As shown in Table \(15.2, K_{p}\) for the equilibrium $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) $$ is \(4.51 \times 10^{-5}\) at \(450^{\circ} \mathrm{C}\). For each of the mixtures listed here, indicate whether the mixture is at equilibrium at \(450^{\circ} \mathrm{C}\). If it is not at equilibrium, indicate the direction (toward product or toward reactants) in which the mixture must shift to achieve equilibrium. (a) \(98 \mathrm{~atm} \mathrm{NH}_{3}, 45 \mathrm{~atm} \mathrm{~N}_{2}, 55 \mathrm{~atm} \mathrm{H}_{2}\) (b) \(57 \mathrm{~atm} \mathrm{NH}_{3}, 143 \mathrm{~atm} \mathrm{~N}_{2},\) no \(\mathrm{H}_{2}\) (c) \(13 \mathrm{~atm} \mathrm{NH}_{2} 27 \mathrm{~atm} \mathrm{~N}_{2} 82 \mathrm{~atm} \mathrm{H}_{2}\)

The equilibrium constant, represented as K_p, is a fundamental concept in chemical equilibrium that provides a quantitative measure of the position of equilibrium in reactions involving gases. It is expressed in terms of partial pressures.

For a general reaction of the form:
\[aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)\] the equilibrium constant, K_p, is given by the following expression: \[ K_{p} = \frac{(P_C)^c(P_D)^d}{(P_A)^a(P_B)^b} \] where P_A, P_B, P_C, and P_D are the partial pressures of the gases A, B, C, and D, respectively, and the lowercase letters represent the stoichiometric coefficients.

A K_p value that is very large suggests that, at equilibrium, there is a greater concentration of products compared to reactants. Conversely, a K_p value that is small indicates an equilibrium with greater concentrations of reactants. Understanding K_p helps us predict how changes in conditions, like pressure and temperature, affect the equilibrium position of a reaction.

The reaction quotient, Q_p, helps us determine the direction in which a reaction mixture will shift in order to reach equilibrium. It is calculated using the same expression as the equilibrium constant but with the initial partial pressures of the reactants and products, not the equilibrium ones.

For the reaction equilibrium equation given earlier, the reaction quotient would be: \[ Q_{p} = \frac{(P_C)^c(P_D)^d}{(P_A)^a(P_B)^b} \] When comparing Q_p to K_p, if Q_p is greater than K_p, the reaction will proceed in the direction of the reactants to reach equilibrium. If Q_p is less than K_p, the reaction will proceed towards the products. If they are equal, the system is at equilibrium.

By calculating Q_p for different mixtures and comparing it to K_p, students can predict which way a reaction will shift. This is particularly valuable in ensuring a reaction is on the right path to achieve the desired outcome.

Le Châtelier's principle is a key idea that predicts how a change in conditions, such as concentration, pressure, or temperature, will affect the position of equilibrium in a closed system. According to this principle, if a system at equilibrium is disturbed by a change in conditions, the system will adjust itself to counteract the effect of that disturbance, and establish a new equilibrium.

For example, if we increase the pressure on a reaction mixture involving gases, the system will respond by shifting the equilibrium position in the direction that produces fewer gas molecules. Similarly, adding more reactant to a system at equilibrium will result in a shift towards the production of more products until a new balance is achieved.

Le Châtelier's principle is essential for understanding how to control the yield of a reaction in industrial processes and for predicting the impact of environmental changes on reactions. This principle empowers students with the ability to hypothesize the behavior of a system when subjected to various perturbations, enabling them to grasp the dynamic nature of chemical equilibria.