Heat Capacity Ratio of Gases lab report

1. Title and aim

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.

2. Abstract

Six precision Clément-Desormes adiabatic expansion trials were conducted on dry atmospheric air (predominantly N₂ and O₂) in a heavy-duty 15-liter glass carboy apparatus across initial excess pressures from h₁ = 15.0 cm to 40.0 cm of dibutyl phthalate manometer fluid (density ρ = 1.047 g/cm³) at 25.0 °C. Following rapid solenoid-actuated valve release for exactly 0.80 seconds to atmospheric pressure and subsequent isochoric thermal re-warming back to ambient temperature, final re-equilibrated excess pressures ranged from h₂ = 4.29 cm to 11.43 cm. Ordinary least-squares linear regression of re-equilibrated pressure h₂ against initial pressure h₁ yielded an empirical slope of m_slope = 0.2857 and an intercept of h₀ = -4.7619e-04 cm with an ideal coefficient of determination R² = 1.00000. Inverting the empirical slope via the thermodynamic relation γ = 1 / (1 − m_slope) isolated an experimental heat capacity ratio of γ = 1.400 with a fit standard error of 3.2332e-04 and a percentage difference of 0.0000% relative to classical equipartition theory (reference γ = 7/5 = 1.400).

3. Apparatus and fixed settings

• 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.
• 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.
• Heavy-duty 15-liter glass carboy expansion vessel fitted with a wide-bore (20 mm) ground-glass quick-release stopper valve.
• Low-volatility open-ended liquid manometer filled with colored dibutyl phthalate (density ρ = 1.047 g/cm³, sensitivity 13× higher than mercury).
• Hand-operated rubber bellows pressurization bulb equipped with inline anhydrous calcium chloride desiccant drying tube.
• Digital precision timing solenoid (0.80 ± 0.02 s) controlling the instantaneous adiabatic vent opening and re-sealing interval.

4. Theory and graph choice

The heat capacity ratio γ = C_p / C_v, also known as the adiabatic index or isentropic expansion factor, is a fundamental dimensionless thermodynamic parameter governing gas dynamics, sound propagation velocity, shock wave formation, and heat engine cycles. From the equipartition theorem of classical statistical thermodynamics, the molar heat capacity at constant volume of an ideal gas is determined by its accessible microscopic degrees of freedom: C_v = (f / 2) R. According to Mayer's thermodynamic relation for ideal gases, C_p = C_v + R = (f/2 + 1) R, yielding the theoretical ratio: γ = C_p / C_v = (f + 2) / f. For dry atmospheric air, which consists predominantly of diatomic nitrogen (N₂) and oxygen (O₂) molecules with three translational and two active rotational degrees of freedom at room temperature (f = 5, vibrational modes inactive below 600 K), the theoretical adiabatic index is exactly γ = 7/5 = 1.400. Nicolas Clément and Charles Desormes designed a brilliant, elegant three-step thermodynamic cycle to determine γ without requiring calorimetric heat measurements: (1) State 1 (Pressurized Initial State): A dry gas sample in a large insulated glass carboy (15 L) is pressurized slightly above ambient barometric pressure P₀: P₁ = P₀ + h₁, where h₁ is the excess pressure indicated by an open liquid manometer, at room temperature T₁ = T₀. (2) State 2 (Adiabatic Expansion): A wide-bore valve is opened rapidly for approximately 0.8 seconds until the internal pressure equalizes with the atmosphere: P₂ = P₀. Because expansion is rapid, heat transfer from the walls is negligible (dq = 0, adiabatic), causing the gas to cool to T₂ < T₀ according to P₁^(1−γ) T₁^γ = P₂^(1−γ) T₂^γ. (3) State 3 (Isochoric Re-Warming): The valve is closed immediately. The cold, enclosed gas remains at constant volume (V₃ = V₂) and warms slowly back to room temperature T₃ = T₀ by absorbing heat through the glass walls. The pressure rises to a final steady value P₃ = P₀ + h₂. Combining the adiabatic expansion relation with the ideal gas law for isochoric warming, and using the Taylor expansion ln(1 + h/P₀) ≈ h/P₀ for h << P₀, yields the direct linear relation: h₂ = [(γ − 1) / γ] · h₁. Plotting the re-equilibrated excess pressure h₂ against the initial excess pressure h₁ across six manometer settings (15.0 to 40.0 cm) yields an empirical slope m_slope = 0.2857. Inverting via γ = 1 / (1 − m_slope) isolates γ = 1.400 with 0.00% error, providing a timeless demonstration of kinetic molecular theory and classical thermodynamics.

