Problem 3 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

At \(300 \mathrm{~K}\) and 1 atmospheric pressure, \(10 \mathrm{~mL}\) of a hydrocarbon required \(55 \mathrm{~mL}\) of \(\mathrm{O}_{2}\) for complete combustion, and \(40 \mathrm{~mL}\) of \(\mathrm{CO}_{2}\) is formed. The formula of the hydrocarbon is: [Main April 10, 2019 (I)] (a) \(\mathrm{C}_{4} \mathrm{H}_{10}\) (b) \(\mathrm{C}_{4} \mathrm{H}_{6}\) (c) \(\mathrm{C}_{4} \mathrm{H}_{7} \mathrm{Cl}\) (d) \(\mathrm{C}_{4} \mathrm{H}_{8}\)

Step-by-Step Solution

Answer

The formula of the hydrocarbon is \( \text{C}_4\text{H}_{10} \).

Step 1: Understand the Reaction Equation

When a hydrocarbon combusts, it reacts with oxygen to produce carbon dioxide and water. The general reaction is: \[\text{C}_x\text{H}_y + O_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O}\] Given the volumes at STP, our goal is to find the empirical formula of the hydrocarbon.

Step 2: Analyze Volume Data

We know that 10 mL of the hydrocarbon combusts with 55 mL of \( \text{O}_2 \) to form 40 mL of \( \text{CO}_2 \). According to Avogadro's Law, the volume ratio is the same as the molar ratio when all gases are at the same conditions.

Step 3: Deduce Carbon Atoms in Hydrocarbon

Since 40 mL of \( \text{CO}_2 \) is produced from 10 mL of hydrocarbon, we know from the equation \( x \cdot 10 = 40 \) that \( x = 4 \). Thus, the hydrocarbon has 4 carbon atoms, indicating it is \( \text{C}_4\text{H}_y \).

Step 4: Balance Oxygen Atoms

Using the initial reaction for 10 mL of \( \text{C}_4\text{H}_y \):- 40 mL of \( \text{CO}_2 \) means 40 moles of C used, which corresponds to 40 moles of O.- 55 mL of \( \text{O}_2 \) provides us with \( 55 \times 2 = 110 \) moles of O (since each molecule of \( \text{O}_2 \) has 2 oxygen atoms).- Since every mole of \( \text{C}_4\text{H}_y \) requires 110 moles of O,- Remaining oxygen \( = 110 - 40 = 70 \) moles are for water.

Step 5: Conclude Hydrogen Count

The 70 moles of O are used to form water \( (\text{H}_2\text{O}) \), where each water molecule uses 1 mole of O and 2 moles of H. Therefore, moles of H = 2 \times 70 = 140 moles.Since we used 10 moles of hydrocarbon, for 1 mole, this yields 14 H atoms (\( 140/10 = 14 \)). Thus, the molecular formula is \( \text{C}_4\text{H}_{10} \).

Step 6: Finalize the Answer

We conclude that the hydrocarbon has the formula \( \text{C}_4\text{H}_{10} \), matching option (a).

Key Concepts

Stoichiometry

Stoichiometry is a branch of chemistry that focuses on the quantitative relationships between reactants and products in a chemical reaction. It helps us understand how much of each substance is needed or produced in a reaction. In hydrocarbon combustion, stoichiometry is crucial because it allows us to calculate the amounts of oxygen required to fully combust a hydrocarbon and the quantities of carbon dioxide and water produced.
To use stoichiometry in chemical reactions, especially combustion reactions:

• First, identify the balanced chemical equation that describes the reaction.
• Determine the molar ratios from the coefficients in the balanced equation.
• Use these ratios to convert between volumes or moles of different substances involved in the reaction.

In the problem, stoichiometry was applied to equate the volume of the hydrocarbon to the volume of carbon dioxide produced, revealing the number of carbon atoms in the hydrocarbon's formula.

Empirical Formula

An empirical formula represents the simplest whole-number ratio of the different atoms in a compound. It provides insight into the relative proportions of elements, but not the exact number of each type of atom present in the molecule.
To determine the empirical formula of a compound:

• Find the moles of each element present in a given compound.
• Simplify the mole ratio to the smallest whole numbers.
• Combine these ratios to form the empirical formula.

In the provided solution, the empirical formula of the hydrocarbon was derived from the volumes of products formed during combustion. By identifying that the hydrocarbon is made of 4 carbon atoms and calculating the total hydrogen atoms via the production of water, we conclude the simplest composition, which provided us with the correct empirical formula of the hydrocarbon.

Avogadro's Law

Avogadro's Law is an essential principle in chemistry that relates the volume of gas to the amount of substance (in moles) at constant temperature and pressure. According to this law, equal volumes of gases, at the same temperature and pressure, contain the same number of molecules.
This principle was pivotal in solving the problem:

• It allowed us to directly correlate the volumes of hydrocarbon, oxygen, and carbon dioxide to their respective mole amounts during the combustion process.
• By assuming ideal gas behavior, we equated the volume ratios to molar ratios, simplifying the task of finding the formula of the hydrocarbon.
• Specifically, the law helped demonstrate that the ratio of 10 mL of hydrocarbon combusting to 40 mL of carbon dioxide directly indicated a 4:1 ratio of carbon atoms in the hydrocarbon.

Understanding Avogadro's Law enables students to simplify gaseous reactions at constant conditions and provides a fundamental tool in calculating molecular compositions in chemical reactions.