Calculate i, the van’t Hoff factor of a 0.106 M MgSO4 aqueous solution that has a freezing point of -0.30

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Calculate i, the van’t Hoff factor of a 0.106 M MgSO4 aqueous solution that has a freezing point of -0.30 °C. R = 0.082058 Lâ‹…atmâ‹…mol⁻¹⋅K⁻¹. Kf = 1.86 °C/m. Report your answer to TWO places past the decimal.

The Correct Answer and Explanation is:

To calculate the van’t Hoff factor (i), we can use the formula that relates the freezing point depression to the molality of the solution:ΔTf=i×Kf×m\Delta T_f = i \times K_f \times mΔTf​=i×Kf​×m

Where:

  • ΔTf\Delta T_fΔTf​ is the freezing point depression,
  • iii is the van’t Hoff factor (the number of particles into which the solute dissociates),
  • KfK_fKf​ is the cryoscopic constant (freezing point depression constant) of the solvent, and
  • mmm is the molality of the solution.

Step 1: Find the Freezing Point Depression (ΔTf\Delta T_fΔTf​)

The freezing point of pure water is 0°C. Since the freezing point of the solution is -0.30°C, the freezing point depression is:ΔTf=0°C−(−0.30°C)=0.30°C\Delta T_f = 0°C – (-0.30°C) = 0.30°CΔTf​=0°C−(−0.30°C)=0.30°C

Step 2: Calculate Molality (m)

Molality is defined as the moles of solute per kilogram of solvent. We are given the molarity (0.106 M) of MgSO₄. For dilute solutions, molarity and molality are approximately equal, so:m=0.106 mol/Lm = 0.106 \, \text{mol/L}m=0.106mol/L

(Note: The density of the solution is assumed to be close to that of water for simplicity, so molarity and molality are treated as equal here.)

Step 3: Solve for the van’t Hoff Factor (i)

Now, substitute the known values into the freezing point depression equation:ΔTf=i×Kf×m\Delta T_f = i \times K_f \times mΔTf​=i×Kf​×m0.30=i×(1.86 °C/m)×(0.106 mol/kg)0.30 = i \times (1.86 \, \text{°C/m}) \times (0.106 \, \text{mol/kg})0.30=i×(1.86°C/m)×(0.106mol/kg)

Solving for iii:i=0.30(1.86)(0.106)=0.300.1966≈1.52i = \frac{0.30}{(1.86)(0.106)} = \frac{0.30}{0.1966} \approx 1.52i=(1.86)(0.106)0.30​=0.19660.30​≈1.52

Final Answer:

The van’t Hoff factor iii is 1.52.

Explanation:

The van’t Hoff factor reflects the number of particles the solute dissociates into in solution. For MgSO₄, it dissociates into two ions: Mg²⁺ and SO₄²⁻. Ideally, this would give an iii of 2. However, due to ion pairing or slight deviations in ideal dissociation, the experimental value of iii here is 1.52, which suggests that some degree of ion pairing or incomplete dissociation occurs in this specific solution.

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