Question

Molar conductivity of a solution of an electrolyte \(A B_{3}\) is \(150 \mathrm{Scm}^{2} \mathrm{~mol}^{-1}\). If it ionises as \(\mathrm{AB}_{3} \longrightarrow A^{3+}+3 B^{-}\), its equivalent conductivity will be : (a) \(150\left(\right.\) in \(\mathrm{Scm}^{2} \mathrm{eq}^{-1}\) ) (b) 75 (in \(\mathrm{Scm}^{2} \mathrm{eq}^{-1}\) ) (c) 50 (in \(\mathrm{Scm}^{2} \mathrm{eq}^{-1}\) ) (d) 80 (in \(\mathrm{Scm}^{2} \mathrm{eq}^{-1}\) )

Short answer

The equivalent conductivity of the electrolyte \( AB_{3} \) is 50 (in \( \text{Scm}^2 \text{eq}^{-1} \)).

Step-by-step solution

Understanding Molar Conductivity

Molar conductivity, represented as \( \Lambda_m \), is defined as the conductivity of a solution corresponding to a concentration of one mole per volume of solution. It's given that the molar conductivity of \( AB_{3} \) is \( 150 \: \text{Scm}^2 \text{mol}^{-1} \).

Understanding Equivalent Conductivity

Equivalent conductivity, represented as \( \Lambda_{eq} \), is the conductance of all the ions produced by one equivalent of an electrolyte in solution. It is related to molar conductivity by the valency of the ion, because one equivalent of the electrolyte is that amount which provides one mole of charge (ions).

Calculating Equivalent Conductivity

Since the electrolyte ionises as \( AB_{3} \longrightarrow A^{3+} + 3 B^{-} \), the valency of the cation \( A \) is 3. Hence, molar conductivity is divided by the charge number (valency) to get equivalent conductivity. \[ \Lambda_{eq} = \frac{\Lambda_m}{z} \] Here, \( z = 3 \) (the valency), so: \[ \Lambda_{eq} = \frac{150 \: \text{Scm}^2 \text{mol}^{-1}}{3} \]

Solving for Equivalent Conductivity

Divide the given molar conductivity by the valency of the cation to get the equivalent conductivity. \[ \Lambda_{eq} = \frac{150}{3} = 50 \: \text{Scm}^2 \text{eq}^{-1} \]

Key concepts

Molar Conductivity

Molar conductivity, often represented by the symbol \( \Lambda_m \), is a measurement of how well an electrolyte conducts electricity when there is exactly one mole of the electrolyte dissolved in a volume of solution such that the total volume of the solution is one liter. It offers a way to compare the conductivity of different electrolytes without the complication of varying concentrations.

In electrochemistry, conducting behavior greatly depends on the number of ions present and the solution's ability to allow those ions to move. As the concentration of ions in solution changes, so does the molar conductivity. Typically, as we dilute a solution, the molar conductivity increases because the ions have more room to move and there is less likelihood of them being nearby to recombine. In the exercise provided, a molar conductivity of \( 150 \text{ S cm}^2 \text{ mol}^{-1} \) suggests quite a good conductive behavior of the electrolyte \( AB_3 \), as for every mole of it dissolved in a liter of solution, we have a substantial measure of conductance.

Ionisation of Electrolytes

Ionisation is the process by which molecules or atoms become charged ions. Electrolytes are compounds which, when dissolved in water, disassociate into positively and negatively charged ions. This disassociation or ionisation of electrolytes is crucial in electrochemistry, as it provides the charge carriers, or ions, that move in the solution to conduct electricity.

In the context of our exercise, the electrolyte \( AB_3 \) ionises to produce one \( A^{3+} \) ion and three \( B^{-} \) ions. This ionisation process is pivotal as it determines the number of charge carriers in solution. In terms of equivalent conductivity, the valency of the resulting ions is essential as it directly informs us about the proportion between the quantity of substance and the amount of charge it provides. The higher the valency of the ions produced, the more charge is carried per mole of particles dissolved, and thus, the equivalence concept becomes fundamental in understanding the relationship between molar and equivalent conductivity.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electrical energy and chemical changes. It covers various phenomena including batteries, corrosion, and electrolysis. One of its core concepts is the study of conductance in electrolyte solutions.

The conductance of an electrolyte solution is its ability to conduct an electric current, which directly depends on the ion concentration and the mobility of these ions. This is where terms like molar and equivalent conductivity come into play. Molar conductivity tells us about the conductive behavior when there's one mole of electrolyte per liter of solution, while equivalent conductivity allows us to normalize this conductive behavior based on the charge of ions an electrolyte provides upon ionisation.

As students learn about electrochemistry, it's essential to grasp these conductivity concepts to understand how and why solutions behave the way they do when current is applied. The problem-solving approach taken for calculating equivalent conductivity from molar conductivity in the exercise is a prime example of how mathematical relationships in electrochemistry can be applied to practical situations.