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

Question

For which of the following reaction, \(K_{p}=K_{c} ?\) (a) \(2 \mathrm{NOCl}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g})\) (b) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})\) (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) has \( K_p = K_c \) because \( \Delta n = 0 \).

Step 1: Identify the Relationship Between Kp and Kc

The relationship between the equilibrium constants expressed in terms of partial pressures (\( K_p \)) and concentrations (\( K_c \)) is given by the equation:\[K_p = K_c(RT)^{\Delta n}\] where \( \Delta n \) is the change in moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Determine Δn for Each Reaction

For each reaction, calculate \( \Delta n \), which is the difference between the moles of gaseous products and the moles of gaseous 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 Kp Equals Kc

\( K_p = K_c \) when \( \Delta n = 0 \), because the expression\[(RT)^{\Delta n} = (RT)^{0} = 1\]makes the product of \( K_c \) and this term equal to \( K_c \).

Step 4: Compare Δn Values and Select the Reaction

From Step 2, the only reaction with \( \Delta n = 0 \) is reaction (c):\( \mathrm{H}_{2}( ext{g}) + \mathrm{Cl}_{2}( ext{g}) \rightleftharpoons 2 \mathrm{HCl}( ext{g}) \).

Step 5: Verify with the Formula Kp = Kc × (RT)^Δn

Since for reaction (c) \( \Delta n = 0 \),\[ K_p = K_c \times (RT)^0 = K_c \times 1 = K_c \]This confirms that for reaction (c), \( K_p = K_c \).

Key Concepts

Equilibrium Constant

The equilibrium constant is a crucial concept in chemistry. It helps us understand and predict the direction of a chemical reaction in a closed system. For a reaction at equilibrium, the equilibrium constant, whether it is expressed in terms of concentration (\(K_c\)) or in terms of partial pressures (\(K_p\)), remains constant at a constant temperature.

- \(K_c\) is used for reactions where the substances are measured in molar concentrations (mol/L).- \(K_p\) is applied to reactions involving gases, and it is expressed as the ratio of the partial pressures.- The relationship between \(K_p\) and \(K_c\) is influenced by changes in moles of gas, or \(\Delta n\).

When \(\Delta n\) is zero, the equilibrium constants \(K_p\) and \(K_c\) become equal because the effect of temperature and gas laws drops out of the equation. This understanding is fundamental when analyzing or designing reactions, particularly those involving gases.

Partial Pressure

Partial pressure is an essential concept when dealing with gaseous reactions. It refers to the pressure that each gas in a mixture would exert if it occupied the entire volume alone. This measurement is crucial when calculating the \(K_p\) for a reaction.

- The total pressure in a system is the sum of the partial pressures of all gases present.- Using partial pressures, chemists can predict how changes in pressure affect a reaction's equilibrium position.

In chemical reactions involving gases, it's often extensive to reach equilibrium in a closed system at a given temperature. The partial pressure of each gas can be thought of independently working within the context of a reaction balanced by its interaction with other gases.

Reaction Stoichiometry

Reaction stoichiometry is the study of the quantitative relationships between reactants and products in a chemical reaction. This includes the ratios of moles in which substances react.

- It provides the coefficients which indicate the proportions of reactants and products involved.- Stoichiometry is crucial for calculating the change in moles of gas (\(\Delta n\)) which is imperative to decipher \(K_p\) and \(K_c\) relationships.

Understanding stoichiometry allows chemists to predict how much of a reactant is necessary to react with a given amount of another reactant, and how much product will be formed under given conditions. In this exercise, calculating \(\Delta n\) is a demonstration of applying stoichiometry within gaseous chemical reactions to identify when \(K_p\) equals \(K_c\).

Gaseous Reactions

Gaseous reactions involve reactants and products that are in the gas phase. These reactions are particularly sensitive to changes in pressure and temperature due to the gas laws.

- The ideal gas law (\(PV=nRT\)) relates pressure, volume, and temperature of a gas to its amount in moles.- Gaseous reactions often have dynamic equilibria and are significantly influenced by partial pressures and changes in volume or pressure.

The chemical equilibrium of gaseous reactions involves not just concentrations but also the effects of pressure changes. Understanding the interplay of these factors is important for predicting the behavior of gases in reactions, optimizing conditions for industrial chemical processes, and designing experiments. Successfully solving problems around gaseous reactions relies on understanding how changing one factor, like pressure, will influence the others due to the interrelated nature of gases.