Problem 99 - A Complete Resource Book in Chemistry for JEE Main

Question

For which of the following reaction, \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{c}\) ? (a) \(2 \mathrm{NOCl}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g})\) (b) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g})=2 \mathrm{NH}_{3}(\mathrm{~g})\) (c) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{HCl}(\mathrm{g})\) (d) \(\mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{5}(\mathrm{~g})\)

Step-by-Step Solution

Answer

For reaction (c), \( K_p = K_c \).

Step 1: Understanding Kp and Kc

The equilibrium constant expressed in terms of partial pressures is denoted by \( K_p \) and in terms of concentration by \( K_c \). The relationship between them is given by the formula: \( K_p = K_c(RT)^{\Delta n} \), where \( \Delta n \) is the change in the number of moles of gas between products and reactants.

Step 2: Calculate \( \Delta n \)

For each reaction, calculate \( \Delta n \), the difference in the number of moles of gas between products and reactants. (a) \( \Delta n = (2+1) - 2 = 1 \) (b) \( \Delta n = 2 - (1+3) = -2 \) (c) \( \Delta n = 2 - (1+1) = 0 \) (d) \( \Delta n = 1 - (1+1) = -1 \)

Step 3: Identify When \( K_p = K_c \)

The condition \( K_p = K_c \) occurs when \( \Delta n = 0 \), meaning there is no change in the number of moles of gas between products and reactants.

Step 4: Compare \( \Delta n \) for Each Option

(a) \( \Delta n = 1 \), so \( K_p eq K_c \).(b) \( \Delta n = -2 \), so \( K_p eq K_c \).(c) \( \Delta n = 0 \), so \( K_p = K_c \).(d) \( \Delta n = -1 \), so \( K_p eq K_c \).

Key Concepts

Kp and Kc relationship

In chemical equilibrium, equilibrium constants can be expressed in two main ways: in terms of concentrations (denoted as \( K_c \)) and in terms of partial pressures (denoted as \( K_p \)). Understanding how these two forms are related is crucial for mastering equilibrium calculations. The relationship is given by the equation:\[ K_p = K_c (RT)^{\Delta n} \]where:

• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.
• \( \Delta n \) represents the change in the number of moles of gas between the products and reactants.

This equation helps us determine when \( K_p \) equals \( K_c \). This equality holds true when \( \Delta n = 0 \), meaning there is no net change in the number of gas moles during the reaction. This condition simplifies calculations significantly, as the complicating factor of temperature dependence is removed. Recognizing when \( K_p = K_c \) allows for straightforward analysis of gas-phase equilibria, especially in closed systems.

mole change in reactions

The concept of mole change, denoted as \( \Delta n \), is central in calculating relationships between \( K_p \) and \( K_c \). \( \Delta n \) is the difference in the number of moles of gaseous products and gaseous reactants. It is crucial for understanding how changes in system conditions affect chemical equilibria.To compute \( \Delta n \), follow these steps:

• Identify the balanced chemical reaction.
• Calculate the total moles of gas on the product side.
• Calculate the total moles of gas on the reactant side.
• Subtract the total moles of reactants from the total moles of products.

For example, consider the reaction: \[ 2 \text{NOCl}(\text{g}) \rightleftharpoons 2 \text{NO}(\text{g}) + \text{Cl}_2(\text{g}) \]Here, \( \Delta n = (2+1) - 2 = 1 \). A positive \( \Delta n \) indicates an increase in gas moles. Understanding \( \Delta n \) is essential, as it affects equilibrium constants and how they change with pressure and temperature.

chemical equilibrium calculations

Chemical equilibrium involves balancing the rate of the forward reaction with the rate of the reverse reaction. In practical terms, this balance is represented by equilibrium constants like \( K_p \) and \( K_c \). To perform equilibrium calculations, follow these guidelines:1. **Write Balanced Chemical Equations**:
Start with writing the balanced equation to ensure stoichiometry is considered.2. **Calculate Equilibrium Constants**:
Use given data or experimentally determined values to find either \( K_p \) or \( K_c \).3. **Assess Conditions Using \( \Delta n \)**:
Determine \( \Delta n \), which tells if \( K_p = K_c \) or if adjustments using the relationship \( K_p = K_c (RT)^{\Delta n} \) are required.4. **Setup and Solve ICE Tables**:
ICE (Initial, Change, Equilibrium) tables are a systematic way to organize and solve equilibrium problems. They help visualize concentrations and changes clearly in relation to equilibrium constants.5. **Check Results for Consistency**:
Always recheck calculated values against system constraints (like mass balance and charge neutrality) to ensure accuracy.Mastery of these principles allows you to predict how reactions will behave under different conditions and understand the delicate balance of chemical systems.