Problem 197 - A Complete Resource Book in Chemistry for JEE Main

Question

Phosphorous pentachloride dissociates as follows, in a closed reaction vessel \(\mathrm{PCI}_{5}(\mathrm{~g}) \longrightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\) If total pressure at equilibrium of the reaction mixture is \(\mathrm{P}\) and degree of dissociation of \(\mathrm{PCl}_{5}\) is \(\mathrm{x}\), the partial pressure of \(\mathrm{PCl}_{3}\) will be (a) \(\left(\frac{x}{(x+1)}\right) \mathrm{P}\) (b) \(\left(\frac{2 x}{(x-1)}\right) \mathrm{P}\) (c) \(\left(\frac{x}{(x-1)}\right) P\) (d) \(\left(\frac{x}{(1-x)}\right) \mathrm{P}\)

Step-by-Step Solution

Answer

a. \( \left(\frac{x}{(x+1)}\right) \mathrm{P} \)

Step 1: Understand the Reaction

The reaction we are considering is the dissociation of phosphorus pentachloride (PCl₅) into phosphorus trichloride (PCl₃) and chlorine gas (Cl₂). This is represented by the equation: \( \text{PCl}_5 (\text{g}) \rightarrow \text{PCl}_3 (\text{g}) + \text{Cl}_2 (\text{g}) \).

Key Concepts

Phosphorus Pentachloride Dissociation

Phosphorus pentachloride, denoted as PCl extsubscript{5}, is a compound known for its role in various chemical reactions, especially in industries dealing with chlorination processes. When PCl extsubscript{5} is subjected to heat in a closed vessel, it dissociates into phosphorus trichloride (PCl extsubscript{3}) and chlorine gas (Cl extsubscript{2}). This process of breaking down into simpler substances is known as dissociation.

The chemical reaction can be expressed as follows: \( \text{PCl}_5 (\text{g}) \rightarrow \text{PCl}_3 (\text{g}) + \text{Cl}_2 (\text{g}) \).

The reaction signifies that one mole of PCl extsubscript{5} breaks down to form one mole each of PCl extsubscript{3} and Cl extsubscript{2}. This indicates a 1:1:1 molar relationship, essential for calculating pressures in equilibrium reactions. This dissociation usually happens at elevated temperatures and in a closed system to ensure that the equilibrium is established.

• The reaction is characterized by a dynamic balance where the rate of dissociation equals the rate of recombination.
• The extent to which this dissociation occurs is pivotal for understanding the degree of dissociation and calculating partial pressures.

Degree of Dissociation

The degree of dissociation, generally represented as \( x \), is a measure of the fraction of the original substance (PCl extsubscript{5} in our case) that has dissociated into its components at equilibrium. It is expressed as the ratio of the amount dissociated to the original amount.

• If \( x \) is the degree of dissociation of PCl extsubscript{5}, then **1-x** would be the fraction that remains undissociated.
• The greater the value of \( x \), the higher the proportion of PCl extsubscript{5} that has dissociated.

Consider the situation where the initial moles of PCl extsubscript{5} is 1 mole. After dissociation, \( x \) moles of PCl extsubscript{3} and \( x \) moles of Cl extsubscript{2} are formed, while the remaining \( (1-x) \) moles are PCl extsubscript{5}.
Consequently, the mole balance at equilibrium in the reaction will be:

• PCl extsubscript{5}: \( 1-x \) moles
• PCl extsubscript{3}: \( x \) moles
• Cl extsubscript{2}: \( x \) moles

Understanding the degree of dissociation provides insight into the extent and direction of chemical reactions, crucial for thermodynamic calculations.

Partial Pressure Calculation

Partial pressure is the pressure that a particular gas in a mixture would exert if it occupied the entire volume alone at the same temperature. In this dissociation reaction of phosphorus pentachloride, calculating the partial pressures of gases involved is essential for understanding the equilibrium state.

At equilibrium, the total pressure of the system \( P \) consists of the pressures due to PCl extsubscript{5}, PCl extsubscript{3}, and Cl extsubscript{2}.
Assuming an initial mole of PCl extsubscript{5}, after dissociation the partial pressures are given by applying Dalton's Law and the ideal gas equation as follows:

• PCl extsubscript{5}: The partial pressure is \( (1-x) \cdot P/(1+x) \).
• PCl extsubscript{3}: Since it forms \( x \) moles, its partial pressure is \( x \cdot P/(1+x) \).
• Cl extsubscript{2}: It similarly follows PCl extsubscript{3} with \( x \cdot P/(1+x) \).

In many cases, students may encounter multiple-choice questions requiring the identification of correct expressions for partial pressures. Recognizing the relationship between the degree of dissociation and partial pressures is crucial. The formula \( \frac{x}{(1-x)} \cdot P \) corresponds to option (d) from the exercise, applicable for PCl extsubscript{3}'s partial pressure.