Problem 88 - A Complete Resource Book in Chemistry for JEE Main

Question

In the reaction \(\mathrm{PCl}_{5}(\mathrm{~g})=\mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\), the equilibrium concentrations of \(\mathrm{PC} 1_{5}\) and \(\mathrm{PCl}_{3}\) are \(0.4\) and \(0.2\) mole/litre respectively. If the value of \(K\) is \(0.5\), what is the concentration of \(\mathrm{Cl}_{2}\) in mole/ litre? (a) \(2.0\) (b) \(1.5\) (c) \(1.0\) (d) \(0.5\)

Step-by-Step Solution

Answer

[Cl2] = 1.0 mol/L.

Step 1: 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 \( K = \frac{[\mathrm{PCl}_{3}][\mathrm{Cl}_{2}]}{[\mathrm{PCl}_{5}]} \). We are given \( K = 0.5 \), \([\mathrm{PCl}_{3}] = 0.2 \, \text{mol/L} \), and \([\mathrm{PCl}_{5}] = 0.4 \, \text{mol/L}\).

Key Concepts

Equilibrium Constant

In a chemical reaction that has reached equilibrium, the concentrations of reactants and products remain constant over time. The equilibrium constant, represented as \( K \), provides insight into the ratio of the concentrations of the products to the reactants at equilibrium. For the reaction \( \mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g}) + \mathrm{Cl}_{2}(\mathrm{~g}) \), the equilibrium constant expression is derived from the balanced equation. It is given by:

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

Plugging in the known values to find the concentration of one of the unknown species (like \([\mathrm{Cl}_{2}]\)) helps us solve this kind of equilibrium problem. In our problem, \( K = 0.5 \), and we must find \([\mathrm{Cl}_{2}]\), knowing \([\mathrm{PCl}_{3}] = 0.2 \text{ mol/L} \) and \([\mathrm{PCl}_{5}] = 0.4 \text{ mol/L} \). This calculated value provides the concentration of \([\mathrm{Cl}_{2}]\) at equilibrium, demonstrating how the reaction components are balanced at a given temperature and pressure. The equilibrium constant does not depend on the initial concentrations, only on the temperature.

Le Chatelier's Principle

Le Chatelier's Principle is a fascinating concept in chemistry that describes how a system at equilibrium responds to external changes. When a dynamic equilibrium is disturbed by changing conditions, the system will adjust itself to partially counteract the effect of the disturbance and restore a new equilibrium.For instance, if we were to manipulate the concentration of \( \mathrm{PCl}_{5} \), \( \mathrm{PCl}_{3} \), or \( \mathrm{Cl}_{2} \) in the given reaction, Le Chatelier's Principle helps predict the direction the reaction will shift to re-establish equilibrium:

• Adding more of a reactant or product generally shifts the reaction toward utilizing that additional concentration.
• Removing a substance will push the equilibrium to replace the missing component.
• Temperature and pressure can also affect equilibrium, causing shifts explained by endothermic or exothermic reaction properties.

Through understanding Le Chatelier's Principle, we gain insights into controlling and predicting the behavior of reactions under various external changes and designing processes that keep reactions within the desired equilibrium state.

Reaction Quotient

The reaction quotient, symbolized as \( Q \), is a vital concept when studying chemical reactions reaching equilibrium. Unlike the equilibrium constant \( K \), which is calculated only when a reaction is at equilibrium, \( Q \) can be determined at any point during the reaction to assess its progress. The reaction quotient is represented similarly to the equilibrium constant:

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

By calculating \( Q \) and comparing it to \( K \), we can determine the direction the reaction needs to shift to achieve equilibrium:

• If \( Q < K \), the forward reaction will proceed to create more products.
• If \( Q > K \), the reaction will shift backward toward the reactants.
• When \( Q = K \), the system is at equilibrium.

Understanding \( Q \) helps in predicting and implementing control over chemical reactions, ensuring they reach equilibrium efficiently and effectively.