Problem 186 - A Complete Resource Book in Chemistry for JEE Main

Question

The limiting molar conductivities \(\Lambda^{\circ}\) for \(\mathrm{NaCl}, \mathrm{KBr}\) and \(\mathrm{KCl}\) are 126,152 and \(150 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) respectively. The \(\Lambda^{\circ}\) for \(\mathrm{NaBr}\) is (a) \(278 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (b) \(178 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (c) \(128 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (d) \(306 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

178 \(\mathrm{S} \cdot \mathrm{cm}^{2} \cdot \mathrm{mol}^{-1}\)

Step 1: Understand the Problem Statement

The problem asks us to find the limiting molar conductivity, \(\Lambda^{\circ}\), for \(\text{NaBr}\). We are given the limiting molar conductivities for \(\text{NaCl}\), \(\text{KBr}\), and \(\text{KCl}\) as 126, 152, and 150 \( \text{S} \cdot \text{cm}^2 \cdot \text{mol}^{-1} \) respectively.

Step 2: Apply Kohlrausch’s Law of Independent Migration

Kohlrausch’s Law states that the limiting molar conductivity of an electrolyte is the sum of the limiting molar conductivities of its cation and anion. Thus, for \(\text{NaCl}, \Lambda^{\circ}_{\text{NaCl}} = \lambda^\circ_{\text{Na}^+} + \lambda^\circ_{\text{Cl}^-}\). Similarly, for \(\text{KCl}, \Lambda^{\circ}_{\text{KCl}} = \lambda^\circ_{\text{K}^+} + \lambda^\circ_{\text{Cl}^-}\) and for \(\text{KBr}, \Lambda^{\circ}_{\text{KBr}} = \lambda^\circ_{\text{K}^+} + \lambda^\circ_{\text{Br}^-}\).

Key Concepts

Limiting Molar Conductivity

Limiting molar conductivity, \(\Lambda^{\circ}\), is a key concept in understanding how ions move in a solution. It refers to the conductivity of an electrolyte when the concentration approaches zero. At this point, ions exert minimal interactions with each other, allowing us to observe their maximum conductance potential.

This concept is crucial because it helps scientists deduce the individual ionic contributions to the overall conductivity. For instance, in a solution of sodium chloride (NaCl), the limiting molar conductivity can be expressed as the sum of the individual conductivities of sodium ions \(\lambda^{\circ}_{\text{Na}^+}\) and chloride ions \(\lambda^{\circ}_{\text{Cl}^-}\).

A practical application is in determining the \(\Lambda^{\circ}\) of compounds like NaBr using known values for similar compounds such as NaCl, KCl, and KBr. By considering the behavior of ions at infinite dilution, it becomes possible to predict the conductivity of unmeasured electrolytes.

Here are a few key points to remember:

• The lower the concentration, the closer the solution is to its limiting molar conductivity.
• Understanding of \(\Lambda^{\circ}\) assists in evaluating an electrolyte's ability to conduct electricity in its most efficient state.
• It is an essential part of electrochemistry and helps in calculating ionic mobilities.

Electrolyte Conductance

Electrolyte conductance refers to the ability of ions in a solution to carry an electrical current. Conductance is influenced by several factors, including the nature and concentration of the electrolyte, temperature, and the presence of different ions.

The equation for calculating conductance \(G\) is:\[G = \frac{1}{R}\]where \(R\) is the resistance of the electrolyte solution.

In combination with Ohm's law, conductance forms an essential part of understanding how ionic solutions react under the influence of potential differences.

• It is measured in Siemens (S).
• Factors like viscosity of the solvent and the size of ions can affect conductance.
• Higher temperatures generally increase ionic mobility, thus increasing conductance.

Understanding conductance at zero concentration, known as limiting molar conductance, helps identify the intrinsic ability of ions to transport charge independently from any molecule interactions. It provides a clearer picture of how well each ion supports conductivity and assists in comparing different electrolytes' capacities to conduct electricity.

Independent Migration of Ions

The principle of independent migration of ions states that in a dilute solution each ion contributes independently to the conductivity of the solution. This is best described by Kohlrausch's Law, which explains that the total molar conductivity of an electrolyte is the sum of the conductivities of the cations and anions.

Thus, for any electrolyte at zero concentration, the molar conductivity can be written as:\[ \Lambda^{\circ} = \lambda^{\circ}_{\text{cation}} + \lambda^{\circ}_{\text{anion}} \]This independence allows chemists to determine the limiting molar conductivities of ions when combined in various electrolytes.

• The movements of ions are considered separately, accounting for their distinct contributions.
• It makes the calculations of molar conductivity additive, simplifying the process of finding unknown conductivities.

This law not only helps in evaluating the conductivities of uncommon electrolytes but also plays a significant role in studies related to ion transportation and diffusion in solutions. Understanding this concept will help you analyze and approach different electrochemical problems effectively.