Question

Consider the following equilibrium in a closed container: \(\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}_{2}(\mathrm{~g})\) At a fixed temperature, the volume of the reaction container is halved. For this change, which of the following statement holds true regarding the equilibrium constant \(\left(K_{\mathrm{P}}\right)\) and degree of dissociation \((\alpha)\) ? (a) neither \(K_{\mathrm{p}}\) nor \(\alpha\) changes (b) both \(K_{\mathrm{p}}\) and \(\alpha\) changes (c) \(K_{\mathrm{p}}\) changes, but \(\alpha\) does not change (d) \(K_{\mathrm{p}}\) does not change, but \(\alpha\) changes

Short answer

(d) \(K_{P}\) does not change, but \(\alpha\) changes.

Step-by-step solution

Understanding Equilibrium Constants

First, understand that the equilibrium constant, here denoted as \(K_{P}\), is a measure of the extent of the reaction and depends only on the temperature for a given reaction. Changes in volume or pressure do not affect the value of \(K_{P}\) as long as the temperature remains constant.

Understanding Degree of Dissociation

The degree of dissociation, denoted as \(\alpha\), is the fraction of the original compound that has dissociated into products. It can be affected by changes in volume, pressure, or concentration, but not by changes in temperature, which affect the equilibrium constant.

Evaluating the Effects of Volume Change on Equilibrium

When volume of a closed container is halved at constant temperature, the pressure will increase due to the principles of ideal gas law, assuming the amount (moles) of gas stays the same. For this reaction, the number of moles of gas increases from 1 mole of \(N_{2}O_{4}\) to 2 moles of \(NO_{2}\), which means that a decreased volume (increased pressure) will promote the reverse reaction according to Le Chatelier's principle, reducing the degree of dissociation, \(\alpha\). However, the equilibrium constant, \(K_{P}\), will not change as the temperature is constant.

Key concepts

Le Chatelier's Principle

Le Chatelier's principle is a foundational concept in chemistry that describes how a dynamic equilibrium adjusts to counteract changes to its system. When a reaction at equilibrium faces a change in concentration, pressure, or temperature, the system will shift to minimize that change.

For instance, in the reaction described in the exercise \( \mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2\mathrm{NO}_{2}(g) \), when the container's volume is reduced, the pressure inside the container increases. According to Le Chatelier's principle, the system will respond by favoring the direction that decreases pressure. Since there are fewer moles of gas on the reactant side \(1 \, mole\, of \, \mathrm{N}_{2}\mathrm{O}_{4}\) compared to the product side \(2 \, moles\, of \, \mathrm{NO}_{2}\), the reaction will shift left towards the reactants to decrease the number of moles, thereby reducing pressure.

This principle can be applied to understand how changing reaction conditions affect chemical equilibria, and predict the direction in which the reaction will shift to reach a new state of balance.

Degree of Dissociation

The degree of dissociation, often represented as \(\alpha\), is a measure of the extent to which a compound breaks down into its components. For gases, it is defined as the fraction of molecules that have dissociated at equilibrium. In the context of the given exercise, \(\alpha\) represents the fraction of \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\) molecules that have dissociated into \(\mathrm{NO}_{2}(g)\) molecules.

When the volume of the container is reduced, which increases the pressure, the reaction is expected to shift towards the side with fewer moles of gas to reduce that pressure. This shift leads to a decrease in \(\alpha\) because fewer \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\) molecules are dissociating into \(\mathrm{NO}_{2}(g)\) molecules. It's important to note that the degree of dissociation \(\alpha\) is affected by changes in pressure and volume, unlike the equilibrium constant, which remains unchanged at constant temperature.

Ideal Gas Law

The ideal gas law, represented by the equation \(PV = nRT\), is a critical equation in chemistry and physics, where \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the gas constant, and \(T\) for temperature. It relates the four properties of a gas in a simple formula that can be used to calculate changes in a gas system under certain conditions, assuming the gas behaves ideally, meaning its particles do not interact and occupy negligible space.

In the exercise, halving the volume \(V\) of the container at constant temperature \(T\) and unchanged moles \(n\) of gas implies, according to the ideal gas law, that the pressure \(P\) will increase. This relationship helps to predict that the pressure increase will cause the reaction to shift to the side with fewer gas molecules to restore equilibrium, as per Le Chatelier's principle. This interplay between volume, pressure, and quantity of gas at equilibrium is a perfect demonstration of the ideal gas law's utility in understanding chemical reactions and their behavior under different conditions.