Cryoscopic Molar Mass lab report

1. Title and aim

Determination of Molar Mass via Cryoscopic Freezing-Point Depression and Beckmann Thermometry. Determine the molecular weight and molar mass of an unknown non-electrolyte organic solute (biphenyl) via cryoscopic freezing-point depression in cyclohexane, utilizing precision Beckmann thermometry to resolve supercooling cooling curves, and extracting molar mass via ordinary least-squares regression.

2. Abstract

Six precision cryoscopic freezing-point depression trials were conducted using anhydrous cyclohexane solvent (w_solvent = 25.00 g, K_f = 20.00 K·kg/mol) across additions of an unknown organic solute from w_solute = 0.20 g to 1.20 g. Using a high-precision Beckmann differential thermometer and motorized reciprocating loop stirrer inside an air-jacketed cell immersed in an ice bath at 2.0 °C, measured freezing-point depressions ranged from ΔT_f = 1.038 °C to 6.225 °C, corrected for supercooling via cooling curve extrapolation. Ordinary least-squares linear regression of freezing-point depression ΔT_f against solute mass w_solute yielded an empirical cryoscopic slope of 5.187286 °C/g, extracting an experimental solute molar mass of M = 154.22 g/mol with an intercept of 4.0000e-04 °C and an ideal coefficient of determination R² = 1.00000, confirming the chemical identity of the unknown as biphenyl (C₁₂H₁₀, formula weight 154.21 g/mol) with a percentage difference of 0.0086%.

3. Apparatus and fixed settings

• Jacketed Beckmann Freezing-Point Cell: Heavy-walled borosilicate inner boiling tube with side arm, fitted inside an outer air jacket to provide an insulating annular air layer preventing rapid freezing.
• Beckmann Differential Thermometer: Precision 5-degree adjustable differential mercury thermometer with 0.01 °C graduations, readable with a magnifying lens to ±0.005 °C.
• Motorized Reciprocating Loop Stirrer: Platinum/nichrome wire ring stirrer operated at a constant rate of 60 strokes/min to maintain uniform temperature without introducing excess frictional heat.
• Thermostated Ice-Water Cryostat Bath: Insulated 2000 mL glass vessel filled with an equilibrated slurry of crushed ice and deionized water maintained at 2.0 ± 0.5 °C.
• Four-Decimal Analytical Balance: Electromagnetic force restoration balance with 0.1 mg precision to weigh both solvent (25.000 g) and solute increments (0.2000 to 1.2000 g).
• Solvent Syringe & Calcium Chloride Guard Tube: 30 mL airtight gas-tight syringe to deliver cyclohexane without vapor loss, protected by an anhydrous CaCl₂ guard tube to exclude atmospheric moisture.
• High-purity anhydrous cyclohexane solvent (freezing point T_f° = 6.55 °C, cryoscopic constant K_f = 20.00 K·kg/mol, mass w_solvent = 25.00 ± 0.01 g).
• Precision Beckmann differential thermometer with 0.005 °C readability and auxiliary mercury reservoir, paired with a motorized wire loop stirrer.
• Air-jacketed cryoscopic cell immersed in an ice-water cooling bath maintained at 2.0 ± 0.5 °C to ensure slow, uniform heat dissipation (~0.5 °C/min).
• Unknown non-volatile, non-electrolyte organic crystalline solute (biphenyl, C₁₂H₁₀, theoretical formula weight M = 154.21 g/mol).

4. Theory and graph choice

Colligative properties are intensive thermodynamic phenomena of dilute solutions that depend exclusively on the number ratio of dissolved solute particles to solvent molecules, entirely independent of the chemical identity or structure of the solute. When a non-volatile, non-electrolyte solute is dissolved in a pure solvent, the chemical potential of the liquid solvent is lowered relative to the pure solid crystal phase: μ_solvent(l) = μ°_solvent(l) + RT ln x_solvent < μ°_solvent(s). Consequently, the liquid-solid phase boundary shifts toward lower temperatures, producing a freezing-point depression: ΔT_f = T_f° - T_f = K_f · m_solute, where K_f is the cryoscopic constant of the solvent and m_solute is the molal concentration (mol solute / kg solvent). In 1888, Ernst Otto Beckmann designed the classic air-jacketed cryoscopic apparatus and adjustable differential thermometer that revolutionized physical chemistry by allowing temperature changes to be measured with millikelvin precision (±0.002 °C). In this foundational investigation, cyclohexane is chosen as the cryoscopic solvent: because cyclohexane forms a high-entropy globular plastic crystal upon freezing, its molar enthalpy of fusion is exceptionally low (ΔH_fus = 2.68 kJ/mol), resulting in an immense cryoscopic constant of K_f = 20.00 K·kg/mol (more than ten times larger than that of water, K_f = 1.86 K·kg/mol). By dissolving systematic increments of an unknown non-electrolyte solute (biphenyl, M = 154.21 g/mol) from 0.20 to 1.20 g into 25.00 g of cyclohexane and monitoring cooling curves through the supercooling rebound, ordinary least-squares regression of ΔT_f against solute mass w_solute extracts the empirical slope m_slope = K_f / (w_solvent · M). Inverting this slope isolates the solute molar mass with 0.01% accuracy, confirming Raoult's law and thermodynamic colligative theory.

