Problem 16 - A Complete Resource Book in Chemistry for JEE Main

Question

The vapour density of a mixture having \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) is \(27.6 .\) The mole fraction \(\mathrm{NO}_{2}\) in the mixture is (a) \(1.6\) (b) \(0.8\) (c) \(2.4\) (d) \(0.6\)

Step-by-Step Solution

Answer

The mole fraction of \( \mathrm{NO}_{2} \) is 0.8.

Step 1: Understanding Vapour Density

Vapour density is defined as the molar mass divided by 2. Therefore, the average molecular mass of the mixture can be calculated as: \[ M = 2 \times 27.6 = 55.2 \] This represents the weighted average molar mass of the mixture of \( \mathrm{NO}_{2} \) and \( \mathrm{N}_{2} \mathrm{O}_{4} \).

Step 2: Setting up Mole Fraction Equation

Let \( x \) be the mole fraction of \( \mathrm{NO}_{2} \), and \( (1-x) \) be the mole fraction of \( \mathrm{N}_{2} \mathrm{O}_{4} \). The molar masses are: \( \mathrm{NO}_{2} = 46 \text{ g/mol} \) and \( \mathrm{N}_{2} \mathrm{O}_{4} = 92 \text{ g/mol} \). Thus, the weighted molecular mass can be written as: \[ 46x + 92(1-x) = 55.2 \]

Step 3: Solving the Equation

Expand the equation: \[ 46x + 92 - 92x = 55.2 \]Combine like terms: \[ -46x = 55.2 - 92 \]This simplifies to: \[ -46x = -36.8 \]Solving for \( x \), we divide both sides by -46: \[ x = \frac{36.8}{46} \approx 0.8 \]

Step 4: Verify Calculations

Recalculate to confirm: \( x \approx 0.8 \). Substitute back into the equation to ensure correctness: \[ 46(0.8) + 92(0.2) = 55.2 \]. Both sides match, confirming the solution.

Key Concepts

Molecular Mass

Molecular mass is a measure of the mass of a given molecule. It is calculated by adding up the atomic masses of all the atoms in the molecule. Understanding molecular mass is crucial when determining properties like vapour density and mole fraction in chemical mixtures. For instance, in the context of the exercise involving a mixture of nitrogen dioxide (\( \mathrm{NO}_{2} \)) and dinitrogen tetroxide (\( \mathrm{N}_{2} \mathrm{O}_{4} \)), the molecular masses are 46 g/mol for \( \mathrm{NO}_{2} \) and 92 g/mol for \( \mathrm{N}_{2} \mathrm{O}_{4} \). These figures are instrumental in calculating the vapour density and finding out the mole fraction in the mixture.

• Knowing the molecular mass provides insights into the composition of the mixture.
• Molecular mass calculations are foundational for understanding chemical reactions and properties.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of one component to the total number of moles in the mixture. In a given chemical context, like our exercise, if we have a mixture of \( \mathrm{NO}_{2} \) and \( \mathrm{N}_{2} \mathrm{O}_{4} \), we define the mole fraction as follows:

• Let \( x \) be the mole fraction of \( \mathrm{NO}_{2} \).
• Then \( 1-x \) will naturally be the mole fraction of \( \mathrm{N}_{2} \mathrm{O}_{4} \).

To find the mole fraction, we set up and solve an equation using the weighted average molar mass as given by the vapour density. Understanding the mole fraction helps determine exactly how much of each substance is present in a mixture, which is key in chemical and physical analysis.

Nitrogen Dioxide (NO2)

Nitrogen dioxide (\( \mathrm{NO}_{2} \)) is a chemical compound with significance in both environmental science and industrial applications. It’s known for its reddish-brown color and sharp, biting odor. \( \mathrm{NO}_{2} \) is a nitrogen oxide and has a molecular mass of 46 g/mol.
In the exercise, we focus on its mole fraction in a mixture with dinitrogen tetroxide. \( \mathrm{NO}_{2} \) is often produced and found in air due to vehicle emissions and industrial processes. It is a precursor to the kind of nitrogen pollution that can lead to smog and acid rain.

• Being aware of the molecular properties of \( \mathrm{NO}_{2} \) helps in environmental chemistry.
• It's crucial for analysis in atmospheric studies.
• Understanding its behavior in a mixture is key for both environmental science and chemical engineering.

Dinitrogen Tetroxide (N2O4)

Dinitrogen tetroxide (\( \mathrm{N}_{2} \mathrm{O}_{4} \)) is a dimer of nitrogen dioxide and is normally colorless or pale yellow. Its molecular structure is such that it is composed as two \( \mathrm{NO}_{2} \) molecules linked together and reversibly dissociates to \( \mathrm{NO}_{2} \) upon heating. Its molecular mass is known to be 92 g/mol.
In the case of the mixture, its mole fraction, like that of \( \mathrm{NO}_{2} \), will change based on factors such as temperature and vapour pressure, which affect its state.

Being able to understand the presence and concentration of \( \mathrm{N}_{2} \mathrm{O}_{4} \) in a mixture is particularly useful in:

• Calibrating industrial chemical processes.
• Ensuring safety and compliance in chemical manufacturing.
• Working with rocket propellants, due to \( \mathrm{N}_{2} \mathrm{O}_{4} \)'s use as an oxidizer in liquid-fueled rockets.