15-Liter Heavy-Walled Glass Carboy Vessel
Provides a large gas expansion volume ensuring that bulk gas in the center undergoes true adiabatic expansion unperturbed by boundary layer heat conduction from the vessel walls.
Chemistry / Heat Capacity Ratio of Gases
FREE TO EXPLORELABORATORY 17 · NATURAL SCIENCES
Determination of the Heat Capacity Ratio of Gases (γ = Cp/Cv) via the Clément-Desormes Adiabatic Expansion Method
Determine the heat capacity ratio (γ = C_p/C_v) and adiabatic index of dry atmospheric air using the classical Clément-Desormes method, analyzing the rapid adiabatic expansion and subsequent isochoric thermal re-warming cycles, and extracting γ via ordinary least-squares linear regression. Explore interactive analytical gravimetry, Class A volumetric glassware and density determination.
MEASURE · CALIBRATE · ANALYSE
Your virtual chemistry laboratory.
FINAL RE-EQUILIBRATED PRESSURE H₂—
MEASUREMENT STATUS
Ready to dispense 25.0 cm of liquid.
Adjust initial excess pressure V, then weigh the dispensed aliquot.
Mass observations are determined by the high-precision thermodynamic equation of state for pure liquid water. Readings reflect thermal equilibrium and analytical balance force restoration.
Physical chemistry assumptions and metrological limits
The Clément-Desormes formula assumes ideal gas behavior, strictly adiabatic expansion of the bulk gas without acoustic wave dissipation, negligible heat transfer during the 0.8 s valve release, and valid first-order Taylor expansion ln(1 + h/P0) ≈ h/P0 for small excess pressures (h/P0 < 0.03). At h = 40 cm DBP (~4 kPa), the second-order Taylor term contributes <0.2% error, which is effectively minimized by multi-point linear regression. The physical simulation model calculates the final isochoric re-equilibrated manometer pressure h₂ via the linearized Clément-Desormes relation h₂ = [ (γ − 1) / γ ] · h₁ + h₀. Thermodynamic parameters: dry diatomic gas (atmospheric air, 78% N₂, 21% O₂); equipartition degrees of freedom f = 5 (3 translational + 2 rotational); accepted theoretical adiabatic index: γ = C_p / C_v = 7/5 = 1.4000. Theoretical regression slope: m_slope = (γ − 1) / γ = (1.400 − 1.0) / 1.400 = 0.400 / 1.400 = 2/7 = 0.285714. For initial excess pressures h₁ from 15.0 cm to 40.0 cm (dibutyl phthalate manometer fluid, density ρ = 1.047 g/cm³), re-equilibrated pressures span h₂ = 4.29 cm to 11.43 cm. Ordinary least-squares linear regression isolates γ = 1 / (1 − m_slope) with R² = 1.00000; the multi-point linear regression method eliminates zero-point parallax errors and valve oscillation offsets into the intercept h₀.
Analytical chemistry apparatus & tools.
Large-Bore Ground-Glass Quick-Release Stopper Valve (20 mm)
Opens rapidly to allow instantaneous pressure equalization with ambient atmosphere (0.8 s) and seals hermetically without gas leakage.
Open-Ended U-Tube Liquid Manometer (Dibutyl Phthalate)
Measures fine excess gas pressures (h₁ and h₂) with 13× greater sensitivity than mercury, containing low-volatility colored fluid (ρ = 1.047 g/cm³).
Hand Rubber Bellows Pump and Inline Desiccant Drying Tube
Pressurizes the carboy with dry air while scrubbing water vapor through anhydrous calcium chloride granules to prevent moisture-induced heat capacity distortions.
Digital Millisecond Solenoid Actuator and Timing Controller
Standardizes the valve opening duration to precisely 0.80 ± 0.02 seconds, eliminating human reaction time variability and preventing acoustic resonance oscillations.
Precision Cathetometer / Optical Mirror Scale (±0.05 cm)
Reads the liquid meniscus height difference across both arms of the U-tube manometer with zero parallax error.
FOLLOW THE METHOD
A reason behind every step.
Determine the heat capacity ratio (γ = C_p/C_v) and adiabatic index of dry atmospheric air using the classical Clément-Desormes method, analyzing the rapid adiabatic expansion and subsequent isochoric thermal re-warming cycles, and extracting γ via ordinary least-squares linear regression. Suggested independent volumetric settings: 15.0, 20.0, 25.0, 30.0, 35.0, 40.0 cm.
01
System Sealing and Desiccant Check
Inspect the 15-liter glass carboy, ground-glass valve, and rubber tubing connections for airtight integrity. Ensure the inline drying tube is filled with active anhydrous calcium chloride granules to guarantee that dry air enters the system.
02
Initial Carboy Pressurization
Close the main release valve. Using the hand bellows pump, gently pump dry air through the desiccant tube into the carboy until the dibutyl phthalate manometer shows an initial excess pressure slightly above the assigned h₁ value (e.g., ~27 cm for h₁ = 25.0 cm). Seal the intake clamp.
03
Thermal Pre-Equilibration
Allow the carboy to stand undisturbed for 3 to 5 minutes. Compression of the gas during pumping causes a slight initial temperature rise; observe the manometer as the gas cools back to room temperature T₀, settling at the exact target initial excess pressure h₁.
04
Rapid Adiabatic Expansion
Trigger the quick-release valve mechanism. The large-bore stopper opens, allowing excess air to rush out until internal pressure drops to atmospheric (P₂ = P₀), and automatically snaps shut after exactly 0.80 seconds. The gas inside cools adiabatically.
05
Isochoric Thermal Re-Warming and Final Pressure Logging
With the vessel sealed at constant volume (V₃ = V₂), observe the manometer liquid level rise as the chilled gas slowly absorbs heat from the room through the glass walls. After 3 to 5 minutes, when the pressure stabilizes at a constant maximum plateau, record the final re-equilibrated excess pressure h₂.
06
Multi-Pressure Series and Linear Regression
Repeat the measurement across six initial excess pressures: h₁ = 15.0, 20.0, 25.0, 30.0, 35.0, and 40.0 cm. Perform ordinary least-squares linear regression of h₂ against h₁, extract the slope m_slope, and compute the adiabatic index γ = 1 / (1 − m_slope).
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into quantitative chemical insight.
Points are your recorded gravimetric observations. The regression line validates the linear mass-volume proportionality and extracts density.
ORDINARY LEAST-SQUARES FIT · m = ρ·V + m₀
h₂ = [(γ − 1) / γ] · h₁ + h₀ ==> h₂ = 0.2857 · h₁
| # | Initial excess gas pressure h₁ (cm) (cm) | Final re-equilibrated pressure h₂ (cm) | Initial excess pressure h₁ (cm) | Final re-equilibrated pressure h₂ (cm) | Remove |
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