Question

If equilibrium constants of reaction: \(\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO}\) is \(\mathrm{K}_{1}\), and \(\frac{1}{2} \mathrm{~N}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{NO}\) is \(\mathrm{K}_{2}\) then (a) \(\mathrm{K}_{1}=\mathrm{K}_{2}\) (b) \(\mathrm{K}_{1}=2 \mathrm{~K}_{2}\) (c) \(\mathrm{K}_{2}=\sqrt{\mathrm{K}}_{1}\) (d) \(\mathrm{K}_{1}=\frac{1}{2} \mathrm{~K}_{2}\)

Short answer

(c) \( \mathrm{K}_2 = \sqrt{\mathrm{K}_1} \).

Step-by-step solution

Understand the Reaction Equations

First, let's write down the reactions and their equilibrium constants. The first reaction is \( \mathrm{N}_2 + \mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} \) with equilibrium constant \( \mathrm{K}_1 \). The second reaction is \( \frac{1}{2} \mathrm{N}_2 + \frac{1}{2} \mathrm{O}_2 \rightleftharpoons \mathrm{NO} \) with equilibrium constant \( \mathrm{K}_2 \).

Relate Equilibrium Constants to Reaction Stoichiometry

For the first reaction, the equilibrium constant \( \mathrm{K}_1 \) is for the formation of 2 moles of NO. For the second reaction, \( \mathrm{K}_2 \) is for the formation of only 1 mole of NO. Reaction stoichiometry affects the expression of equilibrium constants.

Calculate the Relationship Using Powers of Equilibrium

For reactions scaled by an integer, the relationship between their equilibrium constants involves a power. Thus, if you multiply a reaction by \( n \), you raise \( K \) to the power \( n \). As the second reaction is essentially half of the first reaction, the relationship is \( \mathrm{K}_2 = \sqrt{\mathrm{K}_1} \).

Confirm the Correct Answer

The correct option states the relationship between \( \mathrm{K}_1 \) and \( \mathrm{K}_2 \) as \( \mathrm{K}_2 = \sqrt{\mathrm{K}_1} \). This is because the reaction 2 is effectively the square root of the reaction 1 equilibrium process.

Key concepts

Chemical Equilibria

Chemical equilibria occur when a chemical reaction reaches a state where the forward and reverse reaction rates are equal. This results in a stable concentration of reactants and products over time. For the reactions provided, it is important to note how the equilibrium constants, represented by \( K \), describe this balance. The equilibrium constant is a ratio of the concentrations of products to reactants, each raised to the power of their coefficients in the balanced equation.

In the given exercise, the reactions \( \mathrm{N}_2 + \mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} \) and \( \frac{1}{2} \mathrm{N}_2 + \frac{1}{2} \mathrm{O}_2 \rightleftharpoons \mathrm{NO} \) illustrate how different stoichiometric coefficients affect equilibrium constants \( K_1 \) and \( K_2 \). The constant \( K \) thus provides insight into the position of equilibrium, indicating whether the reactants or products are favored under given conditions.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It provides a quantitative basis for understanding reaction dynamics.

In the context of equilibrium constants, stoichiometry plays a critical role because the coefficients in the balanced chemical equations directly impact the expressions for \( K \). For example, in the equation \( \mathrm{N}_2 + \mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} \), the coefficient of 2 for NO means that the concentration of NO is squared in the expression for \( K_1 \).

This is crucial when comparing equilibrium constants of different reactions. In this exercise, since the stoichiometry for the second equation is effectively half, the relation between \( K_1 \) and \( K_2 \) is \( K_2 = \sqrt{K_1} \). This illustrates how stoichiometry determines the form and magnitude of equilibrium constants.

Thermodynamics

Thermodynamics and equilibrium are closely linked concepts in chemistry. They help explain why some reactions reach equilibrium and others do not. In essence, for a reaction at equilibrium, the Gibbs free energy change (\( \Delta G \)) is zero. This stability reflects the balance between energy and entropy in a system.

The equilibrium constant \( K \) is derived from the standard Gibbs free energy change (\( \Delta G^0 \)). The equation \( \Delta G^0 = -RT \ln K \) connects these two. A negative \( \Delta G^0 \) suggests a spontaneous reaction favoring the formation of products, while a positive \( \Delta G^0 \) implies non-spontaneity and favors reactants. Therefore, understanding equilibrium constants like \( K_1 \) and \( K_2 \) involves understanding how these thermodynamic properties dictate the conditions under which a reaction proceeds.

Gaseous Reactions

Gaseous reactions are a fascinating area of study because gases behave differently from solids and liquids. Their volume, pressure, and temperature are closely interconnected, which can significantly impact reactions. Within the realm of equilibrium, the behavior of gases is described by the Ideal Gas Law, \( PV = nRT \).

For the gaseous reactions discussed, pressures and partial pressures are important factors influencing equilibrium constants. The reaction \( \mathrm{N}_2 + \mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} \) occurs under specific conditions of pressure and volume, whereby the concentrations of each participating species in the reaction mixture are used to calculate equilibrium constants. This behavior of gaseous species emphasizes how external conditions can impact the stoichiometry, equilibrium constant, and overall reaction dynamics, allowing us to control and predict reaction behavior effectively.