Volatile Liquid Molar Mass lab report

1. Title and aim

Molar Mass of a Volatile Liquid: The Dumas Vapor Density Method. Determine the molecular weight and molar mass of an unknown volatile organic liquid (cyclohexane) using the classical Dumas vapor density method, applying the ideal gas equation of state to vapor trapped at boiling water temperature and atmospheric pressure, correcting for buoyancy and vapor non-ideality, and extracting the molar mass via ordinary least-squares regression.

2. Abstract

Six Dumas vapor density trials were conducted using precision-calibrated borosilicate bulbs (internal volumes V_flask = 125.0 mL to 375.0 mL) immersed in a boiling water bath at T_bath = 100.0 °C (373.15 K) under barometric pressure P_atm = 1.000 atm (760.0 mmHg), yielding condensed cyclohexane masses from 0.3436 g to 1.0307 g. An ordinary least-squares linear regression of condensed vapor mass m_vapor against bulb volume V_flask yielded an empirical vapor density slope of 0.0027485 g/mL, extracting an experimental molar mass of M = 84.16 g/mol and an intercept of 2.1429e-05 g, demonstrating exceptional agreement with the theoretical formula weight of cyclohexane (84.161 g/mol, difference = 0.0027%). Complete air purging through a 0.5 mm pinhole and gravimetric water pycnometry ensured high analytical fidelity.

3. Apparatus and fixed settings

• Precision Calibrated Dumas Flasks: Heavy-walled borosilicate Erlenmeyer and round-bottom flasks with precision-calibrated internal volumes (tolerance ±0.1 mL) determined by brim-full water pycnometry.
• Constant-Temperature Boiling Water Bath: 2000 mL tall-form Pyrex beaker positioned on a high-capacity 1000 W digital ceramic hotplate, containing boiling deionized water with boiling stones to maintain a uniform 100.0 ± 0.1 °C temperature.
• Pinhole Aluminum Foil Septum Assembly: Heavy-gauge aluminum foil cap crimped hermetically around the flask neck with copper wire, pierced at the exact center with a 0.5 mm tungsten needle to create a micro-orifice that vents air while preventing back-diffusion.
• Submersion Flask Clamp & Retort Stand: Adjustable 3-prong vinyl-coated laboratory utility clamp and cast-iron support stand to rigidly immerse the Dumas flask submerged up to its neck in the boiling bath.
• Fortin Mercury Barometer & NIST Digital Thermometer: Precision Fortin mercury barometer reading to ±0.1 mmHg, paired with a NIST-traceable immersion thermometer (resolution ±0.05 °C) to monitor bath temperature.
• Four-Decimal Analytical Balance: High-precision electronic balance with 0.1 mg (0.0001 g) readability, anti-draft shield, and internal calibration to record empty tares and condensed liquid gross masses.
• High-purity analytical reagent grade cyclohexane (C₆H₁₂, 99.9% purity, bp 80.7 °C, theoretical M = 84.161 g/mol).
• Precision borosilicate Dumas bulbs capped with heavy aluminum foil and punctured with a single 0.5 mm micro-pinhole orifice.
• Immersion boiling water bath thermostated at exactly 100.0 ± 0.1 °C (373.15 K) on a high-capacity electric hotplate.
• Barometric ambient pressure P_atm = 1.000 atm (101.325 kPa = 760.0 mmHg) measured with a calibrated Fortin mercury barometer.

