The equilibrium constant for the reaction \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g})=2 \mathrm{SO}_{3}(\mathrm{~g})\) has the value \(K=2.5 \times 10^{10}\) at \(500 \mathrm{~K}\). Find the value of \(K\) for each of the following reactions at the same temperature. (a) \(\mathrm{SO}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g})=\mathrm{SO}_{3}(\mathrm{~g})\) (b) \(\mathrm{SO}_{3}(\mathrm{~g})=\mathrm{SO}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g})\) (c) \(3 \mathrm{SO}_{2}(\mathrm{~g})+\frac{3}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 3 \mathrm{SO}_{3}(\mathrm{~g})\)

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This means that the concentrations of reactants and products remain constant over time, though they are not necessarily equal. In this balanced state, no net change in the quantity of reactants and products occurs.

Equilibrium does not indicate that the reactants and products are in equal concentrations, but rather that their ratios are constant. The equilibrium constant, represented as K, is a numerical value that expresses this ratio at a given temperature. Specifically, for the reaction at hand, 2 SO_2(g) + O_2(g) = 2 SO_3(g), K represents the concentrations of the products (SO_3) raised to their stoichiometric coefficients divided by the concentrations of the reactants (SO_2 and O_2), each raised to their respective stoichiometric coefficients.

The reaction quotient, Q, is a measure that tells us how close a system is to reaching equilibrium. It is calculated in the same way as the equilibrium constant, using the current concentrations of the products and reactants.

If Q is less than K, the forward reaction is favored, and the concentrations of products will increase until equilibrium is reached. If Q is greater than K, the reverse reaction is favored, and the concentrations of reactants will increase until equilibrium is achieved. When Q equals K, the system is at equilibrium.

Le Chatelier's principle is crucial for understanding how a system at equilibrium responds to changes in concentration, pressure, or temperature. Essentially, it states that if an external stress is applied to a system at equilibrium, the system will adjust itself to minimize the impact of that change.

For instance, increasing the concentration of reactants will cause the equilibrium to shift towards forming more products to reduce the reactant concentration, thus favoring the forward reaction. Conversely, decreasing the pressure by increasing volume will shift the equilibrium towards the side with more gas molecules. Adjusting temperature can also affect the equilibrium; for exothermic reactions, increasing temperature will shift the equilibrium towards the reactants, while for endothermic reactions, it will shift towards the products.

Gas phase reactions are chemical processes that occur between substances in the gaseous state. One characteristic of gas phase reactions is that changes in pressure and volume can significantly affect the point of equilibrium, according to Le Chatelier's principle.

For the given gas phase reaction involving sulfur dioxide and oxygen forming sulfur trioxide, the reaction's equilibrium can be influenced by such changes. As it's a balanced equation with an equal number of gas molecules on both the reactant and product sides, changing the volume or pressure would typically have no net effect on the direction of the shift in equilibrium. However, for gas-phase reactions where there are different numbers of gas molecules on each side, the effects can be quite pronounced.