Animated Solution for Chemistry - Electrochemistry: The limiting molar conductivities ${\Lambda }^{\circ }$ for NaCl, KBr and KCl are $126$, $152$ and $150{\text{ S cm}}^2{\text{ mol}}^{-1}$, respectively. The ${\Lambda }^{\circ }$ for NaBr is

Imagine you are trying to measure how fast a single person can run on a track. If the track is crowded with thousands of other runners, they will bump into each other, slow each other down, and you won't get a true measure of their maximum potential speed.

In electrochemistry, ions behave the same way. In a concentrated solution, positive and negative ions attract and drag each other down. But what happens if we dilute the solution infinitely? The ions get so far apart that they no longer feel each other's presence. They move with absolute freedom. This state is called infinite dilution, and the conductivity measured here is the limiting molar conductivity, denoted by ${\Lambda }^{\circ }$.

Friedrich Kohlrausch made a groundbreaking discovery: at infinite dilution, every ion migrates independently of its co-ion. This means the total limiting molar conductivity of an electrolyte is simply the sum of the individual contributions of its cation and anion.

Mathematically, this is expressed as:

Where ${\lambda }_{+}^{\circ }$ and ${\lambda }_{-}^{\circ }$ are the limiting molar conductivities of the individual cations and anions, respectively.

Think of ions as Lego blocks. We want to build a specific structure: Sodium Bromide ($\text{NaBr}$). To do this, we need a Sodium block (${\text{Na}}^{+}$) and a Bromide block (${\text{Br}}^{-}$).

However, we don't have these blocks individually. We only have pre-assembled structures: 1. Sodium Chloride ($\text{NaCl}$) which gives us ${\text{Na}}^{+}$ and ${\text{Cl}}^{-}$. 2. Potassium Bromide ($\text{KBr}$) which gives us ${\text{K}}^{+}$ and ${\text{Br}}^{-}$. 3. Potassium Chloride ($\text{KCl}$) which gives us ${\text{K}}^{+}$ and ${\text{Cl}}^{-}$.

To get our desired $\text{NaBr}$ structure, we can play a clever algebraic game.

First, let's take the $\text{NaCl}$ structure to get our ${\text{Na}}^{+}$ block:

Next, let's add the $\text{KBr}$ structure to get our ${\text{Br}}^{-}$ block:

Now we have the ${\text{Na}}^{+}$ and ${\text{Br}}^{-}$ blocks we want, but we also have unwanted ${\text{K}}^{+}$ and ${\text{Cl}}^{-}$ blocks. Notice that these unwanted blocks perfectly make up a $\text{KCl}$ structure!

So, to isolate $\text{NaBr}$, we simply subtract the $\text{KCl}$ structure:

Now, it's just a matter of plugging in the numerical values provided in the problem:

Doing the simple arithmetic:

This elegant algebraic manipulation is not just a neat trick for strong electrolytes; it is the only way to determine the limiting molar conductivity of weak electrolytes (like acetic acid), which do not fully dissociate even at high dilutions. By using strong electrolytes as our 'Lego blocks', we can indirectly calculate the properties of weak ones!