Question

In the reaction \(\mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\), the equilibrium concentrations of \(\mathrm{PCl}_{5}\) and \(\mathrm{PCl}_{3}\) are \(0.4\) and \(0.2\) mole/litre respectively. If the value of \(\mathrm{K}_{\mathrm{c}}\) is \(0.5\), what is the concentration of \(\mathrm{Cl}_{2}\) in mole/litre: (a) \(2.0\) (b) \(1.5\) (c) \(1.0\) (d) \(0.5\)

Short answer

The concentration of \(\mathrm{Cl}_{2}\) is 1.0 mole/litre (option c).

Step-by-step solution

Write the Equilibrium Expression

The equilibrium constant expression for the reaction \(\mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\) is given by:\[ K_c = \frac{[\mathrm{PCl}_{3}][\mathrm{Cl}_{2}]}{[\mathrm{PCl}_{5}]} \]where \([\mathrm{PCl}_{3}]\), \([\mathrm{Cl}_{2}]\), and \([\mathrm{PCl}_{5}]\) are the equilibrium concentrations of \(\mathrm{PCl}_{3}\), \(\mathrm{Cl}_{2}\), and \(\mathrm{PCl}_{5}\) respectively.

Substitute the Known Values

From the problem, we have these concentrations: \([\mathrm{PCl}_{5}] = 0.4\), \([\mathrm{PCl}_{3}] = 0.2\), and \(K_c = 0.5\). Plug these values into the equilibrium expression to solve for \([\mathrm{Cl}_{2}]\):\[ 0.5 = \frac{0.2 \times [\mathrm{Cl}_{2}]}{0.4} \]

Solve for \([\mathrm{Cl}_{2}]\)

Rearrange the equation to solve for \([\mathrm{Cl}_{2}]\):\[ [\mathrm{Cl}_{2}] = \frac{0.5 \times 0.4}{0.2} \]Simplify this equation to find \([\mathrm{Cl}_{2}]\).

Calculation

Calculate the value of \([\mathrm{Cl}_{2}]\):\[ [\mathrm{Cl}_{2}] = \frac{0.5 \times 0.4}{0.2} = \frac{0.2}{0.2} = 1.0 \]Thus, the concentration of \(\mathrm{Cl}_{2}\) at equilibrium is 1.0 mole/litre.

Key concepts

Equilibrium Constant (Kc)

The equilibrium constant, often denoted as \( K_c \), is a crucial concept in chemical equilibrium. It provides insight into the position of equilibrium for a given chemical reaction. This constant is calculated at a specific temperature and remains unchanged as long as the temperature remains constant.
For a general reaction of the type, \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant expression is given by:

• \( K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \)

Here, the square brackets denote the concentrations of the respective reactants \( A \) and \( B \), and products \( C \) and \( D \) at equilibrium, measured in moles per liter.
The value of \( K_c \) tells us the extent to which a reaction proceeds to form products at equilibrium. A large \( K_c \) value indicates that the equilibrium lies toward the products, while a small \( K_c \) suggests the equilibrium favors the reactants.

Concentration Calculations

Understanding how to calculate concentrations at equilibrium is essential for working out the amounts of reactants and products present once a reaction has reached equilibrium.
Given the equilibrium expression, you plug in the known concentrations and the \( K_c \) value to solve for the unknown concentration.
In the reaction \( \mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{3} + \mathrm{Cl}_{2} \), assume you know the equilibrium concentrations of \( \mathrm{PCl}_{5} \) and \( \mathrm{PCl}_{3} \) as well as \( K_c \). You can substitute these values into the equilibrium expression to find the concentration of \( \mathrm{Cl}_{2} \).
Using the given data:

• \([\mathrm{PCl}_{5}] = 0.4 \) mole/liter, \([\mathrm{PCl}_{3}] = 0.2 \) mole/liter, and \( K_c = 0.5 \)

You substitute these into the equation and solve for \([\mathrm{Cl}_{2}] \):

• \( 0.5 = \frac{0.2 \times [\mathrm{Cl}_{2}]}{0.4} \)

By rearranging and simplifying, you will find the missing concentration, demonstrating how equilibrium constants relate with concentrations of substances in a chemical reaction.

Equilibrium Expression

An equilibrium expression is a mathematical representation of a chemical reaction at equilibrium, describing how the concentrations of reactants and products relate to each other.
For reversible reactions, this expression helps predict the behavior of the reaction under different conditions.
In the given reaction example, \( \mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{3} + \mathrm{Cl}_{2} \), the equilibrium expression is:

• \( K_c = \frac{[\mathrm{PCl}_{3}][\mathrm{Cl}_{2}]}{[\mathrm{PCl}_{5}]} \)

This expression shows that, to maintain equilibrium, the concentration of products \( [\mathrm{PCl}_{3}] \) and \( [\mathrm{Cl}_{2}] \) multiplied together should align with \([\mathrm{PCl}_{5}]\) divided by the constant \( K_c \).
Each concentration is raised to the power of its stoichiometric coefficient from the balanced chemical equation. Understanding the equilibrium expression grants a deeper understanding of the dynamic nature of chemical equilibria and assists in making informed predictions about reaction behaviors.
This insight is invaluable in fields such as industrial chemistry, where maintaining specific product yields is critical.