Given the following reactions and their equilibrium constants, $$\begin{array}{cl}\mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g) & K=2.4 \times 10^{-9} \\ \mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{Cl}_{2}(g) & K=8.8 \times 10^{-13} \end{array}$$ calculate \(K\) for the reaction $$\mathrm{C}(s)+\mathrm{CO}_{2}(g)+2 \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{COCl}_{2}(g)$$

In the context of chemistry, the term chemical equilibrium refers to a state where the rate of the forward reaction equals the rate of the reverse reaction for a given chemical process. This balance results in stable concentrations of reactants and products over time, although it's important to understand that the reactions are still occurring; they're just happening at the same rate in both directions.

For example, the equation
\(C(s) + CO_2(g) \rightleftharpoons 2 CO(g)\),
with a provided equilibrium constant \(K=2.4 \times 10^{-9}\), tells us that at equilibrium, the forward reaction converting carbon and carbon dioxide into carbon monoxide, and the reverse reaction converting carbon monoxide back into carbon and carbon dioxide, proceed at equal rates. The equilibrium constant \(K\) is a numerical value that represents the ratio of the concentrations (or partial pressures) of the products to the reactants, each raised to the power of their respective coefficients in the balanced equation.

Understanding equilibrium is crucial in calculating the new equilibrium constant as seen in the exercise, where the individual constants \(K_1\) and \(K'_2\) were multiplied to find the overall equilibrium constant for the desired reaction.

Le Chatelier's principle is a critical concept in chemical equilibrium, stating that if an external change is applied to a system at equilibrium, the system will adjust itself to counteract the effect of the change and restore a new equilibrium position. This principle helps us predict the direction in which a reaction's equilibrium will shift when subjected to changes in concentration, temperature, or pressure.

For instance, considering the exercise given, if the system for the reaction \(COCl_2(g) \rightleftharpoons CO(g) + Cl_2(g)\) experiences an increase in the concentration of \(Cl_2(g)\), Le Chatelier's principle suggests that the equilibrium will shift to the left to reduce the concentration of \(Cl_2(g)\) by producing more \(COCl_2(g)\). This principle assists chemists in controlling the yield of reactions, which is vital in industrial chemical processes.

The reaction quotient (\(Q\)) is a concept that plays a pivotal role when dealing with chemical reactions not at equilibrium. It is calculated in the same manner as the equilibrium constant (\(K\)), but it uses the initial concentrations of the reactants and products, rather than the equilibrium concentrations.

The value of the reaction quotient at any point in time can guide us in predicting the direction in which the reaction must shift to reach equilibrium. If \(Q\) is less than \(K\), the forward reaction is favored, and the reaction will proceed to form more products until equilibrium is achieved. Conversely, if \(Q\) is greater than \(K\), the reverse reaction is favored, and the reaction will proceed to form more reactants.

In combination with Le Chatelier's principle, it allows for a dynamic assessment of a reaction's progression towards equilibrium — adding practical value in both laboratory experimentation and industrial applications.