Problem 20 - A Complete Resource Book in Chemistry for JEE Main

Question

The relation between \(K_{0}\) and \(K_{c}\) for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g})\) is (a) \(\mathrm{K}_{\mathrm{P}}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-1}\) (b) \(\mathrm{K}=\mathrm{K}_{\mathrm{p}}\) (c) \(K=K_{c} /(R T)^{2}\) (d) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K} / \mathrm{RT}\)

Step-by-Step Solution

Answer

For the reaction, the relation is \(K_p = K_c (RT)^{-1}\), matching option (a).

Step 1: Understand the Reaction Equation

The given reaction is \(2 \mathrm{NO}(\mathrm{g}) + \mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g})\). This is a gaseous reaction, meaning both \(K_c\) and \(K_p\) can be calculated; \(K_c\) is based on concentrations, and \(K_p\) is based on partial pressures.

Step 2: Relate KP and KC using the Reaction's Stoichiometry

For gaseous reactions, \(K_p\) and \(K_c\) are related by the equation \(K_p = K_c (RT)^{\Delta n}\), where \(\Delta n\) is the change in the number of moles of gas (moles of products minus moles of reactants).

Step 3: Calculate Δn for the Reaction

For our reaction, \(\Delta n = (2) - (2 + 1) = 2 - 3 = -1\). This means that \(\Delta n = -1\).

Step 4: Substitute Δn into the KP and KC relationship

Now, substitute \(\Delta n = -1\) into the relationship: \(K_p = K_c (RT)^{-1}\).

Step 5: Match with the Given Options

The correct relation from the options provided based on our calculation is (a) \( K_p = K_c (RT)^{-1} \).

Key Concepts

Gas Reactions

Gas reactions are chemical processes that involve gaseous reactants and products. They possess unique properties due to the behavior of gases. In chemistry, reactions taking place in the gas phase are important because gas molecules generally mix faster than solids or liquids due to the large spaces between them and their high kinetic energy.
Gas reactions can affect various everyday phenomena, from the air we breathe to industrial processes. Understanding how gases react helps us anticipate and control these reactions better.
In the context of equilibrium, we often look at how gases come to a state where the rate of the forward reaction equals the rate of the reverse reaction. This is when we say the system is at equilibrium, characterized by constants like the equilibrium constant.

Partial Pressure

Partial pressure is a concept used to describe the pressure exerted by a single gas in a mixture of gases. It is crucial when dealing with gas reactions, as it directly influences how calculations of equilibrium involve gaseous substances.
According to Dalton's Law, the total pressure exerted by a gaseous mixture is the sum of the partial pressures of each individual component.

• Partial pressure of a gas is related to its mole fraction and the total pressure of the gas mixture.
• It has the formula: \( P_i = X_i \times P_{\text{total}} \), where \( X_i \) is the mole fraction of the component gas.

Partial pressure is fundamental when calculating the equilibrium constant \( K_p \), which involves pressures in its expression instead of concentrations used in \( K_c \).

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It is based on the premise that matter is conserved, and it provides the quantitative relationships between substances as they participate in chemical reactions.
For gas reactions, stoichiometry helps us understand how individual molecules interact, ensuring that the law of conservation of mass holds true.

• It involves using the coefficients from the balanced chemical equation to determine molar ratios.
• In the given reaction \( 2 \text{NO}(g) + \text{Cl}_2(g) \rightleftharpoons 2 \text{NOCl}(g) \), stoichiometry tells us that two moles of \( \text{NO} \) react with one mole of \( \text{Cl}_2 \) to form two moles of \( \text{NOCl} \).

Change in Moles (Δn)

The change in moles, denoted as \( \Delta n \), is an important factor in understanding how the partial pressures affect the equilibrium constant \( K_p \) relative to \( K_c \).

• For a reaction \( aA + bB \rightleftharpoons cC + dD \), the change in moles of gas is calculated as \( \Delta n = (c + d) - (a + b) \).
• This value tells us how the number of moles of gas changes as the reaction proceeds from reactants to products.

In the example reaction \( 2 \text{NO}(g) + \text{Cl}_2(g) \rightleftharpoons 2 \text{NOCl}(g) \), \( \Delta n = -1 \), meaning that there is a decrease in the total number of gas moles.
This change is used to adjust the equilibrium expression from \( K_c \) to \( K_p \) or vice-versa, using the relationship \( K_p = K_c (RT)^{\Delta n} \).