Problem 80 - A Complete Resource Book in Chemistry for JEE Main

Question

The ratio of \(\mathrm{Kp} / \mathrm{Kc}\) for the reaction \(\mathrm{SO}_{2}(\mathrm{~g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{SO}_{3}(\mathrm{~g})\) is (a) \((\mathrm{RT})^{-1 / 2}\) (b) \((\mathrm{RT})^{1 / 2}\) (c) \(\mathrm{RT}\) (d)

Step-by-Step Solution

Answer

The correct answer is (a) \\( (RT)^{-1/2} \\).

Step 1: Understanding the Relationship

The relationship between the equilibrium constants \(K_p\) and \(K_c\) for a gaseous reaction 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 \(\Delta n\) is the change in moles of gas from reactants to products.

Step 2: Determine \(\Delta n\)

In the given reaction, \(\mathrm{SO}_2(g) + \frac{1}{2} \mathrm{O}_2(g) \rightleftharpoons \mathrm{SO}_3(g),\)we find \(\Delta n = (1) - (1 + \frac{1}{2}) = -\frac{1}{2}.\)

Step 3: Substitute \(\Delta n\)

Substitute the value \(\Delta n = -\frac{1}{2}\) into the relationship \(K_p = K_c(RT)^{\Delta n}\). Therefore, we have \(K_p = K_c(RT)^{-\frac{1}{2}}\).

Step 4: Conclusion

Thus, the ratio \(K_p / K_c\) is equal to \((RT)^{-\frac{1}{2}}\). Therefore, the correct answer is option (a) \((RT)^{-1/2}\).

Key Concepts

Kp and Kc relationship

The relationship between the equilibrium constants \( K_p \) and \( K_c \) is a fundamental concept in chemical equilibrium, specifically for gaseous reactions. When dealing with gases, the equilibrium constant can be expressed in terms of partial pressures (\( K_p \)) or concentrations (\( K_c \)). To understand the connection between these two constants, we use the equation:

• \( K_p = K_c(RT)^{\Delta n} \)

Here, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. The term \( \Delta n \) represents the change in moles of gas as the reaction proceeds from reactants to products.
This relationship allows us to convert between the pressure and concentration expressions of equilibrium constants, taking into account how the reaction shifts based on changes in moles of gas.

Ideal Gas Law

The Ideal Gas Law is an essential tool in understanding the behavior of gases, and it is closely linked to the calculation of equilibrium constants for gaseous reactions. In its simplest form, the Ideal Gas Law is expressed as:

• \( PV = nRT \)

This equation relates four key properties of gases: pressure (\( P \)), volume (\( V \)), amount of substance (\( n \), in moles), and temperature (\( T \)), with \( R \) being the ideal gas constant.
For chemical equilibria, this law is instrumental when determining how changes in these variables affect gaseous reactions. It provides the basis for comprehending how

• temperature
• volume
• pressure

impact the equilibrium state and thus the values of \( K_p \) and \( K_c \).

Equilibrium Constants

Equilibrium constants are critical for predicting the position of equilibrium in a chemical reaction. They help us determine the concentrations or pressures of reactants and products at equilibrium.
For gaseous reactions, the equilibrium constant can be represented as:

• \( K_c \): concentration-based equilibrium constant
• \( K_p \): pressure-based equilibrium constant

The significance of these constants lies in their ability to inform us

• whether the reactants or products are favored
• and how the system will respond to changes in conditions (temperature, pressure, etc.) to re-establish equilibrium.

Knowing the value of \( \Delta n \) helps in converting \( K_c \) to \( K_p \) and vice versa using the relation \( K_p = K_c(RT)^{\Delta n} \), allowing a comprehensive analysis of reaction dynamics.

Gaseous Reactions

Gaseous reactions feature prominently in studies of chemical equilibrium due to their frequent occurrence in both natural and industrial processes. These reactions typically involve substances in the gas phase, where changes in conditions can significantly affect the equilibrium state.
As gases are easily compressed and expand readily, they respond more dramatically to changes in temperature and pressure compared to liquids and solids. Thus, understanding gaseous reactions requires careful consideration of factors like:

• how pressure influences partial pressures of reaction components
• the effect of volume changes on equilibrium
• and how temperature shifts can sway the balance between products and reactants.

This dynamic nature makes them central to discussions on the application of the Ideal Gas Law and the conversion between \( K_p \) and \( K_c \).
Distinguishing between these constants in gaseous reactions is essential for accurate predictions and control of chemical processes.