Let \(\alpha\) be the fraction of \(\mathrm{PCl}_{5}\) molecules that have decomposed to \(\mathrm{PCl}_{3}\) and \(\mathrm{Cl}_{2}\) in the reaction \(\mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+\) \(\mathrm{Cl}_{2}(\mathrm{~g})\) in a constant- volume container; then the amount of \(\mathrm{PCl}_{5}\) at equilibrium is \(n(1-\alpha)\), where \(n\) is the amount present initially. Derive an equation for \(K\) in terms of \(\alpha\) and the total pressure \(P\), and solve it for \(\alpha\) in terms of \(P\). Calculate the fraction decomposed at \(556 \mathrm{~K}\), at which temperature \(K=4.96\), and the total pressure is (a) \(0.50\) bar; (b) \(1.00\) bar.

Understanding the equilibrium constant is essential for students tackling chemical equilibrium problems. The equilibrium constant, represented as K, is a number that reflects the ratio of the concentration of the products to the concentration of the reactants at equilibrium, with each raised to the power of their coefficients from the balanced chemical equation.

For a reaction such as \(\mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+ \mathrm{Cl}_{2}(\mathrm{~g})\), the equilibrium constant would be expressed as \[K = \frac{[\mathrm{PCl}_3][\mathrm{Cl}_2]}{[\mathrm{PCl}_5]}\]. The square brackets indicate the concentration of the species in moles per liter, but when dealing with gases, as in this equation, we often use partial pressure terms.

The significance of K lies in its constant value at a given temperature, meaning that no matter the starting concentrations or pressures, the reaction reaches equilibrium when the ratio of product to reactant concentrations matches K. A high value of K indicates a greater concentration of products at equilibrium, favoring the forward reaction, while a low K value indicates a reaction that favors the reactants.

Partial pressure is another critical concept in the realm of gas phase chemical equilibria. It refers to the pressure that each gas in a mixture would exert if it were alone in the container. In the context of equilibrium, the partial pressures of the reactants and products at equilibrium are used in place of concentrations when calculating the equilibrium constant for reactions involving gases.

In the given problem, the partial pressures of the gases are expressed in terms of the total pressure P and the degree of dissociation \(\alpha\). These relationships are important when calculating the equilibrium constant in a system where the ideal gas law applies. For instance, \(\mathrm{PCl}_5\)'s partial pressure at equilibrium is expressed as \[P(1-\alpha)\], where P is the total pressure exerted by the gas mixture and \(1-\alpha\) represents the fraction of \(\mathrm{PCl}_5\) that has not dissociated. Understanding partial pressure is essential for correctly interpreting and solving gas phase equilibrium problems.

The reaction quotient, Q, plays a pivotal role when determining the direction in which a reaction will proceed to reach equilibrium. Q has the same form as the equilibrium constant K but is calculated using the initial concentrations (or partial pressures) instead of those at equilibrium.

To illustrate, for the reaction of \(\mathrm{PCl}_5\) dissociating into \(\mathrm{PCl}_3\) and \(\mathrm{Cl}_2\), Q would be expressed as \[Q = \frac{[\mathrm{PCl}_3][\mathrm{Cl}_2]}{[\mathrm{PCl}_5]}\] before the system reaches equilibrium. The comparison of Q to K allows us to predict the shift of the equilibrium: if Q < K, the reaction will proceed forward to produce more products, and if Q > K, the reaction will proceed in the reverse direction to produce more reactants.

At equilibrium, Q equals K, and no net change occurs in the concentrations of reactants and products. Understanding the reaction quotient is crucial for students because it facilitates the prediction of reaction behavior in response to changes in conditions, which is the essence of chemical equilibrium.