Question

At constant temperature, the equilibrium constant \(\left(\mathrm{K}_{\mathrm{p}}\right.\) ) for the decomposition reaction: \(\mathrm{N}_{2} \mathrm{O}_{4} \rightleftharpoons 2 \mathrm{NO}_{2}\) is expressed by \(\mathrm{K}_{\mathrm{p}}=\left(4 \mathrm{x}^{2} \mathrm{P}\right) /\left(1-\mathrm{x}^{2}\right)\), where \(\mathrm{P}=\) pressure, \(\mathrm{x}=\) extent of decomposition. Which one of the following statements is true? (a) \(\mathrm{K}_{\mathrm{p}}\) increases with increase of \(\mathrm{P}\) (b) \(\mathrm{K}_{\mathrm{p}}\) increases with increase of \(\mathrm{x}\) (c) \(K_{p}\) increases with decrease of \(x\) (d) \(K_{p}\) remains constant with change in \(P\) and \(x\)

Short answer

The correct statement is (d) \( K_{p} \) remains constant with change in \( P \) and \( x \).

Step-by-step solution

Understand the Equation

The equilibrium constant expression for this decomposition reaction is \( K_{p} = \frac{4x^{2}P}{1-x^{2}} \), where \( P \) is the pressure and \( x \) is the extent of decomposition. Notice that \( K_p \) is dependent on both \( x \) and \( P \), suggesting that changes in these values could affect \( K_p \).

Analyze the Dependence on Pressure (P)

First, consider if \( K_p \) changes with \( P \). Looking at the expression \( K_{p} = \frac{4x^{2}P}{1-x^{2}} \), as \( P \) increases, \( K_{p} \) appears to be directly proportional to \( P \). This suggests that \( K_p \) might increase with an increase in \( P \). However, you must remember that \( K_p \) is defined as a constant under constant temperature for a given reaction, so increasing \( P \) should not affect \( K_p \).

Analyze the Dependence on Extent of Decomposition (x)

Next, explore the dependence on \( x \). The equation \( K_{p} = \frac{4x^{2}P}{1-x^{2}} \) shows that \( K_{p} \) is influenced by \( x \). As \( x \) increases, \( 4x^2P \) increases, while \( 1-x^2 \) decreases (since \( x^2 \) becomes larger), suggesting an increase in \( K_p \). However, theoretically, \( K_p \) should remain constant with the extent of decomposition \( x \) at a given temperature untangling \( x \)'s effect seems incorrect.

Conclusion: Select the Correct Statement

Considering the insights from steps 2 and 3, \( K_p \) should remain constant with changes in \( P \) or \( x \) since it is a temperature-dependent constant for a given reaction. Therefore, the correct choice is (d) \( K_{p} \) remains constant with change in \( P \) and \( x \).

Key concepts

Decomposition Reaction

In chemistry, a decomposition reaction is a type of chemical reaction where a single compound breaks down into two or more simpler substances. For instance, the given reaction \[\text{N}_2\text{O}_4 \rightleftharpoons 2 \text{NO}_2\] demonstrates this concept by showing one compound, dinitrogen tetroxide, decomposing into nitrogen dioxide molecules. Decomposition reactions are essential in both laboratory and industrial processes. They allow chemists to extract elements from compounds and generate new materials.
For these reactions, different factors, including temperature and pressure, can influence the behavior and rate at which the decomposition occurs.
Understanding the workings of decomposition reactions is crucial as they are foundational to many chemical processes.

Pressure Dependence

Pressure can have a significant effect on the equilibrium of reactions, particularly gaseous ones. According to Le Chatelier's Principle, if the pressure of a system at equilibrium changes, the system adjusts in a way that counteracts that change.
In the equation \[K_{p} = \frac{4x^{2}P}{1-x^{2}}\], pressure \(P\) is a variable that influences the equilibrium constant \(K_p\). The inclusion of \(P\) in the numerator suggests that changes in pressure could directly affect the equilibrium constant.
Although theoretically, \(K_p\) is a constant at a given temperature, this consideration highlights the potential impact pressure could have under different conditions. Understanding how pressure affects equilibrium is critical for industrial applications where pressure conditions can change frequently.

Extent of Decomposition

The extent of decomposition, often represented as \(x\), indicates how far the decomposition reaction has progressed. In the formula \(K_{p} = \frac{4x^{2}P}{1-x^{2}}\), x determines the ratio of decomposed to non-decomposed compounds.
As \(x\) increases, more of the initial compound has decomposed, progressing the reaction toward completion.
However, the equilibrium constant \(K_p\) does not actually depend on \(x\) when considering a specific temperature, as \(K_p\) denotes the ratio of products to reactants at equilibrium. Exploring the full extent of decomposition is relevant in designing experiments and industrial procedures to optimize reactions.

Thermodynamics

Thermodynamics plays a crucial role in understanding chemical reactions, including decomposition reactions. It involves the study of energy changes and transformations, mainly focusing on heat and work.
The equilibrium constant \(K_p\) is derived from the thermodynamic concepts that relate to the Gibbs free energy change (\(\Delta G\)) of a reaction at constant temperature and pressure.
The equation:\[\Delta G = -RT\ln K_p\]shows that \(K_p\) is consistent if \(\Delta G\) is unchanged, highlighting its reliance on temperature rather than pressure or extent of decomposition. The study of thermodynamics aids in predicting the direction, spontaneity, and feasibility of chemical reactions, making it pivotal in both academic and practical applications.