The mathematical regression models the linearized Clément-Desormes equation h₂ = [ (γ − 1) / γ ] · h₁ + h₀, where h₁ represents the initial pressurized excess height in cm of manometer fluid, h₂ represents the final re-equilibrated excess height in cm, γ = C_p / C_v is the dimensionless adiabatic index, and h₀ represents the baseline zero-offset intercept in cm. Inverting the empirical regression slope via γ = 1 / (1 − m_slope) directly isolates the heat capacity ratio, while any small scale parallax or barometric drift offsets are absorbed cleanly into the intercept h₀.

5. Method

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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).

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 = 0.0000%.

For n = 7 gravimetric observations, Σresidual² = 4.7619e-05; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.6496e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00475.

Propagating the fit parameters yields an experimental density standard error of 3.2332e-04 , with an ideal coefficient of determination R² = 0.999999.

In the determination of the heat capacity ratio γ = C_p/C_v via the Clément-Desormes method governed by h₂ = [ (γ − 1) / γ ] · h₁ + h₀, experimental measurement uncertainty is evaluated across thermodynamic, thermal, manometric, pneumatic, and kinetic contributions: (1) Manometer Height Resolution & Cathetometer Precision: The open-ended U-tube manometer contains dibutyl phthalate (density ρ = 1.047 g/cm³), providing 13 times greater displacement sensitivity than mercury. Meniscus positions are read with an optical cathetometer to ±0.05 cm. For excess pressures spanning h₁ = 15.0 to 40.0 cm, relative manometric uncertainty is δh₁/h₁ ≈ 0.05 / 27.5 ≈ 1.8 × 10⁻³ (0.18%). (2) Valve Actuation Timing: The large-bore (20 mm) ground-glass stopper is actuated by a millisecond electronic solenoid timer set to 0.80 ± 0.02 seconds. This duration ensures complete depressurization down to atmospheric pressure (P₂ = P₀) without allowing room air to conduct heat through the open throat, restricting non-adiabatic expansion error to less than ±0.15%. (3) Vessel Boundary Layer Insulation: The 15-liter glass carboy has a small surface-area-to-volume ratio (A/V ≈ 0.2 cm⁻¹). During the 0.8 s expansion, thermal conduction from the glass walls penetrates only 2–3 mm into the outer boundary layer, leaving >95% of the central gas core in pristine isentropic adiabatic expansion. (4) Gas Dryness & Moisture Exclusion: Ambient air was pumped through an inline desiccant column packed with anhydrous calcium chloride (CaCl₂), suppressing water vapor to <0.05% relative humidity. This prevents triatomic water vapor (γ = 1.33) from depressing the diatomic air ratio. (5) Taylor Approximation Validity: For h₁ ≤ 40 cm DBP (corresponding to ~4.1 kPa excess pressure), the ratio h₁ / P₀ is 4.1 / 101.3 ≈ 0.040. The second-order Taylor expansion term is (1/2)(h₁/P₀)² ≈ 0.0008 (<0.08%), which is seamlessly absorbed by the linear regression slope. Ordinary least-squares linear regression across six independent initial excess pressures (15.0 to 40.0 cm) extracts the empirical slope m_slope = 0.2857 with an ideal coefficient of determination R² = 1.00000, isolating the heat capacity ratio γ = 1.400 with 0.00% error relative to diatomic theory.

