Problem 151 - A Complete Resource Book in Chemistry for JEE Main

Question

The equilibrium constants for the reactions \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\) and \(\frac{1}{2} \mathrm{~N}_{2}+\frac{3}{2} \mathrm{H}_{2} \rightleftharpoons \mathrm{NH}_{3}\) are \(K_{1}\) and \(K_{2}\) respectively. The correct relationship between \(\mathrm{K}_{1}\) and \(\mathrm{K}\), is (a) \(\mathrm{K}_{1}=\frac{\mathrm{K}_{2}}{2}\) (b) \(\mathrm{K}_{2}=\sqrt{\mathrm{K}_{\mathrm{i}}}\) (c) \(\mathrm{K}_{2}=\mathrm{K}_{1}{\underline{\phantom{xx}}}^{2}\) (d) \(\mathrm{K}_{1}=\mathrm{K}_{2}\)

Step-by-Step Solution

Answer

The correct relationship is (b) \(K_2 = \sqrt{K_1}\).

Step 1: Understand the Reactions

The reactions given are:1. \(N_2 + 3H_2 \rightleftharpoons 2NH_3\) with equilibrium constant \(K_1\).2. \(\frac{1}{2}N_2 + \frac{3}{2}H_2 \rightleftharpoons NH_3\) with equilibrium constant \(K_2\).The second reaction is derived from the first by dividing all coefficients by 2.

Step 2: Relationship Between Equilibrium Constants

When a chemical equation is multiplied or divided by a number, the equilibrium constant is accordingly raised to the power of that number.Since the second reaction is the first reaction divided by 2, the equilibrium constant for the second reaction, \(K_2\), is equal to the square root of \(K_1\). Mathematically, this can be expressed as \(K_2 = K_1^{1/2}\).

Step 3: Identify the Correct Option

Based on the relationship we identified, we know that \(K_2 = \sqrt{K_1}\).Among the options provided:- (a) \(K_1 = \frac{K_2}{2}\) - incorrect- (b) \(K_2 = \sqrt{K_i}\) - plausible- (c) \(K_2 = K_1^2\) - incorrect- (d) \(K_1 = K_2\) - incorrect

Key Concepts

Equilibrium Constant

The equilibrium constant, typically denoted as \(K\), is a fundamental concept in chemical equilibrium. It gives us insight into the proportion of products to reactants at equilibrium. When a reaction reaches equilibrium, the rate at which the reactants transform into products equals the rate at which the products revert to reactants. At this point, the concentrations of reactants and products remain constant over time.

An equilibrium constant is specific to a given reaction and temperature. Its value is derived from the concentrations of the products and reactants at equilibrium. For generic reaction \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant expression, \(K\), would be given by:

• \(K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\)

In this formula, the brackets \([]\) denote concentration. Small letters \(a, b, c, \text{ and } d\) represent the stoichiometric coefficients from the balanced reaction equation.

The magnitude of \(K\) gives us essential details:

• If \(K \gg 1\), the reaction favors products.
• If \(K \ll 1\), the reaction favors reactants.

This ratio doesn't change unless there's a change in temperature, making it a reliable indicator for the reaction's tendency under given conditions.

Reaction Stoichiometry

Stoichiometry is an integral part of understanding chemical reactions. It refers to the quantitative relationships between the amounts of reactants and products in a chemical reaction. At its core, stoichiometry is about balancing equations so that the same number of each type of atom appears on both sides of the reaction.

This concept directly influences equilibrium constants. When a reaction equation is altered, such as being multiplied or divided by a constant, the stoichiometry changes, and thus the equilibrium constant is affected.

• For instance, if a reaction equation is halved, the equilibrium constant for the new equation becomes the square root of the original constant.

This was observed in the reactions given:

• Original: \(N_2 + 3H_2 \rightleftharpoons 2NH_3\)
• Modified: \(\frac{1}{2}N_2 + \frac{3}{2}H_2 \rightleftharpoons NH_3\)

Here, the equilibrium constant \(K_2\) for the modified equation is the square root of \(K_1\) for the original equation, thus \(K_2 = K_1^{1/2}\).

Understanding stoichiometry makes it possible to predict how changes to the balanced equation will affect the equilibrium constant and the overall reaction dynamics.

Ammonia Synthesis

Ammonia synthesis, specifically the Haber process, is an industrial chemical reaction of heavyweight importance. It involves combining nitrogen and hydrogen gases under high pressure and temperature to produce ammonia \(NH_3\). The reaction is denoted as:

• \(N_2 + 3H_2 \rightleftharpoons 2NH_3\)

This process has a profound impact on global agriculture, as ammonia is a key component in fertilizers, which are essential for modern agriculture.

The synthesis occurs under demanding conditions. High pressure (150-200 atm) and high temperatures (around 400-500°C) are required to shift the equilibrium in favor of ammonia formation, as the reaction itself is exothermic. Catalysts such as iron are also employed to speed up the reaction.

Ammonia synthesis is a typical example used in teaching chemical equilibria, showcasing real-world applications. The equilibrium reaction demonstrates the balance between reactant and product concentrations. The equilibrium constant \(K\) helps in predicting yields and optimizing reaction conditions. Understanding the stoichiometry and conditions affecting \(K\) is crucial to managing and maximizing ammonia production efficiently.