Problem 44 - A Complete Resource Book in Chemistry for JEE Main

Question

Consider the following reactions: 1\. \(\mathrm{AB}_{2}(\mathrm{~g})+1 / 2 \mathrm{~B}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{AB}_{3}(\mathrm{~g})\) 2\. \(2 \mathrm{AB}_{3}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{AB}_{2}+\mathrm{B}_{2}(\mathrm{~g})\) If and \(K_{2}\) are the equilibrium constants at \(27^{\circ} \mathrm{C}\) of reactions 1 and 2 respectively, then \(\mathrm{K}\), and \(\mathrm{K}_{2}\) are related as (a) \(\mathrm{K}_{1}^{x}=\mathrm{K}_{2}\) (b) \(\mathrm{K}_{2} \mathrm{~K}_{1}^{x}=1\) (c) \(\mathrm{K}_{1}=2 \mathrm{~K}_{2}\) (d) \(\mathrm{K}, \mathrm{K}_{2}^{2}=1\)

Step-by-Step Solution

Answer

Option (b): \(K_2 K_1 = 1\) (\(x = 1\)).

Step 1: Understand Relation of Reactions

Both reactions represent the same equilibrium but are written differently. Reaction 2 is simply the reverse of Reaction 1. This means their equilibrium constants will be related inversely.

Step 2: Write Expression for K values

Let the equilibrium constant of Reaction 1 be \(K_1\) and Reaction 2 be \(K_2\). For a reverse reaction, \(K_2 = \frac{1}{K_1}\).

Step 3: Identify Correct Option

From the relationship \(K_2 = \frac{1}{K_1}\), we know \(K_1 \cdot K_2 = 1\). This matches the option (b): \(K_2 K_1^x = 1\), with \(x = 1\). Thus, the correct answer is option (b).

Key Concepts

Equilibrium Constants

In chemistry, equilibrium constants are a fundamental aspect of understanding chemical equilibria. These constants, denoted by the symbol \( K \), serve as indicators of the position of equilibrium in reversible reactions. They are derived from the law of mass action and are calculated using concentrations of the reactants and products. For a given reversible reaction at a specific temperature, the equilibrium constant remains constant. This implies that no matter the initial concentrations of the reactants or products, the ratio of product concentrations to reactant concentrations raised to their stoichiometric coefficients remains equal to \( K \).

It's important to understand that equilibrium constants are dimensionless numbers in simple systems. This dimensionlessness arises from the use of activities (or effective concentrations), which are unitless entities in equilibrium expressions. The value of \( K \) can help predict the direction of a reaction:

• If \( K > 1 \), products are favored at equilibrium.
• If \( K < 1 \), reactants are favored.
• If \( K = 1 \), neither reactants nor products are favored, indicating a dynamic balance.

Reversible Reactions

Reversible reactions are reactions where the reactants form products, which can then react to form the reactants again. This feature contrasts with irreversible reactions, which complete to a single direction. In reversible reactions, there is a point at which both the forward and reverse reactions occur at the same rate. This establishment of dynamic equilibrium means that the concentrations of reactants and products remain constant over time.

For a simple chemical system represented by the reaction \( A + B \rightleftharpoons C + D \), the forward reaction \( A + B \rightarrow C + D \) and reverse reaction \( C + D \rightarrow A + B \) perpetually compete. This competition leads to the premise of chemical equilibrium.

Understanding reversibility is crucial in chemical processes and industries where yields of products need optimization. By manipulating conditions such as concentration, pressure, and temperature, chemists can shift the equilibrium position to favor either products or reactants, a principle guided by Le Chatelier's Principle.

Thermodynamics

Thermodynamics provides the underlying principles that explain why reactions occur and how conditions affect their directions. It's a branch of physics that studies heat, temperature, and their relation to energy and work. Two key concepts in thermodynamics relateto chemical reactions: enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)).

Enthalpy refers to the heat absorbed or released during a reaction. If a reaction releases heat (exothermic), it's more likely to occur spontaneously; if it absorbs heat (endothermic), it might require external energy to proceed. Entropy, on the other hand, is a measure of disorder or randomness.

The Second Law of Thermodynamics states that systems tend to move towards maximum entropy, or disorder. In a chemical equilibrium context, the Gibbs free energy change (\( \Delta G \)) combines both enthalpy and entropy to predict spontaneity. The equation \( \Delta G = \Delta H - T\Delta S \) (where \( T \) is temperature in Kelvin) reveals that a negative \( \Delta G \) indicates a spontaneous process, while a positive \( \Delta G \) suggests non-spontaneity.

• When \( \Delta G = 0 \), the system is at equilibrium;
• disorder is maximized, and the process no longer proceeds in any net direction.

Understanding these factors allows chemists to predict how changes in pressure, temperature, and concentration will affect the position and behavior of equilibrium in reversible reactions.