Animated Solution for Chemistry - Solutions: The elevation of boiling point of $0.10\text{ m}$ aqueous ${\text{CrCl}}_3\cdot x{\text{NH}}_3$ solution is two times that of $0.05\text{ m}$ aqueous ${\text{CaCl}}_2$ solution. The value of $x$ is ....... . [Assume 100% ionisation of the complex and ${\text{CaCl}}_2$, coordination number of Cr as 6, and that all ${\text{NH}}_3$ molecules are present inside the coordination sphere]

Imagine you are a chemical detective, and you are handed two mysterious vials. The first vial contains a simple, well-known salt: calcium chloride. The second vial is where the real mystery lies—a coordination complex of chromium with an unknown number of ammonia molecules. Your mission is to deduce the exact structure of this complex using nothing but the boiling points of the two solutions. Let's dive into this fascinating intersection of colligative properties and coordination chemistry.

When you dissolve a non-volatile solute in a solvent, the boiling point of the solution increases. This phenomenon is known as the elevation in boiling point, and it is governed by a beautifully simple equation:

Here, $\Delta T_b$ is the elevation in boiling point, $m$ is the molality of the solution, and $K_b$ is the molal elevation constant of the solvent. But the real star of the show in this problem is $i$, the van't Hoff factor. This factor tells us exactly how many particles a single formula unit of the solute breaks into when it dissolves. For electrolytes, this number is crucial because colligative properties depend strictly on the number of particles, not their identity.

Let's start with the known entity: the $0.05\text{ m}$ aqueous ${\text{CaCl}}_2$ solution. We know that calcium chloride is a strong electrolyte that completely dissociates in water:

One calcium ion and two chloride ions give us a total of 3 ions. Therefore, the van't Hoff factor for ${\text{CaCl}}_2$ is $i=3$. Plugging this into our boiling point elevation formula, we get:

Now, let's turn our attention to the mysterious chromium complex, ${\text{CrCl}}_3\cdot x{\text{NH}}_3$. We are given that its molality is $0.10\text{ m}$. Since we don't yet know how it dissociates, let's represent its van't Hoff factor simply as $i$. Its boiling point elevation is:

The problem provides a critical clue: the elevation of the complex solution is exactly two times that of the calcium chloride solution. Let's set up the equation:

Notice how elegantly the solvent constant $K_b$ cancels out from both sides. We are left with a simple algebraic equation:

Solving for $i$, we find that $i=3$. This is a massive breakthrough! It tells us that our mysterious chromium complex dissociates into exactly 3 ions in solution.

What does a van't Hoff factor of 3 mean for a coordination complex? It means that when the complex dissolves, it breaks apart into one large complex cation and two simple counter anions. Since the only available anions are chloride ions, there must be two chloride ions outside the coordination sphere acting as counter ions.

This leaves one chloride ion trapped inside the coordination sphere, directly bonded to the central chromium atom. The general formula of our complex is now taking shape:

The final piece of the puzzle is the coordination number, which the problem states is 6. The coordination number is the total number of ligand attachment points to the central metal atom. Inside our square brackets, we currently have one chloride ligand. To reach a total of 6 ligands, the remaining 5 spots must be occupied by the ammonia molecules.

Therefore, $x$ must be equal to 5. The complete, true identity of our complex is $[\text{Cr}({\text{NH}}_3{)}_5\text{Cl}]{\text{Cl}}_2$, known as pentaamminechloridochromium(III) chloride. The value of $x$ is 5.