At \(25^{\circ} \mathrm{C}\), a mixture of \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) gases are in equilibrium in a cylinder fitted with a movable piston. The concentrations are: \(\left[\mathrm{NO}_{2}\right]=0.0475 \mathrm{M}\) and \(\left[\mathrm{N}_{2} \mathrm{O}_{4}\right]=0.491 \mathrm{M}\). The volume of the gas mixture is halved by pushing down on the piston at constant temperature. Calculate the concentrations of the gases when equilibrium is reestablished. Will the color become darker or lighter after the change? [Hint: \(K_{\mathrm{c}}\) for the dissociation of \(\mathrm{N}_{2} \mathrm{O}_{4}\) is \(4.63 \times\) \(10^{-3} . \mathrm{N}_{2} \mathrm{O}_{4}(g)\) is colorless and \(\mathrm{NO}_{2}(g)\) has a brown color.]

Understanding the concept of equilibrium concentrations is key to unlocking the behavior of chemical reactions that have reached a state of balance. Equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, and thus the concentrations of the reactants and products remain constant over time.

For example, consider the dissociation of dinitrogen tetroxide (2O4) into nitrogen dioxide (NO2). At equilibrium, even though both the forward and reverse reactions are still occurring, the concentration of N2O4 and NO2 does not change. When an external pressure is applied, such as by halving the volume of the gas mixture, it's crucial to understand that the concentrations of the gases double as a response. By systematically calculating the change in these concentrations, one can predict the behavior of the system when it reaches a new equilibrium state.

Le Chatelier's Principle provides a qualitative prediction of how a system at equilibrium responds to changes in concentration, temperature, and pressure. It states that if an external change is applied to a system at equilibrium, the system will adjust itself in order to counteract that change and a new equilibrium will be established.

In the context of the dissociation equilibrium of N2O4 and NO2, applying pressure by reducing the volume causes the equilibrium to shift in the direction that would reduce pressure. Since 2 moles of NO2 are produced for each mole of N2O4 that dissociates, the system can decrease pressure by consuming NO2 and producing N2O4. However, the external pressure change due to the volume reduction encourages the system to shift towards the side with more gas particles—toward NO2 in this case—thus, the concentration of NO2 will increase.

The equilibrium constant Kc is a numerical value that characterizes the ratio of the concentrations of products to reactants at equilibrium for a reversible chemical reaction at constant temperature. For the given reaction, the equilibrium constant is expressed as \(K_c=\frac{[NO_{2}]^{2}}{[N_{2}O_{4}]}\). It reflects the extent to which a reaction will proceed before reaching equilibrium.

When the volume of the reaction vessel is halved, we face a change in concentration. The mathematical expression for Kc allows us to set up an equation to solve for the new equilibrium concentrations. Despite the change in volume, Kc remains constant for a given temperature—a crucial point allowing us to predict the concentrations of reactants and products after any disturbance to the system.

The dissociation of N2O4 into NO2 is an equilibrium reaction that can be represented as \(N_{2}O_{4} (g) \leftrightarrow 2 NO_{2} (g)\). This reaction is endothermic, meaning that it absorbs heat. As N2O4 gas dissociates, it produces NO2 gas, which is brown in color, thereby increasing the color intensity of the gas mixture.

When the dissociation is subjected to a change in conditions, such as an increase in volume or a decrease in pressure, the dissociation reaction's direction will shift to re-establish equilibrium. Since N2O4 is colorless, its increase leads to a lighter color, whereas an increase in NO2 concentration results in a darker color. Calculating the changes in the equilibrium concentrations can also help in predicting the visual color change, as observed in our exercise.