At \(450^{\circ} \mathrm{C}\) the value of the equilibrium constant for the following system is \(6.59 \times 10^{-3}\) . If \(\left[\mathrm{NH}_{3}\right]=1.23 \times 10^{-4}\) and \(\left[\mathrm{H}_{2}\right]=2.75 \times 10^{-3}\) at equilibrium, determine the concentration of \(\mathrm{N}_{2}\) at that point. $$\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \rightleftarrows 2 \mathrm{NH}_{3}(\mathrm{g})$$

The law of mass action lays the foundation for understanding how chemical reactions behave at equilibrium. In simple terms, it states that at a given temperature, the rates at which reactions occur depend directly on the concentration of the reactants and products, each raised to the power of their coefficients in the balanced chemical equation.
For the reaction
\(N_2(g) + 3H_2(g) \rightleftarrows 2NH_3(g)\),
the law of mass action can be applied to express the equilibrium condition as an equation. This equation is derived from the ratios of the concentrations of the products to the reactants, with each concentration raised to the power of the respective stoichiometric coefficient. Specifically, for this reaction, the equilibrium constant (\(K_c\)) expression is
\(K_c = \frac{[NH_3]^2}{[N_2][H_2]^3}\).
This equation crucially assumes the reaction has reached a state where the rate of the forward reaction, forming ammonia (\(NH_3\)), is equal to the rate of the reverse reaction, breaking down ammonia back into nitrogen (\(N_2\)) and hydrogen gas (\(H_2\)). Therefore, the concentrations of the reactants and products no longer change with time.

Chemical equilibrium represents a dynamic but stable state of a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. At equilibrium, the concentrations of the reactants and products remain constant over time, not because the reactions have stopped, but because the reactions continue at an equal rate in both directions.
Understanding equilibrium is crucial for calculating concentrations of substances in a chemical system, as demonstrated in the provided exercise. By using the equilibrium constant (\(K_c\)), which numerically defines the ratio of product concentrations to reactant concentrations at equilibrium, we were able to determine the unknown concentration of \(N_2\).
However, it’s important to realize this state of balance does not imply that reactants and products are present in equal amounts but rather that their ratios do not change. This concept allows chemists to predict the extent of a reaction and how it can be influenced by changing conditions such as concentration, temperature, or pressure.

Stoichiometry is the mathematical relationship between the quantities of reactants and products in a chemical reaction. It is derived from the balanced chemical equation and allows for the precise calculation of the amounts of substances involved.
In the context of equilibrium constant calculations, stoichiometry comes into play by determining how the concentrations of reactants and products are related to each other through the coefficients in the chemical equation. In the provided exercise, for instance, every mole of \(N_2\) reacts with three moles of \(H_2\) to produce two moles of \(NH_3\).
When calculating the equilibrium constant, the stoichiometric coefficients provide the exponents for the concentrations in the equilibrium constant expression. The mathematics of stoichiometry helps in understanding the proportional relationships in reactions, which is essential when predicting the amounts of products formed from given quantities of reactants or vice versa. In practical applications, such as the synthesis of chemicals, stoichiometry aids in optimizing reactant use and avoiding excess waste.