Problem 10 - A Complete Resource Book in Chemistry for JEE Main

Question

In the reaction, \(\mathrm{N}_{2}+3 \mathrm{H}_{2}=2 \mathrm{NH}_{3}+\) heat, relationship between \(K_{p}\) and \(K\) is (a) \(\mathrm{K}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-2}\) (b) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{2}\) (c) \(K_{p}=K_{c}(R T)^{-3}\) (d) \(\mathrm{K}_{\mathrm{c}}=\mathrm{K}_{\mathrm{p}}(\mathrm{RT})^{3}\)

Step-by-Step Solution

Answer

The correct answer is (c) \(K_p = K_c (R T)^{-3}\).

Step 1: Understand the Reaction Equation

The given reaction is \( \mathrm{N}_{2}(g) + 3 \mathrm{H}_{2}(g) \rightarrow 2 \mathrm{NH}_{3}(g) \). It's crucial to note that this is a gaseous reaction, which allows us to use the concepts of \( K_c \) and \( K_p \).

Step 2: Calculate Change in Moles (\(\Delta n\))

To find the relationship between \( K_c \) and \( K_p \), use the change in moles. In the reaction, reactants have a total of 4 moles \((1 + 3)\) and products have 2 moles. Thus, \( \Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} = 2 - 4 = -2 \).

Step 3: Use the Relation Between \(K_p\) and \(K_c\)

The relationship between \( K_p \) and \( K_c \) is given by the formula \( K_p = K_c (RT)^{\Delta n} \). Substitute \( \Delta n = -2 \) into this equation to get \[ K_p = K_c (RT)^{-2} \].

Step 4: Match the Equation with Options

Compare the derived equation \( K_p = K_c (RT)^{-2} \) with the given multiple-choice options. The correct match for \( K_p = K_c (RT)^{-2} \) is option (c).

Key Concepts

Equilibrium Constant

The Equilibrium Constant, often represented as either \( K_c \) or \( K_p \), is a fundamental concept in chemistry that describes the state of balance achieved in a reversible chemical reaction. It is important to understand that this equilibrium is dynamic, meaning that the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time.

In any given chemical reaction, the equilibrium constant is calculated based on the concentrations or partial pressures of the reactants and products at equilibrium. For reactions involving gases, the equilibrium constant can be expressed in terms of concentrations (\( K_c \)) or partial pressures (\( K_p \)). Each of these expressions plays a critical role in predicting how a reaction behaves under different conditions.

For example, in a gaseous reaction like \( N_{2}(g) + 3 H_{2}(g) \rightarrow 2 NH_{3}(g) \), the equilibrium constant \( K_c \) would be calculated using the formula: \[ K_c = \frac{[NH_{3}]^2}{[N_2][H_2]^3} \] and \( K_p \) would consider the partial pressures. This allows chemists to determine how far the reaction has proceeded and to predict the concentrations of species at equilibrium.

Gas Reactions

Gas reactions involve reactants and products in the gaseous state. They are significant in studying chemical equilibrium because pressure and temperature can greatly influence the reaction. For a balanced gaseous reaction, like the synthesis of ammonia: \( N_{2}(g) + 3 H_{2}(g) \rightarrow 2 NH_{3}(g) \), understanding the contribution of each component to the overall reaction is crucial.

When dealing with gas reactions,:

• The behavior and movement of the molecules are dictated by the pressure and temperature conditions within the system.
• These reactions are usually expressed in terms of partial pressures (\( K_p \)) instead of concentrations (\( K_c \)), since gases expand to fill their containers, unlike solids and liquids.

Gas reactions often require careful monitoring and control of these variables to achieve and maintain equilibrium. Recognizing how changes in these conditions might shift equilibrium is essential in processes like industrial chemical manufacturing.

Kp and Kc Relationship

The relationship between \( K_p \) and \( K_c \) is a crucial part of understanding chemical equilibria in systems involving gases. This relationship is defined mathematically to account for the difference in how equilibrium is expressed.

• \( K_c \) is used for concentrations in molarity.
• \( K_p \) refers to partial pressures.

For reactions involving gases, the two constants are related by the equation:\[ K_p = K_c (RT)^{\Delta n} \]where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in moles of gas from reactants to products. This formula allows conversion between \( K_p \) and \( K_c \) by incorporating the effect of pressure due to a change in mole numbers.

For the ammonia synthesis reaction: \(N_{2}(g) + 3 H_{2}(g) \rightarrow 2 NH_{3}(g) \), \( \Delta n \) is calculated as \(2 - 4 = -2\). Inserting this \( \Delta n \) value into the equation gives:\[ K_p = K_c (RT)^{-2} \] This shows how the two constants are interconnected and helps predict how a system behaves under different pressures and temperatures.