Problem 110 - A Complete Resource Book in Chemistry for JEE Main

Question

One mole of \(\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g})\) at 300 is kept in a closed container under one atmosphere. It is heated to 600 when \(20 \%\) by mass of \(\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g})\) decomposes to \(\mathrm{NO}_{2}\) (g). The resultant pressure is (a) \(1.2 \mathrm{~atm}\) (b) \(2.4 \mathrm{~atm}\) (c) \(2.0 \mathrm{~atm}\) (d) \(1.0 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The resultant pressure is 1.2 atm (option a).

Step 1: Initial Moles of N₂O₄

Initially, we have 1 mole of \(\mathrm{N_2O_4}\) gas.

Step 2: Determine Moles Decomposed

20% decomposition of \(\mathrm{N_2O_4}\) means that \(0.2 \times 1 = 0.2\) moles of \(\mathrm{N_2O_4}\) decompose.

Step 3: Reacting Moles and Products Formed

The decomposition reaction of \(\mathrm{N_2O_4}\) is \(\mathrm{N_2O_4(g)} \rightarrow 2\,\mathrm{NO_2(g)}\). Therefore, 0.2 moles of \(\mathrm{N_2O_4}\) produces \(2 \times 0.2 = 0.4\) moles of \(\mathrm{NO_2}\).

Step 4: Calculate Remaining Moles of N₂O₄

After decomposition, the remaining \(\mathrm{N_2O_4}\) is \(1 - 0.2 = 0.8\) moles.

Step 5: Total Moles of Gas After Reaction

The total moles of gas are the remaining \(\mathrm{N_2O_4}\) moles plus the \(\mathrm{NO_2}\) moles: \(0.8 + 0.4 = 1.2\) moles.

Step 6: Calculate Resultant Pressure

According to the ideal gas law, the pressure is proportional to the number of moles of gas when volume and temperature are constant. Thus, the resultant pressure is \(1 \text{ atm} \times 1.2 = 1.2\, \text{atm}\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry that helps us understand the behavior of gases under different conditions. The formula for the Ideal Gas Law is \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of the gas.
• \(R\) is the universal gas constant.
• \(T\) is the temperature of the gas in Kelvin.

This law assumes that the gas behaves ideally, meaning the gas particles do not have interactions with each other, and they occupy no volume. When a gas is heated or decomposed, like in this exercise, the ideal behavior allows us to calculate changes in pressure if volume and temperature are kept constant.

Decomposition Reaction

A decomposition reaction is a type of chemical reaction where one compound breaks down into two or more simpler substances. In this exercise, the compound \(\mathrm{N_2O_4}\) decomposes into \(\mathrm{NO_2}\), following the equation: \[\mathrm{N_2O_4 (g) \rightarrow 2\, NO_2 (g)}\] This reaction shows that every mole of \(\mathrm{N_2O_4}\) produces two moles of \(\mathrm{NO_2}\). Decomposition reactions often require energy input, through heat, to occur. That's why heating the \(\mathrm{N_2O_4}\) causes it to decompose, altering the balance of molecules in the container.

Mole Concept

The mole concept is a method in chemistry for expressing amounts of a chemical substance. One mole is equivalent to Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles, be it atoms, ions, or molecules.In the given problem:

• We begin with 1 mole of \(\mathrm{N_2O_4}\).
• 20% of this mole, or 0.2 moles, decomposes to form new substances.
• The decomposed \(\mathrm{N_2O_4}\) generates 0.4 moles of \(\mathrm{NO_2}\).
• Remaining \(\mathrm{N_2O_4}\) is 0.8 moles.

Understanding moles enables us to accurately describe reactions and predict the amounts of products formed and reactants used during chemical transformations.

Pressure Calculation

Pressure calculation in the context of gases often involves using the relation of pressure to moles, derived from the Ideal Gas Law. Here, the initial pressure was given as 1 atm for 1 mole of \(\mathrm{N_2O_4}\).After the decomposition reaction:

• The total moles of gas become 1.2 moles: 0.8 moles of \(\mathrm{N_2O_4}\) + 0.4 moles of \(\mathrm{NO_2}\).
• The pressure in the container will increase proportionately with the increase in moles, since the volume and temperature are constant.
• Thus, if the initial pressure is 1 atm, the resultant pressure is \(1 \times 1.2 = 1.2\) atm.

Understanding how to calculate pressure changes with varying moles of gas is crucial in predicting the behavior of a system under chemical reactions.