Pressure Coefficient of a Gas lab report

1. Title and aim

Determination of the Thermal Pressure Coefficient of a Gas at Constant Volume (Charles’ Law). Determine the thermal pressure coefficient of dry air at constant volume (α_v) across temperatures from 0.0 °C to 80.0 °C using a constant-volume air thermometer (Jolly bulb) with a mercury manometer, verify the linear pressure-temperature relation P = P_0(1 + α_v t), and extrapolate to calculate the absolute zero of temperature.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Thermal pressure coefficient α_v of 0.003660 K⁻¹, differing by 0.013% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Constant-Volume Air Thermometer (Jolly Bulb Apparatus): A rigid spherical borosilicate glass bulb of nominal capacity 100.00 cm³ terminating in a narrow capillary stem, enclosing a fixed dry air charge and designed for total immersion in the variable-temperature bath.
• Precision Vertical Mercury Manometer with Fiducial Index Pointer: Two parallel glass manometer tubes: a closed limb containing a sharp downward-pointing glass fiducial index tip defining the fixed reference volume, and an open limb connected via flexible tubing to register gauge pressure Δh.
• Adjustable Vertical Rack-and-Pinion Leveling Reservoir Mount: Provides fine vertical translation of the open manometer tube across a 600 mm range, allowing the operator to adjust the mercury column height precisely until the closed-tube meniscus touches the index pointer.
• Thermostatically Controlled Stirred Heating Bath with Mechanical Agitator: A transparent cylindrical jacket surrounding the bulb, equipped with an electric heating element and a motorized propeller stirrer to ensure homogeneous temperature distribution throughout the liquid volume without spatial gradients.
• Precision Laboratory Thermometer (0.0 °C to 100.0 °C, 0.1 °C Divisions): Calibrated mercury-in-glass secondary standard thermometer suspended directly adjacent to the Jolly bulb to register the bath thermal equilibrium temperature to ±0.05 °C.
• Fortin Mercurial Precision Laboratory Barometer: Measures the prevailing ambient atmospheric pressure B in millimeters of mercury (mmHg) and kilopascals (kPa), essential for converting measured manometer gauge pressure Δh into absolute gas pressure P = B + Δh.
• Confined dry atmospheric air mass held at constant volume V = 100.00 cm³ via fiducial pointer index alignment.
• Ambient atmospheric barometric pressure P_0 = B = 760.0 mmHg = 101.325 kPa verified by precision Fortin laboratory barometer.
• Borosilicate glass bulb envelope with linear thermal expansion coefficient α_g = 3.3 × 10⁻⁶ K⁻¹ (cubical expansion γ = 9.9 × 10⁻⁶ K⁻¹).
• Triple-distilled liquid mercury manometer fluid (density ρ_Hg = 13,595 kg m⁻³ at 0 °C, local gravitational acceleration g = 9.80665 m s⁻²).
• Theoretical reference thermal pressure coefficient for an ideal gas: α_v_ref = 1 / 273.15 K⁻¹ = 0.00366099 K⁻¹ (3.6610 × 10⁻³ K⁻¹).

4. Theory and graph choice

Charles' law at constant volume—frequently designated Gay-Lussac's law of pressure or Amontons' law of pressure-temperature dependence—is a cornerstone principle of classical thermodynamics and kinetic gas theory. It asserts that for a fixed mass of an ideal gas maintained at constant volume, the absolute pressure exerted by the gas against the boundaries of its containment vessel increases linearly with thermodynamic temperature. Preliminary investigations into the pressure of heated air were conducted as early as 1699 and 1702 by the French mechanical philosopher Guillaume Amontons, who designed a constant-volume air thermometer and posited that the pressure of air would diminish to zero at an extreme minimum temperature, providing the first historical inkling of an absolute zero. Exactly one century later, between 1802 and 1808, Joseph Louis Gay-Lussac and John Dalton conducted comprehensive quantitative experiments demonstrating that all permanent gases, when confined to an unvarying geometric volume, exhibit a uniform relative rate of pressure increase per unit temperature rise. The mathematical formulation of this isochoric relationship, referenced to the ice point at 0.00 °C, is P_t = P_0 (1 + α_v t), where P_0 represents the absolute gas pressure at 0.00 °C, t is the Celsius temperature, and α_v is the thermal pressure coefficient of the gas at constant volume (often called the isochoric pressure-temperature coefficient). For an ideal gas adhering to the equation of state PV = nRT, the pressure coefficient α_v is numerically identical to the isobaric volume expansion coefficient β, both being precisely equal to the reciprocal of the absolute temperature of the ice point: α_v = β = 1 / T_0 = 1 / 273.15 K⁻¹ ≈ 0.003661 K⁻¹. In the classical precision laboratory, this relationship is investigated using a constant-volume air thermometer, historically perfected by the German physicist Philipp von Jolly in 1874 (the Jolly bulb air thermometer). The apparatus consists of a rigid spherical glass bulb containing dry air, immersed completely in a thermostatically regulated, stirred liquid bath. The bulb neck is fused to a narrow capillary stem leading to a closed vertical manometer tube featuring a sharp, downward-pointing glass or ivory fiducial index tip (the index pointer). This closed limb is connected via thick flexible rubber tubing to an open, vertically movable manometer tube (the leveling reservoir). At every operating temperature, thermal expansion would tend to push the mercury column downward away from the pointer; however, the experimenter restores the gas to its exact original reference volume by raising the open leveling tube until the summit of the mercury meniscus in the closed capillary just touches the tip of the fiducial index pointer. Once this constant-volume condition is established, the absolute internal pressure of the confined air is determined directly by summing the local barometric atmospheric pressure B with the difference in mercury column elevations: P = B + Δh = B + (h_open − h_closed). Plotting the resulting absolute pressure P against Celsius temperature t yields a linear curve whose slope m = P_0 α_v and vertical intercept b = P_0 provide a direct determination of the pressure coefficient α_v = m / P_0. Furthermore, extrapolating this regression line to the point where absolute pressure theoretically vanishes (P → 0) determines the absolute zero of temperature: T_0 = −b / m = −1 / α_v ≈ −273.15 °C. This elegant measurement confirms that the gas thermometer is not merely an empirical sensor, but a thermodynamic instrument that defines the fundamental absolute temperature scale.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units Pa K⁻¹.

