Question

For which of the following reaction, \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}:\) (a) \(2 \mathrm{NOCl}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g})\) (b) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})\) (c) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{HCl}(\mathrm{g})\) (d) \(\mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{5}(\mathrm{~g})\)

Short answer

Reaction (c) is where \(K_p = K_c\).

Step-by-step solution

Understand the Relationship Between Kp and Kc

The relationship between the equilibrium constants \(K_p\) and \(K_c\) for gaseous reactions is given by the formula: \[K_p = K_c(RT)^{\Delta n}\] where \(\Delta n\) is the change in the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. \(K_p = K_c\) when \(\Delta n = 0\).

Calculate Δn for Each Reaction

For each reaction, calculate \(\Delta n\), which represents the change in the number of moles of gases.- Reaction (a): \(2\,\text{NOCl(g)} \rightleftharpoons 2\,\text{NO(g)} + \text{Cl}_2\,(\text{g})\), \(\Delta n = (2 + 1) - 2 = 1\).- Reaction (b): \(\text{N}_2\,(\text{g}) + 3\,\text{H}_2\,(\text{g}) \rightleftharpoons 2\,\text{NH}_3\,(\text{g})\), \(\Delta n = 2 - 4 = -2\).- Reaction (c): \(\text{H}_2\,(\text{g}) + \text{Cl}_2\,(\text{g}) \rightleftharpoons 2\,\text{HCl}\,(\text{g})\), \(\Delta n = 2 - 2 = 0\).- Reaction (d): \(\text{PCl}_3\,(\text{g}) + \text{Cl}_2\,(\text{g}) \rightleftharpoons \text{PCl}_5\,(\text{g})\), \(\Delta n = 1 - 2 = -1\).

Identify Reaction with Δn = 0

Find the reaction for which \(\Delta n = 0\). For \(K_p = K_c\), the change in moles \(\Delta n\) must be zero. From the calculations:- Reaction (c) \(\text{H}_2\,(\text{g}) + \text{Cl}_2\,(\text{g}) \rightleftharpoons 2\,\text{HCl}\,(\text{g})\) has \(\Delta n = 0\).

Key concepts

Kp and Kc relationship

In the study of chemical equilibria, understanding the relationship between the equilibrium constants for gases is crucial. The two main constants are Kp, which is used for partial pressures, and Kc, which is for concentrations. These constants are related mathematically for gaseous reactions involving gases through the equation: \[ K_p = K_c (RT)^{\Delta n} \] Here, \(R\) represents the ideal gas constant, and \(T\) denotes the temperature in Kelvin. The term \(\Delta n\) is the key as it refers to the change in the number of moles of gas as the reaction progresses from reactants to products. To have \(K_p = K_c\), the exponent \((RT)^{\Delta n}\) should equal 1. This happens when \(\Delta n = 0\). To simplify further: - If \(\Delta n = 0\), then \(K_p = K_c\). - If \(\Delta n > 0\), \(K_p\) will typically be greater than \(K_c\). - If \(\Delta n < 0\), \(K_p\) would be less than \(K_c\). Understanding this relationship helps in predicting how equilibrium constants can vary with changes in pressure and concentration.

Delta n calculation

Calculating \(\Delta n\), or the change in the number of moles, is an essential step in analyzing gaseous reactions at equilibrium. This term determines how the shift from initial to final states affects the relationship between \(K_p\) and \(K_c\). To compute \(\Delta n\), use the following formula: \[ \Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} \] Here's how it works for some reactions: - For reaction (a): \(2\, \text{NOCl(g)} \rightleftharpoons 2\, \text{NO(g)} + \text{Cl}_2\, \text{(g)}\), \(\Delta n = (2 + 1) - 2 = 1\). It implies an increase in moles, suggesting that \(K_p > K_c\). - For reaction (b): \(\text{N}_2\, \text{(g)} + 3\, \text{H}_2\, \text{(g)} \rightleftharpoons 2\, \text{NH}_3\, \text{(g)}\), \(\Delta n = 2 - 4 = -2\). The decrease in moles indicates \(K_p < K_c\). - For reaction (c): \(\text{H}_2\, \text{(g)} + \text{Cl}_2\, \text{(g)} \rightleftharpoons 2\, \text{HCl}\, \text{(g)}\), \(\Delta n = 2 - 2 = 0\). This neutral change means \(K_p = K_c\). Quickly determining \(\Delta n\) based on stoichiometric coefficients aids significantly in chemical equilibrium analysis.

Gaseous reactions equilibrium

Gaseous reactions at equilibrium involve a dynamic system where rates of forward and reverse reactions are equal, leading to constant macroscopic properties in the system. For chemists, understanding this equilibrium provides valuable insights into reaction dynamics and the influence of external factors. Key factors influencing gaseous equilibria include: - **Concentration**: Altering the concentration of reactants or products can shift the equilibrium position according to Le Chatelier’s principle. Adding more reactants tends to drive the equilibrium towards the products, while increasing product concentration pushes it back towards reactants. - **Pressure**: Changes in pressure especially affect reactions involving gases due to their volume-sensitive nature. Generally, increasing the pressure shifts the equilibrium to the side with fewer moles of gas, while decreasing pressure shifts it to the side with more moles of gas. - **Temperature**: Temperature changes can affect the equilibrium position depending on whether the reaction is endothermic or exothermic. Raising the temperature favors the endothermic process, while lowering it favors the exothermic one. Equilibrium constants \(K_p\) and \(K_c\) help quantify these equilibria. Knowing how these constants relate helps in predicting system responses to shifts in pressure, temperature, and concentration. Mastering equilibria for gaseous reactions guides successful manipulations and predictions of chemical processes.