Problem 76 - A Complete Resource Book in Chemistry for JEE Main

Question

The equivalent conductance of \(\mathrm{Ba}^{2+}\) and \(\mathrm{Cl}^{-}\)are respectively 127 and \(76 \mathrm{ohm}^{-1} \mathrm{~cm}^{2}\) equiv \(^{-1}\) at infinite dilution. The equivalent conductance (in \(\mathrm{ohm}^{-1} \mathrm{~cm}^{2}\) equiv \(^{-1}\) ) of \(\mathrm{BaCl}_{2}\) at infinite dilution will be (a) \(139.5\) (b) 203 (c) 279 (d) \(101.5\)

Step-by-Step Solution

Answer

The equivalent conductance of \(\text{BaCl}_2\) at infinite dilution is 279 \(\mathrm{ohm}^{-1}\,\mathrm{cm}^2\,\mathrm{equiv}^{-1}\).

Step 1: Understand the Concept of Equivalent Conductance

Equivalent conductance at infinite dilution, represented by \(\Lambda^0\), is the sum of the equivalent conductances of the constituent ions of the electrolyte when the solution is sufficiently dilute such that there is no interaction between the ions. In mathematical terms: \(\Lambda^0 = \lambda^0_{ion 1} + \lambda^0_{ion 2} + ... \lambda^0_{ion n}\).

Step 2: Identify the Ions in BaCl2

In the compound \(\text{BaCl}_2\), the ions present are \(\text{Ba}^{2+}\) and two \(\text{Cl}^-\) ions. Since each \(\text{BaCl}_2\) dissociates into one \(\text{Ba}^{2+}\) ion and two \(\text{Cl}^-\) ions, their contributions to the equivalent conductance need to be considered in these proportions.

Step 3: Substitute the Known Values

The given equivalent conductances are: \(\lambda^0_{\text{Ba}^{2+}} = 127 \ \text{ohm}^{-1} \ \text{cm}^2 \ \text{equiv}^{-1}\) and \(\lambda^0_{\text{Cl}^-} = 76 \ \text{ohm}^{-1} \ \text{cm}^2 \ \text{equiv}^{-1}\).

Step 4: Calculate the Equivalent Conductance of BaCl2

Using the formula \(\Lambda^0_{\text{BaCl}_2} = \lambda^0_{\text{Ba}^{2+}} + 2 \times \lambda^0_{\text{Cl}^-}\), substitute the known values: \(\Lambda^0_{\text{BaCl}_2} = 127 + 2 \times 76\).

Step 5: Perform the Calculations

First, calculate the contribution from the \(\text{Cl}^-\) ions: \(2 \times 76 = 152\). Then add the contribution of \(\text{Ba}^{2+}\): \(127 + 152 = 279\). The equivalent conductance of \(\text{BaCl}_2\) at infinite dilution is therefore \(279 \ \text{ohm}^{-1} \ \text{cm}^2 \ \text{equiv}^{-1}\).

Key Concepts

Equivalent Conductance

Equivalent conductance is an important concept in electrochemistry, especially when analyzing how substances behave in a solution. When considering equivalent conductance, we often think about how much charge can move through the solution because of its ions. This is particularly useful when you want to understand solutions of electrolytes at different concentrations.

To break it down further, imagine equivalent conductance (\( \Lambda \)) as a measure of how well ions can carry an electric current. It's expressed in units like \(\text{ohm}^{-1} \text{cm}^{2} \text{equiv}^{-1}\).

• It represents the conductance of all ions produced by the dissociation of one equivalent of the electrolyte at infinite dilution.
• At infinite dilution, it's noticeable that ions are far apart and thus do not interact or hinder each other's movement, making the conductance measurements reliable and straightforward.

Therefore, by understanding equivalent conductance, you get insights into how effectively ions manage to move through a solution, which is crucial in predicting electrolyte behavior under diverse conditions.

Infinite Dilution

Infinite dilution is a term that often comes up in discussions about equivalent conductance. It refers to a hypothetical state where the concentration of a solute is so low that its ions are effectively non-interacting. This situation allows each ion to exhibit its own full potential of charge transport without interference from other ions.

When chemists talk about infinite dilution:

• They assume that all interactions between ions can be neglected. This means ions have ideal mobilities, exhibiting maximum conductance capabilities.
• Important calculations can be simplified to determine intrinsic properties like equivalent conductance.

Consider it like having each ion moving freely amidst a vast ocean of solvent, unaffected by the presence of other similar or charged entities. This is why measurements at this point are integral to understanding the intrinsic properties of ions, which helps in constructing accurate models of electrolyte behavior in real-world applications.

Ionic Contributions

In the realm of electrochemistry, it's crucial to understand ionic contributions. When a compound like \( \text{BaCl}_2 \) gets dissolved into a solution, it splits into its respective ions, namely \(\text{Ba}^{2+} \) and \(\text{Cl}^- \). Each ion plays its part in determining the overall conductive behavior of the solution at infinite dilution.

Keep in mind:

• The contributions of each type of ion to the overall conductance depend on their concentration and individual mobilities in the solution.
• For example, \(\text{BaCl}_2 \) dissolves to provide one \(\text{Ba}^{2+} \) ion and two \(\text{Cl}^- \) ions. The chloride ions will each have their own contribution, multiplied by two due to their presence in twos, to the final conductance.

This is foundational for the calculation of equivalent conductance, as understanding the specific role of each ion allows us to predict and assess the conductive properties of solutions accurately in laboratory and industrial processes.