To solve the problem, we can use the concept of freezing point depression, which is described by the formula:

$\Delta T_f = K_f \cdot m$

where:

• $\Delta T_f$ is the freezing point depression.
• $K_f$ is the cryoscopic constant (freezing point depression constant) for the solvent.
• $m$ is the molality of the solution.

Given Information

• $\Delta T_f = 1.88 \, \text{K}$
• Mass of compound = $2.177 \, \text{g}$
• Mass of $HgI_2$ = $100.0 \, \text{g}$
• Enthalpy of fusion of $HgI_2$ = $41.6 \, \text{J/g}$
• Melting point of $HgI_2$ = $532 \, \text{K}$

Steps to Solve

1. Calculate the moles of $HgI_2$:

   • The molar mass of $HgI_2$ is calculated as: $M_{HgI_2} = M_{Hg} + 2 \cdot M_{I} = 200.59 \, \text{g/mol} + 2 \cdot 126.90 \, \text{g/mol} = 454.39 \, \text{g/mol}$
   • Moles of $HgI_2$: $n_{HgI_2} = \frac{100.0 \, \text{g}}{454.39 \, \text{g/mol}} \approx 0.220 \, \text{mol}$

2. Calculate the molality of the solution:

   • Since molality $m$ is defined as: $m = \frac{n_{solute}}{mass_{solvent \, (kg)}}$
   • Here, the solute is $HgxIx$ and the solvent is $HgI_2$, which we will treat as the solvent mass in kg: $m = \frac{n_{HgxIx}}{0.1 \, \text{kg}} = \frac{n_{HgxIx}}{0.1}$

3. Calculate the moles of the solute $HgxIx$:

   • The molar mass of $HgxIx$ can be represented as: $M_{HgxIx} = M_H + x \cdot M_I = 1.008 \, \text{g/mol} + x \cdot 126.90 \, \text{g/mol}$
   • Moles of $HgxIx$: $n_{HgxIx} = \frac{2.177 \, \text{g}}{M_{HgxIx}} = \frac{2.177}{1.008 + x \cdot 126.90}$

4. Plug values into the freezing point depression formula:

   • Set $\Delta T_f$ equal to $K_f \cdot m$: $1.88 = K_f \cdot \left(\frac{n_{HgxIx}}{0.1}\right)$
   • Assuming $K_f$ is derived from the enthalpy of fusion: $K_f = \frac{R \cdot T^2}{\Delta H_{fus}} \quad (R = 8.314 \, \text{J/(mol K)}, \Delta H_{fus} = 41.6 \, \text{J/g})$

5. Substituting and solving for $x$:

   • Find $n_{HgxIx}$ in terms of $x$ and set it equal to the previously established relationship. After substituting and simplifying, you can solve for $x$.

Final Calculation

1. Calculate $K_f$ using the values provided:

   • This calculation will provide you with the precise $K_f$ value to plug into the depression formula and help isolate $x$.

2. Isolate and solve for $x$ in the equation set above.

By completing these calculations, you will arrive at the integer value of $x$.

If you have any questions or need further details, feel free to ask! Here are some additional questions you might consider:

1. What is the significance of the cryoscopic constant in this calculation?
2. How would changing the mass of $HgI_2$ affect the freezing point depression?
3. Can you explain how the enthalpy of fusion relates to the freezing point depression?
4. What would happen if more $HgxIx$ were added to the solution?
5. How do real solutions differ from ideal solutions in terms of colligative properties?

Tip: Always ensure to keep track of units during calculations to avoid confusion.