Boyle’s Law lab report

1. Title and aim

Verification of Boyle’s Law and Determination of the Isothermal State Constant for an Ideal Gas. Verify Boyle’s law for a confined sample of dry air at constant room temperature by measuring absolute gas pressure P as a function of volume V using a mercury J-tube manometer, and determine the isothermal Boyle state constant k = P · V via linear regression of absolute pressure P against reciprocal volume 1/V.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Boyle isothermal state constant k of 3.0399 J, differing by 0.005% 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 Closed Manometer Tube (Specimen Air Column): Hermetically confines the dry air sample of variable volume V over a liquid mercury column, featuring precision linear graduations etched along a bore of uniform internal diameter to permit direct volumetric readings in cm³ (±0.1 cm³).
• Open Movable Leveling Tube and Vertical Slider Carriage: Provides an adjustable atmospheric mercury column that can be elevated or lowered along a sturdy vertical rack-and-pinion sliding mount to vary the hydrostatic head pressure Δh applied to the trapped gas column.
• Flexible Heavy-Walled Mercury U-Tube Reservoir Hose: Fluidly interconnects the base of the closed specimen tube with the movable open leveling reservoir, containing triple-distilled mercury while resisting hydrostatic expansion, leakage, and wall distortion under pressure.
• Fortin Precision Laboratory Mercury Barometer: Measures the prevailing ambient atmospheric pressure B in millimeters of mercury (mmHg) and kilopascals (kPa) with vernier accuracy (±0.1 mmHg) to establish the barometric reference datum for absolute pressure calculations.
• Central Metric Height Scale and Parallax Index: A rigid brass or steel millimeter scale mounted symmetrically between the two glass tubes, equipped with sliding hairline vernier indices to measure the exact vertical liquid meniscus heights h_closed and h_open without optical parallax.
• Precision Digital Thermometer and Environmental Monitor: Continuously monitors ambient air and mercury column temperatures to ensure that isothermal conditions (ΔT < 0.2 °C) are strictly preserved throughout the compression and expansion measurement sequence.
• Confined dry air specimen at room temperature T_0 = 20.00 °C (293.15 K) with molar mass M_air = 28.966 g mol⁻¹.
• Calibrated precision closed glass tube with uniform internal cross-sectional bore A = 1.000 cm² and graduated metric volume scale (15.0 to 50.0 cm³).
• Standard ambient barometric atmospheric pressure B = 101.325 kPa (760.0 mmHg = 1.01325 bar).
• High-density triple-distilled liquid mercury manometer fluid (density ρ_Hg = 13,546 kg m⁻³ at 20.00 °C; local gravitational acceleration g = 9.80665 m s⁻²).
• Reference isothermal state constant for nominal charge V_0 = 30.0 cm³ at atmospheric pressure: k = P_0 V_0 = 101,325 Pa × 30.0 × 10⁻⁶ m³ = 3.03975 J.

4. Theory and graph choice

Boyle's law represents the foundational empirical cornerstone of pneumatic physics and classical thermodynamics, enunciating the fundamental inverse proportionality between the absolute pressure and the volume of a confined mass of gas held at invariant thermodynamic temperature. First published in 1662 by the Anglo-Irish natural philosopher Robert Boyle in his seminal treatise 'New Experiments Physico-Mechanicall, Touching the Spring of the Air, and its Effects'—and independently verified with rigorous mathematical formalism in 1676 by the French physicist Edme Mariotte (whence it is frequently termed Boyle-Mariotte's law)—this governing physical principle dismantled the Aristotelian notion of horror vacui and established gases as elastic fluids composed of particulate entities whose macroscopic mechanical manifestations derive from microscopic momentum exchanges. Under isothermal thermodynamic conditions (T = constant), the mechanical work performed upon a gas during quasi-static compression increases its volumetric energy density: P · V = n R T = k, where n is the molar quantity of gas, R = 8.31446 J mol⁻¹ K⁻¹ is the universal gas constant, and k is the isothermal Boyle state constant (possessing units of joules, equivalent to Pa m³ or N m). In the classical laboratory configuration, a clean, dry air sample is hermetically confined within a calibrated uniform glass bore manometer tube closed at the upper extremity, bounded inferiorly by an adjustable liquid mercury meniscus column. The lower end of the graduated closed tube communicates through heavy-walled flexible rubber or elastomeric tubing with an open leveling tube mounted upon a vertical metric slider. By adjusting the vertical elevation of the open reservoir relative to the closed specimen tube, the experimenter imposes variable hydrostatic mercury head pressures Δh = h_open − h_closed upon the trapped air column. The absolute pressure P exerted upon the gas is established by summing the local barometric atmospheric pressure B (measured with an auxiliary Fortin mercury barometer) and the differential manometer column height: P = B + Δh. By plotting the resulting absolute pressure P against the reciprocal of the gas volume (1/V), the curvilinear hyperbolic Boyle isotherm P(V) is linearized into a strictly origin-intercepting linear relationship y = m x whose gradient m directly delivers the physical state constant k = n R T. High-precision least-squares regression across systematic compression and expansion intervals provides an exemplary verification of ideal gas behavior, elucidating the macroscopic operational envelope within which atmospheric air strictly approximates the thermodynamic ideal gas.

