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

Question

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

Reaction (c), \(\mathrm{H}_2(g) + \mathrm{Cl}_2(g) \rightleftharpoons 2\ \mathrm{HCl}(g)\), has \(\Delta n = 0\), so \(K_p = K_c\).

Step 1: Understand the relationship between Kc and Kp

The relationship between the equilibrium constant in terms of concentration, \(K_c\), and pressure, \(K_p\), 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, and \(\Delta n\) is the change in moles of gas (moles of gaseous products minus moles of gaseous reactants). If \(\Delta n = 0\), then \(K_c = K_p\).

Step 2: Calculate Δn for each reaction

For each reaction provided, calculate \(\Delta n\). \(\Delta n\) is the difference between the moles of gaseous products and moles of gaseous reactants:(a) \(2\ \mathrm{NOCl}(g) \rightleftharpoons 2\ \mathrm{NO}(g) + \mathrm{Cl}_2(g)\): \(\Delta n = (2 + 1) - 2 = 1\).(b) \(\mathrm{N}_2(g) + 3\ \mathrm{H}_2(g) \rightleftharpoons 2\ \mathrm{NH}_3(g)\): \(\Delta n = 2 - (1 + 3) = -2\).(c) \(\mathrm{H}_2(g) + \mathrm{Cl}_2(g) \rightleftharpoons 2\ \mathrm{HCl}(g)\): \(\Delta n = 2 - (1 + 1) = 0\).(d) \(\mathrm{PCl}_3(g) + \mathrm{Cl}_2(g) \rightleftharpoons \mathrm{PCl}_5(g)\): \(\Delta n = 1 - (1 + 1) = -1\).

Step 3: Identify reaction with Δn = 0

Since \(K_p = K_c\) when \(\Delta n = 0\), identify the reactions that satisfy this condition. From our calculations in Step 2, Reaction (c) has \(\Delta n = 0\).

Step 4: Confirm the correct answer

The reaction for which \(K_p = K_c\) is reaction (c): \(\mathrm{H}_2(g) + \mathrm{Cl}_2(g) \rightleftharpoons 2\ \mathrm{HCl}(g)\).

Key Concepts

Kc and Kp relationship

In chemical equilibrium reactions, we are often interested in understanding how to compare the equilibrium constants given in terms of concentration or pressure. These are represented as \( K_c \) and \( K_p \) respectively. The relationship between these two constants can be described by the equation:\[ K_p = K_c (RT)^{\Delta n} \]Here, \( R \) stands for the universal gas constant and \( T \) is the temperature in Kelvin. The exponent \( \Delta n \) represents the change in moles of gas, calculated as the difference between the moles of gaseous products and the moles of gaseous reactants.

• If \( \Delta n = 0 \), it simplifies the relationship to \( K_p = K_c \), meaning that the equilibrium constants in terms of concentration and pressure are identical.
• If \( \Delta n \) differs from zero, \( K_p \) and \( K_c \) will not be equal, thus affected by the temperature and the specific \( \Delta n \) value.

Understanding this relationship is crucial when dealing with gas phase reactions, as the choice of using \( K_c \) or \( K_p \) depends on the conditions and specifics of the chemical reaction being analyzed.

Delta n calculation

The calculation of \( \Delta n \) is an essential step in determining whether \( K_p \) is equal to \( K_c \). It reveals the stoichiometric change in moles between the reactants and products in a gas-phase reaction. For each reaction, you calculate \( \Delta n \) as follows:

• Count the total moles of gaseous products.
• Count the total moles of gaseous reactants.
• Subtract the moles of reactants from the moles of products to find \( \Delta n \).

For example, consider the reaction: \( \mathrm{H}_2(g) + \mathrm{Cl}_2(g) \rightleftharpoons 2 \mathrm{HCl}(g) \)In this case, the product side has 2 moles of HCl, whereas the reactant side has 1 mole each of H2 and Cl2, summing up to 2 moles. Calculating \( \Delta n \) gives:\[ \Delta n = 2 - (1 + 1) = 0 \]Thus, \( K_p = K_c \) as \( \Delta n \) equals zero. This illustrates how to determine if the equilibrium constants for pressure and concentration are equivalent based on the stoichiometry.

Chemical equilibrium reactions

Chemical equilibrium reactions involve a delicate balance between the forward and reverse processes. When a reaction reaches equilibrium, the rate at which reactants are converted to products equals the rate at which products revert to reactants.

• At equilibrium, the composition of the reaction mixture remains constant over time.
• The equilibrium constant \( K_c \) is derived from concentrations, while \( K_p \) is from partial pressures of gases involved.

An example is the reaction: \( \mathrm{H}_2(g) + \mathrm{Cl}_2(g) \rightleftharpoons 2 \mathrm{HCl}(g) \)In chemical equilibrium reactions, it's important to note that the equilibrium position depends on various factors such as pressure, concentration, and temperature. Le Chatelier's Principle provides insight into how the system responds to these changes to maintain balance.By understanding these dynamics, one can predict changes in a system at equilibrium or how modifications to the system will affect the equilibrium position.