Introduction & Context

The Freezing Point Depression calculation is a fundamental thermodynamic analysis used in Process Engineering, particularly within the food and dairy industries, and it also serves as a basis for eutectic point determination, which identifies the temperature at which a mixture solidifies as a single phase.

In the context of a scraped‑surface heat exchanger (ice cream freezer), this calculation is critical for establishing the lower boundary for the refrigerant temperature. By identifying the initial freezing point—often determined from water activity measurements—engineers can ensure an adequate thermal driving force for efficient heat transfer and controlled crystallization. For a detailed method on deriving the freezing point from water activity, consult the related guide.

Methodology & Formulas

The calculation relies on the colligative property of freezing point depression, where the presence of solute particles lowers the chemical potential of the solvent. The step-by-step physics are defined as follows:

1. Calculate Moles of Solute: For each component i, the molar quantity is determined by the mass of the solute divided by its molecular weight:

\[ n_{i} = \frac{m_{i}}{MW_{i}} \]

2. Adjust for Dissociation: To account for the number of particles in solution, the moles are multiplied by the van't Hoff factor (i), which represents the number of ions or particles produced per formula unit:

\[ n_{total} = \sum \left( n_{i} \cdot i_{i} \right) \]

3. Compute Molality: The molality (m) of the solution is the ratio of total particle moles to the mass of the solvent (water) in kilograms:

\[ m = \frac{n_{total}}{m_{water}} \]

4. Determine Freezing Point Depression: The depression of the freezing point (\(\Delta T_{f}\)) is calculated using the cryoscopic constant for water (\(K_{f}\)), and this value can then be used in the unfrozen water content calculation to assess the residual liquid present in frozen systems.

\[ \Delta T_{f} = K_{f} \cdot m \]

5. Calculate Initial Freezing Point: The final freezing point (\(T_{f}\)) is the shift from the pure solvent freezing point (0°C):

\[ T_{f} = 0 - \Delta T_{f} \]

Regime	Condition	Applicability
Ideal Dilute	\(m \leq 2.0\ \text{mol/kg}\)	Linear model (\(K_{f} \cdot m\)) is valid.
Concentrated	\(m > 2.0\ \text{mol/kg}\)	Model invalid; requires activity-based correction.
Solvent Presence	\(m_{water} > 0\)	Required for valid molality calculation.

The cryoscopic constant, denoted as \(K_f\), is an intrinsic property of the solvent. To determine it for your process calculations, you should:

Consult established thermodynamic databases or chemical engineering handbooks for the specific solvent.
Calculate it using the formula \(K_f = \dfrac{R \cdot T_f^2 \cdot M}{1000 \cdot \Delta H_{fus}}\), where \(R\) is the gas constant, \(T_f\) is the freezing point of the pure solvent, \(M\) is the molar mass of the solvent, and \(\Delta H_{fus}\) is the enthalpy of fusion.
Verify that the units for enthalpy and molar mass are consistent with the SI system to ensure accuracy in your process model.

The standard equation \(\Delta T_f = K_f \cdot m\) is an approximation that assumes ideal solution behavior. Process engineers must account for the following limitations:

It is only accurate for dilute solutions where the solute concentration is low.
It does not account for non-ideal interactions between solute and solvent molecules.
It assumes the solute is non-volatile and does not dissociate or associate in the solvent.
For high-concentration industrial streams, you must apply activity coefficients to correct for deviations from ideality.

If your solute dissociates in the solvent, such as an electrolyte in water, the freezing point depression will be greater than predicted by the standard molality calculation. To adjust your process calculations:

Incorporate the van't Hoff factor, denoted as \(i\), into your equation: \(\Delta T_f = i \cdot K_f \cdot m\).
Determine the value of \(i\) based on the degree of dissociation for the specific solute at the operating temperature.
Note that for strong electrolytes, \(i\) is approximately equal to the number of ions produced per formula unit.

Worked Example: Freezing Point Depression of a Simple Ice Cream Mix

A standard soft-serve ice cream mix contains 15% w/w sucrose dissolved in water. The objective is to estimate the temperature at which the first ice crystals form, which sets the minimum refrigerant temperature for the scraped-surface freezer. Assume the solution is dilute and ideal, with no other solutes present.

Knowns:

Cryoscopic constant for water, \(K_f = 1.86\ \text{°C·kg/mol}\)
Mass of sucrose (per 1 kg mix), \(m_{\text{sucrose}} = 0.150\ \text{kg}\)
Molar mass of sucrose, \(M_{\text{sucrose}} = 0.3423\ \text{kg/mol}\)
Mass of water, \(m_{\text{water}} = 0.850\ \text{kg}\)
Van’t Hoff factor for sucrose (non-electrolyte), \(i = 1.0\)

Step-by-Step Calculation:

Moles of sucrose: \[ n_{\text{sucrose}} = \frac{m_{\text{sucrose}}}{M_{\text{sucrose}}} = \frac{0.150}{0.3423} = 0.4382\ \text{mol} \]
Total moles of particles after dissociation: Since sucrose does not dissociate, the total particle moles equal the moles of sucrose. \[ n_{\text{total}} = n_{\text{sucrose}} \cdot i = 0.4382 \cdot 1.0 = 0.4382\ \text{mol} \]
Molality of the solution: Molality is the moles of particles per kilogram of solvent. \[ m = \frac{n_{\text{total}}}{m_{\text{water}}} = \frac{0.4382}{0.850} = 0.5155\ \text{mol/kg} \]
Freezing point depression: Apply the linear relation for dilute solutions. \[ \Delta T_f = K_f \cdot m = 1.86 \cdot 0.5155 = 0.9588\ \text{°C} \approx 0.959\ \text{°C} \]
Initial freezing point: The freezing point of pure water is 0°C. The depression lowers it below zero. \[ T_f = 0 - \Delta T_f = -0.959\ \text{°C} \]

Final Answer: The initial freezing point of the ice cream mix is -0.959 °C. This value is within the typical range for simple soft-serve mixes and provides the necessary refrigerant temperature target for the freezer.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle