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 \(\mathrm{PCl}_{3}(g)\) and \(\mathrm{Cl}_{2}(g)\), which is allowed to equilibrate at \(450 \mathrm{~K}\). At equilibrium the partial pressures of the three gases are \(P_{\mathrm{PCl}_{3}}=0.124 \mathrm{~atm}, \quad P_{\mathrm{Cl}_{2}}=0.157 \mathrm{~atm}\), and \(P_{\mathrm{PCl}}=1.30 \mathrm{~atm}\) (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 the equilibrium constant (Kp) at this temperature is approximately 52.9. (b) The equilibrium favors the products.

Step-by-step solution

Write the balanced chemical equation

The balanced chemical equation for the given reaction is: \[ PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g) \]

Write the expression for Kp

The expression for Kp based on the partial pressures of reactants and products is given by: \[ K_p = \frac{P_{PCl_5}}{P_{PCl_3} \cdot P_{Cl_{2}}} \]

Substitute the given partial pressures into the equation

Now substitute the given partial pressures of each species at equilibrium into the expression for Kp: \[ K_p = \frac{1.30 \, \mathrm{atm}}{(0.124 \, \mathrm{atm})\cdot(0.157 \, \mathrm{atm})}\]

Compute the value of Kp

Calculate the value of Kp using the given partial pressures: \[ K_p = \frac{1.30}{(0.124) \cdot (0.157)} \approx 52.9 \]

Determine if the equilibrium favors reactants or products

We can tell if the equilibrium favors reactants or products by looking at the value of Kp: - If \(K_p > 1\), the equilibrium favors products. - If \(K_p < 1\), the equilibrium favors reactants. Since \(K_p \approx 52.9 > 1\), the equilibrium favors products. In conclusion, (a) the value of the equilibrium constant (Kp) at this temperature is approximately 52.9, and (b) the equilibrium favors the products.

Key concepts

Equilibrium Constant (Kp)

In chemical reactions, the equilibrium constant (Kp) is a number that provides us with a ratio of the concentration (expressed in terms of partial pressures in gas-phase reactions) of products to reactants at equilibrium. Higher values of Kp indicate a greater concentration of products, suggesting the reaction tends to form products under the given conditions. In the provided exercise, Kp for the phosphorus trichloride and chlorine gas reaction is calculated using the formula of Kp for a reaction where the stoichiometry is 1:1:1. The formula is:
\[ K_p = \frac{P_{\text{PCl}_5}}{P_{\text{PCl}_3} \cdot P_{\text{Cl}_2}} \]
The partial pressures substituted into this formula result in a Kp value of approximately 52.9. Since Kp is significantly greater than 1, it suggests that the reaction heavily favors the formation of products (phosphorus pentachloride) at 450 K.

Partial Pressures

The term partial pressure refers to the pressure that each gas in a mixture exerts if it were alone in the container. It is directly proportional to its mole fraction in the mixture and the total pressure. This concept is vital when analyzing gas-phase equilibrium reactions. In the exercise, partial pressures were used to calculate Kp. Since equilibrium concerns the balance between reactants and products, understanding partial pressures is crucial. For each gas, such as \( P_{\text{PCl}_3} \), \( P_{\text{Cl}_2} \), and \( P_{\text{PCl}_5} \), their respective partial pressures can indicate how far the reaction has progressed towards equilibrium.

Le Chatelier's Principle

When you perturb an equilibrium system, the system will adjust to minimize the effect of the change, a concept known as Le Chatelier's principle. It can be understood in terms of changes in concentration, pressure, or temperature.

• If you add more reactants, the system shifts to produce more products.
• If you increase the pressure, the system shifts to the side with fewer moles of gas.
• Raising the temperature of an exothermic reaction shifts the balance towards the reactants, as heat is a product.

Applying this principle to the textbook problem, if the temperature were raised, the equilibrium might shift towards the reactants, because the formation of \( PCl_{5} \) from \( PCl_{3} \) and \( Cl_{2} \) is exothermic.

Reaction Quotient

The reaction quotient (Q) is a measure like Kp but for a system that is not at equilibrium. It uses the same formula as the Kp but with the initial concentrations or partial pressures. The relationship between Q and Kp can tell us the direction in which the reaction will move to reach equilibrium.

• If \( Q < K_p \), the system will proceed in the forward direction, producing more products.
• If \( Q = K_p \), the system is at equilibrium.
• If \( Q > K_p \), the system will shift to produce more reactants.

By comparing Q and Kp, one can predict how a reaction would shift if any changes occur, allowing us to anticipate how the system will react before reaching equilibrium.