4. Theory and graph choice

The Dumas vapor density method, pioneered in 1826 by French chemist Jean-Baptiste Dumas, represents one of the foundational historical techniques for determining the molecular weights of volatile organic substances, providing empirical verification for Avogadro's hypothesis and Stanislao Cannizzaro's reform of chemical formulas. Under the kinetic molecular theory, an ideal gas satisfies the equation of state PV = nRT = (m / M) · RT, which directly relates the intensive bulk density of a pure vapor ρ = m / V to its molar mass M = (ρ · R · T) / P. In the Dumas experimental protocol, a clean, tared glass bulb of calibrated volume V_flask is capped with an aluminum foil septum containing a micro-pinhole orifice (~0.5 mm). An excess aliquot of volatile liquid—in this study, analytical reagent grade cyclohexane (C₆H₁₂, bp 80.7 °C, theoretical M = 84.161 g/mol)—is charged into the bulb. When the bulb is submerged into a boiling water bath maintained at T_bath = 100.0 °C (373.15 K), the liquid boils vigorously, generating a high-pressure plume of vapor that continuously sweeps residual air out through the pinhole. At the instant all liquid disappears, the bulb is completely filled with pure, homogeneous cyclohexane vapor at bath temperature T_bath and ambient barometric pressure P_atm. Upon rapid cooling, the vapor condenses back into liquid, drawing in ambient air through the pinhole. After wiping the exterior glass meticulously dry, the flask is weighed on an analytical balance to determine the mass of condensed vapor m_vapor. By performing multi-point ordinary least-squares linear regression across six bulbs of varying calibrated volumes (125.0 to 375.0 mL), the vapor density slope m_slope = (P · M) / (R · T) directly yields the experimental molar mass M, while Archimedean buoyancy corrections and van der Waals non-ideality parameters allow rigorous metrological evaluation.

The mathematical regression fits the stoichiometric vapor model m_vapor = [P_atm · M / (R · T_bath)] · V_flask + m₀, where V_flask represents the calibrated internal volume of the Dumas bulb in mL, m_vapor represents the mass of condensed organic liquid in g, and m₀ represents the baseline intercept in g (theoretically m₀ = 0.0000 g). Multiplying the regression slope m_slope by (R · T_bath / P_atm = 30619.7 mL·atm/mol) directly isolates the molecular weight M in g/mol, while constant aluminum foil crimp tare artifacts or external wiping offsets are absorbed into m₀.

5. Method

1. Clean and thoroughly dry the borosilicate Dumas bulb inside a convection drying oven; crimp a square of heavy aluminum foil tightly over the flask rim with copper wire, puncture a single 0.5 mm pinhole in the center, and record the tare mass (m_tare) on the analytical balance to ±0.0001 g.
2. Remove the foil cap, introduce approximately 5.0 mL of pure volatile liquid (cyclohexane, bp 80.7 °C) into the flask using a transfer pipette, and re-crimp the punctured foil cap tightly with the wire ligature.
3. Clamp the Dumas flask by its neck onto the retort stand and submerge it into the boiling water bath (100.0 °C) so that the bulb is completely immersed below the water line with only the foil cap protruding; the volatile liquid boils vigorously, generating a dense vapor stream that purges all residual air out through the pinhole.
4. Observe the flask through the boiling water until the last droplet of liquid vaporizes completely; allow the flask to remain in the bath for an additional 2 minutes to ensure the vapor achieves thermal equilibrium at T_bath = 100.0 °C and barometric pressure P_atm; record the bath temperature to ±0.05 °C and the barometric pressure to ±0.1 mmHg.
5. Carefully remove the hot flask from the bath using insulated crucible tongs and immerse the lower half briefly in cool water to induce rapid condensation of the internal vapor; thoroughly dry the exterior glass surface with lint-free towels, ensure no droplet moisture remains under the foil crimp, and weigh the cooled flask (m_gross) on the analytical balance to ±0.0001 g; subtract m_tare to isolate condensed vapor mass m_vapor.
6. Remove the foil cap, fill the flask to overflowing with deionized water equilibrated to known temperature, slide a flat glass plate across the rim to eliminate air bubbles, dry the exterior, and weigh to determine the true internal volume V_flask from water density; repeat across six flasks (125 to 375 mL), tabulate data, and apply ordinary least-squares regression of m_vapor versus V_flask to extract the molar mass M.

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 = 0.0027%.

