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Step-by-Step Solution
Step 1: Write down the known data
• The 0.10 m aqueous solution of CrCl3·xNH3 shows an elevation of boiling point (ΔTb′) that is twice the elevation of boiling point (ΔTb) of a 0.05 m aqueous CaCl2 solution.
• Both solutions are assumed to be 100% ionized.
• Coordination number of Cr is 6 (meaning there are 6 total ligands—ammonia + chloride—around the Cr center in the complex).
Step 2: Calculate the van ’t Hoff factor (i) for CaCl₂
For CaCl2 undergoing 100% ionization:
\text{CaCl}_{2} \rightarrow \text{Ca}^{2+} + 2\,\text{Cl}^{-} \\
\implies i_{\text{CaCl}_{2}} = 1 + 2 = 3
Step 3: Express the boiling point elevation for CaCl₂
The elevation of boiling point ΔTb for a solute is given by:
\Delta T_{b} = i \, K_{b} \, m
Here, for the 0.05 m solution of CaCl2,
\Delta T_{b} = 3 \times K_{b} \times 0.05 = 0.15 \, K_{b}.
Step 4: Write the expression for the complex solution
For the 0.10 m solution of CrCl3·xNH3, let the van ’t Hoff factor be i_{\text{complex}} .
\Delta T_{b}' = i_{\text{complex}} \, K_{b} \times 0.10.
We are given that:
\Delta T_{b}' = 2 \times \Delta T_{b}.
Hence,
i_{\text{complex}} \, K_{b} \times 0.10 = 2 \times 0.15 \, K_{b}.
Step 5: Solve for the van ’t Hoff factor of the complex
Canceling K_{b} and solving for i_{\text{complex}} :
i_{\text{complex}} \times 0.10 = 0.30 \\
\implies i_{\text{complex}} = 3.
Step 6: Relate the van ’t Hoff factor to the coordination complex formula
The coordination number of Cr is 6. So inside the coordination sphere, we have x NH3 molecules and (6 – x) Cl– ions. The overall compound is CrCl3·xNH3.
If the complex ion forms one charged particle, and the remaining chlorides outside the sphere contribute to the total ion count, then the van ’t Hoff factor i_{\text{complex}} is given by:
i_{\text{complex}} = 1 \; (\text{for the complex ion}) + \bigl(\text{number of chloride ions outside the sphere}\bigr).
There are 3 total chlorides in the formula; (6 – x) are inside the sphere, so the number of outside chlorides is:
3 - (6 - x) = x - 3.
Thus,
i_{\text{complex}} = 1 + (x - 3).
We know i_{\text{complex}} = 3 , so:
3 = 1 + (x - 3) \\
3 = x - 2 \\
x = 5.
Step 7: Conclude the value of x
The number of ammonia molecules in the coordination sphere is 5.
