Charles’ Law (Volume Expansion) lab report

1. Title and aim

Determination of the Thermal Expansion Coefficient of a Gas at Constant Pressure (Charles’ Law). Determine the thermal coefficient of volume expansion of dry air at constant atmospheric pressure (β) by measuring the expansion volume V of a bulb-confined air sample across temperatures from 0.0 °C to 80.0 °C using a constant-pressure air dilatometer with a mercury leveling reservoir, and extrapolate to determine 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 Volume expansion coefficient β of 0.003661 K⁻¹, differing by 0.014% 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

• Calibrated Borosilicate Glass Air Bulb (Specimen Vessel T₁): Hermetically encloses the nominal 100.00 cm³ dry air sample within a spherical thin-walled glass chamber designed for high thermal conductance and uniform immersion within the variable-temperature liquid heating bath.
• Thermostated Liquid Bath with Submersible Heating Element and Stirrer: Provides a stable, uniform temperature enclosure around the specimen bulb, equipped with a mechanical motor-driven paddle stirrer to eliminate thermal stratification and an electric heater to control temperatures from 0.0 °C to 80.0 °C.
• Precision Graduated Gas Dilatometer Tube (Manometer Limb T₂): A uniform-bore precision glass tube with metric millimeter graduations etched along its stem, receiving the gas expanding out of the bulb to measure volumetric dilatation ΔV to ±0.05 cm³.
• Adjustable Mercury Leveling Reservoir and Vertical Rack Slider: An open glass mercury reservoir mounted upon a smooth vertical rack-and-pinion sliding carriage, elevated or lowered by the operator to match the mercury level in tube T₂ so that gas pressure remains strictly isobaric at barometric pressure P = B.
• Precision Digital Immersion Thermometer (RTD Sensor): Monitors the uniform water bath temperature directly adjacent to the glass bulb with a resolution of ±0.05 °C, ensuring thermal equilibrium between the bath fluid and enclosed air.
• Flexible Heavy-Duty Elastomeric Interconnecting U-Tube: Connects the bottom of the graduated measuring tube to the movable leveling reservoir, securely containing triple-distilled liquid mercury without kinking or volume compliance under hydrostatic head changes.
• Confined dry air specimen of nominal volume V_0 = 100.00 cm³ (1.000 × 10⁻⁴ m³) at the ice point (0.00 °C = 273.15 K).
• Constant ambient atmospheric barometric pressure P = B = 101.325 kPa (760.0 mmHg) verified by Fortin laboratory barometer.
• Heavy-walled borosilicate glass bulb (T₁) with volumetric expansion coefficient α_v = 9.9 × 10⁻⁶ K⁻¹ immersed in stirred water bath.
• High-density triple-distilled liquid mercury leveling reservoir maintaining zero hydrostatic head (h_open = h_closed) across all temperature stations.
• Accepted standard isobaric expansion coefficient for an ideal gas: β_ref = 1 / 273.15 K⁻¹ = 0.00366099 K⁻¹ (3.6610 × 10⁻³ K⁻¹).

4. Theory and graph choice

Charles' law—frequently designated the law of Charles and Gay-Lussac or Gay-Lussac's law of isobaric expansion—constitutes one of the foundational empirical Pillars of classical pneumatics and thermodynamic science, enunciating that under constant absolute pressure, the volume of a confined mass of an ideal gas expands linearly with increasing thermodynamic temperature. While preliminary qualitative observations were recorded in 1699 and 1702 by Guillaume Amontons, the quantitative relationship was experimentally formulated in unpublished work around 1787 by the French aeronautical pioneer and physicist Jacques Alexandre César Charles, who demonstrated that equal volumes of hydrogen, oxygen, nitrogen, and carbon dioxide undergo identical volumetric expansion when warmed from the freezing point to the boiling point of water. In 1802, the brilliant French chemist and physicist Joseph Louis Gay-Lussac published the definitive quantitative treatise establishing that all common atmospheric gases exhibit a universal isobaric volume expansion coefficient of approximately 1/267 (later refined by Henri Victor Regnault in 1842 to the modern standard value of 1/273.15 K⁻¹ ≈ 0.003661 K⁻¹). The mathematical formulation of isobaric dilatation relative to the ice-point volume V_0 at 0 °C is expressed as V_t = V_0 (1 + β t), where t is the Celsius temperature, V_0 is the initial volume occupied by the gas at 0.00 °C, and β is the thermal volume expansion coefficient at constant pressure. In the classical laboratory constant-pressure air thermometer (or dilatometer), a dry air sample is enclosed within a spherical glass bulb of calibrated volume V_0 completely submerged within a stirred, variable-temperature liquid bath. The bulb neck communicates via an unheated capillary tube with a precision graduated glass manometer limb and an adjustable liquid mercury leveling reservoir. By systematically shifting the vertical elevation of the leveling reservoir after every temperature station so that the mercury menisci in both limbs stand at exactly equal horizontal elevations, the internal gas pressure is held strictly invariant at prevailing ambient atmospheric pressure (P = B). Plotting the measured gas volume V_t against Celsius temperature t produces an exceptionally straight line whose slope m = V_0 β and y-intercept b = V_0 directly yield the thermal expansion coefficient β = m / b. Crucially, extrapolating the linear regression line leftward to the point of theoretical vanishing volume (V_t → 0) determines the absolute zero of temperature: T_0 = −1 / β ≈ −273.15 °C, establishing the fundamental physical concept of the absolute thermodynamic Kelvin 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 m³ K⁻¹.

