For the system $$\mathrm{SO}_{3}(g) \rightleftharpoons \mathrm{SO}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)$$ at \(1000 \mathrm{~K}, K=0.45 .\) Sulfur trioxide, originally at \(1.00 \mathrm{~atm}\) pressure, partially dissociates to \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) at \(1000 \mathrm{~K}\). What is its partial pressure at equilibrium?

The equilibrium constant, symbolized as K, is a numerical value that provides a measure of the extent of a chemical reaction at equilibrium. For gas-phase reactions, like the one involving sulfur trioxide and oxygen, the equilibrium constant is expressed in terms of partial pressures. This is designated as Kp.

When dealing with these reactions, it’s crucial to understand that a large Kp value indicates that the reaction favors the formation of products, while a small Kp value suggests that reactants are favored. In the exercise provided, K equals 0.45, which implies that the reaction does not heavily favor either the reactants or the products at 1000 K.

The equilibrium expression for a chemical reaction is derived by writing the concentration (or pressure) of products, raised to the power of their coefficients, divided by the concentration (or pressure) of reactants, similarly raised to the power of their coefficients. These expressions allow chemists to calculate the equilibrium constant for a reaction.

For the given reaction equilibrium, \(SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g)\),the equilibrium expression is \(K = \frac{[SO_2][O_2]^{1/2}}{[SO_3]}\),which reflects the stoichiometry of the reaction. Partial pressures are used in the expression because the substances involved are gases.

An ICE table, which stands for Initial, Change, and Equilibrium, is a systematic way to organize and calculate changes in concentration or pressure for a reaction reaching equilibrium. It is particularly useful to track the progress of the reaction and deduce unknown values.

To set up an ICE table, you list the initial pressures or concentrations of reactants and products (which may often be zero for products in a reaction starting from reactants only). Then you denote the changes that occur as the reaction moves towards equilibrium and finally compute the equilibrium concentrations or pressures. The exercise we're examining uses an ICE table effectively to approach the problem methodically.

The partial pressure of a gas is the pressure that it would exert if it occupied the whole volume of the mixture alone. In a mixture of gases, the total pressure is the sum of the partial pressures of all the individual gases. It’s an important concept in gas-phase equilibrium chemistry because it reflects the concentration of a gas within a mixture.

In our sample exercise, sulfur trioxide dissociates to sulfur dioxide and oxygen, and the individual partial pressures of these gases at equilibrium are crucial in calculating the equilibrium constant. The partial pressure is directly used in the equilibrium expression for gases, and the understanding of partial pressures is key to establishing the condition of the system at equilibrium.