Problem 139 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reversible system \(\mathrm{PCl}_{3}+\mathrm{Cl}_{2} \rightleftharpoons \mathrm{PCl}_{5}\), the relationship between degree of dissociation \((\alpha)\) of \(\mathrm{PCl}_{5}\) and the equilibrium constant \(K_{p}\) of the above equilibrium is (a) \(\frac{\mathrm{K}_{\mathrm{p}}}{\sqrt{\mathrm{P}+\mathrm{K}_{\mathrm{p}}}}\) (b) \(\sqrt{\frac{\mathrm{K}_{\mathrm{p}}}{1+\mathrm{P} \cdot \mathrm{K}_{\mathrm{p}}}}\) (c) \(\frac{1}{\sqrt{\mathrm{P}+\mathrm{K}_{\mathrm{P}}}}\) (d) \(\sqrt{\frac{1}{1+P \cdot K_{P}}}\)

Step-by-Step Solution

Answer

The correct answer is option (a).

Step 1: Write the Initial Reaction and Conditions

Start by writing the balanced chemical equation for the equilibrium reaction given: \[ \text{PCl}_3 + \text{Cl}_2 \rightleftharpoons \text{PCl}_5 \]. Assume that the initial pressure of PCl\(_5\) is \( P \), and initially, the concentrations of \( \text{PCl}_3 \) and \( \text{Cl}_2 \) are zero.

Step 2: Set Up Expressions for Equilibrium Concentrations

Assuming \( \alpha \) is the degree of dissociation of PCl\(_5\), the equilibrium concentrations are: - \([\text{PCl}_5] = P(1 - \alpha)\) - \([\text{PCl}_3] = \alpha P\) - \([\text{Cl}_2] = \alpha P\).

Step 3: Express Equilibrium Constant in Terms of \( \alpha \)

The equilibrium constant \( K_p \) is defined as: \[ K_p = \frac{[\text{PCl}_3] \cdot [\text{Cl}_2]}{[\text{PCl}_5]} = \frac{(\alpha P)^2}{P(1-\alpha)} \]Simplify this to: \[ K_p = \frac{\alpha^2 P}{1-\alpha} \].

Step 4: Solve for \( \alpha \) in Terms of \( K_p \) and \( P \)

From \[ K_p = \frac{\alpha^2 P}{1-\alpha} \], rearranging gives: \[ \alpha^2 P = K_p (1-\alpha) \]. After simplifying, find: \[ \alpha^2 = \frac{K_p}{P + K_p} \]. Thus, \[ \alpha = \sqrt{\frac{K_p}{P + K_p}} \].

Step 5: Match Result with Given Options

The derived expression \( \alpha = \sqrt{\frac{K_p}{P + K_p}} \) matches with option (a): \( \frac{K_{p}}{\sqrt{P + K_{p}}} \).

Key Concepts

Degree of Dissociation

The degree of dissociation, denoted by the symbol \( \alpha \), represents how much a compound, such as a gas or solution, has dissociated into its components. It is particularly useful in understanding chemical reactions reaching equilibrium, especially reversible ones like the equilibrium between phosphorus pentachloride (\( \text{PCl}_5 \)), phosphorus trichloride (\( \text{PCl}_3 \)), and chlorine (\( \text{Cl}_2 \)).

In the case of the reaction \( \text{PCl}_3 + \text{Cl}_2 \rightleftharpoons \text{PCl}_5 \), \( \alpha \) denotes the fraction of \( \text{PCl}_5 \) that has dissociated into \( \text{PCl}_3 \) and \( \text{Cl}_2 \). Put simply, if \( \alpha = 0.1 \), it means 10% of \( \text{PCl}_5 \) has dissociated while the remaining 90% has not.

The degree of dissociation helps to establish a connection between individual species' concentrations and the overall equilibrium constant, providing insight into reaction progress and conditions at equilibrium. An increase in \( \alpha \) typically implies greater dissociation, changing the pressures and concentrations involved in the equilibrium state.

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, resulting in a stable condition where the concentrations of reactants and products remain constant over time.

In the reaction \( \text{PCl}_3 + \text{Cl}_2 \rightleftharpoons \text{PCl}_5 \), equilibrium is reached when the rates of the forward reaction \( \text{PCl}_3 + \text{Cl}_2 \to \text{PCl}_5 \) and the reverse reaction \( \text{PCl}_5 \to \text{PCl}_3 + \text{Cl}_2 \) are equal.

At equilibrium, the equilibrium constant \( K_p \) can be expressed in terms of partial pressures of the components in the gas phase, revealing how favorably products or reactants are formed. The constants like equilibrium constant are critical for calculating the position of equilibrium and assessing reaction yield. It helps determine reactant conversion levels and predict how changes in conditions (like pressure and temperature) affect the system.

Reaction Kinetics

Reaction kinetics is the study of the rates of chemical processes, including the factors that influence these rates.

In our reversible system with \( \text{PCl}_3 \), \( \text{Cl}_2 \), and \( \text{PCl}_5 \), kinetics focuses on how quickly equilibrium is achieved and what factors, such as temperature or catalyst presence, might change the speed of reaching equilibrium.

Understanding kinetics allows chemists to manipulate reaction conditions to achieve desired outcomes faster. For example, an increase in temperature generally increases reaction rates, speeding up the time to reach equilibrium. In some reactions, catalysts are used to lower activation energy requirements, allowing reactions to proceed more swiftly.

By studying reaction kinetics, students can predict how long a reaction will take to reach equilibrium and how to optimize conditions to achieve the best reaction efficiency.