5. Method

1. Fill the thermal bath with finely shaved pure distilled ice and water, ensuring the glass bulb T₁ is completely submerged up to the index mark on the capillary stem. Vigorously agitate the slurry for several minutes until the thermometer reads 0.00 °C. Adjust the movable leveling reservoir until the mercury meniscus in the open tube aligns precisely with the meniscus in the graduated tube T₂ (isobaric condition, Δh = 0). Record the initial ice-point volume V_0 = 100.00 cm³.
2. Replace the ice slurry with distilled water and energize the bath heater, setting the thermal controller to warm the bath to the first elevated target temperature of 20.0 °C. Maintain continuous mechanical stirring to prevent vertical convection currents or thermal boundary layers around the glass bulb.
3. Upon reaching each target temperature station (20.0 °C, 40.0 °C, 60.0 °C, 80.0 °C), reduce heater power and maintain a dwell time of at least three to four minutes. Confirm that the thermometer reading remains steady within ±0.05 °C and the expanding mercury meniscus in tube T₂ has ceased downward movement.
4. Gently manipulate the rack-and-pinion thumb drive on the movable reservoir slider to lower the open mercury tube until its liquid meniscus matches the exact horizontal elevation of the meniscus in graduated tube T₂. Sight across the horizontal level index with an optical magnifier to verify zero differential height (Δh = h_open − h_closed = 0).
5. Read the total gas volume V_t on the graduated scale of tube T₂ at the crest of the convex mercury meniscus, recording the volume to the nearest 0.05 cm³. Subtract the initial ice-point volume V_0 to compute the net thermal expansion volume ΔV = V_t − V_0.
6. Compile the data table of volume V (in cm³ and SI units of m³) versus temperature t (in °C). Perform an ordinary least-squares linear regression fitting V = m · t + b, compute the isobaric volume expansion coefficient β = m / b, evaluate its statistical standard error, and project the regression line to V = 0 to determine the experimental absolute zero 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 = 0.0139%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.0000e-17; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 5.0000e-11. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.4495e-09.

Propagating only the relevant fit coefficient gives a fit-only standard error of 5.0000e-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 water heating bath during continuous mechanical stirring (±0.2 °C), thermal response lag of the precision thermometer, capillary stem volume exclusion (a small fraction of the gas column in the capillary remains at intermediate temperature between bath and room air, ~0.8%), meniscus parallax error across the graduated leveling tube (±0.05 cm³), atmospheric barometric fluctuations over the heating cycle (±0.2 mmHg), and slight thermal expansion of the glass bulb envelope (borosilicate glass volume expansion α_v ≈ 10 × 10⁻⁶ K⁻¹) contribute experimental uncertainty. The OLS regression standard error reflects random scatter across temperature stations, but does not capture unmodelled capillary dead-space gradients or glass envelope 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

• Ensure the air charged into the glass bulb is thoroughly dry and desiccated, as the presence of condensed water vapor will vaporize at elevated temperatures, superimposing non-linear vapor pressure increases upon the ideal gas expansion.
• Keep the water bath vigorously and continuously stirred at all times, as unagitated liquid baths develop substantial vertical thermal stratification where the surface can be several degrees warmer than the base.
• Always re-level the mercury columns before recording the volume reading at each temperature station, as failure to eliminate differential hydrostatic head Δh introduces unwanted pressure variations that violate isobaric constraints.
• Ensure the unheated capillary stem connecting the bulb to the measuring tube is of minimal bore diameter, minimizing the unheated dead-space volume of gas that remains outside the thermal bath.
• Handle the glass bulb and mercury apparatus with care, avoiding rapid thermal shocks that could crack the thin glass envelope or dislodge the flexible mercury tubing connections.

The teaching simulation models an ideal gas under isobaric conditions at constant atmospheric pressure B = 101.325 kPa (760.0 mmHg) and initial bulb volume V_0 = 100.00 cm³ at 0.00 °C. In physical laboratory experiments, temperature non-uniformity across the heated water bath introduces minor temperature measurement offsets (±0.2 °C), thermal response latency in the thermometer sensor causes reading delays during heating, capillary stem volume exclusion (a small fraction of the gas column in the capillary stem remains outside the bath at room temperature, ~0.8%), meniscus leveling parallax error across the graduated tube (±0.05 cm³), atmospheric barometric drift over the heating run (±0.2 mmHg), and slight thermal expansion of the borosilicate glass bulb envelope (borosilicate volume expansion α_v ≈ 9.9 × 10⁻⁶ K⁻¹) contribute experimental uncertainty. The simulation evaluates confined gas volume using V_t = V_0 (1 + β_ref · t), where V_0 = 100.00 cm³ is the ice-point volume, β_ref = 1 / 273.15 K⁻¹ is the ideal gas volume expansion coefficient, and t is the bath temperature in °C. The volume in cm³ is rounded to 0.01 cm³ to simulate precision burette dilatometry. Ordinary least-squares regression of volume V (m³) against temperature t (°C) across five temperature stations (0.0, 20.0, 40.0, 60.0, 80.0 °C) yields the reported thermal volume expansion coefficient β = m / b and extrapolated absolute zero T_0 = −b / m.

12. Conclusion

The worked example illustrates how to obtain Volume expansion coefficient β from measurements, graph variables and a fitted coefficient. The estimate is 0.003661 K⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.