Question

You are given a box in which \(\mathrm{PCl}_{5}(\mathrm{~g}), \mathrm{PCl}_{3}(\mathrm{~g})\), and \(\mathrm{Cl}_{2}(\mathrm{~g})\) are in equilibrium with each other at \(546^{\circ} \mathrm{K}\). Assuming that the decomposition of \(\mathrm{PCl}_{5}\) to \(\mathrm{PCl}_{3}\) and \(\mathrm{Cl}_{2}\) is endothermic, what effect would there be on the concentration of \(\mathrm{PCl}_{5}\) in the box if each of the following changes were made? (a) Add \(\mathrm{Cl}_{2}\) to the box, (b) Reduce the volume of the box, and (c) Raise the temperature of the system.

Short answer

In summary, (a) Adding Cl2 to the box increases the concentration of PCl5. (b) Reducing the volume of the box increases the concentration of PCl5. (c) Raising the temperature of the system decreases the concentration of PCl5.

Step-by-step solution

Identify the equilibrium reaction

First, we can write down the equilibrium reaction: \[ PCl_{5(g)} \rightleftharpoons PCl_{3(g)} + Cl_{2(g)} \]

Establish the equation representing the equilibrium constant

Now, we can establish the equilibrium constant (K) equation for the given reaction: \[ K = \frac{[PCl_{3}][Cl_{2}]}{[PCl_{5}]} \] Now, we will consider each change separately and use Le Chatelier's Principle to determine the effect on the concentration of PCl5.

(a) Adding Cl2 to the box

When Cl2 is added to the box, according to Le Chatelier's Principle, the system will shift to counteract this change by moving towards the side that consumes Cl2. In this case, this will favor the reverse reaction, and the concentration of PCl5 will increase.

(b) Reducing the volume of the box

When the volume of the box is reduced, the pressure increases. According to Le Chatelier's Principle, the equilibrium will shift to minimize the pressure by favoring the side with fewer moles of gas. In the given reaction, the forward reaction has 1 mole of PCl5 being converted to 2 moles of gas (PCl3 and Cl2). Therefore, the reverse reaction will be favored, and the concentration of PCl5 will increase.

(c) Raising the temperature of the system

Since the decomposition of PCl5 is endothermic, increasing the temperature will favor the forward reaction as the system absorbs the heat to counteract the temperature change. According to Le Chatelier's Principle, the equilibrium will shift to favor the forward reaction. Consequently, the concentration of PCl5 will decrease. In summary: -Adding Cl2 to the box and reducing the volume of the box both lead to an increase in the concentration of PCl5. -Raising the temperature of the system leads to a decrease in the concentration of PCl5.

Key concepts

Le Chatelier's Principle

When a chemical system at equilibrium is disturbed by an external change, Le Chatelier's Principle provides a way to predict how the system will respond to re-establish equilibrium. It essentially says that the system will adjust in a way that partially counteracts the change.

Consider a seesaw balanced evenly at first; if a weight is added on one side, the seesaw will tilt. If we want to balance it again, we could add weight to the other side. Similarly, in chemical reactions, the 'weight' can be changes in concentration of reactants or products, changes in volume (and thus pressure), or changes in temperature.

In the case of our exercise involving (PCl_5), (PCl_3), and Cl_2, adding more Cl_2 is like adding extra weight on one side of the seesaw. To re-balance, the equilibrium shifts to the left to produce more PCl_5, 'adding weight' to the opposite side. Similarly, reducing volume is akin to forcing the molecules closer together, increasing pressure. The system adapts by favoring the side with fewer molecules, which also leads to the formation of more PCl_5.

Equilibrium Constant

The equilibrium constant (K) quantifies the ratio of product concentrations to reactant concentrations at equilibrium, with each concentration raised to the power of its coefficient in the balanced equation. This constant is pivotal for understanding at which point a reaction is at equilibrium.

In mathematical terms, for our given reaction, it's described by the equation:
\[ K = \frac{[PCl_{3}][Cl_{2}]}{[PCl_{5}]} \]

It's important to note that K is constant only at a given temperature. If the temperature changes, K changes. When we perform manipulations such as adding Cl_2 or reducing the volume, the equilibrium shifts, but the value of K remains the same, assuming the temperature is constant. Interestingly, if the temperature changes, that's when K itself changes - reflecting a new balance point for the reaction.

Endothermic Reactions

In an endothermic reaction, the system absorbs heat from its surroundings; it essentially 'takes in energy'. This means that we need to provide heat for the reaction to proceed forward. Think of it as a sponge soaking up water; the reaction soaks up heat.

Our exercise states that the decomposition of PCl_5 into PCl_3 and Cl_2 is endothermic. Therefore, raising the temperature supplies energy to the system, which favors the formation of products. For endothermic reactions like this, the value of the equilibrium constant K will increase with temperature because heat is essentially a reactant.

Intuitively, when you heat a system that absorbs heat during its reaction, it's like giving it more of what it needs to proceed; in our case, the reaction shifts to form more PCl_3 and Cl_2, thus reducing the concentration of PCl_5.