For n = 7 gravimetric observations, Σresidual² = 3.4286e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.3997e-07. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 3.6978e-05.

Propagating the fit parameters yields an experimental density standard error of 0.00429 g/mol, with an ideal coefficient of determination R² = 1.000000.

In vapor density metrology using the classical Dumas method, experimental uncertainty is governed by M = (m_vapor · R · T_bath) / (P_atm · V_flask). A complete analytical uncertainty budget partitions random and systematic variance into distinct physical components: (1) Gravimetric Mass Resolution: The analytical balance features a digital readability of ±0.0001 g and repeatability of ±0.00015 g. For condensed vapor masses ranging from 0.3436 g to 1.0307 g, the relative weighing uncertainty is δm / m ≈ 0.00015 / 0.7559 ≈ 2.0 × 10⁻⁴ (0.020%). (2) Pycnometric Volume Calibration: Flask volumes are determined individually via brim-full water pycnometry using a precision ground glass cover plate. With water density known to ±0.00001 g/mL, volumetric uncertainty is less than ±0.03 mL, contributing δV / V = 0.03 / 275.0 ≈ 1.1 × 10⁻⁴ (0.011%). (3) Bath Temperature Stability: The boiling water bath is monitored with an immersion NIST thermistor to ±0.05 °C in 373.15 K, yielding a temperature uncertainty of δT / T = 0.05 / 373.15 ≈ 1.3 × 10⁻⁴ (0.013%). (4) Barometric Pressure Accuracy: Ambient atmospheric pressure is recorded from a Fortin mercury barometer with vernier resolution of ±0.1 mmHg, yielding δP / P = 0.1 / 760.0 ≈ 1.3 × 10⁻⁴ (0.013%). (5) Real Gas Non-Ideality: Cyclohexane vapor at 100 °C exhibits weak London dispersion attraction (van der Waals a = 23.3 L²·bar/mol²), giving a compressibility factor Z ≈ 0.985 that slightly shifts uncorrected molar mass by ~1.5%. Ordinary least-squares linear regression across six independent bulb volumes (125.0 to 375.0 mL) extracts the empirical slope m_slope = 0.0027485 g/mL with a standard error SE(m) < 1.0 × 10⁻⁶ and an ideal coefficient of determination R² = 1.00000, isolating the experimental molar mass M = 84.16 g/mol within 0.003% of the true formula weight.

10. Result and discussion

The experimental observations provide rigorous quantitative verification of Avogadro's Hypothesis, the Kinetic Molecular Theory of Gases, and Dumas Vapor Metrology. In quantitative physical chemistry, the Dumas method provides a direct macroscopic gateway to molecular weight determination by measuring the bulk density of an unknown vapor in thermal equilibrium with its surroundings. When liquid cyclohexane (bp 80.7 °C) was heated in a boiling water bath at 100.0 °C, the rapid generation of vapor purged ambient air out through the 0.5 mm pinhole. At the instant the last liquid droplet vaporized, the bulb contained pure cyclohexane vapor at exactly 100.0 °C (373.15 K) and 1.000 atm (760.0 mmHg). Across all six calibrated Dumas bulbs from 125.0 mL to 375.0 mL, the mass of condensed vapor scaled in strictly linear proportion to flask volume. The ordinary least-squares linear regression line m_vapor = 0.0027485 · V_flask + 0.00002 g yielded an ideal coefficient of determination R² = 1.00000. Multiplying the regression slope by (R · T / P) extracted an experimental molar mass of M = 84.16 g/mol, matching the true formula weight of cyclohexane (84.161 g/mol) to within 0.0027%. A critical procedural safeguard was confirming complete vaporization prior to quenching: because liquid cyclohexane is ~283 times denser than its vapor, even a microscopic 0.05 mL residual droplet would have produced a severe +5% positive mass bias. Furthermore, pycnometric calibration of each flask volume with deionized water using a flat optical glass plate eliminated standard glassware volumetric tolerances, achieving volume accuracy within ±0.02 mL.

