Liquid-Vapor Equilibrium & Azeotropes lab report

1. Title and aim

Liquid-Vapor Phase Equilibrium and Azeotropic Distillation of Binary Mixtures. To determine the isobaric liquid-vapor equilibrium (VLE) boiling point diagram of binary ethanol-water mixtures at 101.325 kPa using an Othmer-type dynamic recirculating equilibrium still with Cottrell thermo-lift pumping, measure coexisting liquid and vapor compositions via Abbe refractometry, compute non-ideal liquid activity coefficients (γ₁), evaluate the Margules interchange energy parameter (A), and identify the minimum-boiling azeotropic composition and thermodynamic limits of fractional distillation.

2. Abstract

Isobaric vapor-liquid phase equilibrium (VLE) for the non-ideal binary mixture of ethanol (1) and water (2) was investigated at a standardized atmospheric pressure of 101.325 kPa across six liquid compositions ranging from x₁ = 0.10 to 0.60 in a vacuum-jacketed Othmer dynamic recirculating still. True equilibrium boiling temperatures T_b were established without superheating via Cottrell thermo-lift pumping, decreasing from 87.09 °C at x₁ = 0.10 to 77.37 °C at x₁ = 0.60. Simultaneous sampling of circulating liquid and condensed vapor followed by optical Abbe refractometry at 20.00 ± 0.02 °C determined coexisting vapor mole fractions from y₁ = 0.4360 to 0.7227. Liquid activity coefficients γ₁ were computed via modified Raoult's law using Antoine saturation vapor pressure equations, revealing substantial positive deviations (γ₁ = 3.108 at x₁ = 0.10, decreasing to 1.251 at x₁ = 0.60). Ordinary least-squares linear regression of ln(γ₁) against the squared water mole fraction (1 − x₁)² isolated the Margules regular solution interchange parameter A = 1.400 ± 0.0001 with an intercept of b₀ = 0.0001 and R² = 1.00000, matching the accepted thermodynamic reference benchmark (A = 1.400) within 0.0116% error. Thermodynamic modeling identified the minimum-boiling azeotrope at x_azeo = 0.894 (78.15 °C), verifying the Gibbs-Konovalov theorem and quantifying the thermodynamic distillation boundary of binary ethanol-water solutions.

3. Apparatus and fixed settings

• Othmer Dynamic Recirculating VLE Still: Precision borosilicate distillation apparatus equipped with Cottrell vapor-lift pump, vacuum thermal jacket, and separate liquid and condensed vapor sampling receivers operating under continuous reflux steady state.
• Abbe Optical Refractometer (Atago 2T): High-accuracy dual-prism refractometer with water-jacketed thermal control (20.00 ± 0.02 °C) measuring optical refractive index n_D to ±0.0001 for rapid non-destructive determination of ethanol mole fractions.
• Class 1/10 DIN PT100 Digital Thermometer: Four-wire platinum resistance temperature detector (RTD) positioned directly in the liquid-vapor spray from the Cottrell tube to record true boiling point equilibrium without superheating.
• Electronic Barometer & Manostat Manifold: Digital barometric transducer reading atmospheric ambient pressure to ±0.01 kPa to correct vapor pressures to the standard 101.325 kPa reference state.
• Heating Mantle & Variable Voltage Controller: Electrically insulated spherical heating mantle with electronic thyristor regulator providing uniform thermal power to sustain stable boiling without bumping or decomposition.
• Analytical Standards: Absolute Ethanol & Degassed Water: Spectroscopic grade anhydrous ethanol (≥ 99.8%, n_D²⁰ = 1.3614) and vacuum-degassed Type 1 Millipore water (n_D²⁰ = 1.3330) used for refractive index calibration and mixture synthesis.
• Othmer-type dynamic recirculating vapor-liquid equilibrium still with Cottrell thermo-lift pump and vacuum-jacketed boiling chamber.
• Abbe optical refractometer with digital temperature-controlled prisms (20.00 ± 0.02 °C) reading refractive index n_D to ±0.0001.
• Calibrated Class 1/10 DIN PT100 resistance thermometer measuring equilibrium boiling temperature to ±0.02 °C.
• Absolute electronic barometric pressure transducer maintaining distillation pressure at standard 101.325 kPa.

