Problem 9 - A Complete Resource Book in Chemistry for JEE Main

Question

\(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})\) in the above reaction \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) are related as (a) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}} \times(\mathrm{RT})\) (b) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}} \times(\mathrm{RT})^{-1}\) (c) \(\mathrm{K}_{\mathrm{c}}=\mathrm{K}_{\mathrm{p}} \times(\mathrm{RT})^{2}\) (d) \(K_{p}=K_{c} \times(R T)^{-2}\)

Step-by-Step Solution

Answer

The correct relationship is (b) \(K_p = K_c \times (RT)^{-1}\).

Step 1: Understanding the Reaction Equation

The balanced chemical equation given is \(2 \mathrm{SO}_{2}(\mathrm{~g}) + \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})\). This is an example of a gas-phase equilibrium reaction.

Step 2: Recall the Relationship between Kp and Kc

For a general reaction \(aA + bB \rightleftharpoons cC + dD\) in the gas phase, the relation between \(K_p\) and \(K_c\) is given by: \[K_p = K_c (RT)^{\Delta n}\] where \(\Delta n = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})\).

Step 3: Calculate Δn

For the given reaction, \(\Delta n = (2) - (2 + 1) = 2 - 3 = -1\).

Step 4: Apply Kp and Kc Relationship

Substituting \(\Delta n = -1\) into the equation, we get: \[K_p = K_c (RT)^{-1}\]. This represents the relationship between \(K_p\) and \(K_c\) for the given reaction.

Step 5: Match the Equation to the Options

Compare the expression \(K_p = K_c (RT)^{-1}\) with the given options. The correct option is (b) \(K_p = K_c \times (RT)^{-1}\).

Key Concepts

Gas Phase Equilibrium

Gas phase equilibrium involves a dynamic balance between reactants and products in their gaseous states, meaning the rates of the forward and reverse reactions are equal. In such systems, the concentrations of all species remain constant over time because the rates at which they are consumed and produced are the same.
In our example reaction \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})\), the equilibrium is established when sulfur dioxide \((\mathrm{SO}_{2})\) and oxygen \((\mathrm{O}_{2})\) gases react to form sulfur trioxide \((\mathrm{SO}_{3})\) and the rate of formation of \(\mathrm{SO}_{3}\) is balanced by its decomposition back into \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\).
Understanding gas phase equilibrium is crucial as it allows chemists to control conditions to favor the formation of desired products through factors such as pressure, temperature, and concentration adjustments. A key feature of these reactions is that they are easily influenced by external changes, a point crucial in industrial applications, like the Haber-Bosch process, where optimization of equilibrium conditions can increase yields.

Relation between Kp and Kc

The equilibrium constant for reactions in the gas phase can be expressed in two different ways: \(K_p\) and \(K_c\). These expressions differ because \(K_p\) is based on partial pressures of the gases, while \(K_c\) is based on molar concentrations.
The relationship between \(K_p\) and \(K_c\) for a reaction is key for converting between these two expressions, and is given by the equation:

• \(K_p = K_c (RT)^{\Delta n}\)

Here, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in moles of gas, calculated as the difference between the moles of gaseous products and reactants.
By applying this formula, chemists can determine how changes in pressure or concentration affect the equilibrium position and can adjust conditions to obtain a larger amount of desired product. For example, in our given reaction, \(\Delta n\) was calculated to be \(-1\). Substituting into the relation gives \(K_p = K_c (RT)^{-1}\), helping to choose the right conditions for experiments or processes.

Chemical Equilibrium Calculations

Chemical equilibrium calculations are essential for predicting the behavior of systems in equilibrium. They allow chemists to understand and quantify how a system responds to various changes in conditions. This particular type of calculation involves using the equilibrium constants \(K_p\) and \(K_c\) to determine concentrations and pressures at equilibrium.
In chemical equilibrium calculations, a systematic approach is needed:

• First, write the balanced chemical equation.
• Determine the expression for the equilibrium constant \(K_c\) or \(K_p\).
• Calculate initial moles or concentrations of the reactants and products.
• Use an ICE table (Initial, Change, Equilibrium) to visualize changes in concentrations or pressures.
• Apply the equilibrium constant expression and solve for the unknowns using algebraic methods.

This method allows us to accurately predict how the equilibrium will shift under different scenarios. In our example, by knowing \(K_p = K_c (RT)^{-1}\), we can predict how the gas partial pressures influence the equilibrium position, offering insights for laboratory experimentation and industrial process optimization.