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

Question

The relation between \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{c}\) 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) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}\) (c) \(K_{P}-K_{e} /(R T)^{2}\) (d) \(K_{p}=K_{c} / R T\)

Step-by-Step Solution

Answer

The correct answer is (a): \(\mathrm{K}_{\mathrm{p}} = \mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-1}\).

Step 1: Recall the formula for Kp and Kc

The relationship between the equilibrium constants for gases, \(K_{p}\) and \(K_{c}\), is given by the formula: \[ K_{p} = K_{c} (RT)^{\Delta n} \] where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) represents the change in the number of moles of gas between the products and the reactants.

Step 2: Calculate Δn

For the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g})\), the change in the number of moles of gas \(\Delta n\) can be calculated. - Moles of gaseous products: 2 (from 2 NOCl)- Moles of gaseous reactants: 2 (from 2 NO) + 1 (from Cl3) - Thus, \(\Delta n = 2 - (2 + 1) = 2 - 3 = -1\).

Step 3: Substitute Δn into the relation

Now, substitute \(\Delta n = -1\) into the relation between \(K_{p}\) and \(K_{c}\): \[ K_{p} = K_{c} (RT)^{-1} \] This shows the direct relationship between \(K_{p}\) and \(K_{c}\) for the given reaction under the assumption of a constant temperature \(T\).

Step 4: Identify the correct option

From the options given, the formula that matches the derived relationship is option (a): \[ \mathrm{K}_{\mathrm{p}} = \mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-1} \].

Key Concepts

Equilibrium constant

In the study of chemical reactions, the equilibrium constant is a vital concept. It provides a numerical value that represents the ratio of the concentrations of products to reactants at equilibrium. For reactions involving gases, the equilibrium constant can be expressed in two main forms: \(K_c\) and \(K_p\).
\(K_c\) is utilized when dealing with concentrations in molarity, while \(K_p\) is used for partial pressures of gases. These constants are central to understanding the position of equilibrium and predict whether a reaction will favor the production of reactants or products.
When using these constants, it's essential to note that their values depend on the temperature of the reaction. Therefore, any change in temperature will alter the equilibrium constant and, consequently, the reaction's equilibrium position.

Ideal gas law

The ideal gas law is a fundamental equation in chemistry linking pressure, volume, temperature, and moles of a gas. It is often represented by the formula \(PV = nRT\).
Here, \(P\) is the pressure of the gas, \(V\) is its volume, \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.
This equation helps in understanding how gaseous substances behave under various conditions of temperature and pressure. In the context of reactions, the ideal gas law plays a crucial role in relating \(K_p\) and \(K_c\) through the gas constant \(R\), as seen in the equation \(K_p = K_c (RT)^{\Delta n}\). It provides the mathematical link needed to interconvert these equilibrium constants when dealing with gases.

Chemical equilibrium

Chemical equilibrium occurs when a reversible reaction reaches a state where the rates of the forward and backward reactions are equal. At equilibrium, the concentrations of reactants and products remain constant over time.
This state does not imply that the concentrations are equal but that their ratios, as given by the equilibrium constant, remain the same. Reaching equilibrium means the reaction has achieved maximum efficiency under the given conditions, making it a state of balance where energy is effectively distributed.
Understanding equilibrium is essential for predicting the yield of chemical products and for designing industrial processes where reaction conditions can be controlled to optimize product formation.

Change in moles (Δn)

The concept of the change in moles, often represented by \(\Delta n\), is integral to relating \(K_p\) and \(K_c\) for reactions involving gases. \(\Delta n\) is calculated by subtracting the total moles of gaseous reactants from the total moles of gaseous products.
This figure helps in expressing how the equilibrium constants are affected by differences in mole quantities on either side of a reaction. For instance, in the reaction \(2 \text{NO}_\text{(g)} + \text{Cl}_2\text{(g)} \rightleftharpoons 2 \text{NOCl}_\text{(g)}\), \(\Delta n = -1\) because there is one less mole of gas in the products compared to the reactants.
The value of \(\Delta n\) directly impacts the exponent on the \((RT)\) term when converting between \(K_p\) and \(K_c\). A negative \(\Delta n\) indicates a reduction in gas volume, which mathematically scales the equilibrium constant downwards with respect to \(R\) and \(T\).