Problem 205 - A Complete Resource Book in Chemistry for JEE Main

Question

The equivalent conductance of \(\mathrm{NaCl}\) at concentration \(\mathrm{C}\) and at infinite dilution are \(\lambda_{\mathrm{C}}\) and \(\lambda_{m}\), respectively. The correct relationship between \(\lambda_{\mathrm{C}}\) and \(\lambda_{\infty}\) is given as: (where the constant B is positive) (a) \(\lambda_{\mathrm{C}}=\lambda_{\infty}-(\mathrm{B}) \sqrt{\mathrm{C}}\) (b) \(\lambda_{\mathrm{c}}=\lambda_{\infty}+(\mathrm{B}) \sqrt{\mathrm{C}}\) (c) \(\lambda_{\mathrm{C}}=\lambda_{m}+(\mathrm{B}) \mathrm{C}\) (d) \(\lambda_{\mathrm{C}}=\lambda_{-\infty}-(\mathrm{B}) \mathrm{C}\)

Step-by-Step Solution

Answer

(a) \( \lambda_{C} = \lambda_{\infty} - (B) \sqrt{C} \)

Step 1: Understand Equivalent Conductance

Equivalent conductance (\( \lambda \)) describes how well ions conduct electricity in solution, and it depends on concentration \( C \) of the electrolyte and a term known as limiting or infinite dilution conductance (\( \lambda_{\infty} \)).

Step 2: Kohlrausch's Law Explanation

According to Kohlrausch's Law for strong electrolytes, the equivalent conductance at any concentration \( C \) is related to the conductance at infinite dilution. The law states \( \lambda_{C} = \lambda_{\infty} - B \sqrt{C} \), where \( B \) is a constant valid for NaCl.

Step 3: Analyze the Options

We analyze each option for correctness in terms of Kohlrausch's Law. Option (a) fits \( \lambda_{C} = \lambda_{\infty} - B \sqrt{C} \), which follows directly from the law. This suggests option (a) demonstrates the correct relationship.

Step 4: Verify the Correctness Based on Theory

We verify that option (a) aligns with theoretical expectations for how conductance decreases as concentration increases due to ion pair formation at higher concentrations. Therefore, \( \lambda_{C} \) should decrease as \( C \) increases, matching the negative relationship in option (a).

Key Concepts

Kohlrausch's Law

Kohlrausch's Law is a fundamental principle in electrochemistry that deals with the behavior of strong electrolytes in solution. This law gives us a way to quantify how the conductance of ions changes with concentration. More specifically, it describes how the equivalent conductance of an ion pair in solution decreases as the concentration of the solution increases. This behavior stems from the increased interactions between ions at higher concentrations. Kohlrausch's Law states that the equivalent conductance (\( \lambda_{C} \)) at a certain concentration (\( C \)) is related to the conductance at infinite dilution (\( \lambda_{\infty} \)), using the formula:\[\lambda_{C} = \lambda_{\infty} - B \sqrt{C}\]where \( B \) is a constant that depends on the specific electrolyte. This relationship reflects the fact that as ions are more closely packed together, there are more opportunities for ion-pair interactions, which in turn reduces their mobility and the conductance. This mathematical expression helps to quantify how ions behave in varying concentrations and is particularly useful for understanding strong electrolytes like sodium chloride (\( NaCl \)).

Strong Electrolytes

Strong electrolytes are substances that are completely dissociated into ions when dissolved in a solvent, typically water. This means that they produce a significant number of ions, enhancing the solution's ability to conduct electricity. Examples include salts like sodium chloride (\( NaCl \)), potassium nitrate (\( KNO_3 \)), and strong acids such as hydrochloric acid (\( HCl \)).Characteristics of Strong Electrolytes:

• Complete ionization in solution.
• High electrical conductivity.
• Dependable relationships as predicted by Kohlrausch's Law.

Due to their complete dissociation, the conductance behavior of strong electrolytes can be reliably predicted by Kohlrausch's Law. This law describes how even though a strong electrolyte is fully dissociated, the interaction between ions becomes more pronounced as concentration increases, thereby affecting conductivity. This makes strong electrolytes particularly interesting for the study of electrolytic conductivity and distinguishing their behavior from weak electrolytes.

Limiting Conductance

Limiting conductance, also known as conductance at infinite dilution, is a key point of reference in the study of electrolyte solutions. It represents the maximum conductance value that an electrolyte solution can achieve. At infinite dilution, the ions are so spaced out that they behave independently without interacting with each other. Important Attributes:

• Occurs when the concentration approaches zero.
• Indicates maximum ion mobility.
• Utilized as a baseline for comparing conductance at higher concentrations.

This concept is essential for understanding the effects of concentration on conductance. For example, at limiting conductance, ions don't hinder each other's movement, so they travel freely throughout the solution, resulting in high conductivity. This is why limiting conductance serves as a critical parameter in Kohlrausch's Law, providing a benchmark against which the effects of concentration on ion mobility and conductance can be measured. Studying limiting conductance allows us to isolate the intrinsic conductance ability of ions without the influence of concentration-dependent interactions.

Electrolyte Solution Conductivity

Electrolyte solution conductivity is the measure of a solution's ability to conduct electricity, which occurs due to the movement of charged ions. The greater the number of ions and their mobility, the higher the conductivity of the solution. This property is vital for applications across chemistry, physics, and engineering. Key Factors Affecting Conductivity:

• Concentration of ions: More ions usually mean higher conductivity.
• Type of ions: Different ions conduct electricity to varying degrees.
• Temperature: Increasing temperature generally increases conductivity due to enhanced ion mobility.

Conductivity not only depends on these factors but is also influenced by whether an electrolyte is strong or weak. In strong electrolytes, complete dissociation produces many ions, boosting conductivity. In contrast, weak electrolytes only partially dissociate, resulting in lower conductivity. This is why understanding electrolyte solution conductivity is crucial for tasks such as quality control, determining solution purity, and other analytical purposes.