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

Question

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

Step-by-Step Solution

Answer

(a) \(K_{p} = K_{c}(RT)^{-2}\)

Step 1: Identify Reaction Coefficients

For the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}+\) heat, identify the stoichiometric coefficients: 1 for \(\mathrm{N}_2\), 3 for \(\mathrm{H}_2\), and 2 for \(\mathrm{NH}_3\).

Step 2: Determine ∆n (Change in Moles)

Calculate the change in moles between products and reactants: \(\Delta n = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants}) = 2 - (1+3) = 2 - 4 = -2\).

Step 3: Apply Formula to Relate Kp and Kc

Use the formula \(K_{p} = K_{c}(RT)^{\Delta n}\) where \(RT\) is the product of the gas constant and temperature in Kelvin, and \(\Delta n = -2\) for this reaction.

Step 4: Substitute Values into the Formula

Substitute \(\Delta n = -2\) into the formula, yielding \(K_{p} = K_{c}(RT)^{-2}\). This matches option (a).

Key Concepts

Kp and Kc relationship

In chemical reactions involving gases, the equilibrium constant can be expressed in two forms: \( K_c \) and \( K_p \). These constants are essential for understanding how the concentrations (K_c") and pressures (K_p") of reactants and products relate at equilibrium.

The relationship between these two constants is established through the equation:\[ \[\begin{equation} K_p = K_c(RT)^{\Delta n} \end{equation}\] \]where:

• \( R \) is the universal gas constant
• \( T \) is the absolute temperature (in Kelvin)
• \( \Delta n \) represents the change in moles of gas between products and reactants

This equation helps convert concentration-based expressions to pressure-based ones, or vice versa, depending on what is needed for calculations. Understanding how to use this equation is key for solving equilibrium problems in gaseous systems.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers in front of molecules in a balanced chemical equation. They indicate the ratio in which substances react or are produced. For instance, in the reaction \( \mathrm{N}_2 + 3 \mathrm{H}_2 \rightleftharpoons 2 \mathrm{NH}_3 \), the coefficients are \( 1, 3, \) and \( 2 \).

These coefficients not only help balance the equation, ensuring mass conservation, but also play a critical role in determining the change in moles of gas, known as \( \Delta n \). The value of \( \Delta n \) is derived by subtracting the sum of the coefficients of the products from the sum of the coefficients of the reactants.

Knowing the stoichiometric coefficients is essential when applying the equilibrium constant relationship because \( \Delta n \) directly impacts the behavior of \( K_p \) and \( K_c \) in the equation.

Change in Moles

The term \( \Delta n \) refers to the difference in the number of moles of gaseous products and reactants in a balanced equation. To compute \( \Delta n \): calculate the total moles in products and subtract those in reactants.

In the reaction \( \mathrm{N}_2 + 3 \mathrm{H}_2 \rightleftharpoons 2 \mathrm{NH}_3 \), this works out to:\[ \Delta n = (2) - (1 + 3) = -2 \]
This negative change indicates fewer gas moles at equilibrium in the products compared to reactants. The sign and magnitude of \( \Delta n \) are crucial for understanding how pressure and concentrations are interrelated. They also determine the exponent on \( RT \) when converting \( K_c \) to \( K_p \).

Gas Constant and Temperature

The gas constant \( R \) and temperature \( T \) are integral parts of the relationship between \( K_p \) and \( K_c \). \( R \) is a universal constant that connects the energy scale with the temperature scale, commonly valued at 0.0821 \( \text{L atm K}^{-1} \text{mol}^{-1} \), though other units can be used depending on the context.

Temperature \( T \), in Kelvin, influences the positions of equilibrium since it impacts reaction rates and equilibrium constants. When converting between \( K_p \) and \( K_c \), \( RT \) raised to the power of \( \Delta n \) allows adjustment for these variables, reflecting changes in pressure and volume associated with temperature shifts.

Understanding how \( R \) and \( T \) interact gives insights into thermodynamic principles that govern chemical reactions, providing a comprehensive picture of how equilibrium states alter under varying conditions.