For the reaction \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})\) at \(400 . \mathrm{K}\), \(K=41\). Find the value of \(K\) for each of the following reactions at the same temperature: (a) \(2 \mathrm{NH}_{3}\) (g) \(\rightleftharpoons \mathrm{N}_{2}\) (g) \(+3 \mathrm{H}_{2}\) (g) (b) \(\frac{1}{2} \mathrm{~N}_{2}(\mathrm{~g})+\frac{3}{2} \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{~g})\) (c) \(2 \mathrm{~N}_{2}(\mathrm{~g})+6 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 4 \mathrm{NH}_{3}(\mathrm{~g})\)

In the study of chemical reactions, the state of chemical equilibrium is a crucial concept that occurs when the rate of the forward reaction equals the rate of the reverse reaction. At this point, the concentrations of reactants and products remain constant over time, although both reactions continue to occur. It's important to note that this does not necessarily mean the reactants and products are present in equal amounts, but rather that their ratios stay constant.

Equilibrium can be disrupted by changes in conditions such as pressure, temperature, or concentration. However, the system will attempt to restore a new equilibrium in response to these changes. Understanding equilibrium is essential in fields such as chemistry and engineering where reactions are optimized for industry or research purposes.

One of the key principles in chemical equilibrium is Le Chatelier's principle. This principle states that when a system at equilibrium is subjected to a change in concentration, pressure, or temperature, the system will adjust itself to counteract the imposed change and a new equilibrium will be established. For example, if you increase the concentration of a reactant, the system will shift to produce more product, aiming to reduce the concentration of the added reactant.

Le Chatelier's principle helps predict the direction a reaction will shift when disturbed and is invaluable for industrial processes that require precise conditions to optimize yield and for understanding natural systems like blood oxygenation in the body.

The reaction quotient (Q) is akin to the equilibrium constant but is applicable to any point in time during a reaction, not just at equilibrium. It's calculated using the same formula as the equilibrium constant but using the current concentrations of the reactants and products. The reaction quotient is essential when determining the direction in which a reaction will proceed to reach equilibrium.

By comparing the reaction quotient (Q) to the equilibrium constant (K), you can discern the reaction's direction. If Q < K, the reaction will proceed in the forward direction, converting reactants into products. Conversely, if Q > K, the reaction will proceed in the reverse direction, converting products into reactants.

The equilibrium constant expression is a mathematical representation of the state of a chemical reaction at equilibrium. It is denoted by K and is calculated using the molar concentrations of the reactants and products, each raised to the power of their respective coefficients in the balanced equation. Distinct for every reaction, the value of K provides insight into the relative proportions of reactants and products at equilibrium.

The equilibrium constant expression for the reaction N₂(g) + 3H₂(g) ⇌ 2NH₃(g) is given by: \[ K = \frac{[NH_3]^2}{[N_2][H_2]^3} \]
When the reaction is reversed or the stoichiometry is altered (as seen in the exercise), the expression and value of K will change accordingly. These changes follow specific rules, like taking the reciprocal for the reverse reaction (part a) or raising to a power when coefficients are altered (parts b and c), which are fundamental in solving equilibrium problems.