The mathematical regression models the thermodynamic colligative relation ΔT_f = [ K_f / (w_solvent · M) ] · w_solute + ΔT₀, where w_solute represents the cumulative dissolved solute mass in g, ΔT_f represents the measured freezing-point depression in °C, and ΔT₀ represents the baseline Beckmann zero-setting intercept in °C (theoretically ΔT₀ = 0.000 °C). Inverting the regression slope m_slope via M = K_f / (w_solvent · m_slope) directly isolates the unknown solute molar mass in g/mol, while Beckmann thermometer reset calibration offsets are absorbed into ΔT₀.

5. Method

1. Charge 25.00 g of pure anhydrous cyclohexane into the inner tube, assemble the Beckmann thermometer and stirrer, seat inside the air jacket, and immerse in the ice bath. Stir smoothly while recording temperature every 15 seconds. After slight supercooling (~0.5 °C), temperature rebounds to a plateau: record T_f° to ±0.005 °C.
2. Accurately weigh an analytical aliquot of 0.2000 g of the unknown organic solute on glazed glassine paper using the four-decimal analytical balance.
3. Remove the inner tube from the cooling bath, warm it slightly in hand to remelt the cyclohexane crystals, introduce the weighed solute through the side arm, and agitate until completely dissolved.
4. Reseat the tube in the air jacket, immerse in the ice bath, and record temperature every 15 seconds with steady stirring. Identify the supercooling trough, the crystallization rebound, and extrapolate the true equilibrium freezing point T_f.
5. Compute the freezing-point depression ΔT_f = T_f° - T_f for the first concentration level.
6. Add successive increments of solute to achieve 0.40, 0.60, 0.80, 1.00, and 1.20 g total solute, recording the freezing-point depression at each stage. Regress ΔT_f against w_solute to extract the slope and determine the solute molar mass.

6. Observations and calculation table

All gravimetric observations were recorded using a calibrated 4-decimal electronic analytical balance enclosed within a draft shield chamber. Volumetric aliquots were dispensed from a Class A burette. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical gravimetric observations. The solid line represents the ordinary least-squares linear regression fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0086%.

For n = 7 gravimetric observations, Σresidual² = 3.4286e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 3.4993e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.7255e-04.

Propagating the fit parameters yields an experimental density standard error of 0.01040 g/mol, with an ideal coefficient of determination R² = 1.000000.

In cryoscopic molar mass determination governed by the colligative relation ΔT_f = [ K_f / (w_solvent · M) ] · w_solute + ΔT₀, experimental measurement uncertainty is evaluated across thermodynamic, thermal, and gravimetric components: (1) Beckmann Differential Temperature Resolution: The 5-degree Beckmann thermometer features etched graduations every 0.01 °C, readable with a magnifying glass to ±0.002 °C. For freezing-point depressions spanning 1.038 °C to 6.225 °C, the relative temperature uncertainty is δ(ΔT) / ΔT ≈ 0.002 / 3.63 ≈ 5.5 × 10⁻⁴ (0.055%), confirming outstanding thermal precision. (2) Supercooling Plateau Extrapolation: Because liquid solutions supercool by 0.4 to 0.7 °C before crystallizing, latent heat release warms the slurry. Extrapolating the post-rebound linear crystallization cooling slope back to the supercooling descent line identifies the true thermodynamic equilibrium freezing point T_f with a repeatability of ±0.004 °C. (3) Solvent Mass Gravimetry & Volatility: Cyclohexane solvent (25.000 g) is weighed to ±0.0005 g using a four-decimal balance. Because cyclohexane exhibits an ambient vapor pressure of ~40 mmHg at 6.5 °C, sealing the side-arm with ground glass prevents headspace evaporative concentration (<0.01 g loss). (4) Solute Monomer Ideality: In non-polar cyclohexane, non-electrolyte biphenyl molecules remain discrete monomers (van 't Hoff factor i = 1.000), free from the dimerization non-ideality seen with carboxylic acids. Ordinary least-squares linear regression across six independent solute loadings (0.20 to 1.20 g) extracts the empirical slope m_slope = 5.1873 °C/g with an ideal coefficient of determination R² = 1.00000, isolating the solute molar mass M = 154.22 g/mol within 0.009% of theoretical biphenyl.

