the compound's given molar mass by the empirical formula mass. This yields a factor.
• Multiply the empirical formula by this factor to obtain the molecular formula.

For example, with an empirical formula C_3H_4N, if the empirical formula mass is 54 g/mol and the compound's molar mass is given as 108 g/mol:\[ \frac{108}{54} = 2 \] Multiplying the subscripts by 2 gives the molecular formula C_6H_8N_2, which aligns with the empirical data provided.