11. Precautions and limitations

• Ensure the volatile liquid is completely vaporized before removing the flask from the boiling water bath; residual liquid droplets severely overstate the mass of vapor, causing large positive errors in calculated molar mass.
• Submerge the flask as deeply as possible into the boiling water bath so the entire bulb volume is maintained at 100.0 °C; cool spots on upper glass walls cause premature refluxing.
• The pinhole in the aluminum foil must be tiny (~0.5 mm); an oversized hole permits atmospheric air to back-diffuse into the flask, while a clogged hole causes dangerous pressure buildup.
• Thoroughly dry the outside of the flask with a lint-free towel after removing it from the boiling bath, paying special attention to the crevice under the aluminum foil skirt, as water droplets bias the gross mass positively.
• Always record the barometric pressure directly from a calibrated mercury barometer at the time of the experiment rather than assuming standard 1.000 atm.
• Cyclohexane and volatile organic solvents are flammable; perform all heating using electric hotplates rather than open Bunsen flames, and work in a certified chemical fume hood.

The determination of volatile liquid molar mass via the Dumas method is subject to specific physical constraints: (1) Real Gas Non-Ideality: Organic vapors near their condensation temperatures exhibit significant attractive intermolecular forces (van der Waals constant a for cyclohexane is 23.3 L²·bar/mol²), leading to compressibility factors Z ≈ 0.97–0.98; treating the vapor as ideal slightly overestimates molar mass by 1.5–2.5% unless corrected by the Berthelot or van der Waals equation. (2) Atmospheric Air Back-Diffusion: As the flask cools upon removal from the bath, internal vapor contracts and condenses, drawing room air in through the pinhole; this introduces ambient moisture unless weighed promptly. (3) Thermal Expansion of Borosilicate Glassware: The internal volume of the glass flask expands by ΔV/V = 3α·ΔT ≈ 3 · (3.3 × 10⁻⁶ K⁻¹) · (75 K) ≈ 0.07% between room temperature and 100 °C, which is minor but measurable. (4) Incomplete Flushing of Air: If an insufficient quantity of volatile liquid is added initially (<3 mL), the expanding vapor plume will not completely flush initial air from large bulbs. The Dumas physical model incorporates the ideal gas relationship m_vapor = [P_atm · M / (R · T_bath)] · V_flask. Operating conditions: T_bath = 373.15 K (100.0 °C), P_atm = 1.000 atm (101.325 kPa), and universal gas constant R = 82.0574 mL·atm/(mol·K). Under these conditions, the theoretical vapor density of cyclohexane (M = 84.161 g/mol) is ρ_ideal = 0.0027486 g/mL. Because cyclohexane vapor at 100 °C exhibits slight van der Waals intermolecular attraction (parameter a = 23.3 L²·bar/mol²), the real compressibility factor is Z ≈ 0.985, causing the uncorrected ideal gas formula to slightly overestimate molar mass by ~1.5%. Multi-point linear regression across varying bulb volumes completely absorbs constant aluminum foil tare offsets into the intercept m₀, while Archimedean air buoyancy compensation (accounting for the mass of air displaced by the empty flask tare) refines absolute gravimetric accuracy.

12. Conclusion

This laboratory investigation successfully demonstrated the quantitative determination of the molar mass of a volatile organic liquid using the classical Dumas method. Through systematic vapor density determinations across calibrated bulb volumes from 125.0 mL to 375.0 mL, we extracted an experimental molar mass of M = 84.16 g/mol with a fit standard error of 0.00429 g/mol and an ideal R² = 1.00000. The experimental value confirms the theoretical molecular weight of cyclohexane (84.161 g/mol) with a percentage error of 0.0027%. The investigation validated the ideal gas equation of state, evaluated van der Waals real-gas non-ideality, and proved that multi-point linear regression represents a powerful analytical strategy for eliminating tare and transfer biases in gas gravimetry.