Animated Solution for Chemistry - Solutions: If the freezing point of a $0.01$ molal aqueous solution of a cobalt(III) chloride-ammonia complex (which behaves as a strong electrolyte) is $-0.0558^{\circ }\text{C}$, the number of chloride(s) in the coordination sphere of the complex is [${\text{K}}_{\text{f}}$ of water = $1.86{\text{ K kg mol}}^{-1}$]

The journey to solving this problem is a beautiful intersection of physical chemistry and coordination chemistry. We are tasked with finding the structural formula of a coordination complex just by looking at how it affects the freezing point of water. Imagine being a chemical detective, using the macroscopic property of freezing point depression to peek into the microscopic arrangement of atoms!

Let's start by laying out the clues we have. We are given a $0.01$ molal aqueous solution of a cobalt(III) chloride-ammonia complex. The freezing point of this solution is depressed by $0.0558^{\circ }\text{C}$. We also know the molal freezing point depression constant for water, $K_f$, is $1.86{\text{ K kg mol}}^{-1}$.

The most crucial piece of information here is that the complex behaves as a strong electrolyte. This means it completely dissociates into its constituent ions when dissolved in water.

To connect the macroscopic freezing point depression to the microscopic dissociation, we use the formula for the colligative property:

Here, $i$ is the van't Hoff factor, which represents the total number of moles of particles produced per mole of solute dissolved. Since our complex is a strong electrolyte, $i$ will be exactly equal to the number of ions one formula unit breaks into.

Let's substitute our known values into the equation. We have:

Now, we just need to solve for $i$. First, let's multiply the terms on the right side:

So, the equation becomes:

Dividing both sides by $0.0186$, we get:

This is a massive breakthrough! An $i$ value of $3$ tells us that every single molecule of our cobalt complex splits into exactly three separate ions in the solution.

Now comes the coordination chemistry part. A coordination complex consists of a central metal ion surrounded by ligands (the coordination sphere), and potentially some counter ions outside the sphere. When it dissolves, the coordination sphere remains intact as a single complex ion, while the counter ions dissociate.

Since $i=3$, the complex must dissociate into one complex cation and two counter anions.

What are these counter anions? The problem states it's a cobalt(III) chloride-ammonia complex. Ammonia (${\text{NH}}_3$) is a neutral molecule, so it can't be a counter ion. The only possible counter ions are the chloride (${\text{Cl}}^{-}$) ions. Therefore, there must be exactly two chloride ions outside the coordination sphere.

We know the oxidation state of cobalt is $+3$. To balance this $+3$ charge, there must be a total of three chloride ions (each with a $-1$ charge) in the entire formula.

If two of these chloride ions are outside the coordination sphere acting as counter ions, how many are left inside?

Exactly one chloride ion must be trapped inside the coordination sphere, directly bonded to the cobalt atom.

The structural formula of the complex is therefore $[\text{Co}({\text{NH}}_3{)}_5\text{Cl}]{\text{Cl}}_2$. The number of chlorides in the coordination sphere is 1.