Question

Phosphorus trichloride gas and chlorine gas react to form phosphorus pentachloride gas: \(\mathrm{PCl}_{3}+\mathrm{Cl}_{2}(g) \rightleftharpoons\) \(\mathrm{PCl}_{5}(g) .\) A gas vessel is charged with a mixture of \(\mathrm{PCl}_{3}(\mathrm{~g})\) and \(\mathrm{Cl}_{2}(\mathrm{~g})\), which is allowed to equilibrate at \(450 \mathrm{~K}\). At equilibrium the partial pressures of the three gases are \(P_{\mathrm{PCl}_{3}}=0.124 \mathrm{~atm}, P_{\mathrm{Cl}_{2}}=0.157 \mathrm{~atm}\), and \(P_{\mathrm{PCl}_{5}}=1.30 \mathrm{~atm}\). (a) What is the value of \(K_{p}\) at this temperature? (b) Does the equilibrium favor reactants or products?

Short answer

(a) The value of Kp at 450 K is 66.98. (b) The equilibrium favors products.

Step-by-step solution

Write the expression for Kp

For the given equilibrium reaction: PCl3(g) + Cl2(g) ⇌ PCl5(g) The equilibrium constant Kp is expressed as the ratio of the products to the reactants, raised to the corresponding stoichiometric coefficients. In this case, the coefficients are 1. So, the expression for Kp becomes: Kp = (P_PCl5) / (P_PCl3 * P_Cl2)

Calculate the value of Kp using the provided partial pressures

We are given the equilibrium partial pressures for PCl3, Cl2, and PCl5: P_PCl3 = 0.124 atm P_Cl2 = 0.157 atm P_PCl5 = 1.30 atm Substitute the values into the Kp expression: Kp = (1.30) / (0.124 * 0.157) = 66.98

Determine if the equilibrium favors reactants or products

Compare the calculated Kp value to 1: - If Kp > 1, the equilibrium favors products. - If Kp < 1, the equilibrium favors reactants. - If Kp = 1, the equilibrium lies in the middle. We got Kp = 66.98, which is greater than 1. Hence, the equilibrium favors the products. #Output#: (a) The value of Kp at 450 K is 66.98. (b) The equilibrium favors products.

Key concepts

Equilibrium Constant (Kp)

Understanding the equilibrium constant, represented as Kp, is crucial for students studying chemical equilibrium. Kp is a number that provides the ratio of the concentrations of the products to the reactants, each raised to the power of their coefficients in the balanced chemical equation, at equilibrium condition for reactions involving gases. The 'p' indicates that we are dealing with partial pressures. It is essential to note that Kp is temperature-dependent and does not change unless the temperature changes.

To determine Kp, you would use the equation: $$ Kp = \frac{{(P_\text{{product 1}})^{\text{{coefficient 1}}} \cdots}}{{(P_\text{{reactant 1}})^{\text{{coefficient 1}}} \cdots}} $$where P represents partial pressure, and the coefficients come from the balanced equation. When Kp is greater than 1, the formation of products is favored. A value less than 1 signifies a reaction that favors reactants.

Partial Pressure

In the context of chemical equilibrium, the term partial pressure refers to the pressure exerted by a specific gas in a mixture of gases. It is proportional to the mole fraction of the gas in the mixture and the total pressure of the mixture. The partial pressure of a gas can be calculated using the ideal gas law, given by: $$ P = \frac{{nRT}}{V} $$where P is the partial pressure, n is the number of moles of the gas, R is the universal gas constant, T is the temperature in Kelvin, and V is the volume of the gas.

In the exercise given, the partial pressures of the reactants and products are directly provided. It's important to use these values as they are, and not to convert them into concentrations, when calculating Kp.

Le Chatelier's Principle

When studying chemical equilibrium, we must pay attention to Le Chatelier's Principle, which states that if an external change is applied to a system at equilibrium, the system adjusts in such a way as to partially counteract the change. This can include changes in concentration, pressure, volume, or temperature.

For instance, if you increase the pressure of the system by reducing its volume, a reaction that produces fewer moles of gas will be favored. Conversely, decreasing the pressure can shift the equilibrium to favor the production of more moles of gas. Temperature changes can also affect the equilibrium, as endothermic reactions absorb heat, and exothermic reactions release it. By understanding this principle, students can predict how changes to the conditions will influence the position of equilibrium in a chemical reaction.

Reaction Quotient

The reaction quotient, Q, is a measure similar to the equilibrium constant, Kp, but for conditions that may not be at equilibrium. It uses the same formula as Kp but with the initial concentrations or partial pressures of the reactants and products:

$$ Q = \frac{{(P_\text{{product 1}})^{\text{{coefficient 1}}} \cdots}}{{(P_\text{{reactant 1}})^{\text{{coefficient 1}}} \cdots}} $$
The comparison between Q and Kp gives an insight into the direction in which the reaction will proceed to reach equilibrium.

• If Q < Kp, the reaction will proceed forward, forming more products.
• If Q > Kp, the reaction will proceed backward, forming more reactants.
• If Q = Kp, the system is already at equilibrium.

Understanding Q allows students to predict the immediate behavior of a reaction mixture before reaching equilibrium.