Animated Solution for Chemistry - Electrochemistry: The equation that is incorrect is

The beauty of electrochemistry often lies in the elegant simplicity of its fundamental laws. One such masterpiece is Kohlrausch's Law of Independent Migration of Ions. It tells us a fascinating story about how ions behave when they are completely free from each other's influence.

Imagine a solution so incredibly dilute that the cations and anions are miles apart (on an atomic scale, of course!). At this state of infinite dilution, they don't attract or repel each other. They just migrate independently towards their respective electrodes. Kohlrausch realized that because of this independence, the total limiting molar conductivity of an electrolyte is simply the sum of the individual contributions of its ions.

Mathematically, for an electrolyte $AB$, it is expressed as:

Armed with this powerful tool, let's dissect the options given in the problem to find the imposter!

Let's take a close look at option (a):

We can expand both sides using Kohlrausch's Law. On the left-hand side (LHS), we have:

Notice how the ${\lambda }_m^{\circ }(N{a}^{+})$ terms beautifully cancel each other out! We are left with:

Now, let's do the same for the right-hand side (RHS):

Here, the ${\lambda }_m^{\circ }(K^{+})$ terms cancel out, leaving us with:

Since LHS exactly matches the RHS, this equation is perfectly correct.

Moving on to option (b):

Expanding the LHS:

The chloride ions cancel, giving:

Expanding the RHS:

The bromide ions cancel, giving:

Once again, LHS equals RHS. This equation is also correct.

Now, let's scrutinize option (c):

Let's expand the LHS:

The sodium ions cancel out:

Now, let's expand the RHS:

The bromide ions cancel out:

Look at that! The LHS is ${\lambda }_m^{\circ }(B{r}^{-})-{\lambda }_m^{\circ }(I^{-})$, while the RHS is ${\lambda }_m^{\circ }(K^{+})-{\lambda }_m^{\circ }(N{a}^{+})$. These two expressions are fundamentally different. Therefore, LHS $ eq$ RHS, making this the incorrect equation we were looking for!

Just to be absolutely thorough, let's verify option (d). This equation represents one of the most famous applications of Kohlrausch's Law: finding the limiting molar conductivity of a weak electrolyte (water) using strong electrolytes.

Let's expand the RHS:

Watch the magic happen as the spectator ions cancel out. The ${\lambda }_m^{\circ }(N{a}^{+})$ and ${\lambda }_m^{\circ }(C{l}^{-})$ terms vanish, leaving only:

And what is the sum of the conductivities of $H^{+}$ and $O{H}^{-}$? It is exactly the limiting molar conductivity of water, $({\Lambda }_m^{\circ }{)}_{H_{2}O}$! So, this equation is beautifully correct.

By systematically applying Kohlrausch's Law and expanding each term, we successfully identified that the equation in option (c) is mathematically inconsistent. This problem is a great reminder to always trust the fundamental principles and work through the algebra step-by-step!