Problem 20 - A Complete Resource Book in Chemistry for JEE Main

Question

The relation between \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g})\) is (a) \(\mathrm{K}_{\mathrm{n}}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-1}\) (b) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}\) (c) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}} /(\mathrm{RT})^{2}\) (d) \(K_{p}^{p}=K_{c} / R T\)

Step-by-Step Solution

Answer

The correct relation is (a): \(K_p = K_c (RT)^{-1}\).

Step 1: Understand the relation between Kp and Kc

For any equilibrium reaction, the relationship between the equilibrium constants in terms of pressure (\(K_p\)) and concentration (\(K_c\)) is given by the formula: \[K_p = K_c (RT)^{\Delta n}\] where \(\Delta n\) is the change in moles of gas (moles of products - moles of reactants), \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Calculate \(\Delta n\) for the reaction

For the reaction \(2 \mathrm{NO}(\mathrm{g}) + \mathrm{Cl}_2(\mathrm{g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g})\), calculate \(\Delta n\).

Step 2.1: Count moles of gas for reactants

Add up the moles of the gaseous reactants. Here, there are 2 moles of \(\mathrm{NO}\) and 1 mole of \(\mathrm{Cl}_2\), totaling 3 moles of reactants.

Step 2.2: Count moles of gas for products

Add up the moles of the gaseous products. Here, there are 2 moles of \(\mathrm{NOCl}\) in the products.

Step 2.3: Determine \(\Delta n\)

Subtract the total moles of gaseous reactants from the total moles of gaseous products: \(\Delta n = 2 - 3 = -1\).

Step 3: Apply \(\Delta n\) to calculate the relation

Substitute \(\Delta n = -1\) into the equation: \[K_p = K_c (RT)^{-1}\]. This is the correct relationship between \(K_p\) and \(K_c\) for this reaction.

Key Concepts

Relation between Kp and Kc

It is essential to understand the equilibrium constants for gases in terms of pressure and concentration. These constants help predict how an equilibrium will behave under different conditions. The equilibrium constant expressed in terms of pressure is denoted as \( K_p \), while \( K_c \) represents the equilibrium constant in terms of concentration. The relationship between \( K_p \) and \( K_c \) is given by the equation: \[ K_p = K_c (RT)^{\Delta n} \] where:

• \( R \) is the ideal gas constant.
• \( T \) is the temperature in Kelvin.
• \( \Delta n \) is the change in moles of gas, calculated as the difference between moles of gaseous products and moles of gaseous reactants.

In the exercise, we substituted a \( \Delta n \) of \(-1\) due to the reaction stoichiometry, giving us the formula \( K_p = K_c (RT)^{-1} \). This indicates that the equilibrium constant expressed in terms of pressure decreases with increasing temperature for this particular reaction.

Chemical Equilibrium

Chemical equilibrium is a state in which the rates of the forward and reverse reactions in a chemical process are equal. When a reaction reaches equilibrium, the concentrations of the reactants and products remain constant over time. This does not mean that the reactions have stopped, but rather that they occur at the same rate, achieving a dynamic balance. At equilibrium:

• The reaction quotient (Q) equals the equilibrium constant (K).
• The ratio of the concentration of products to reactants raised to the power of their stoichiometric coefficients is constant.

In practical terms, understanding chemical equilibrium allows chemists to control and manipulate conditions to optimize the yield of products. Handling equilibrium situations correctly is crucial in industries like pharmaceuticals and materials manufacturing, where product purity and yield are vital.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationship between the amounts of reactants and products in a chemical reaction. It provides the foundation for calculating how much of each substance is consumed and produced. To determine stoichiometry:

• Identify the balanced chemical equation, which reflects the proportions of reactants and products.
• Calculate molar relationships using the coefficients in the equation, as they indicate the ratio of how substances react with each other.
• Use the reaction coefficients to determine changes in moles of reactants and products when calculating \( \Delta n \).

In our exercise, the stoichiometry was used to determine \( \Delta n = -1 \) for the reaction \( 2 \text{NO}(\text{g}) + \text{Cl}_2(\text{g}) \rightleftharpoons 2 \text{NOCl}(\text{g}) \). Correct stoichiometry ensures the proper calculation of the changes in pressure, concentration, and ultimately, equilibrium constants.