Problem 200 - A Complete Resource Book in Chemistry for JEE Main

Question

For the following three reactions \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), equilibrium constants are given: \(\quad\) (a) \(\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) ; \mathrm{K}_{1}\) (b) \(\mathrm{CH}_{4}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{~g}) ; \mathrm{K}_{2}\) (c) \(\mathrm{CH}_{4}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}_{2}(\mathrm{~g})+4 \mathrm{H}_{2}(\mathrm{~g}) ; \mathrm{K}_{3}\) Which of the following relation is correct? (a) \(\mathrm{K}_{1} \sqrt{\mathrm{K}}_{2}=\mathrm{K}_{2}\) (b) \(\mathrm{K}_{2} \mathrm{~K}_{3}=\mathrm{K}_{1}\) (c) \(\mathrm{K}_{3}=\mathrm{K}_{1} \mathrm{~K}_{2}\) (d) \(\mathrm{K}_{3} \cdot \mathrm{K}_{2}^{3}=\mathrm{K}_{1}^{2}\)

Step-by-Step Solution

Answer

The correct relation is not listed; none of the options match \( \mathrm{K}_3 = \frac{\mathrm{K}_2}{\mathrm{K}_1} \).

Step 1: Understand the Reactions

Examine each equilibrium reaction given: 1. Reaction (a): \( \mathrm{CO} + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{CO}_2 + \mathrm{H}_2 \), with equilibrium constant \( \mathrm{K}_1 \).2. Reaction (b): \( \mathrm{CH}_4 + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{CO} + 3\mathrm{H}_2 \), with equilibrium constant \( \mathrm{K}_2 \).3. Reaction (c): \( \mathrm{CH}_4 + 2\mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{CO}_2 + 4\mathrm{H}_2 \), with equilibrium constant \( \mathrm{K}_3 \).

Step 2: Combine Reactions

Combine Reactions (a) and (b) to form Reaction (c):- Reaction (a) reversed gives \( \mathrm{CO}_2 + \mathrm{H}_2 \rightleftharpoons \mathrm{CO} + \mathrm{H}_2\mathrm{O} \), with equilibrium constant \( \frac{1}{\mathrm{K}_1} \).- Adding Reaction (b) and the reversed Reaction (a) yields Reaction (c): \( \mathrm{CH}_4 + 2\mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{CO}_2 + 4\mathrm{H}_2 \).

Step 3: Calculate Resultant Equilibrium Constant

Use the rule for combining equilibrium constants: when reactions are added, multiply their equilibrium constants. For combined Reaction (c), the equilibrium constant \( \mathrm{K}_3 \) is:\[ \mathrm{K}_3 = \mathrm{K}_2 \times \frac{1}{\mathrm{K}_1} = \frac{\mathrm{K}_2}{\mathrm{K}_1} \]

Step 4: Verify Possible Options

Review each option and calculate to see which fits:- (a) \( \mathrm{K}_1 \sqrt{\mathrm{K}_2} = \mathrm{K}_2 \) does not hold.- (b) \( \mathrm{K}_2 \mathrm{~K}_3 = \mathrm{K}_1 \) implies \( \mathrm{K}_3 = \frac{\mathrm{K}_1}{\mathrm{K}_2} \) which is incorrect.- (c) \( \mathrm{K}_3 = \mathrm{K}_1 \mathrm{~K}_2 \) also incorrect since it does not match derived \( \frac{\mathrm{K}_2}{\mathrm{K}_1} \).- (d) \( \mathrm{K}_3 \cdot \mathrm{K}_2^{3} = \mathrm{K}_1^{2} \) doesn't fit derived relation either.

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K \), plays a crucial role in chemical reactions at equilibrium. It is a numerical value that describes the ratio of the concentrations of products to reactants. This ratio is specific to a particular reaction at a given temperature. The equilibrium constant is an indicator of the extent to which a reaction proceeds.
- For a general reaction, \( aA + bB \rightarrow cC + dD \), the equilibrium constant \( K \) is given by: \[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]- A large \( K \) value means the reaction favors the formation of products, while a small \( K \) indicates the reactants are favored.
- It's important to note that \( K \) changes with temperature but stays constant under a given set of conditions for a reaction.Understanding \( K \) helps predict the direction of the reaction and allows chemists to manipulate conditions to achieve desired outcomes.

Reversible Reactions

Reversible reactions are those that can proceed in both forward and backward directions. This implies that the products can convert back into the original reactants, reaching a state of equilibrium over time. Here's what you need to know:
- **Dynamic Equilibrium**: In reversible reactions, equilibrium is dynamic, meaning reactants and products continuously change, but their concentrations remain constant over time because the forward and reverse reaction rates are equal.
- **Example**: For the equation \( A + B \rightleftharpoons C + D \), equilibrium is achieved when the rate of the forward reaction \( A + B \rightarrow C + D \) equals that of the reverse reaction \( C + D \rightarrow A + B \).
Reversible reactions are common in nature and industry, allowing equilibrium manipulation for desired chemical processes.

Chemical Reaction Combination

Chemical reaction combination involves adding or modifying reactions to form new equations. When combining reactions, we look at both the reactions' stoichiometry and their equilibrium constants:
- **Combining Reactions**: Reactions can be added together to form a composite reaction. The equilibrium constant for the resultant reaction is the product of the equilibrium constants of the individual reactions, taking into account any reversals or modifications. This principle helps in predicting and calculating the equilibrium constants for complex reactions.
- **Example**: If Reaction 1 is combined with the reverse of Reaction 2, we multiply the equilibrium constant of Reaction 1 by the reciprocal of Reaction 2's constant. - **Practical Use**: This is useful in constructing balanced reaction pathways in industrial processes and research settings to control product formations and yields.

Reaction Stoichiometry

Stoichiometry in chemistry involves the quantitative relationships between reactants and products in a chemical reaction. Understanding stoichiometry is essential for calculating the amounts of substances consumed and produced:
- **Balances and Ratios**: Each chemical equation is a balanced representation of a reaction. Stoichiometric coefficients indicate the proportional amounts of each substance.- **Examples**: In the reaction \( aA + bB \rightarrow cC + dD \), the numbers \( a, b, c \), and \( d \) are stoichiometric coefficients that determine the ratios of reactants and products.- **Applications**: Stoichiometry is used to calculate the masses, volumes, and moles in reactions. It's fundamental for laboratory work, engineering problems, and interpreting the quantitative results of chemical processes.