Question

\(\Lambda_{\mathrm{AgCl}}^{\infty}\) can be obtained : (a) by extraplotation of the graph \(\Lambda\) and \(\sqrt{C}\) to zero concentration (b) by known values of \(\Lambda^{\infty}\) of \(\mathrm{AgNO}_{3}, \mathrm{HCl}\) and \(\mathrm{HNO}_{3}\) (c) both (a) and (b) (d) None of these

Short answer

\(\Lambda_{\mathrm{AgCl}}^{\infty}\) can be obtained by both (a) extrapolating \(\Lambda\) vs \(\sqrt{C}\) to zero concentration, and (b) using known \(\Lambda^{\infty}\) values of \(\mathrm{AgNO}_{3}, \mathrm{HCl}\), and \(\mathrm{HNO}_{3}\).

Step-by-step solution

Understanding the concept of Molar Conductivity at Infinite Dilution

Molar conductivity at infinite dilution \(\Lambda_{M}^{\infty}\) for an electrolyte is the conductivity value attained when the electrolyte is diluted such that there is no interaction between the ions. It is typically determined by extrapolating the plot of molar conductivity \(\Lambda\) against the square root of concentration \(\sqrt{C}\) to zero concentration.

Using extrapolation to find \(\Lambda_{\mathrm{AgCl}}^{\infty}\)

For option (a), we would draw a graph of \(\Lambda\) versus \(\sqrt{C}\) for \(\mathrm{AgCl}\) and extrapolate to \(\sqrt{C} = 0\) to find \(\Lambda_{\mathrm{AgCl}}^{\infty}\). The y-intercept of this extrapolated line would give us the required molar conductivity at infinite dilution.

Using known values for related substances to find \(\Lambda_{\mathrm{AgCl}}^{\infty}\)

For option (b), the molar conductivity at infinite dilution for \(\mathrm{AgCl}\) can be calculated if we know \(\Lambda^{\infty}\) for \(\mathrm{AgNO}_{3}, \mathrm{HCl}\), and \(\mathrm{HNO}_{3}\). Since \(\mathrm{AgCl}\) is a strong electrolyte, its \(\Lambda^{\infty}\) can be represented as the sum of its constituent ions. The values for \(\mathrm{Ag}^{+}\) can be taken from \(\mathrm{AgNO}_{3}\) and for \(\mathrm{Cl}^{-}\) from \(\mathrm{HCl}\), while subtracting the value for the common ion \(\mathrm{NO}_{3}^{-}\) from \(\mathrm{HNO}_{3}\).

Deciding the correct option

Since both methods (a) and (b) described above are valid approaches to determine \(\Lambda_{\mathrm{AgCl}}^{\infty}\), the correct answer is option (c): both (a) and (b).

Key concepts

Electrolyte Conductivity

Electrolyte conductivity is a measure of a solution's ability to conduct electricity, which depends on the presence of ions that can move to conduct electric current. These ions originate from dissolved electrolytes, which are substances that dissociate into ions when they dissolve in a solvent such as water.

In simple terms, think of electrolyte conductivity as the ease with which electricity can flow through a liquid. This conductivity increases with the concentration of ions up to a certain limit. However, beyond a certain point, the ions' natural repulsion and hydration shells prevent a further increase in conductivity.

Therefore, to understand an electrolyte's true potential for conducting electricity, scientists refer to the molar conductivity at infinite dilution, denoted as \( \Lambda_{M}^{\infty} \). This is a hypothetical scenario where the concentration of an electrolyte is so low that the ions do not interact with each other, and their movement is only restricted by the solvent molecules.

Extrapolation Technique

Extrapolation is a mathematical technique used to predict, or 'extend', a known data set to infer values beyond the range of the provided data. In the context of molar conductivity, this technique is applied to plot molar conductivity (\( \Lambda \) against the square root of electrolyte concentration (\( \sqrt{C} \)).

As this relationship is typically linear for strong electrolytes at lower concentrations, the line can be extended to intercept the y-axis, which represents the condition of infinite dilution (zero concentration). This y-intercept value gives us the molar conductivity at infinite dilution \( \Lambda_{M}^{\infty} \).

It seems almost like working backward: instead of measuring at zero concentration, which is virtually impossible, scientists measure at various higher concentrations and then use the extrapolation method to arrive at the theoretical zero-concentration value.

Ionic Interaction in Solutions

Ionic interaction becomes increasingly important as we deal with solutions containing ions. In a solution, ions do not exist in isolation but interact with other ions and the solvent molecules. These interactions can affect properties like conductivity.

At a basic level, ions in a solution can attract or repel each other due to their charges, similar to magnets. They also form hydration shells as water molecules align around them due to polarity. These phenomena can lead to the reduced mobility of ions as the solution becomes more concentrated, hence impacting the conductivity.

Understanding these interactions is key when predicting the conductivity behavior of electrolytes in solutions. The concept becomes exceptionally relevant when discussing the molar conductivity at infinite dilution, as it depicts the conductivity of ions free from interactive forces with other ions.