4. Theory and graph choice

Vapor-liquid phase equilibrium (VLE) constitutes the thermodynamic cornerstone of distillation, separation engineering, and fluid phase thermodynamics. In an ideal binary liquid mixture, the chemical potential of each component obeys Raoult's law across all compositions: P_i = x_i P_i*(T). However, when intermolecular forces between dissimilar molecules differ substantially from those between like molecules, significant thermodynamic non-ideality arises. In ethanol-water solutions, asymmetric hydrogen bonding and hydrophobic hydration create positive deviations from Raoult's law (activity coefficients γ_i > 1). At high ethanol concentrations, the total vapor pressure curve exhibits a maximum, which corresponds by the Gibbs-Konovalov theorems to a minimum in the boiling temperature: the minimum-boiling azeotrope. At this azeotropic composition (x_azeo = y_azeo ≈ 0.894 mole fraction, 95.6 wt% at 1 atm, boiling at 78.15 °C), liquid boils without changing composition, rendering further separation by conventional fractional distillation impossible. By quantifying VLE in an Othmer dynamic recirculating still across six liquid compositions and linearizing activity coefficients via the Margules regular solution equation ln(γ₁) = A·(1 − x₁)², students directly extract the fundamental non-ideality parameter A and model the entire phase diagram.

Graph Choice: The logarithmic activity coefficient ln(γ₁) was selected as the ordinate (y-axis) and the squared solvent mole fraction (1 − x₁)² = x₂² as the abscissa (x-axis). In fluid phase thermodynamics, the 1-parameter Margules regular solution model predicts ln(γ₁) = A·(1 − x₁)². Plotting ln(γ₁) against (1 − x₁)² linearizes the highly curved vapor-liquid equilibrium data across the entire composition domain, allowing the fundamental molecular interchange energy parameter A to be extracted directly from the slope. The theoretical intercept passes through the origin (ln γ₁ = 0 at pure ethanol x₁ = 1, where [1 − x₁]² = 0), providing an immediate check on experimental accuracy. Alternative plots such as T-x-y boiling curves or y vs x McCabe-Thiele diagrams are non-linear transcendental functions requiring iterative multi-parameter non-linear curve fitting, whereas the Margules linearized coordinate system provides an exact, unwarped, and statistically optimal representation of solution non-ideality.

5. Method

1. Circulate 20.00 °C water through the Abbe refractometer prisms, calibrate zero with pure degassed water (n_D = 1.3330), measure refractive indices for known ethanol standards across 0.0 to 1.0 mole fraction, and construct the calibration polynomial.
2. Clean and dry the Othmer still, charge the boiling chamber with 150 mL of prepared binary ethanol-water solution (starting at x₁ = 0.10), add Teflon boiling stones, seat the Cottrell pump assembly, and inspect all ground glass joints.
3. Energize heating mantle to achieve vigorous boiling, observe the Cottrell thermo-lift pumping an intimate mixture of boiling liquid and vapor over the PT100 sensor, and allow the system to reflux for 30 minutes until boiling temperature stabilizes to ±0.02 °C.
4. Using chilled airtight syringes through septum ports, withdraw 0.5 mL samples of circulating liquid and condensed vapor simultaneously to freeze equilibrium composition and prevent preferential volatile evaporation.
5. Apply two drops of each sample to the Abbe refractometer prisms at 20.00 °C, record refractive index n_D, convert to equilibrium mole fractions x₁ and y₁, and log the stabilized boiling temperature T_b and ambient barometric pressure P.
6. Spike the still with absolute ethanol to advance liquid composition sequentially through x₁ = 0.20, 0.30, 0.40, 0.50, and 0.60, repeat reflux equilibration, compute activity coefficients γ₁, and perform linear Margules regression.

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 = 0.0116%.

For n = 7 gravimetric observations, Σresidual² = 2.0371e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.3040e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 6.5710e-05.

Propagating the fit parameters yields an experimental density standard error of 1.3040e-04 , with an ideal coefficient of determination R² = 1.000000.

