At \(2000^{\circ} \mathrm{C}\) the equilibrium constant for the reaction $$2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g)$$ is \(K_{c}=2.4 \times 10^{3}\). If the initial concentration of \(\mathrm{NO}\) is \(0.175 \mathrm{M}\) what are the equilibrium concentrations of \(\mathrm{NO}, \mathrm{N}_{2},\) and \(\mathrm{O}_{2} ?\)

The equilibrium constant, denoted as \( K_c \), is a numerical value that provides a measure of the relative amounts of products to reactants at equilibrium for a particular reaction at a constant temperature. In the given exercise, \( K_c \) is \(2.4 \times 10^3\) for the reaction \(2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g) + \mathrm{O}_{2}(g)\). This high value indicates that at equilibrium, the reaction mixture will mainly consist of the products. To calculate the equilibrium constant, the concentrations of the products are divided by the concentrations of the reactants, each raised to the power of their respective stoichiometric coefficients in the balanced equation.

During the calculation, it's essential to ensure that all concentrations are in molarity (M) and that solids and pure liquids are excluded since their concentrations do not change during the reaction. Furthermore, equilibrium constants are specific to a particular temperature and can vary significantly with temperature changes. Therefore, any discussion involving \( K_c \) must also take into account the temperature of the system.

Equilibrium concentrations are the concentrations of reactants and products at the point when a chemical reaction reaches equilibrium. At this stage, the rate of the forward reaction equals the rate of the reverse reaction, and no net change is observed in the concentrations of reactants and products. These concentrations remain constant over time unless the system is disturbed. The approach taken in the exercise to find equilibrium concentrations is to begin with the initial concentrations and account for the changes as the reaction moves towards equilibrium.

The changes in concentrations are typically represented by \(x\), indicating how much the concentration of each substance has shifted from its initial value. Once the value of \(x\) is found by using the equilibrium constant expression, it can be substituted into the expressions for equilibrium concentrations to determine the actual concentrations of each species in the equilibrium mixture.

The reaction quotient, \( Q_c \), is a ratio similar to the equilibrium constant except that it applies to concentrations of reactants and products that are not necessarily at equilibrium. It is calculated in the same way as \( K_c \), using the current concentrations of the reactants and products. The value of \( Q_c \) is then compared to \( K_c \) to predict the direction in which the reaction will proceed to achieve equilibrium.

If \( Q_c < K_c \), the system will shift towards the products to reach equilibrium. If \( Q_c > K_c \), the reaction will proceed towards the reactants. If \( Q_c = K_c \), the system is already at equilibrium. This concept helps us understand how changes in concentrations affect the position of equilibrium without necessarily having to calculate the new equilibrium concentrations.

Le Chatelier's Principle gives us a qualitative prediction of how a system at equilibrium will respond to an external change. According to this principle, if a stress such as change in concentration, pressure, or temperature is applied to a system at equilibrium, the system will adjust itself to oppose the effect of the stress and restore a new equilibrium.

For example, if more reactants are added to the system, the equilibrium will shift towards the products to reduce the concentration of the added substances. Conversely, removing a product will result in the equilibrium shifting to replace the lost product. Changes in pressure and temperature also affect the equilibrium state. This principle is vital in industrial chemistry, such as in the synthesis of ammonia using the Haber process, where manipulating conditions can maximize product yields.