Problem 3 - A Complete Resource Book in Chemistry for JEE Main

Question

The value of \(K_{p}\) in the reaction \(\mathrm{MgCO}_{3}(\mathrm{~s}) \rightleftharpoons \mathrm{MgO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{~g})\) is (a) \(\mathrm{K}_{\mathrm{p}}=\mathrm{P}\left(\mathrm{CO}_{2}\right)\) (b) \(\mathrm{K}_{\mathrm{p}}=\frac{\mathrm{P}\left(\mathrm{MgCO}_{3}\right)}{\mathrm{P}\left(\mathrm{CO}_{2}\right) \times \mathrm{P}(\mathrm{Mg} \mathrm{O})}\) (c) \(K_{p}=\frac{P\left(C O_{2}\right) \times P\left(C O_{2}\right) \times P(M g O)}{P(M g C O)_{3}}\) (d) \(\mathrm{K}_{\mathrm{p}}=\frac{\mathrm{P}\left(\mathrm{CO}_{2}\right) \times \mathrm{P}(\mathrm{Mg} \mathrm{O})}{\mathrm{P}(\mathrm{MgCO})_{3}}\)

Step-by-Step Solution

Answer

The correct answer is (a): \( \mathrm{K}_{p} = \mathrm{P}(\mathrm{CO}_{2}) \).

Step 1: Identify Reaction Components

The given reaction is \( \mathrm{MgCO}_{3} \rightleftharpoons \mathrm{MgO} + \mathrm{CO}_{2} \). It involves the decomposition of magnesium carbonate into magnesium oxide and carbon dioxide gas. Only \( \mathrm{CO}_{2} \) is a gas.

Step 2: Recall Equation for \( K_p \)

The equilibrium constant \( K_p \) for a reaction that involves gases is expressed in terms of partial pressures of the gaseous products and reactants. Since only \( \mathrm{CO}_{2} \) is in gaseous state, \( K_p = P(\mathrm{CO}_2) \).

Step 3: Match with Given Options

Compare the expression \( K_p = P(\mathrm{CO}_2) \) with the given options. The correct match is option (a): \( \mathrm{K}_{p} = \mathrm{P}(\mathrm{CO}_{2}) \).

Key Concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K_p \) when dealing with gases, is a crucial concept in chemical equilibrium. It helps us understand and quantify the balance between reactants and products in a chemical reaction at equilibrium. For reactions involving gases, \( K_p \) is expressed in terms of the partial pressures of the gases involved. It provides a way to predict the direction in which a reaction will proceed under certain conditions.In our given decomposition reaction, \( \mathrm{MgCO}_3 \left( \mathrm{s} \right) \rightleftharpoons \mathrm{MgO} \left( \mathrm{s} \right) + \mathrm{CO}_2 \left( \mathrm{g} \right) \), the equilibrium constant \( K_p \) is particularly simple because it involves only one gas: carbon dioxide. The equation for \( K_p \) simplifies to \( K_p = P(\mathrm{CO}_2) \). Here, \( P(\mathrm{CO}_2) \) represents the partial pressure of carbon dioxide. Thus, \( K_p \) is directly equal to the partial pressure of \( \mathrm{CO}_2 \), as there are no other gaseous species in the reaction.

Partial Pressure

Partial pressure is an important concept in understanding how gases contribute to the total pressure of a system. In any gas mixture, the partial pressure is the pressure that one of the gases in the mixture would exert if it occupied the entire volume on its own. It's a way to measure the contribution of a single gas to the overall pressure.In our specific reaction, \( \mathrm{MgCO}_3 \left( \mathrm{s} \right) \rightleftharpoons \mathrm{MgO} \left( \mathrm{s} \right) + \mathrm{CO}_2 \left( \mathrm{g} \right) \), only carbon dioxide exists as a gas. Therefore, the partial pressure of \( \mathrm{CO}_2 \) becomes vital in determining the equilibrium constant \( K_p \). In this scenario, the total pressure in the system is essentially the partial pressure of \( \mathrm{CO}_2 \) alone, as no other gases are involved.Understanding partial pressures allows us to predict how changes in conditions like volume or temperature might affect the reaction and its equilibrium position. This is particularly important in industrial processes where controlling the composition of gas mixtures is crucial.

Decomposition Reaction

A decomposition reaction is a type of chemical reaction in which a single compound breaks down into two or more simpler substances. The given exercise is a perfect example of such a reaction:\[ \mathrm{MgCO}_3 \left( \mathrm{s} \right) \rightleftharpoons \mathrm{MgO} \left( \mathrm{s} \right) + \mathrm{CO}_2 \left( \mathrm{g} \right) \]Here, magnesium carbonate (\( \mathrm{MgCO}_3 \)) decomposes into magnesium oxide (\( \mathrm{MgO} \)) and carbon dioxide gas (\( \mathrm{CO}_2 \)). Decomposition reactions are common in many chemical processes, both in the laboratory and in nature.For example, heating magnesium carbonate causes it to decompose, which is exactly what's represented in our exercise. Key characteristics of decomposition reactions include:

• They often require energy input, such as heat.
• They result in the release of gas, which in this case is \( \mathrm{CO}_2 \).
• They transform a single reactant into multiple products, simplifying complex compounds into simpler substances.

Understanding decomposition reactions is essential for fields like chemistry and materials science, where manipulating reactions and understanding how substances decompose is crucial.