Problem 76 - A Complete Resource Book in Chemistry for JEE Main

Question

In which of the following gaseous reaction, \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) have the same values? (a) \(2 \mathrm{Hl} \rightleftharpoons \mathrm{H}_{2}+\mathrm{I}_{2}\) (b) \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\) (c) \(2 \mathrm{SO}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{SO}_{3}\) (d) \(\mathrm{PCI}_{5} \rightleftharpoons \mathrm{PCI}_{3}+\mathrm{Cl}_{2}\)

Step-by-Step Solution

Answer

Reaction (a) \( 2 \mathrm{Hl} \rightleftharpoons \mathrm{H}_{2} + \mathrm{I}_{2} \) has \( K_{p} = K_{c} \).

Step 1: Understand the Relationship Between Kp and Kc

For any gaseous reaction, the equilibrium constant in terms of partial pressure \( K_{p} \) and the equilibrium constant in terms of concentration \( K_{c} \) are related by the expression: \[ K_{p} = K_{c} (RT)^{\Delta n} \]where \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in the number of moles of gas.

Step 2: Determine \( \Delta n \) for Each Reaction

Calculate the change in the number of moles of gas \( \Delta n \) for each reaction:(a) \( 2 \text{Hl} \rightleftharpoons \text{H}_2 + \text{I}_2 \): \( \Delta n = 2 - 2 = 0 \)(b) \( \text{N}_2 + 3\text{H}_2 \rightleftharpoons 2\text{NH}_3 \): \( \Delta n = 2 - 4 = -2 \)(c) \( 2\text{SO}_2 + \text{O}_2 \rightleftharpoons 2\text{SO}_3 \): \( \Delta n = 2 - 3 = -1 \)(d) \( \text{PCl}_5 \rightleftharpoons \text{PCl}_3 + \text{Cl}_2 \): \( \Delta n = 2 - 1 = 1 \).

Step 3: Identify Reaction with \( \Delta n = 0 \)

The equilibrium constants \( K_{p} \) and \( K_{c} \) will have the same value when the change in moles of gas, \( \Delta n \), equals zero, because the term \( (RT)^{\Delta n} \) becomes \( (RT)^0 = 1 \). In the list of reactions:- Only reaction (a) has \( \Delta n = 0 \): \( 2\text{Hl} \rightleftharpoons \text{H}_2 + \text{I}_2 \).

Step 4: Conclusion

Based on the calculations, the reaction in which \( K_{p} \) and \( K_{c} \) have the same value is reaction (a) because its \( \Delta n = 0 \).

Key Concepts

Kp and Kc relationship

In chemistry, we often encounter reactions that occur in the gas phase. For these gaseous reactions, we use different ways to express equilibrium constants. Two common expressions are the equilibrium constant in terms of partial pressure, denoted as \( K_p \), and the equilibrium constant in terms of concentration, denoted as \( K_c \). These constants provide valuable information about the position of equilibrium in a reaction.

The mathematical relationship between \( K_p \) and \( K_c \) is given by:

• \[ K_p = K_c (RT)^{\Delta n} \]

This formula shows how \( K_p \) and \( K_c \) are connected through the ideal gas constant \( R \), the temperature \( T \) in Kelvin, and \( \Delta n \), which represents the change in moles of gas during the reaction.

Ultimately, the relationship between these constants helps determine if the equilibrium concentrations or pressures of reactants and products will change appreciably with temperature, giving us insights into their dependence on experimental conditions.

Gaseous Reactions

Gaseous reactions are chemical reactions where all reactants and products are in the gas phase. Understanding gaseous reactions involves considering both the concentrations and pressures of involved gases since these factors can influence the direction and extent of the reaction. Here are some critical points about gaseous reactions:

• They are highly influenced by changes in pressure and temperature because gases expand or compress easily.
• The dynamics of the reaction can be described by the equilibrium constant, either \( K_c \) for concentrations or \( K_p \) for partial pressures.

In the specific context of equilibrium, gaseous reactions are represented by balanced chemical equations that detail how molecules rearrange. For instance, in the reaction of nitrogen and hydrogen forming ammonia, \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \), there is a precise balance between the formation and decomposition of ammonia at equilibrium. This dynamic balance underpins the behavior of gases during a reaction, where conditions such as pressure and number of moles play crucial roles.

Change in Moles (Δn)

The term \( \Delta n \) is significant in the context of gaseous reactions because it directly affects the relationship between \( K_p \) and \( K_c \).

• \( \Delta n \) represents the change in the number of moles of gas as the reaction progresses from reactants to products.
• It is calculated as the difference between the total moles of gaseous products and the total moles of gaseous reactants.

The value of \( \Delta n \) plays a crucial role in determining how \( K_p \) and \( K_c \) relate to each other. When \( \Delta n = 0 \), the expression \( (RT)^{\Delta n} \) simplifies to \( 1 \), meaning \( K_p = K_c \). This happens because there’s no net change in the number of gas molecules, thus leading to no dependency on volume changes induced by pressure or temperature variations.

Understanding \( \Delta n \) is essential for predicting how gases will behave under different conditions, allowing chemists to control reactions more effectively by adjusting pressures and temperatures as needed.