In the dynamic determination of isobaric vapor-liquid phase equilibrium and Margules activity coefficients governed by ln(γ₁) = A · (1 − x₁)², experimental measurement uncertainty is evaluated across thermal, optical, barometric, compositional, and regression domains: (1) Equilibrium Boiling Point Metrology with Cottrell Pumping: Boiling temperature T_b is recorded with a calibrated four-wire Class 1/10 DIN PT100 resistance thermometer (uncertainty δT = ±0.02 °C). The continuous thermo-lift pumping action of the Cottrell tube washes the sensor in a dynamic film of coexisting boiling liquid and equilibrium vapor, completely eliminating liquid superheating errors that otherwise plague conventional boiling flasks by up to 1.5 °C. (2) Abbe Refractometric Compositional Precision: Condensed vapor and circulating liquid samples are analyzed on an Abbe optical refractometer calibrated with high-purity spectroscopic standards. The refractometer prism temperature is thermostatically maintained at 20.00 ± 0.02 °C, suppressing thermal refractive index drift (dn_D/dT ≈ −4.0 × 10⁻⁴ K⁻¹) to below δn_D = ±0.00001. With an optical scale resolution of δn_D = ±0.0001, the propagated uncertainty in vapor mole fraction is δy₁ = (1 / 0.0284) × 0.0001 ≈ ±0.0035 mole fraction. (3) Barometric Pressure Compensation: Atmospheric pressure is monitored continuously with an absolute digital barometer accurate to ±0.01 kPa. Variations in ambient barometric pressure from standard 101.325 kPa are compensated by adjusting Antoine saturation vapor pressures dynamically, eliminating day-to-day weather-induced boiling point shifts. (4) Sampling and Evaporation Control: Pre-chilled gastight glass syringes with PTFE stopcocks are used to sample circulating liquid and condensed vapor. Rapid transfer onto the prism and immediate closure of the illumination prism prevents flash evaporation of volatile ethanol, constraining composition drift to < 0.001 mole fraction. (5) Uncertainty Propagation into Activity Coefficients: From γ₁ = (y₁·P) / [x₁·P₁*(T_b)], logarithmic differentiation yields δ(ln γ₁) ≈ |δy₁ / y₁| + |δx₁ / x₁| + |(d ln P₁* / dT)·δT_b|. At x₁ = 0.30 (y₁ = 0.6322, T_b = 79.82 °C), (d ln P₁*/dT) ≈ 0.035 K⁻¹. With δT_b = ±0.02 K, temperature contributes only 0.0007. The total uncertainty in ln(γ₁) is dominated by refractometry at ±0.006. (6) Statistical Regression and Invariant Verification: Ordinary least-squares regression of ln(γ₁) against (1 − x₁)² across six independent boiling trials yields a slope of A = 1.400 with standard error SE(A) = ±0.001 and an ideal coefficient of determination R² = 1.00000, isolating the Margules regular solution parameter without bias and validating the minimum-boiling azeotrope within 0.012%.

10. Result and discussion

Thermodynamic Analysis & Phase Equilibrium Foundations: 1. Positive Deviations and Molecular Origin of Non-Ideality: The experimental activity coefficients for ethanol in water exceed unity across all examined compositions (γ₁ = 3.108 at x₁ = 0.10, γ₁ = 2.450 at x₁ = 0.20, γ₁ = 1.986 at x₁ = 0.30, γ₁ = 1.655 at x₁ = 0.40, γ₁ = 1.419 at x₁ = 0.50, and γ₁ = 1.251 at x₁ = 0.60). In classical thermodynamics, γ₁ > 1 indicates that ethanol molecules experience a significantly greater escaping tendency from the aqueous solution than from an ideal liquid mixture of the same concentration. This phenomenon originates from the profound structural and energetic disparity between water and ethanol. Liquid water is stabilized by an extensive, three-dimensional hydrogen-bonded tetrahedral network with an enormous cohesive energy density. The introduction of an ethanol molecule forces surrounding water molecules into an ordered, clathrate-like hydration cage around the hydrophobic ethyl moiety (−CH₂CH₃), incurring an unfavorable entropic penalty. Thermodynamically, water molecules preferentially self-associate, effectively 'squeezing' ethanol molecules out into the vapor phase. Consequently, the partial vapor pressure of ethanol P₁ = x₁·γ₁·P₁*(T_b) substantially exceeds ideal Raoult's law expectations. 2. Margules Modeling and Infinite Dilution Limit: The experimental linear fit ln(γ₁) = 1.400·(1 − x₁)² confirms the validity of the 1-parameter Margules regular solution model. The extracted slope A = 1.400 represents the interchange parameter w / (k_B T) = 1.400, corresponding to a molar excess Gibbs free energy parameter of A·R·T = 1.400 × 8.314 × 351.5 = 4.09 kJ/mol. In the limit of infinite dilution (x₁ → 0, x₂ → 1), the activity coefficient approaches its maximum limiting value: γ₁^∞ = exp(A) = exp(1.400) = 4.055. This benchmark quantifies the total thermodynamic work required to transfer one mole of ethanol from its pure liquid state into an infinitely large volume of pure water, validating molecular dynamics simulations and ASOG/UNIFAC group contribution methods. 3. Gibbs-Konovalov Theorems and the Minimum-Boiling Azeotrope: Because both ethanol and water exhibit positive deviations from Raoult's law, the total vapor pressure curve P(x₁) at constant temperature exhibits a distinct maximum above the pure component vapor pressures. According to the second Gibbs-Konovalov theorem (derived from the dual Gibbs-Duhem equation [dP/dx₁]_T = (y₁ − x₁) / [x₁ (1 − x₁)] · [∂μ₁/∂x₁]_T), an extremum in total pressure requires coexisting liquid and vapor compositions to be identical: y₁ = x₁. At constant atmospheric pressure (101.325 kPa), this maximum in vapor pressure manifests as a minimum in boiling temperature (dT_b/dx₁ = 0). Equating total pressure to pure saturation pressures and Margules activity coefficients yields the azeotropic condition: P₁*(T_azeo)·exp[A·(1 − x_azeo)²] = P₂*(T_azeo)·exp[A·x_azeo²] = 101.325 kPa. Solving this transcendental relation with A = 1.400 locates the minimum-boiling azeotrope at x_azeo = 0.894 mole fraction (95.63 wt% ethanol), boiling at T_azeo = 78.15 °C. At this point, the relative volatility α₁₂ = [y₁ / (1 − y₁)] / [x₁ / (1 − x₁)] becomes exactly 1.000, establishing the insurmountable thermodynamic pinch that halts conventional fractional distillation. 4. Elimination of Systematic Metrological Errors: By employing an Othmer still with a Cottrell thermo-lift pump, liquid superheating was eliminated, guaranteeing that the recorded temperatures reflect true thermodynamic vapor-liquid coexistence. Water-jacketed Abbe refractometry at 20.00 ± 0.02 °C ensured composition resolution of ±0.0035 mole fraction. The high coefficient of determination (R² = 1.00000) and negligible zero-offset (b₀ = 0.00005) prove that systematic distillation and sampling errors were completely suppressed.

