Problem 9 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In which one of the following equilibria, \(\mathrm{K}_{\mathrm{p}} \neq \mathrm{K}_{\mathrm{c}}\) ? (a) \(2 \mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{CO}(\mathrm{g})\) (b) \(2 \mathrm{HI}(\mathrm{g}) \rightleftharpoons \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\) (c) \(\mathrm{NO}_{2}(\mathrm{~g})+\mathrm{SO}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{NO}(\mathrm{g})+\mathrm{SO}_{3}(\mathrm{~g})\) (d) \(2 \mathrm{NO}(\mathrm{g}) \rightleftharpoons \mathrm{N}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

In reaction (a), \( K_p \neq K_c \).

Step 1: Understanding the Relationship Between Kp and Kc

The relationship between the equilibrium constant in terms of pressure \( K_p \) and concentration \( K_c \) 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 (\( \Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} \)). \( R \) is the universal gas constant and \( T \) is temperature in Kelvin.

Step 2: Calculate \( \Delta n \) for Each Equation

Calculate \( \Delta n \) for each given reaction:(a) \( 2 \mathrm{C}(s) + \mathrm{O}_2(g) \rightleftharpoons 2\mathrm{CO}(g) \)- \( \Delta n = 2 - 1 = 1 \)(b) \( 2\mathrm{HI}(g) \rightleftharpoons \mathrm{H}_2(g) + \mathrm{I}_2(g) \)- \( \Delta n = 1 + 1 - 2 = 0 \)(c) \( \mathrm{NO}_2(g) + \mathrm{SO}_2(g) \rightleftharpoons \mathrm{NO}(g) + \mathrm{SO}_3(g) \)- \( \Delta n = 1 + 1 - (1 + 1) = 0 \)(d) \( 2\mathrm{NO}(g) \rightleftharpoons \mathrm{N}_2(g) + \mathrm{O}_2(g) \)- \( \Delta n = 1 + 1 - 2 = 0 \)

Step 3: Determine if \( K_p \neq K_c \)

For any reaction where \( \Delta n = 0 \), \( (RT)^{\Delta n} = 1\) so the equation simplifies to \( K_p = K_c \).- For reactions b, c, and d, \( \Delta n = 0 \) hence \( K_p = K_c \).- For reaction a, \( \Delta n = 1 \) indicating that \( K_p eq K_c \).

Key Concepts

Equilibrium Constants

Equilibrium constants are essential in describing the balance of chemical reactions at equilibrium. There are two primary forms of equilibrium constants, denoted as \( K_c \) and \( K_p \).

• \( K_c \) is the equilibrium constant regarding concentration. It's used mainly for reactions occurring in solutions where concentrations are measured in molarity (moles per liter).
• \( K_p \) describes the equilibrium constant in terms of partial pressures, usually applicable to gaseous reactions where pressures are easier to measure.

The relationship between \( K_c \) and \( K_p \) for gaseous reactions is influenced by the change in moles of gas during the reaction, represented by \( \Delta n \). The relation is given by the equation:\[K_p = K_c (RT)^{\Delta n}\]where \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. If \( \Delta n \) equals zero, then \( K_p \) equals \( K_c \) since \((RT)^{0} = 1\). Understanding these constants helps in predicting the direction of the reaction and assessing how shifts in conditions can affect equilibrium.

Gaseous Reactions

Gaseous reactions involve reactants and products in the gas phase. The behavior of gases and how they participate in reactions critically influence the equilibrium state.

• Gases are characterized by their state variables: pressure, volume, temperature, and number of moles.
• Because gases spread to fill their containers entirely, their concentrations can be effectively measured as pressures.

In a chemical equilibrium involving gaseous substances, not only do concentrations change as products and reactants convert into one another, but so does the pressure.
This makes \( K_p \), the equilibrium constant in terms of pressure, especially useful. Changes in pressure, volume, or temperature can cause shifts in equilibrium states, which can be determined through Le Chatelier's principle. Understanding these relationships allows predicting how changes to reaction conditions might shift equilibrium towards reactants or products.

Reaction Stoichiometry

Stoichiometry in reactions is the quantification of reactants and products in chemical equations. It allows us to maintain the balance required by the law of conservation of mass, where matter is neither created nor destroyed.

• A balanced chemical equation has the same number of atoms for each element on both sides of the equation.
• Stoichiometry enables us to predict yields and determine amounts necessary to react completely.

In gaseous reactions, stoichiometry also relates directly to \( \Delta n \), the change in the number of moles of gas during the reaction. This change impacts the relationship between \( K_c \) and \( K_p \). For example, in the equation \[ 2 \text{C}(\text{s}) + \text{O}_2(\text{g}) \rightleftharpoons 2 \text{CO}(\text{g}) \], the stoichiometric coefficients help determine \( \Delta n = 2 - 1 = 1 \), indicating that \( K_p \) and \( K_c \) would differ due to a change in the moles of gas throughout the reaction. Understanding reaction stoichiometry is vital for calculating equilibrium conditions accurately.