The equilibrium constant \(K_{P}\) for the reaction $$2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{SO}_{3}(g)$$ is \(5.60 \times 10^{4}\) at \(350^{\circ} \mathrm{C}\). The initial pressures of \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) in a mixture are 0.350 atm and 0.762 atm, respectively, at \(350^{\circ} \mathrm{C}\). When the mixture equilibrates, is the total pressure less than or greater than the sum of the initial pressures \((1.112 \mathrm{~atm}) ?\)

Chemical equilibrium is a dynamic state where the rate of the forward reaction equals the rate of the reverse reaction. This balance occurs in a closed system and results in no net change in the concentrations of reactants and products. However, it's crucial to understand that the reactions are still occurring, just at equal rates. When a system reaches equilibrium, the mixture of reactants and products remains consistent over time.

In the context of our exercise, the equilibrium is established between sulfur dioxide (SO2), oxygen (O2), and sulfur trioxide (SO3) gases. At this point, the amount of SO2 reacting with O2 to form SO3 will be equal to the amount of SO3 decomposing to form SO2 and O2. The equilibrium constant \(K_{P}\) quantifies the concentrations of these gases at equilibrium.

Partial pressure is a fundamental concept when dealing with gases in a mixture. It refers to the pressure that each gas in a mixture would exert if it occupied the entire volume of the mixture alone, at the same temperature. Partial pressures are crucial when discussing gaseous equilibria because gases behave independently and contribute to the total pressure of the system proportionally to their mole fraction.

For the given problem, the initial partial pressures of SO2 and O2 are provided. When the chemical reaction reaches equilibrium, the partial pressures change according to the stoichiometry of the reaction and the consumption or production of each gaseous substance. These changes are denoted by 'x', which represents the change in pressure as the system shifts to reach equilibrium.

Equilibrium calculations involve finding the value of the change in pressure 'x' that satisfies the equilibrium constant expression. In our example, we begin by denoting the initial pressures and then express how they change as the reaction proceeds. The next step is to substitute these expressions into the equilibrium constant expression for \(K_{P}\).

Often, the resulting equation is a quadratic that requires algebraic manipulation to solve. In our case, solving the quadratic equation for 'x' provides the equilibrium pressures of the gases. With these pressures, we can then calculate the total pressure at equilibrium and answer the question comparing it to the sum of initial pressures. This process demonstrates how equilibrium is not only a conceptual state but also a quantifiable one.

Le Chatelier's Principle is crucial to predicting how a system at equilibrium responds to changes in conditions such as concentration, pressure, and temperature. According to this principle, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. It's nature's way of balancing effects to maintain equilibrium.

In our exercise, if we were to change the pressure or temperature of the system, Le Chatelier's Principle would predict in which direction the system would shift to re-establish equilibrium. For instance, increasing the pressure of the system would shift the equilibrium towards the side with fewer moles of gas, which, in the case of our reaction, is the production of SO3. This principle helps chemists understand and control chemical reactions in various applications.