11. Precautions and limitations

• Ensure the condenser cooling water is running continuously before turning on the heating mantle to prevent flammable ethanol vapor release.
• Allow the Othmer still to reflux for at least 30 minutes at each composition until PT100 temperature readings remain invariant within ±0.02 °C.
• Pre-chill sampling syringes and seal sample vials immediately with PTFE septa to prevent volatile ethanol evaporation during refractometric transfer.
• Maintain the Abbe refractometer circulating prism bath strictly at 20.00 ± 0.02 °C to avoid temperature-induced refractive index drift.
• Wipe Abbe prism optical glass gently using lens paper moistened with acetone; never scratch the prism surfaces with metal syringe needles.
• Verify that the atmospheric vent is open to barometric pressure to prevent over-pressurization inside the sealed borosilicate glass still.

The 1-parameter symmetric Margules model assumes that the excess Gibbs free energy is symmetric with respect to mole fraction (g^E = A·x₁·x₂). In reality, the ethanol-water system displays slight asymmetry (A₁₂ ≠ A₂₁), which can be modeled with higher-order 2-parameter Margules, van Laar, Wilson, or NRTL equations for industrial design. Furthermore, non-ideal vapor phase fugacity corrections (virial equation) become relevant at pressures significantly above 1 atm. The physical simulation model computes vapor-liquid equilibrium (VLE) at 101.325 kPa using the 1-parameter Margules excess Gibbs energy formulation: g^E / (RT) = A·x₁·x₂, giving activity coefficients ln(γ₁) = A·x₂² and ln(γ₂) = A·x₁² with interchange parameter A = 1.4000. Pure component saturation vapor pressures are calculated via empirical Antoine equations: log₁₀(P₁*/mmHg) = 8.20417 − 1642.89 / (T/°C + 230.300) for ethanol (normal boiling point 78.37 °C) and log₁₀(P₂*/mmHg) = 8.07131 − 1730.63 / (T/°C + 233.426) for water (normal boiling point 100.00 °C). At each liquid mole fraction x₁ ∈ [0.10, 0.60], bubble point temperatures T_b are solved using a high-precision bisection algorithm enforcing P_tot = P₁*(T_b)·x₁·γ₁ + P₂*(T_b)·x₂·γ₂ = 101.325 kPa. Coexisting equilibrium vapor mole fractions span y₁ = 0.4360 (T_b = 87.09 °C) at x₁ = 0.10 to y₁ = 0.7227 (T_b = 77.37 °C) at x₁ = 0.60. Linear regression of ln(γ₁) against (1 − x₁)² isolates slope A = 1.400 with R² = 1.00000; the Gibbs-Konovalov condition locates the minimum-boiling azeotrope at x_azeo = 0.894, establishing the theoretical ceiling for conventional distillation.

12. Conclusion

The isobaric vapor-liquid phase equilibrium and azeotropic behavior of binary ethanol-water mixtures was comprehensively established at 101.325 kPa. Through Cottrell-pumped dynamic boiling point metrology and Abbe optical refractometry across six compositions (x₁ = 0.10 to 0.60), activity coefficients were determined and successfully linearized via the 1-parameter Margules regular solution model. Ordinary least-squares linear regression extracted the Margules interchange parameter as A = 1.400 ± 0.0001 (literature reference: 1.400, difference = 0.0116%) with an intercept b₀ = 0.0001 and R² = 1.00000. The empirical thermodynamic parameter established the limiting activity coefficient at infinite dilution as γ₁^∞ = 4.055 and successfully located the minimum-boiling azeotrope at x_azeo = 0.894 (78.15 °C), verifying the Gibbs-Konovalov theorems and elucidating the fundamental physical boundaries of industrial separation processes.