Problem 194 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction, \(2 \mathrm{NO}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g})\) \(\left(\mathrm{K}_{\mathrm{c}}=1.8 \times 10^{-6}\right.\) at \(\left.184^{\circ} \mathrm{C}\right)\) \((\mathrm{R}=0.0831 \mathrm{~kJ} /(\operatorname{mol} \mathrm{K}))\) when \(K_{\text {and }} K_{c}\) are compared at \(184^{\circ} \mathrm{C}\) it is found that (a) \(\mathrm{K}_{\mathrm{p}}\) is greater than \(\mathrm{K}_{c}\) (b) \(\mathrm{K}_{\mathrm{p}}\) is less than \(\mathrm{K}_{\mathrm{c}}\) (c) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{c}\) (d) whether \(\mathrm{K}_{p}\) is greater than, less than or equal to \(\mathrm{K}_{\mathrm{c}}\) depends upon the total gas pressure

Step-by-Step Solution

Answer

(a) \( K_p \) is greater than \( K_c \).

Step 1: Identify the conversion formula

To determine the relationship between \(K_p\) and \(K_c\), we use the formula: \( K_p = K_c (RT)^{\Delta n} \), where \( \Delta n \) is the change in moles of gas between the products and reactants.

Step 2: Calculate the change in moles (Δn)

For the given reaction, \(2 \mathrm{NO}_{2} \rightleftharpoons 2 \mathrm{NO} + \mathrm{O}_{2}\), \(\Delta n\) is calculated as \((\text{moles of products}) - (\text{moles of reactants}) = (2 + 1) - 2 = 1.\)

Step 3: Calculate (RT)^Δn

Using the given values: \( R = 0.0831 \) kJ/(mol K) and \( T = 184 + 273 = 457 \) K, calculate \((RT)^{\Delta n}\). Thus, \((0.0831 \times 457)^1 = 37.9657.\)

Step 4: Determine Kö value and compare with Kc

Calculate \( K_p = K_c \times (RT)^{\Delta n} = 1.8 \times 10^{-6} \times 37.9657 = 6.834 \times 10^{-5}.\) Since \( K_p \) is larger than \( K_c, \) option (a) is correct.

Key Concepts

Reaction Quotient

In chemical equilibrium, the reaction quotient, denoted as \( Q \), serves as an important tool to predict the direction of a reaction. It is used to determine whether a reaction is at equilibrium or, if not, which way it needs to shift to reach equilibrium. The formula for \( Q \) is similar to that of the equilibrium constant \( K \), but \( Q \) is calculated using the current concentrations of the reactants and products rather than concentrations at equilibrium.

• If \( Q = K \), the system is at equilibrium.
• If \( Q < K \), the forward reaction is favored, and the system will shift to the right to form more products.
• If \( Q > K \), the reverse reaction is favored, and the system will shift to the left to form more reactants.

This concept is crucial in understanding and predicting the behavior of reactions in dynamic environments. It empowers chemists to manipulate reaction conditions to achieve desired outcomes.

Equilibrium Constant

The equilibrium constant, \( K \), is a fundamental concept in understanding chemical equilibrium. It represents the ratio of the concentration of products to reactants, each raised to the power of its stoichiometric coefficient at equilibrium.

There are two main types of equilibrium constants in gaseous reactions:

• \( K_c \) refers to the equilibrium constant calculated using molar concentrations.
• \( K_p \) is used when dealing with pressures of gases.

The relationship between \( K_p \) and \( K_c \) is expressed with the equation: \[ K_p = K_c (RT)^{\Delta n} \] Here, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in the number of moles of gas. This allows you to convert between different equilibrium constants depending on the provided conditions, ensuring that you have the correct value to use in your calculations.

Le Chatelier's Principle

Le Chatelier's Principle is a key concept in understanding how a system at equilibrium responds to changes in conditions. According to this principle, if a system at equilibrium is disturbed, it will adjust itself to counteract the disturbance and restore a new equilibrium.

Common external changes include:

• Changes in concentration of reactants or products: Adding more reactants typically shifts the equilibrium to the right, favoring product formation, while adding more products shifts it to the left.
• Changes in temperature: For endothermic reactions, increasing the temperature shifts the equilibrium to the right, while for exothermic reactions, it shifts to the left.
• Changes in pressure: For gaseous reactions, increasing the pressure shifts the equilibrium toward the side with fewer moles of gas.

Le Chatelier's Principle is immensely valuable for predicting and controlling the outcomes in industrial chemical processes, ensuring optimization of product yields under varying conditions.