Question

Gaseous phosphorus pentachloride decomposes according to the reaction $$ \mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) $$ The equilibrium system was analyzed at a particular temperature, and the concentrations of the substances present were determined to be \(\left[\mathrm{PCl}_{5}\right]=1.1 \times 10^{-2} \mathrm{M}\) \(\left[\mathrm{PCl}_{3}\right]=0.325 \mathrm{M},\) and \(\left[\mathrm{Cl}_{2}\right]=3.9 \times 10^{-3} M .\) Calculate the value of \(K\) for the reaction.

Short answer

The value of the equilibrium constant, \(K\), for the given reaction at this particular temperature is approximately 11.475, as calculated by the formula \(K = \frac{[\mathrm{PCl}_3][\mathrm{Cl}_2]}{[\mathrm{PCl}_5]} = \frac{(0.325)(3.9 \times 10^{-3})}{(1.1 \times 10^{-2})}\).

Step-by-step solution

Write down the balanced chemical equation

Write down the balanced chemical equation for the reaction given. \[ \mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \]

Write down the expression for the equilibrium constant

Write down the expression for the equilibrium constant, K, for the reaction. K is the ratio of the product concentrations to the reactant concentrations, each raised to the power of their respective coefficients in the balanced chemical equation. \[ K = \frac{[\mathrm{PCl}_{3}][\mathrm{Cl}_{2}]}{[\mathrm{PCl}_{5}]} \]

Plug in the given concentrations

Plug in the given equilibrium concentrations of PCl5, PCl3, and Cl2 into the expression for the equilibrium constant. \[ K = \frac{(0.325)(3.9 \times 10^{-3})}{(1.1 \times 10^{-2})} \]

Calculate the value of K

Multiply and divide the numbers to find the value of K. \[ K = \frac{(0.325)(3.9 \times 10^{-3})}{(1.1 \times 10^{-2})} = 11.475 \] The value of the equilibrium constant, K, for the reaction at this particular temperature is approximately 11.475.

Key concepts

Chemical Equilibrium

Understanding the nature of chemical equilibrium is crucial when it comes to predicting how a chemical reaction proceeds. Chemical equilibrium occurs in a closed system when the rates of the forward and reverse reactions are equal, leading to no observable changes in the amounts of reactants and products over time. Despite this apparent standstill, the reaction is dynamically ongoing, with reactants converting to products and vice versa at the same rate.

For the equation
\[ \mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g) + \mathrm{Cl}_{2}(g) \],
the point of equilibrium is reached when the decomposition of phosphorus pentachloride to phosphorus trichloride and chlorine proceeds at the same rate as their recombination to form phosphorus pentachloride. Equilibrium doesn’t mean the concentrations are equal, but rather that they remain constant over time.

Reaction Quotient

The reaction quotient, denoted as Q, is a measure that tells us how far a system is from reaching equilibrium at any given moment during a reaction. Like the equilibrium constant, K, it is calculated using the same expression involving the reactant and product concentrations. However, Q can be determined at any point during the reaction, not just at equilibrium.

The formula for the reaction quotient is identical to the equilibrium constant expression:
\[ Q = \frac{[\mathrm{PCl}_{3}][\mathrm{Cl}_{2}]}{[\mathrm{PCl}_{5}]} \]
By comparing the value of Q to the equilibrium constant K, predictions can be made about which direction the reaction will proceed to reach equilibrium. If Q < K, the forward reaction is favored; if Q > K, the reverse reaction is favored; and if Q = K, the system is already at equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle provides a qualitative understanding of how a system at equilibrium responds to changes in concentration, temperature, or pressure. It states that when a stress is applied to a system at equilibrium, the system adjusts to minimize that stress and restore equilibrium.

If the concentration of a substance in the system is increased, the system responds by favoring the reaction that consumes that substance. Conversely, if the concentration of a substance is decreased, the reaction that produces it is favored. In the context of our example with phosphorus pentachloride, if additional PCl5 were added to the equilibrium mixture, the system would respond by forming more PCl3 and Cl2, consequently shifting the position of equilibrium.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of reactants and products in a reaction mixture when the reaction has reached equilibrium. These concentrations remain constant over time as long as the system is undisturbed. Determining these values is vital for calculating the equilibrium constant K, which quantifies the position of equilibrium.

In the case of the decomposition of PCl5, the equilibrium concentrations are given as:
\[ [\mathrm{PCl}_{5}] = 1.1 \times 10^{-2} \text{M} \]
\[ [\mathrm{PCl}_{3}] = 0.325 \text{M} \]
\[ [\mathrm{Cl}_{2}] = 3.9 \times 10^{-3} \text{M} \]
These concentrations are plugged into the equilibrium expression to calculate K, thus indicating the extent to which a reaction proceeds before achieving equilibrium.