Animated Solution for Chemistry - Electrochemistry: The equivalent conductance of NaCl at concentration $C$ and at infinite dilution are ${\lambda }_c$ and ${\lambda }_{\infty }$, respectively. The correct relationship between ${\lambda }_c$ and ${\lambda }_{\infty }$ is given as

Understanding the Debye-Hückel-Onsager Equation

Imagine you are observing a bustling dance floor. When the floor is relatively empty, the dancers can move freely and swiftly across the room. However, as more and more people crowd the floor, they start bumping into each other, and their overall movement slows down.

This is exactly what happens to ions in a solution! In electrochemistry, the ability of ions to carry an electric current is measured by their equivalent conductance (${\lambda }_c$). For a strong electrolyte like $\text{NaCl}$, which dissociates completely into ${\text{Na}}^{+}$ and ${\text{Cl}}^{-}$ ions, you might expect the conductance to remain constant regardless of concentration. But nature has a subtle catch.

As the concentration $C$ of the solution increases, the ions are forced closer together. Because they carry opposite charges, they begin to attract each other. A positive ion moving towards the negative electrode is constantly pulled back by the negative ions surrounding it (the ionic atmosphere). This interionic attraction creates a 'drag' that slows the ions down, causing the equivalent conductance to decrease slightly.

To mathematically describe this phenomenon, scientists Debye, Hückel, and Onsager developed a brilliant empirical relationship known as the Debye-Hückel-Onsager equation:

Let's break down the anatomy of this beautiful equation:

${\lambda }_c$: This is the equivalent conductance at a specific concentration $C$. ${\lambda }_{\infty }$: This is the limiting equivalent conductance. Imagine a solution so infinitely dilute that the ions are miles apart and don't interact at all. This is the theoretical maximum conductance. $B$: This is a constant that depends on the physical properties of the solvent (like its dielectric constant and viscosity) and the temperature. $\sqrt{C}$: The square root of concentration. The theoretical derivation shows that the retarding effect of the ionic atmosphere is proportional to the square root of the ionic strength, not the concentration itself.

When we plot ${\lambda }_c$ against $\sqrt{C}$ for a strong electrolyte, we get a straight line with a negative slope ($-B$) and a y-intercept of ${\lambda }_{\infty }$.

Looking at the options provided in the question, we need to find the exact match for this linear relationship.

Option (c) states: ${\lambda }_c={\lambda }_{\infty }-(B)\sqrt{C}$

This perfectly mirrors our master equation. Therefore, the correct relationship is indeed given by option (c). Always remember, this linear dance is strictly for strong electrolytes; weak electrolytes have a completely different, non-linear rhythm!