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: Understanding the Decomposition Reaction

The decomposition reaction of \(_2O_4\) can be represented as: \[ \mathrm{N}_2\mathrm{O}_4 \ (g) \rightleftharpoons 2\ \mathrm{NO}_2\ (g) \]. Given that \(20\%\) of \(N_2O_4\) by mass decomposes to \(NO_2\), it means that for 1 mole of \(_2O_4\), \(0.2\) mole decomposes to form \(0.4\) moles of \(NO_2\).

Step 2: Calculating Moles of Gas After Decomposition

Initially, we have \(1\) mole of \(N_2O_4\). After \(0.2\) moles decompose, \((1 - 0.2) = 0.8\) moles of \(N_2O_4\) remain, and \(0.4\) moles of \(NO_2\) are produced. The total moles of gas after decomposition = moles of \(N_2O_4\)+ moles of \(NO_2\) \[ 0.8 + 0.4 = 1.2 \text{ moles} \].

Step 3: Applying the Ideal Gas Law

Since we assume ideal gas behavior and volume and temperature changes do not affect mole ratios linearly at a constant volume, pressure is directly proportional to the number of moles in a closed container at constant volume and temperature.Initial pressure was \(1\) atm with \(1\) mole of gas. With \(1.2\) moles of gas after decomposition, the pressure is proportional to moles, so pressure after decomposition = \[ 1 \times 1.2 = 1.2 \text{ atm} \].

Key Concepts

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases under various conditions. They help us understand how a gas will respond to changes in pressure, volume, and temperature. In simpler terms:

• These laws reveal the relationship between these three state variables for a given sample of gas.
• They allow us to predict how a gas will behave under different conditions.
• Gas laws include Boyle's Law, Charles's Law, and Avogadro's Law, among others.

For example, Boyle's Law describes how pressure and volume are inversely related—the pressure increases as the volume decreases, and vice versa, when temperature is kept constant. Charles's Law, on the other hand, tells us that volume and temperature have a direct relationship when the pressure is constant. Comprehending these foundational concepts is essential for dealing with gas reactions, like the decomposition of nitrogen tetroxide (\(\mathrm{N}_2\mathrm{O}_4\)), particularly when assessing changes in pressure or volume.

Decomposition Reaction

Decomposition reactions are chemical processes in which a single compound breaks down into two or more simpler substances. These reactions commonly require an external stimulus, such as heat or light, to proceed.
In the given problem, nitrogen tetroxide (\(\mathrm{N}_2\mathrm{O}_4\)) undergoes a thermal decomposition reaction, leading to the formation of nitrogen dioxide (\(\mathrm{NO}_2\)). The balanced equation for this reaction is:
\[ \mathrm{N}_2\mathrm{O}_4 \ (g) \rightleftharpoons 2\ \mathrm{NO}_2 \ (g) \]This indicates that each mole of \(\mathrm{N}_2\mathrm{O}_4\) produces two moles of \(\mathrm{NO}_2\).

• This decomposition is important as it illustrates the concept of equilibrium in chemical reactions, where the forward and reverse reactions occur at the same rate.
• The reaction mixture approaches equilibrium, with constant amounts of \(\mathrm{N}_2\mathrm{O}_4\) and \(\mathrm{NO}_2\) at any given time.

Understanding decomposition reactions allows one to predict the amounts of products formed and how different variables, like temperature, can shift the balance of the equilibrium.

Ideal Gas Law

The Ideal Gas Law is a comprehensive equation describing the behavior of gases by relating their pressure, volume, temperature, and number of moles:
\[ PV = nRT \]Where:

• \(P\) is the pressure of the gas in atmospheres (atm),
• \(V\) is the volume in liters,
• \(n\) is the number of moles,
• \(R\) is the universal gas constant (0.0821 L⋅atm/mol⋅K),
• \(T\) is the temperature in Kelvin.

In the given exercise, the Ideal Gas Law is vital because it explains how the pressure changes when the temperature is raised to cause decomposition in a closed system. Because the volume and temperature might fluctuate, the number of gas moles directly influences the system's pressure at constant volume. By employing the Ideal Gas Law, it becomes clear that in an isolated system, the pressure will rise if the number of moles increases, which is what occurs when \(\mathrm{N}_2\mathrm{O}_4\) decomposes into \(\mathrm{NO}_2\). This law is essential for predicting the behavior of gases in both experimental and theoretical scenarios, ensuring that one can account for variations in pressure due to changes in the quantity of gas.