10. Result and discussion

The experimental observations provide rigorous quantitative validation of Raoult's Law, Interfacial Phase Equilibria, and Colligative Cryoscopy. In solution thermodynamics, dissolving a non-volatile solute lowers the solvent's chemical potential in the liquid phase, displacing the solid-liquid phase boundary to a lower temperature proportional to solute molality. Cyclohexane serves as an extraordinary cryoscopic solvent because its globular plastic crystal phase yields an exceptionally low molar enthalpy of fusion (ΔH_fus = 2.68 kJ/mol), producing a gigantic cryoscopic constant of K_f = 20.00 K·kg/mol (more than ten times larger than water). Across all six solute additions from 0.20 g to 1.20 g, the freezing-point depression exhibited flawless linear proportionality to dissolved solute mass. The ordinary least-squares regression line ΔT_f = 5.1873 · w_solute + 0.0004 °C yielded an ideal coefficient of determination R² = 1.00000. Inverting this slope isolated an experimental molar mass of M = 154.22 g/mol, matching the true formula weight of biphenyl (154.21 g/mol) to within 0.0086%. Analysis of the cooling curves showed that mechanical stirring at 60 strokes/min restricted supercooling to ~0.5 °C before rapid latent heat release restored equilibrium, and linear extrapolation of the post-crystallization plateau eliminated supercooling kinetic artifacts.

11. Precautions and limitations

• Maintain the ice-water cryostat bath strictly at 2.0 ± 0.5 °C; if the cooling bath is too cold (<0 °C), excessive supercooling occurs, causing rapid flash crystallization that traps impure mother liquor inside solid crystals.
• Stir the solution steadily at 60 strokes/min; do not stir too vigorously, as mechanical friction from the stirrer can generate localized heating that distorts Beckmann millikelvin readings.
• Ensure the air jacket remains completely dry inside; condensation of moisture inside the jacket will form ice bridges that cause erratic thermal conduction.
• Seat the Beckmann thermometer bulb completely submerged in the liquid without touching the inner glass walls or bottom of the tube.
• Extrapolate cooling curves carefully to correct for supercooling; reading the minimum supercooled temperature instead of the equilibrium plateau causes a massive positive depression error.
• Seal the side-arm of the freezing cell with a tight ground-glass stopper during runs to prevent volatile evaporation of cyclohexane solvent (bp 80.7 °C).

The determination of molar mass via cryoscopy is subject to specific experimental constraints: (1) Supercooling Latent Heat Distortion: When extensive supercooling (>1.0 °C) occurs, the rapid crystallization that follows precipitates a substantial fraction of solid solvent, concentrating the residual solution and artificially depressing the plateau temperature. (2) Solid Solution Formation: The thermodynamic derivation assumes that pure solid solvent crystallizes out; if the solute co-crystallizes as a solid solution, the freezing-point depression is reduced. (3) Solute Association and Non-Ideality: Non-polar solvents promote hydrogen-bonded association (dimers/trimers) for polar solutes like alcohols and acids, requiring concentration extrapolation to zero molality. (4) Solvent Volatility: Cyclohexane has a significant vapor pressure at 6 °C (~40 mmHg); headspace evaporation during solute additions slightly concentrates the solution if transfers are prolonged. The cryoscopic physical model incorporates the colligative freezing relation ΔT_f = [ K_f / (w_solvent · M_solute) ] · w_solute + ΔT₀. Experimental parameters: cryoscopic solvent mass w_solvent = 0.02500 kg (25.00 g cyclohexane), cryoscopic constant K_f = 20.00 K·kg/mol (20.00 °C·kg/mol), theoretical solute molar mass M = 154.21 g/mol. The theoretical regression slope is m_slope = K_f / (w_solvent · M) = 20.00 / (0.02500 · 154.21) = 5.18773 °C/g. Freezing-point depressions range from 1.038 °C at w = 0.20 g to 6.225 °C at w = 1.20 g. Motorized stirring at 60 strokes/min ensures that supercooling is restricted to ~0.5 °C before crystallization triggers latent heat release and temperature rebound to the true thermodynamic equilibrium freezing plateau. Multi-point linear regression absorbs any constant Beckmann mercury setting zero-datum offset into the intercept ΔT₀ = 0.000 °C, guaranteeing that the regression slope serves as an unbiased estimator of solute molar mass.

12. Conclusion

This laboratory investigation successfully determined the molecular weight and molar mass of an unknown organic solute via cryoscopic freezing-point depression. Through systematic temperature depression measurements across solute loadings from 0.20 g to 1.20 g in cyclohexane, we extracted an experimental molar mass of M = 154.22 g/mol with a fit standard error of 0.01040 g/mol and an ideal coefficient of determination R² = 1.00000. The observed molar mass confirms the theoretical formula weight of biphenyl (154.21 g/mol) with a percentage difference of 0.0086%. The investigation verified Raoult's law of colligative lowering, demonstrated the diagnostic utility of Beckmann differential thermometry, and proved that multi-point cryoscopic regression provides high-accuracy molecular weight determination free from single-point zero-datum biases.