A sealed glass bulb contains a mixture of \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) gases. Describe what happens to the following properties of the gases when the bulb is heated from \(20^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}:\) (a) color, (b) pressure, (c) average molar mass, (d) degree of dissociation (from \(\mathrm{N}_{2} \mathrm{O}_{4}\) to \(\mathrm{NO}_{2}\) ), (e) density. Assume that volume remains constant. (Hint: \(\mathrm{NO}_{2}\) is a brown gas; \(\mathrm{N}_{2} \mathrm{O}_{4}\) is colorless.)

Le Chatelier's Principle predicts how a chemical system at equilibrium reacts to disturbances or external changes. It states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. When the glass bulb in our exercise is heated, the principle foresees that the system will adjust. In this case, the increased temperature causes the equilibrium to shift so that the exothermic reaction, the formation of N2O4 from NO2, is reduced, and the endothermic reaction, the dissociation of N2O4 to NO2, is favored. This response to an external temperature change results in a higher concentration of brown NO2 gas, leading to the observed color change in the bulb.

Understanding Le Chatelier's Principle helps students predict the outcome of altering variables such as temperature, pressure, or concentration in a chemical reaction at equilibrium. This also informs us that heating the bulb not only affects the color but also has implications for pressure, molecular properties, and gas density, which are interconnected through the behavior of the gases under equilibrium.

The ideal gas law is a critical equation in chemistry and physics, represented as PV=nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature. This equation describes the state of a hypothetical 'ideal' gas, and although real gases do not always behave perfectly, the ideal gas law provides a good approximation under many conditions.

Applying the ideal gas law to the problem at hand, when the temperature (T) inside the bulb increases, and since the volume (V) remains constant, either the pressure (P) must increase, or the number of moles (n) must decrease. In our exercise, as the temperature rises, the dissociation of N2O4 into NO2 increases the number of gas molecules, thereby increasing the pressure inside the sealed bulb given that the volume remains constant.

Molecular dissociation occurs when molecules break apart into smaller molecules or atoms, which is common in reversible reactions like the one between NO2 and N2O4. The forward reaction is the association of NO2 molecules to form N2O4, while the reverse reaction is the dissociation of N2O4 to form NO2. When the equilibrium between these two is disturbed by a temperature change, the balance between association and dissociation shifts.

In our scenario, heating promotes the endothermic dissociation of N2O4 into NO2, leading to an increase in the degree of dissociation as more N2O4 is broken down. This concept is crucial to understand as it not only explains the changes in color and pressure but is also directly related to changes in average molar mass and density.

Gas density is a measure of mass per unit volume of a gas. It can be influenced by temperature, pressure, and molecular weight of the gas particles. In the context of the exercise, the density change of the gas mixture when heated can be counterintuitive. As N2O4 dissociates into NO2, the average molar mass decreases because we now have a higher proportion of the lighter NO2 molecules.

However, what may seem unexpected is that the density of the gas mixture increases, despite the lower average molar mass. This increase in density can be attributed to the rise in pressure within the fixed volume of the bulb - a result of the increased number of NO2 molecules produced from the dissociation of N2O4. The combined understanding of temperature effects on chemical equilibrium, and the use of the ideal gas law, helps to fully grasp this sophisticated behavior of gas density in response to temperature changes.