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

The limiting molar conductivity for \( \mathrm{NaBr} \) is \( 128 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1} \).

Step 1: Understanding Limiting Molar Conductivity

The limiting molar conductivity \( \Lambda^{\circ} \) is the sum of the individual contributions from the anion and cation at infinite dilution. For example, \( \Lambda^{\circ}_{\text{NaCl}} = \lambda^{\circ}_{\text{Na}^+} + \lambda^{\circ}_{\text{Cl}^-} \).

Step 2: Setting Up the Equations

We have the equations for the given substances as follows:1. \( \Lambda^{\circ}_{\text{NaCl}} = \lambda^{\circ}_{\text{Na}^+} + \lambda^{\circ}_{\text{Cl}^-} = 126 \, \text{S cm}^2 \, \text{mol}^{-1} \)2. \( \Lambda^{\circ}_{\text{KBr}} = \lambda^{\circ}_{\text{K}^+} + \lambda^{\circ}_{\text{Br}^-} = 152 \, \text{S cm}^2 \, \text{mol}^{-1} \) 3. \( \Lambda^{\circ}_{\text{KCl}} = \lambda^{\circ}_{\text{K}^+} + \lambda^{\circ}_{\text{Cl}^-} = 150 \, \text{S cm}^2 \, \text{mol}^{-1} \)

Step 3: Isolating Ion Contributions

We can find the contributions of the individual ions by solving two of the equations. Subtract equation 1 from equation 3 to find \( \lambda^{\circ}_{\text{K}^+} - \lambda^{\circ}_{\text{Na}^+} \): \( 150 = \lambda^{\circ}_{\text{K}^+} + \lambda^{\circ}_{\text{Cl}^-} \) \(-126 = -\lambda^{\circ}_{\text{Na}^+} - \lambda^{\circ}_{\text{Cl}^-} \)\( \hline \24 = \lambda^{\circ}_{\text{K}^+} - \lambda^{\circ}_{\text{Na}^+} \)

Step 4: Solving for \( \lambda^{\circ}_{\text{K}^+} \) and \( \lambda^{\circ}_{\text{Na}^+} \)

Use the result from Step 3 and equation 1:- We know \( 126 = \lambda^{\circ}_{\text{Na}^+} + \lambda^{\circ}_{\text{Cl}^-} \).- Assume \( \lambda^{\circ}_{\text{Na}^+} + 24 = \lambda^{\circ}_{\text{K}^+} \). - Solve for \( \lambda^{\circ}_{\text{Na}^+} \) using equation 1 and substitute back to find \( \lambda^{\circ}_{\text{K}^+} \): 1. \( \lambda^{\circ}_{\text{Na}^+} = 126 - \lambda^{\circ}_{\text{Cl}^-} \)Substitute in Step 5.

Step 5: Calculating \( \Lambda^{\circ}_{\text{NaBr}} \)

We use the general formula again:\[ \Lambda^{\circ}_{\text{NaBr}} = \lambda^{\circ}_{\text{Na}^+} + \lambda^{\circ}_{\text{Br}^-} \] Using equations already solved:- \( \lambda^{\circ}_{\text{K}^+} + \lambda^{\circ}_{\text{Br}^-} = 152 \) (Equation 2)- Substituting \( \lambda^{\circ}_{\text{K}^+} = \lambda^{\circ}_{\text{Na}^+} + 24 \) Therefore, \( \Lambda^{\circ}_{\text{NaBr}} = \lambda^{\circ}_{\text{Na}^+} + (\lambda^{\circ}_{\text{K}^+} - \lambda^{\circ}_{\text{Na}^+}) + \lambda^{\circ}_{\text{Br}^-} \) Which simplifies to \( 152 = \lambda^{\circ}_{\text{Na}^+} + \lambda^{\circ}_{\text{Br}^-} \), and solve for the final value using these unknown terms.

Step 6: Final Calculation and Selection

The computed value from substitutions gives \( \Lambda^{\circ}_{\text{NaBr}} \) as \( 152 \) (indicating a mistake here, doubling checked interpretations to validate). Reevaluate values and recompute.Based on correct solve methods, find it's option, but recognize corrections by setting prosthetic calculation accurately it's \( 126 + 152 - 150 = 128 \) implying (c) is correct due to readjustment of errors in values.

Key Concepts

Ionic Conductivity

Ionic conductivity is a fundamental property of ions in solution, which describes how well ions can carry electrical charge. It is essentially a measure of an ion's ability to move through a solution when a voltage is applied. Several factors influence ionic conductivity, including the size of the ions, the viscosity of the solution, and the temperature.

To understand ionic conductivity, it is crucial to know that each type of ion, such as sodium (Na⁺) or chloride (Cl^-), contributes to the overall conductivity of the solution. This is why the individual ionic conductivities, often referred to as \(\lambda^{\circ}\), are considered when calculating the limiting molar conductivity. At infinite dilution, where ion-ion interactions are negligible, each ion's contribution is taken into account to determine the solution's ability to conduct electricity effectively.

Notably, smaller and more mobile ions tend to have higher conductivity because they can move more freely through the solution, contributing more significantly to the overall conductivity. In electrochemistry, understanding ionic conductivity is critical for designing electrolytes in batteries and fuel cells, where efficient charge transport is crucial.

Dilute Solutions

Dilute solutions are those where the concentration of solute particles, such as ions, is relatively low compared to the solvent. In electrochemistry, the behavior of ions in dilute solutions is of particular interest because the interactions between ions are minimal. This minimal interaction allows for a clearer study of each ion's individual contribution to properties like conductivity.

In the context of limiting molar conductivity, we evaluate ionic contributions under the assumption of infinite dilution. This means that any interference from neighboring ions is effectively disregarded. At such dilution, the ions behave independently, and each ion's unique properties, like size and charge, determine their mobility and thus their contribution to overall conductivity.

There are several practical reasons to study dilute solutions. For one, they provide a foundational understanding of more complex systems. Moreover, knowing how substances behave in dilute solutions is vital for industries such as pharmaceuticals and environmental sciences, where precise dilution is often necessary for accurate measurements and reactions.

Electrolytic Conductance

Electrolytic conductance refers to the ability of an electrolyte solution to conduct electricity. When dissolved in a solvent like water, electrolytes dissociate into positive and negative ions, which carry electric current through the solution. Electrolytic conductance depends on several factors, including the type and concentration of ions present, the nature of the solvent, and the temperature.

It's important to note that conductance increases with more ions or higher mobility of ions in the solution. In particular, at lower concentrations (dilute solutions), the total ionic concentration contributes more directly, as ions are further apart and less likely to interact with one another. Therefore, each ion's full potential to conduct can be more fully realized.

Electrolytic conductance is often measured in terms of either conductance (7) or conductivities (
4A). The latter is considered when evaluating limiting molar conductivities, reflecting how well the whole solution can sustain an electrical current, which is vital in processes like electrolysis and other industrial applications. Understanding these concepts is key to developing better electrolyte systems for various applications.