Problem 10 - A Complete Resource Book in Chemistry for JEE Main

Question

In the reaction, \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}+\) heat, relation- ship between \(K_{p}\) and \(K_{c}\) is (a) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-2}\) (b) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{c}(\mathrm{RT})^{2}\) (c) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-3}\) (d) \(\mathrm{K}_{\mathrm{c}}=\mathrm{K}_{\mathrm{p}}(\mathrm{RT})^{3}\)

Step-by-Step Solution

Answer

The relationship is (a) \(K_{p} = K_{c}(RT)^{-2}\).

Step 1: Identify the Reaction Type

The given reaction is \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}+\) heat, which is a gaseous equilibrium reaction where \(\Delta n\) (change in moles of gas) is calculated by subtracting the moles of gaseous reactants from the moles of gaseous products.

Step 2: Calculate \(\Delta n\)

For the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\), we calculate \(\Delta n = \text{moles of products} - \text{moles of reactants} = 2 - (1 + 3) = 2 - 4 = -2\).

Step 3: Use the Relationship Formula

The general relationship between \(K_{p}\) and \(K_{c}\) is given by the formula \(K_{p} = K_{c}(RT)^{\Delta n}\), where \(R\) is the gas constant and \(T\) is the temperature in Kelvin.

Step 4: Substitute \(\Delta n\) Into the Formula

Substituting \(\Delta n = -2\) into the formula gives us \(K_{p} = K_{c}(RT)^{-2}\).

Step 5: Match with the Given Options

Compare the derived formula \(K_{p} = K_{c}(RT)^{-2}\) with the provided options to find the matching one: (a) \(\mathrm{K}_{p}= ext{K}_{c}(RT)^{-2}\).

Key Concepts

Gaseous Reactions

Gaseous reactions involve substances that are in the gas phase during the course of the reaction. In these types of reactions, reactants and products are gaseous, allowing them to freely collide and interact with each other. This makes comprehension of gaseous reactions essential in physical chemistry.

• One critical factor in gaseous reactions is the pressure of the system, which can influence the reaction rate and equilibrium.
• These reactions often follow the ideal gas law, which links the pressure, volume, temperature, and amount of gas in the system.
• Equilibria in gaseous reactions are represented using expressions like the equilibrium constant, which depicts the balance between reactants and products.

Knowing these principles associated with gaseous reactions arms students with the foundational knowledge needed to understand more complex aspects of chemical equilibria.

Change in Moles (Δn)

The concept of change in moles, expressed as \(\Delta n\), is crucial when analyzing gaseous equilibrium reactions. It refers to the difference in the amount of gaseous products and reactants.

When calculating \(\Delta n\), you subtract the total moles of gaseous reactants from the total moles of gaseous products:

• In the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\), \(\Delta n = 2 - (1 + 3) = -2\).
• \(\Delta n\) can be positive, negative, or zero, and it significantly affects the relationship between different equilibrium constants.

Understanding \(\Delta n\) is valuable because it helps determine how variables like pressure and volume impact gaseous reactions, and it is pivotal to establish the relationship between \(K_p\) and \(K_c\).

Kp and Kc Relationship

The relationship between \(K_p\) and \(K_c\) links the equilibrium constants for reactions involving gases. \(K_p\) represents the equilibrium constant in terms of partial pressures, while \(K_c\) is in terms of concentrations.

The formula that connects \(K_p\) and \(K_c\) is:

• \(K_p = K_c (RT)^{\Delta n}\), where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in moles.
• A negative \(\Delta n\) indicates the reaction will shift toward producing fewer moles of gas, while a positive \(\Delta n\) signifies more moles will be produced.

This formula demonstrates not only the interdependence of \(K_p\) and \(K_c\) but also underscores the influential role that temperature and moles play in balancing gaseous equilibria. Understanding this relationship allows chemists to predict how changes in pressure or temperature might influence a reaction direction.