Problem 185 - A Complete Resource Book in Chemistry for JEE Main

Question

For the following reaction in gaseous phase \(\mathrm{CO}+1 / 2 \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\) \(\mathrm{K}_{c} / \mathrm{K}_{\mathrm{p}}\) is (a) \((\mathrm{RT})^{1 / 2}\) (b) \((\mathrm{RT})^{-1 / 2}\) (c) (RT) (d) \((\mathrm{RT})^{-1}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the relationship between Kc and Kp

For a given equilibrium reaction in gaseous phase, the relationship between equilibrium constants \(K_c\) and \(K_p\) is given by: \[ K_p = K_c(RT)^{\Delta n} \] where \(\Delta n\) is the change in the number of moles of gases during the reaction, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.

Step 2: Determine \(\Delta n\) for the reaction

The given reaction is \( \mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2} \). Calculate \(\Delta n\) by subtracting the sum of moles of gaseous products from the sum of moles of gaseous reactants. \[ \Delta n = 1 - (1 + \frac{1}{2}) = 1 - 1.5 = -0.5 \]

Step 3: Apply the value of \(\Delta n\)

Substitute \(\Delta n = -0.5\) into the equation \( K_p = K_c(RT)^{\Delta n} \): \[ K_p = K_c(RT)^{-0.5} \] Rearranging, we find: \[ \frac{K_c}{K_p} = (RT)^{-1/2} \]

Step 4: Choose the correct option

The expression \(\frac{K_c}{K_p} = (RT)^{-1/2}\) corresponds to option (b).

Key Concepts

Understanding Gaseous Reactions

When we discuss gaseous reactions, we are talking about chemical processes where all reactants and products are in the gas phase. These reactions can easily change volume and pressure due to the nature of gases.
For example, in the reaction \(\mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\), all components are gases. This makes calculations such as predicting how the reaction will behave under different conditions possible by using equilibrium constants like \(K_c\) (concentration) and \(K_p\) (pressure).
Gaseous reactions are significant in various fields, like synthesizing industrial chemicals or analyzing atmospheric changes. Understanding these can help predict how changes in conditions might shift the balance of the reaction.

Kc and Kp Relationship

The relationship between the equilibrium constants \(K_c\) and \(K_p\) is crucial for gaseous reactions. It allows for conversions between concentration-based equilibrium expressions and pressure-based ones.
The formula linking them is \(K_p = K_c(RT)^{\Delta n}\). Here, \(K_p\) represents equilibrium in terms of partial pressures, while \(K_c\) uses molar concentrations.
The symbols in the equation are:

• \(R\): Ideal gas constant.
• \(T\): Temperature in Kelvin.
• \(\Delta n\): Change in moles of gas between reactants and products.

This equation is invaluable in determining how changes in pressure and temperature influence the reaction and helps chemists translate data between different setups effectively.

Change in Moles

To utilize the formula \(K_p = K_c(RT)^{\Delta n}\), calculating \(\Delta n\) is essential. This represents the net change in moles of gas from reactants to products, which directly affects equilibrium behavior.
In the given reaction, \(\Delta n\) is derived from subtracting the sum of moles of products from reactants. For \(\mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\), we find \(\Delta n = 1 - (1 + \frac{1}{2}) = -0.5\).
The negative \(\Delta n\) indicates a reduction in the total number of gas molecules, usually implying that the pressure may decrease as the reaction proceeds towards equilibrium. This change in moles is a pivotal aspect in accurately predicting pressure shifts during the reaction.

Ideal Gas Constant

The ideal gas constant, denoted \(R\), is a fundamental entity in gas law equations and all calculations involving gases.
Its value is generally \(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1}\) when using atmospheres for pressure, liters for volume, and Kelvin for temperature.
In the context of equilibrium constants, \(R\) plays a role in converting concentration expressions into pressure ones, ensuring that we can apply the ideal gas law seamlessly.
Understanding \(R\) is crucial because it allows us to handle reactions under different conditions and standards, thus bridging the gap between laboratory and real-world applications effectively. This role in \(K_c\) to \(K_p\) conversions highlights its importance for chemists working with gaseous equilibria.