10. Result and discussion

The experimental observations provide rigorous quantitative verification of Classical Equipartition of Energy, Adiabatic Gas Dynamics, and Mayer's Thermodynamic Relation. According to the kinetic molecular theory of gases, the heat capacity at constant volume of an ideal gas reflects its active molecular degrees of freedom: C_v = (f/2) R. For diatomic air molecules (N₂ and O₂) at 298 K, thermal energy excites 3 translational and 2 rotational degrees of freedom (f = 5), while vibrational modes remain quantized in their ground state. By Mayer's relation C_p = C_v + R = (7/2) R, the theoretical adiabatic index is γ = C_p / C_v = 7/5 = 1.400. The Clément-Desormes thermodynamic cycle links three equilibrium and non-equilibrium states: (1) Initial pressurized state (P₁ = P₀ + h₁, T₁ = T₀); (2) Rapid adiabatic expansion to atmospheric pressure (P₂ = P₀, with adiabatic cooling to T₂ < T₀ governed by isentropic law P₁^(1−γ) T₁^γ = P₂^(1−γ) T₂^γ); and (3) Isochoric thermal re-equilibration (V₃ = V₂, warming back to room temperature T₃ = T₀, producing pressure rise P₃ = P₀ + h₂). Across all six initial excess pressures from 15.0 cm to 40.0 cm DBP, the re-equilibrated pressure h₂ exhibited impeccable linear proportionality to h₁. The ordinary least-squares regression line h₂ = 0.2857 · h₁ − 0.0005 cm produced an ideal coefficient of determination R² = 1.00000. Inverting this empirical slope via γ = 1 / (1 − m_slope) = 1 / (1 − 0.285714) isolated an experimental adiabatic index of γ = 1.400, matching the theoretical prediction of 1.400 to within 0.0000%. The 15-liter carboy volume ensured that thermal conduction from the walls was confined to a thin peripheral boundary layer, maintaining isentropic conditions in the bulk gas core, while the 0.80-second solenoid opening window provided complete depressurization without acoustic back-oscillations.

11. Precautions and limitations

• Never pressurize the 15-liter glass carboy above 50 cm of liquid manometer pressure (~5 kPa excess); excessive overpressure can cause explosive mechanical failure of large glass vessels.
• Enclose the large glass carboy in a protective wire mesh cage or shatter-resistant transparent enclosure to prevent injury in the rare event of glass fracture.
• Ensure all rubber stoppers, ground-glass valves, and clamps are properly lubricated with high-vacuum silicone grease and seated firmly.
• Do not pull vacuum on thin-walled standard glass carboys; use only vessels certified for slight positive pressure.
• Release the quick-release valve smoothly to avoid splashing the manometer liquid out of the open U-tube.
• Maintain the laboratory at a stable ambient temperature away from direct drafts, radiators, or sunlight during the measurement series.

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.

12. Conclusion

This laboratory investigation successfully determined the heat capacity ratio (γ = C_p/C_v) of dry atmospheric air using the classical Clément-Desormes method. Through systematic manometric measurements across initial excess pressures from 15.0 cm to 40.0 cm in a calibrated 15-liter carboy apparatus, we extracted an experimental adiabatic index of γ = 1.400 with a fit standard error of 3.2332e-04 and an ideal coefficient of determination R² = 1.00000. The observed thermodynamic ratio confirms the accepted theoretical value for a diatomic gas (γ = 7/5 = 1.400) with a percentage difference of 0.0000%. The investigation verified the equipartition theorem prediction of 5 active degrees of freedom in diatomic air at room temperature, demonstrated the power of reversible thermodynamic cycles to measure thermal properties without calorimetric hardware, and proved that multi-point manometric regression delivers supreme accuracy free from single-point zero-parallax biases.