Given the following data at \(25^{\circ} \mathrm{C}\), $$\begin{array}{cl}2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) & K=1 \times 10^{-30} \\ 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{NOBr}(g) & K=8 \times 10^{1} \end{array}$$ calculate \(K\) for the formation of one mole of NOBr from its elements in the gaseous state at \(25^{\circ} \mathrm{C}\).

The equilibrium constant, represented by the symbol K, is fundamental in understanding chemical reactions that can occur in both forward and reverse directions, ultimately reaching a state of balance. In such a state, the concentrations of reactants and products remain constant over time. The equilibrium constant quantifies the ratio of the concentrations of products to reactants, each raised to the power of their respective coefficients in the balanced equation.

In the context of the given problem, the equilibrium constant for each reaction was used to determine the overall equilibrium constant for the formation of one mole of NOBr from its elements. This involved manipulating known equilibrium constants based on adjustments to the reaction stoichiometry, illustrating the mathematical relationship between changes to the reaction equation and the corresponding equilibrium constant value. The final calculation, combining individual constants, gave a quantifiable measure of the reaction's tendency to proceed under specific conditions, reflecting the dynamic balance between all involved species.

In chemistry, reaction stoichiometry refers to the quantitative relationship between reactants and products in a chemical reaction. It is crucial for understanding how much of a substance is consumed or produced and for predicting the outcomes of reactions. The balanced chemical equation provides the stoichiometric coefficients that represent the molar ratios of each reactant and product.

When solving the problem of finding the equilibrium constant for a new reaction constructed from known reactions, we used the stoichiometry of the initial reactions to manipulate them into the desired form. We then adjusted the equilibrium constants according to the stoichiometric changes we made. In essence, reaction stoichiometry acts as a bridge that allows us to accurately connect different chemical equations and their equilibrium constants, showcasing the precise and calculable nature of chemical reactions.

Le Chatelier's Principle is a qualitative tool that helps predict how a system at equilibrium reacts to external changes such as concentration, temperature, and pressure. According to this principle, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. It’s a way for the system to re-establish balance.

For students seeking a deeper understanding, it's essential to recognize that while Le Chatelier's principle does not directly impact the mathematical calculation of equilibrium constants, it provides insight into the directional shifts of reactions when subjected to different stresses. Each change in conditions can lead to a predictable shift in the system, which is why mastering this principle is invaluable for both conceptual comprehension and practical application in chemistry.