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) \(\left(\frac{x}{(x-1)}\right) \mathrm{P}\) (d) \(\left(\frac{x}{(1-x)}\right) \mathrm{P}\)

Step-by-Step Solution

Answer

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

Step 1: Define Initial Conditions

Initially, consider that the number of moles of \( \text{PCl}_5 \) is 1 mole, and at the start, the moles of \( \text{PCl}_3 \) and \( \text{Cl}_2 \) are zero.

Step 2: Account for Degree of Dissociation

Let the degree of dissociation of \( \text{PCl}_5 \) be \( x \). This means \( x \) moles of \( \text{PCl}_5 \) dissociate into \( x \) moles of \( \text{PCl}_3 \) and \( x \) moles of \( \text{Cl}_2 \).

Step 3: Calculate Equilibrium Moles

At equilibrium, the moles of \( \text{PCl}_5 \) = \( 1 - x \), moles of \( \text{PCl}_3 \) = \( x \), and moles of \( \text{Cl}_2 \) = \( x \).

Step 4: Write Total Moles at Equilibrium

The total number of moles at equilibrium is \( (1 - x) + x + x = 1 + x \).

Step 5: Express Pressure in Terms of Partial Pressure

The total pressure \( P \) is related to the total number of moles at equilibrium. Therefore, the partial pressure of \( \text{PCl}_3 \) is proportional to its mole fraction: \( \frac{x}{1+x} \times P \).

Step 6: Compare With Given Options

From Step 5, the partial pressure of \( \text{PCl}_3 \) is \( \left( \frac{x}{1+x} \right) P \). The correct option resembling this calculation is (a).

Key Concepts

Phosphorus Pentachloride

Phosphorus pentachloride (PCl\(_5\)) is an important chemical compound used in various industrial applications. It is a pale yellow crystalline solid that sublimates, turning into a gas without passing through a liquid phase under atmospheric conditions. One of the notable characteristics of PCl\(_5\) is its ability to undergo a reversible chemical reaction, where it dissociates into phosphorus trichloride (PCl\(_3\)) and chlorine gas (Cl\(_2\)).

This dissociation process is critical in understanding the behavior of PCl\(_5\) in different chemical settings. The equilibrium state of such a reaction is established when the rate of decomposition of PCl\(_5\) equals the rate at which PCl\(_3\) and Cl\(_2\) recombine to form PCl\(_5\) again. This equilibrium is influenced by temperature, pressure, and the degree to which PCl\(_5\) dissociates.

Understanding the dissociation of PCl\(_5\) is essential for predicting the concentration of the gases involved at equilibrium, which is a valuable skill in chemical engineering and industrial chemistry. The knowledge of equilibrium in such a reversible reaction provides insights that are crucial for controlling reaction conditions to favor the production of desired chemicals.

Degree of Dissociation

The degree of dissociation is a vital concept in chemical reactions, particularly in equilibrium reactions involving gases like phosphorus pentachloride. It measures the extent to which a compound separates into its constituent parts in a chemical reaction.

To understand the degree of dissociation for PCl\(_5\), consider the reaction it undergoes: PCl\(_5\) breaks down into PCl\(_3\) and Cl\(_2\). Let's denote the initial moles of PCl\(_5\) as 1 mole. If we assume that 'x' is the degree of dissociation, it implies that 'x' moles of PCl\(_5\) dissociate to produce 'x' moles of PCl\(_3\) and 'x' moles of Cl\(_2\).

This means that at equilibrium, the moles of PCl\(_5\) remaining are \(1 - x\), while PCl\(_3\) and Cl\(_2\) each are 'x' moles. The concept of degree of dissociation helps in evaluating how much of the original compound remains un-reacted at equilibrium, and contributes to understanding the dynamics of pressure and concentration in the reaction mixture.

Partial Pressure Calculation

In a gaseous equilibrium reaction involving phosphorus pentachloride, understanding how to calculate the partial pressure of each component is crucial. The partial pressure of a gas is its contribution to the total pressure in a mixture of gases. For a gas like PCl\(_3\), which forms as a result of the dissociation of PCl\(_5\), its partial pressure can be calculated using the concept of mole fraction in combination with the ideal gas law relations.

As the dissociation of PCl\(_5\) occurs, the total moles at equilibrium becomes \(1 + x\), where 'x' is the degree of dissociation. The mole fraction of PCl\(_3\) is then \(\frac{x}{1 + x}\).

To determine the partial pressure of PCl\(_3\), this mole fraction is multiplied with the total pressure P. Therefore, the partial pressure for PCl\(_3\) is given by \[\frac{x}{1 + x}\] \( \times P \). This calculation is fundamental in predicting how the pressure of each gas in the system alters with time and helps in adjusting industrial processes to maintain optimal reactions conditions.