Animated Solution for Chemistry - Electrochemistry: The molar conductivities ${\Lambda }_{\text{NaOAc}}^{\circ }$ and ${\Lambda }_{\text{HCl}}^{\circ }$ at infinite dilution in water at $25^{\circ }\text{C}$ are $91.0$ and $426.2{\text{ S cm}}^2/\text{mol}$, respectively. To calculate ${\Lambda }_{\text{HOAc}}^{\circ }$, the additional value required is

Imagine you are a chemist tasked with finding the limiting molar conductivity of acetic acid ($\text{HOAc}$). You set up your conductivity cell, prepare a series of increasingly dilute solutions, and plot the molar conductivity (${\Lambda }_m$) against the square root of concentration ($\sqrt{C}$).

For strong electrolytes like $\text{NaCl}$ or $\text{HCl}$, this plot yields a beautiful, predictable straight line. You simply grab a ruler, extrapolate the line to the y-axis (where concentration is zero), and boom—you have your limiting molar conductivity (${\Lambda }_m^{\circ }$).

But acetic acid is a weak electrolyte. It stubbornly refuses to fully dissociate. As you dilute it, the degree of dissociation shoots up exponentially, causing the graph to curve sharply upwards, running almost parallel to the y-axis. It never touches! Direct extrapolation is mathematically impossible.

Are we defeated? Not at all. Enter the genius of Friedrich Kohlrausch.

The Master Key

Kohlrausch discovered a profound truth about ions at infinite dilution: they stop caring about their partners. At zero concentration, the ions are so far apart that inter-ionic attractions vanish completely.

This led to Kohlrausch's Law of Independent Migration of Ions, which states that the limiting molar conductivity of an electrolyte is simply the sum of the individual contributions of its constituent ions.

Mathematically, for our elusive acetic acid, this means:

This equation is our target. If we can somehow find the individual values for the ${\text{H}}^{+}$ ion and the ${\text{OAc}}^{-}$ ion, we can just add them up. But how do we get those individual values? We borrow them from strong electrolytes!

The problem provides us with the limiting molar conductivities of two strong electrolytes: sodium acetate ($\text{NaOAc}$) and hydrochloric acid ($\text{HCl}$). Because they are strong electrolytes, their ${\Lambda }_m^{\circ }$ values are easily measured experimentally.

Let's write out their Kohlrausch equations:

Look closely at these two equations. Do you see the pieces of our target? $\text{HCl}$ contains the ${\text{H}}^{+}$ ion we need, and $\text{NaOAc}$ contains the ${\text{OAc}}^{-}$ ion we need.

What happens if we add these two equations together? Let's do the algebra:

Rearranging the terms to group our desired ions together:

We have successfully created our target, $({\lambda }_{{\text{H}}^{+}}^{\circ }+{\lambda }_{{\text{OAc}}^{-}}^{\circ })$, which is exactly ${\Lambda }_{\text{HOAc}}^{\circ }$.

However, there is a catch. We have also generated some unwanted "spectator" ions: ${\text{Na}}^{+}$ and ${\text{Cl}}^{-}$.

To isolate the molar conductivity of acetic acid, we must mathematically eliminate the extra $({\lambda }_{{\text{Na}}^{+}}^{\circ }+{\lambda }_{{\text{Cl}}^{-}}^{\circ })$.

What compound is made of exactly one ${\text{Na}}^{+}$ ion and one ${\text{Cl}}^{-}$ ion? Sodium chloride ($\text{NaCl}$)!

According to Kohlrausch's Law:

Therefore, to perfectly balance our equation and remove the spectator ions, we simply subtract the limiting molar conductivity of $\text{NaCl}$.

The final, elegant algebraic expression becomes:

This reveals that to calculate the value for acetic acid, the additional value required is ${\Lambda }_{\text{NaCl}}^{\circ }$.

This method is a cornerstone of electrochemistry. By cleverly adding and subtracting the easily measurable properties of strong electrolytes, we can unlock the hidden properties of weak electrolytes. It is a beautiful demonstration of how algebraic logic can overcome physical experimental limitations.