Question

In which of the following gaseous reaction, \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) have the same values: (a) \(2 \mathrm{Hl} \rightleftharpoons \mathrm{H}_{2}+\mathrm{I}_{2}\) (b) \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\) (c) \(2 \mathrm{SO}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{SO}_{3}\) (d) \(\mathrm{PCI}_{5} \rightleftharpoons \mathrm{PCI}_{3}+\mathrm{Cl}_{2}^{3}\)

Short answer

\( K_p \) and \( K_c \) have the same values in reaction (a): \( 2 \, \mathrm{HI} \rightleftharpoons \mathrm{H}_2 + \mathrm{I}_2 \).

Step-by-step solution

Understand the Relationship Between \( K_p \) and \( K_c \)

The relationship between the equilibrium constant in terms of partial pressures \( K_p \) and in terms of concentration \( K_c \) for a gaseous reaction is given by the formula: \[ K_p = K_c (RT)^{\Delta n} \]where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in the number of moles of gas, calculated as final moles minus initial moles.

Calculate \( \Delta n \) for Each Reaction

Calculate \( \Delta n \) for each reaction:- Reaction (a): \( 2 \, \mathrm{HI} \rightleftharpoons \mathrm{H}_2 + \mathrm{I}_2 \) - \( \Delta n = (1 + 1) - 2 = 0 \)- Reaction (b): \( \mathrm{N}_2 + 3 \, \mathrm{H}_2 \rightleftharpoons 2 \, \mathrm{NH}_3 \) - \( \Delta n = 2 - (1 + 3) = -2 \)- Reaction (c): \( 2 \, \mathrm{SO}_2 + \mathrm{O}_2 \rightleftharpoons 2 \, \mathrm{SO}_3 \) - \( \Delta n = 2 - (2 + 1) = -1 \)- Reaction (d): \( \mathrm{PCl}_5 \rightleftharpoons \mathrm{PCl}_3 + \mathrm{Cl}_2 \) - \( \Delta n = (1 + 1) - 1 = 1 \)

Determine When \( K_p = K_c \)

For the values of \( K_p \) and \( K_c \) to be the same, \( (RT)^{\Delta n} \) must equal 1. This occurs only when \( \Delta n = 0 \). Thus, we need to identify the reaction (or reactions) for which \( \Delta n = 0 \). From Step 2, only reaction (a) has \( \Delta n = 0 \).

Key concepts

Equilibrium Constant

In chemistry, the equilibrium constant is a vital parameter that helps us understand the behavior of reversible chemical reactions at equilibrium. There are two main types of equilibrium constants used, namely

• `K_c`: Refers to the equilibrium constant when dealing with concentrations in a reaction solution, and
• `K_p`: This is applicable when the components are gases, and it involves their partial pressures.

The equilibrium constant is fundamental because it gives insight into the position of equilibrium, determining whether reactants or products are favored under given conditions. A large value of the equilibrium constant (k will appear in a larger font) indicates a greater concentration of products, emphasizing the completion of the reaction, whereas a small value suggests reactants predominate at equilibrium. Understanding the equilibrium constant assists in predicting and controlling the reactions in chemical industries, environmental systems, and many other scientific fields.

Gaseous Reactions

Gaseous reactions are transformations involving gases as reactants and/or products. These reactions play a significant role in various natural and industrial processes. For students studying chemistry, understanding gaseous reactions involves recognizing that these reactions can occur under various pressures and temperatures.
Gaseous reactions are characterized by changes in pressure and volume, a consequence of the behavior of gases according to the ideal gas law, \( PV = nRT \). This equation relates the pressure (\(P\)), volume (\(V\)), number of moles of gas (\(n\)), gas constant (\(R\)), and temperature (\(T\)) in Kelvin. The understanding of gaseous reactions is crucial, as it forms the foundation for predicting reaction behavior and calculating equilibrium in gas phase chemical processes. The balanced chemical equation indicates the stoichiometry, which helps in calculating the changes in variables, like concentrations or pressures of the components involved.

Partial Pressure

Partial pressure is an important concept when analyzing gaseous reactions. It refers to the pressure exerted by an individual gas in a mixture of gases. In the context of chemical equilibrium, it helps in calculating the equilibrium constant for gaseous reactions (`K_p`). Dalton's Law of Partial Pressures states that:"The total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of each individual component in the gas mixture."Mathematically, this can be expressed as:\[ P_{ ext{total}} = P_1 + P_2 + P_3 + ext{...} \]Understanding partial pressures enables us to measure and manipulate gas reaction systems effectively, offering a way to determine the effect of changing conditions on the system's equilibrium. This is especially significant when conducting experiments and industrial processes, ensuring reactions proceed safely and efficiently.

Reaction Stoichiometry

Reaction stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. Understanding stoichiometry is vital, as it provides a framework for calculating the amounts of reactants or products involved. In gaseous reactions, stoichiometry is expressed in terms of moles of reaction participants, determined by the balanced chemical equation. The stoichiometric coefficients in the equation indicate the proportionate quantities of substances involved in the reaction. These coefficients are essential in determining how the reaction progresses and calculating both the equilibrium constants `K_c` and `K_p` . The concept becomes even more crucial when analyzing the change in the number of moles during a reaction, denoted as Δn , which is essential for equating `K_p` and `K_c` . Reaction stoichiometry is a pivotal concept, acting as a bridge between theoretical chemistry and practical application, facilitating the optimization of chemical reactions in various domains.