Problem 75 - Chemistry: The Central Science

Question

The reaction \(\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g)\) has \(K_{p}=0.0870\) at \(300^{\circ} \mathrm{C}\). A flask is charged with \(0.50\) atm \(\mathrm{PCl}_{3}, 0.50 \mathrm{~atm} \mathrm{Cl}_{2}\), and \(0.20 \mathrm{~atm} \mathrm{PCl}_{5}\) at this tempera- ture. (a) Use the reaction quotient to determine the direction the reaction must proceed to reach equilibrium. (b) Calculate the equilibrium partial pressures of the gases. (c) What effect will increasing the volume of the system have on the mole fraction of \(\mathrm{Cl}_{2}\) in the equilibrium mixture? (d) The reaction is exothermic. What effect will increasing the temperature of the system have on the mole fraction of \(\mathrm{Cl}_{2}\) in the equilibrium mixture?

Step-by-Step Solution

Answer

In summary, for the reaction PCl₃(g) + Cl₂(g) ⇌ PCl₅(g) with initial pressures 0.50 atm PCl₃, 0.50 atm Cl₂, and 0.20 atm PCl₅, the reaction proceeds in the reverse direction to reach equilibrium. The equilibrium partial pressures for the gases are approximately: PCl₃ = 0.619 atm, Cl₂ = 0.619 atm, PCl₅ = 0.081 atm. When the system's volume is increased, the mole fraction of Cl₂ in the equilibrium mixture will increase. Similarly, when the temperature is increased for this exothermic reaction, the mole fraction of Cl₂ will also increase.

Step 1: Write down the equilibrium expression for the reaction

We know that for the given reaction: PCl₃(g) + Cl₂(g) ⇌ PCl₅(g) The equilibrium constant expression (\(K_p\)) can be written as: \[K_p = \frac{P_{PCl_5}} {P_{PCl_3} \cdot P_{Cl_2}}\]

Step 2: Calculate the reaction quotient (Q)

Now we use the given initial pressures to calculate the reaction quotient (Q): \[Q = \frac{P_{PCl_5}} {P_{PCl_3} \cdot P_{Cl_2}} = \frac{0.20}{0.50 \cdot 0.50} = 0.80\]

Step 3: Compare Q with Kp

Now we compare the calculated Q with the given \(K_p\): Since \(Q > K_p (0.80 > 0.0870)\), the reaction will proceed in the reverse direction to reach equilibrium, meaning that PCl₅ will dissociate into PCl₃ and Cl₂. #b: Calculate equilibrium partial pressures#

Step 1: Set up an ICE table

Set up the ICE table as follows, with x representing the change in pressure for each species as the reaction proceeds: | PCl₃ | Cl₂ | PCl₅ --------------------------------------- Initial(atm) | 0.50 | 0.50 | 0.20 Change(atm) | +x | +x | -x Equilibrium(atm) | 0.50+x | 0.50+x | 0.20-x

Step 2: Substitute values from the ICE table into the equilibrium expression

Substitute the values from the ICE table into the equilibrium expression and solve for x: \[K_p = \frac{(0.20-x)} {(0.50+x) \cdot (0.50+x)}\]

Step 3: Solve for x

Solve this equation for x (either by making approximations or using a quadratic formula). In this case, x is approximately 0.119 atm.

Step 4: Calculate equilibrium pressures

Use the calculated value of x to determine the equilibrium pressures for each gas: \[P_{PCl_3} = 0.50 + x = 0.50 + 0.119 = 0.619\, atm\] \[P_{Cl_2} = 0.50 + x = 0.50 + 0.119 = 0.619\, atm\] \[P_{PCl_5} = 0.20 - x = 0.20 - 0.119 = 0.081\, atm\] #c: Effect of increasing volume on the mole fraction of Cl₂#

Step 1: Apply Le Chatelier's Principle with respect to volume change

According to Le Chatelier's Principle, increasing the volume of the system will cause the reaction to shift in the direction of the side with more moles of gas. In this case, since the dissociation side (PCl₃ and Cl₂) has more moles of gas, the reaction will shift towards it.

Step 2: Predict the effect on Cl₂'s mole fraction

Since the reaction will shift towards the side of PCl₃ and Cl₂, the mole fraction of Cl₂ in the equilibrium mixture will increase. #d: Effect of increasing temperature on the mole fraction of Cl₂#

Step 1: Apply Le Chatelier's Principle with respect to temperature change

According to Le Chatelier's Principle, for an exothermic reaction, increasing the temperature will cause the reaction to shift in the direction of the endothermic (reverse) reaction. In this case, the formation of PCl₃ and Cl₂ is the endothermic process (reverse reaction).

Step 2: Predict the effect on Cl₂'s mole fraction

Since the reaction will shift towards the side of PCl₃ and Cl₂, the mole fraction of Cl₂ in the equilibrium mixture will increase when the temperature is increased.

Key Concepts

Reaction Quotient (Q)

To understand chemical equilibrium, the reaction quotient (\( Q \)) is a valuable tool. It helps us predict the direction in which a reaction should proceed to reach equilibrium. In simple terms, it's a snapshot of the current state of a reaction, similar to the equilibrium constant (\( K \)), but not necessarily at equilibrium.

• Calculate \( Q \) using the initial concentrations or pressures of the species involved in the reaction, plugged into the equilibrium expression.
• Compare \( Q \) to \( K \): - If \( Q < K \), the reaction will shift to the right (towards products). - If \( Q > K \), the reaction will shift to the left (towards reactants).
• A balanced equilibrium occurs when \( Q = K \).

In our specific example, \( Q \) is calculated to be 0.80, whereas \( K_p \) is 0.087. Since \( Q > K_p \), the reaction must shift towards the formation of reactants, PCl₃ and Cl₂, to reach equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle provides insight into how a chemical system at equilibrium will respond to various changes. It's like predicting how an ecosystem might adjust when external factors change. When a change is imposed on a system at equilibrium, the system will find a way to counteract that change and establish a new equilibrium position. Here are a few common scenarios:

• Concentration Changes: Adding or removing reactants/products will shift the equilibrium to oppose the change.
• Pressure/Volume Changes: For gaseous reactions, increasing the system's volume decreases pressure. The reaction will shift towards the side with more gas molecules to increase pressure.
• Temperature Changes: For exothermic reactions, increasing temperature shifts equilibrium towards the endothermic (reverse) direction.

For our reaction, increasing the volume shifts the equilibrium towards the side with more molecules, PCl₃ and Cl₂, increasing their mole fractions in the mixture.

Exothermic Reactions

Exothermic reactions release heat, interacting interestingly with temperature changes according to Le Chatelier's Principle. These reactions are heat-expelling, and their equilibrium behavior changes with temperature alterations.Here's what happens with temperature changes:

• Increase in Temperature: For an exothermic reaction, adding heat causes the system to shift towards the endothermic direction to absorb the extra heat.
• Decrease in Temperature: The reaction shifts towards the exothermic side, producing more heat.

In our discussed chemical system, the reaction \(\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g)\) is exothermic. Raising the temperature encourages the system to shift towards the reverse reaction, PCl₃ and Cl₂ formation, thus increasing the mole fraction of Cl₂.