At \(900 \mathrm{~K}\) the following reaction has \(K_{p}=0.345\) : $$2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{SO}_{3}(g)$$ In an equilibrium mixture the partial pressures of \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) are 0.135 atm and 0.455 atm, respectively. What is the equilibrium partial pressure of \(\mathrm{SO}_{3}\) in the mixture?

Understanding chemical equilibrium is crucial in the study of reactions and their behavior under different conditions. Chemical equilibrium refers to the state of a reaction wherein the rate of the forward reaction equals the rate of the reverse reaction, leading to no net change in the concentration of reactants and products over time. It's a dynamic condition, meaning that the molecules are constantly reacting, but the overall concentrations remain constant.

At equilibrium, the system has reached a state of balance, but it doesn't mean that the reactants and products are at equal concentrations. Instead, they are at a ratio that will remain consistent as long as external conditions such as temperature or pressure do not change. External changes can 'shift' the equilibrium position according to Le Chatelier's Principle, which predicts how the position of equilibrium will change to counteract the effect of the disturbance.

The equilibrium constant expression for a reaction is a way of quantifying the position of equilibrium. It is symbolized by the letter 'K'. For reactions involving gases, we use partial pressures and represent the constant as \(K_p\). The equilibrium constant is a reflection of the proportions of the concentrations of the products to the reactants at equilibrium and each is raised to the power of their coefficients in the balanced chemical equation.

To construct an equilibrium constant expression, the concentrations of gaseous products are divided by the concentrations of gaseous reactants, taking into account their stoichiometric coefficients from the balanced equation. The value of \(K\) is constant for a given reaction at a specific temperature. When the system is at equilibrium, the constant expression will yield the equilibrium constant's value, which can provide insights into the extent of the reaction and the proportions of reactants and products.

Partial pressure is a measure of the individual pressure exerted by a specific gas in a mixture of gases. It is proportional to the mole fraction of the gas in the total mixture. To calculate partial pressures, we make use of Dalton's Law, which states that the total pressure of a mixture of gases is the sum of the partial pressures of each individual gas.

In an equilibrium situation involving gases, the equilibrium constant expression \(K_p\) is particularly useful. By rearranging the equilibrium expression and substituting the known values of partial pressures for the reactants, as well as the value of \(K_p\), it is possible to calculate the partial pressure of an unknown product. This involves algebraic manipulation to first isolate the term that includes the unknown partial pressure and then solving for that unknown, often by taking square roots if the term is squared, as in the case of our given exercise.