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 \(\mathrm{K}_{2}\) is (a) \(\mathrm{K}_{\mathrm{t}}=\frac{\mathrm{K}_{2}}{2}\) (b) \(\mathrm{K}_{2}=\sqrt{\mathrm{K}_{1}}\) (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: Write the Reactions and Equilibrium Constants

The first reaction is given by \( N_2 + 3H_2 \rightleftharpoons 2NH_3 \) with equilibrium constant \( K_1 \). The second reaction \( \frac{1}{2} N_2 + \frac{3}{2} H_2 \rightleftharpoons NH_3 \) has equilibrium constant \( K_2 \).

Step 2: Relate the Two Reactions

Observe that the second reaction is half of the first reaction. Each coefficient in the first reaction is divided by 2 to get the second reaction.

Step 3: Apply the Rules for Equilibrium Constants

When reaction coefficients are divided by a number, the equilibrium constant becomes the original constant raised to the power of the reciprocal of that number. So, if we divide reaction coefficients by 2, then \( K_{2} = (K_{1})^{1/2} \).

Step 4: Choose the Correct Relationship

Based on the rule applied in Step 3, the correct relationship between \( K_1 \) and \( K_2 \) is \( K_2 = \sqrt{K_1} \). This corresponds to option (b).

Key Concepts

Equilibrium Constant

In any balanced chemical reaction that attains equilibrium, the equilibrium constant (\( K \)) is a critical component. It helps in understanding how far a reaction progresses before reaching a state of balance. The equilibrium constant is derived from the concentrations of the reactants and products at equilibrium. For a general reaction,\( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant (\( K \)) is expressed as:
\[K = \frac{[C]^c [D]^d}{[A]^a [B]^b}\]It's important to note that:

• The equilibrium constant is specific to a particular reaction at a given temperature.
• A large \( K \) indicates that the reaction predominantly favors the formation of products, while a small \( K \) suggests it favors reactants.
• Reactions with high equilibrium constants progress nearly to completion before reaching equilibrium.

Remember, equilibrium does not mean that the reactants and products are equal, but their concentrations remain constant over time.

Haber Process

The Haber Process is an industrial method for synthesizing ammonia from nitrogen and hydrogen gas. This reaction is significant in industries for the production of fertilizers. The general reaction for the Haber Process is given by\( N_2 + 3H_2 \rightleftharpoons 2NH_3 \).
Several important aspects characterize the Haber Process:

• Catalyst Usage: Typically, an iron catalyst is used to speed up the reaction, making it more feasible on a large scale.
• Pressure and Temperature: It operates at high pressures (approximately 200 atm) and moderately high temperatures (about 400-500°C) to favor the forward reaction, forming ammonia.
• Equilibrium Consideration: While high pressure favors the synthesis of ammonia, the reaction must balance kinetics and thermodynamics to ensure efficient production.

This process underscores the importance of balancing economic and chemical efficiencies to optimize ammonia production on an industrial scale.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationships between reactants and products in a chemical reaction. This fundamental concept in chemistry helps chemists calculate how much reactants are needed or how much products will be produced.
Using the balanced equation for the Haber Process as an example,\( N_2 + 3H_2 \rightleftharpoons 2NH_3 \), we observe:

• The ratio of nitrogen to hydrogen is 1:3.
• The ratio of nitrogen to ammonia is 1:2.
• Hydrogen combines with nitrogen in a 3:1 molar ratio to produce ammonia.

Understanding stoichiometry ensures that reactions proceed with the correct proportions of reactants, preventing wastage and optimizing yield. In practical scenarios, it allows us to adjust the quantities used in reaction to control the amount of product formed.

Equilibrium Calculations

Equilibrium calculations involve computing the concentrations of reactants and products when a reaction reaches equilibrium. This calculation is crucial for predicting the position of equilibrium and the extent of a reaction under certain conditions.
For example, in the Haber Process,\( N_2 + 3H_2 \rightleftharpoons 2NH_3 \), if given the initial moles of nitrogen and hydrogen, and the equilibrium concentration of ammonia, one can calculate the other concentrations using the expression for the equilibrium constant\( K_1 \).

• Determine the change in concentration of reactants and products using stoichiometry of the balanced equation.
• Substitute these values into the equilibrium expression to solve for unknown concentrations.
• Use the relationship\( K = \frac{[NH_3]^2}{[N_2][H_2]^3}\), to find the equilibrium state from known or necessary values.

These calculations provide insights into how reactions can be shifted by altering concentrations, pressure, or temperature, crucial for optimizing industrial processes like the Haber Process.