Question

In which of the following gaseous reaction, the value of \(K_{p}\) is less than \(K_{c}:\) (a) \(\mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCI}_{3}+\mathrm{Cl}_{2}\) (b) \(2 \mathrm{SO}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{SO}_{3}\) (c) \(2 \mathrm{HI} \rightleftharpoons \mathrm{H}_{2}+\mathrm{I}_{2}\) (d) \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\)

Short answer

Reactions (b) and (d) have \(K_p < K_c\).

Step-by-step solution

Understand the Factors that Affect \( K_p \) and \( K_c \)

The relationship between the equilibrium constants \( K_p \) (pressure) and \( K_c \) (concentration) for a gaseous reaction is given by the equation:\[K_p = K_c (RT)^{\Delta n}\]where \( \Delta n \) is the change in moles of gas between the products and reactants, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. If \( \Delta n = 0 \), then \( K_p = K_c \). If \( \Delta n > 0 \), then the value of \( K_p \) can be greater than \( K_c \), and if \( \Delta n < 0 \), \( K_p \) can be less than \( K_c \) assuming \( RT \) is greater than 1.

Calculate \( \Delta n \) for Each Reaction

Calculate the change in moles of gas \( \Delta n \) for each reaction option. This involves subtracting the sum of the stoichiometric coefficients of the reactants from those of the products:- (a) \( \Delta n = (1 + 1) - 1 = 1 \)- (b) \( \Delta n = 2 - (2 + 1) = -1 \)- (c) \( \Delta n = (1 + 1) - 2 = 0 \)- (d) \( \Delta n = 2 - (1 + 3) = -2 \)

Compare \( K_p \) and \( K_c \) Based on \( \Delta n \)

Examine \( \Delta n \) calculated for each option:- For reaction (a), since \( \Delta n = 1 \), \( K_p > K_c \).- For reaction (b), since \( \Delta n = -1 \), \( K_p < K_c \).- For reaction (c), since \( \Delta n = 0 \), \( K_p = K_c \).- For reaction (d), since \( \Delta n = -2 \), \( K_p < K_c \).

Identify the Appropriate Reaction

Identify which reactions have \( K_p < K_c \) based on \( \Delta n < 0 \). According to Step 3, reactions (b) and (d) have \( K_p < K_c \) because \( \Delta n < 0 \).

Key concepts

Gaseous Reactions

Gaseous reactions involve substances that are in the gas phase, meaning their molecules move freely and are spread out. These types of reactions often occur in an open environment, like the atmosphere, and are governed by variables such as pressure and temperature. The way gas molecules interact is crucial to understanding how gaseous reactions work.

In a gaseous reaction, molecules collide with each other and may break apart to form new substances. Because gases can be compressed or expanded, changes in pressure and temperature can significantly influence these reactions. For instance, increased pressure may push molecules closer, enhancing the likelihood of collisions which may speed up the reaction.

The study of gaseous reactions helps us understand processes like combustion in engines or atmospheric reactions. Gaseous reactions also play a pivotal role in industrial applications, such as the synthesis of ammonia in the Haber process.

Equilibrium Constant Relationship

The equilibrium constant (\( K \)) represents the state of a reaction when both the forward and reverse reactions happen at the same rate, yielding a balance of products and reactants. Each equilibrium constant depends on the reaction condition, such as concentration (\( K_c \)) or pressure (\( K_p \)).

For gaseous reactions, the relationship between \( K_p \) and \( K_c \) is described by the equation: \[ K_p = K_c (RT)^{\Delta n} \]
- \( R \) is the ideal gas constant.- \( T \) is the absolute temperature.- \( \Delta n \) is the difference in moles of gas between products and reactants.

This equation highlights how changes in pressure or concentration affect the equilibrium position. If \( \Delta n = 0 \), the transition from concentration to pressure equilibrium constants does not change, \( K_p = K_c \). If \( \Delta n > 0 \), \( K_p \) is typically greater than \( K_c \). Conversely, if \( \Delta n < 0 \), \( K_p \) can be less than \( K_c \).

Understanding this relationship allows scientists and engineers to predict how a reaction will shift under various conditions, such as changes in pressure or temperature.

Stoichiometry in Chemical Reactions

Stoichiometry is the area of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. This concept is fundamental when calculating how much of each substance is involved in a reaction.

In gaseous reactions, stoichiometry helps in determining the \( \Delta n \) value, critical to understanding the equilibrium constants \( K_p \) and \( K_c \). To find \( \Delta n \), one subtracts the number of moles of gaseous reactants from the number of moles of gaseous products. This particular aspect is key to determining whether \( K_p \) is greater than, less than, or equal to \( K_c \).

For example, consider the reaction: \[ 2 \text{SO}_2 + \text{O}_2 \rightleftharpoons 2 \text{SO}_3 \] Here, the total moles of reactants add up to 3 (2 moles of \( \text{SO}_2 \) and 1 mole of \( \text{O}_2 \)), while the products are only 2 moles \( \text{SO}_3 \), resulting in \( \Delta n = -1 \).

Stoichiometry ensures we know exactly how much reactants and products to expect, which is crucial for effective planning and analysis in chemical processes involving gases.