Question

Explain, using its effect on the reaction quotient, why adding a reactant to the following equilibrium shifts the position of equilibrium to the right. $$ \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g) $$

Short answer

Adding a reactant to an equilibrium system decreases the reaction quotient (Q), causing the reaction to shift to the right in order to increase Q back to the equilibrium constant (K).

Step-by-step solution

Understand Le Chatelier's Principle

Le Chatelier's Principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. In the context of a chemical reaction, adding more reactant will generally cause the equilibrium to shift to the right, favoring the production of more products.

Define the Reaction Quotient

The reaction quotient, Q, is a measure of the relative amounts of products and reactants present during a reaction at any given point in time. It is calculated similarly to the equilibrium constant, K, but is not necessarily at the equilibrium state. For the reaction \( \mathrm{PCl}_3(g) + \mathrm{Cl}_2(g) \rightleftharpoons \mathrm{PCl}_5(g) \), the reaction quotient is given by \( Q = \frac{[\mathrm{PCl}_5]}{[\mathrm{PCl}_3][\mathrm{Cl}_2]} \).

Explain the Shift in Equilibrium

When the concentration of a reactant, such as \( \mathrm{PCl}_3 \) or \( \mathrm{Cl}_2 \) is increased, the value of Q decreases because the concentrations of the reactants appear in the denominator of the reaction quotient formula. Since Q is now less than the equilibrium constant, K, the reaction will shift to the right to reach equilibrium again, moving towards the formation of more \( \mathrm{PCl}_5 \) (the product). This shift is in response to the principle of Le Chatelier, as the system moves to restore equilibrium.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a reversible chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction, leading to no overall change in the concentrations of reactants and products. It's crucial to recognize that equilibrium does not mean the reactants and products are in equal concentrations; rather, it implies that their concentrations have stabilized in a fixed ratio.

For instance, in the reaction involving phosphorus trichloride (PCl₃), chlorine gas (Cl₂), and phosphorus pentachloride (PCl₅), the equilibrium is reached when the rate at which PCl₃ and Cl₂ combine to form PCl₅ is equal to the rate at which PCl₅ decomposes back into PCl₃ and Cl₂. The constant forward and backward movement of molecules in this state comprises the dynamic aspect of chemical equilibrium.

Reaction Quotient Q

The reaction quotient, Q, plays an important role in determining the direction in which a reaction mixture will proceed to reach equilibrium. It is defined as the ratio of the concentrations of the products raised to the power of their stoichiometric coefficients to the concentrations of the reactants raised to the power of their coefficients, at any point in time before the system has reached equilibrium.

For the given chemical equation, Q is represented by the formula:
\[ Q = \frac{[\text{PCl}_5]}{[\text{PCl}_3][\text{Cl}_2]} \]
This snapshot of the reaction's progress tells us how close or far the system is from equilibrium. If Q is less than the equilibrium constant K, the system is not in equilibrium, and the reaction will move forward, producing more products. Conversely, if Q is greater than K, the reaction will proceed in the reverse direction to produce more reactants. Thus, altering the concentrations of reactants or products will change the value of Q, prompting the reaction to shift in order to restore equilibrium.

Equilibrium Shift

An equilibrium shift refers to the change in the concentrations of reactants and products when a chemical system at equilibrium is subjected to a stress. This can be due to changes in concentration, pressure, volume, or temperature. According to Le Chatelier’s Principle, the system will respond to this stress by shifting the equilibrium in a direction that counteracts the change.

In the context of the exercise involving PCl₃ and Cl₂, when more reactant is added, the system counters this by producing more product, which results in a shift to the right. This restores the balance by using up the added reactants and increases the amount of PCl₅ until the system reaches a new equilibrium point. If a reactant or product is removed, the equilibrium will shift towards the opposite side to fill up the 'gap' created by the loss.

Equilibrium Constant K

The equilibrium constant, K, is an expression that sets the standard for whether a given chemical reaction at equilibrium favors the reactants or products. It is calculated similarly to the reaction quotient Q, but specifically when the system is at equilibrium, thus the concentrations are stable and do not change over time.

The equilibrium constant for our reaction is given by the expression:
\[ K = \frac{[\text{PCl}_5]}{[\text{PCl}_3][\text{Cl}_2]} \]
When K is greater than 1, there is a greater concentration of products at equilibrium, indicating a product-favored reaction. If K is less than 1, the reactants are favored. It's important to note that K is only affected by temperature and remains constant under constant temperature regardless of changes in concentrations or pressure. Understanding K helps predict the direction of the equilibrium shift when any of the reaction conditions are altered.