5. Method

1. Read the laboratory Fortin mercurial barometer to 0.1 mmHg, applying brass scale and temperature reductions to determine the true ambient atmospheric pressure B (nominal baseline B = 760.0 mmHg = 101.325 kPa). Inspect the Jolly bulb capillary and manometer tubes to verify that the mercury surfaces are clean, free of oxidation films, and that no air bubbles reside in the flexible connecting hose.
2. Fill the bath vessel surrounding the Jolly bulb with finely crushed pure melting ice and distilled water. Agitate the slurry thoroughly for at least five minutes until the immersion thermometer stabilizes at 0.00 °C. Adjust the movable open manometer tube until the apex of the convex mercury meniscus in the closed capillary just touches the sharp tip of the fiducial index pointer. Sighting with an optical magnifier, confirm zero differential head between the two tubes (Δh = 0) at 0 °C, establishing the baseline absolute pressure P_0 = 101.325 kPa.
3. Drain the ice water and fill the jacket with tap water. Energize the electric heater to raise the bath temperature progressively in 20.0 °C increments (20.0 °C, 40.0 °C, 60.0 °C, 80.0 °C). Run the motorized stirrer continuously at moderate speed to circulate the water around the spherical bulb and prevent thermal stratification.
4. As the gas temperature rises, the increased internal thermal pressure tends to depress the mercury in the closed capillary. Use the fine rack-and-pinion control to raise the open leveling tube until the mercury meniscus in the closed limb is restored exactly to touch the fiducial index pointer tip. Confirm that no gap and no meniscus deformation exists at the contact point.
5. Once thermal equilibrium is established (constant temperature for at least 3 minutes), record the bath temperature t to the nearest 0.1 °C. Read the heights of the mercury columns in the closed tube (h_closed) and open tube (h_open) on the mirror-backed millimeter scale. Calculate the gauge head Δh = h_open − h_closed and compute the absolute pressure P = B + Δh in kilopascals (kPa).
6. De-energize the heater and allow the bath to cool slowly, recording repeat pressure readings at the same temperature stations with falling temperatures to check for thermal hysteresis or gas leakage. Plot the equilibrium absolute pressure P against Celsius temperature t, compute the best-fit slope m and intercept b, and extrapolate the line to P = 0 to calculate the absolute zero of temperature T_0 = −b / m.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0135%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 40.00000; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.05774. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.82843.

Propagating only the relevant fit coefficient gives a fit-only standard error of 5.6980e-07 K⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Temperature gradients throughout the stirred heating bath (±0.2 °C), emergent thermometer stem cooling correction, dead space in the capillary connection between the glass bulb and the closed manometer tube (unheated gas fraction ~0.6%), parallax in leveling the mercury column precisely to the index tip, barometric pressure drift during heating (±0.2 mmHg), and cubical thermal expansion of the borosilicate glass bulb (γ_glass ≈ 10 × 10⁻⁶ K⁻¹, causing apparent gas pressure to be slightly lower than in a rigid vessel) contribute experimental uncertainty. The OLS regression standard error reflects random scatter across temperature stations, but does not capture unmodelled capillary dead-space gradients or bulb volumetric expansion.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Always lower the movable leveling reservoir well down before heating or cooling the bath to prevent sudden thermal contractions from drawing mercury up into the capillary bulb.
• Ensure the water bath is agitated continuously with the mechanical stirrer for at least three minutes prior to reading, eliminating convective thermal gradients around the Jolly bulb.
• Use an optical magnifying lens and align your line of sight horizontally with the mirror backing on the millimeter scale to eliminate optical parallax when reading mercury levels.
• Confirm that the mercury meniscus touches the fiducial index pointer tip precisely—neither leaving a gap nor engulfing the tip—before recording the differential manometer height.

The teaching model assumes dry air behaves as a classical ideal gas with negligible intermolecular van der Waals forces and zero dead-space volume in the capillary stem. In a physical laboratory, small corrections must be applied for capillary dead space (~0.6% unheated volume), cubical thermal expansion of the borosilicate glass envelope (γ ≈ 9.9 × 10⁻⁶ K⁻¹), and real-gas virial departures. The linear relationship P = P_0(1 + α_v t) directly bridges classical pneumatic manometry with Kelvin’s absolute thermodynamic temperature scale. The experimental pressure coefficient α_v = 1/273.15 K⁻¹ demonstrates that gas pressure approaches zero at −273.15 °C, establishing the physical foundation for the absolute scale of temperature where T = t + 273.15.

12. Conclusion

The worked example illustrates how to obtain Thermal pressure coefficient α_v from measurements, graph variables and a fitted coefficient. The estimate is 0.003660 K⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.