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 J.

5. Method

1. Read the prevailing ambient laboratory atmospheric pressure B from the Fortin precision mercury barometer to ±0.1 mmHg, applying standard capillary and brass-scale temperature corrections to establish the exact reference datum B = 101.325 kPa (760.0 mmHg). Record the baseline room temperature T_0 = 20.00 °C.
2. Adjust the height of the movable open leveling tube until the mercury meniscus in the open tube aligns precisely in the same horizontal plane as the mercury meniscus in the closed specimen tube (h_open = h_closed), establishing a differential hydrostatic head of Δh = 0. Verify that the enclosed air volume corresponds to the nominal baseline volume V_0 = 30.0 cm³.
3. Execute the compression series by systematically raising the open leveling tube above the closed tube along its vertical rack slider, imposing positive hydrostatic mercury heads to compress the air sample to designated volumes of V = 25.0 cm³ and V = 20.0 cm³. Align the sliding vernier index with the apex of each meniscus to eliminate parallax.
4. Following each vertical adjustment of the leveling tube, pause for a dwell period of at least two to three minutes before recording pressure values. Observe the thermometer immersed in the adjacent thermal buffer to verify that the air column returns to baseline temperature T_0 = 20.00 °C.
5. Execute the expansion series by lowering the open leveling tube beneath the closed specimen tube, applying negative hydrostatic heads to expand the confined air column to volumes of V = 35.0 cm³ and V = 40.0 cm³. Record the open tube height h_open and closed tube height h_closed, computing the differential head Δh = h_open − h_closed.
6. Compute the absolute pressure P = B + Δh (in pascals and kilopascals) and the reciprocal volume X = 1/V (in m⁻³ and display L⁻¹) for each of the five observation stations. Perform an ordinary least-squares linear regression of P against 1/V, evaluating the slope m = k, the y-intercept b, the statistical standard error, and the coefficient of determination R².

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 = 0.0055%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 31.68089; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.6384e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 5.97638.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1.6384e-04 J. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Barometric pressure fluctuations during the observation period (±0.2 mmHg), capillary depression at the mercury-glass menisci in unequal tube diameters (±0.3 mm), room temperature drift affecting isothermal conditions (±0.2°C), non-uniformity in the internal cross-section of the graduated closed tube, parallax errors in reading the mercury columns, and slight trace moisture in the confined air contributing water vapor pressure departures from ideal behavior contribute experimental uncertainty. The OLS regression standard error reflects random reading scatter, but does not capture unmodelled capillary curvature offsets or ambient barometric drift.

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 enclosed within the closed manometer tube is thoroughly dry, as water vapor condensible at room temperature violates ideal gas behavior by exhibiting non-ideal vapor pressure fluctuations during compression.
• Allow sufficient equilibration time (2 to 3 minutes minimum) following every vertical height adjustment to ensure the gas returns to ambient room temperature, dissipating transient adiabatic heating or cooling.
• Read the mercury meniscus height at the extreme crest of its convex curvature in both tubes, aligning the eye with the sighting index to eliminate optical parallax errors across the millimeter scale.
• Maintain the flexible connecting tubing free from kinking, pinching, or trapped air bubbles, which distort hydrostatic pressure transmission and disrupt leveling column continuity.
• Handle liquid mercury with rigorous environmental hygiene protocols, keeping the apparatus within a deep spill-containment tray and ensuring all tube vents are shielded against accidental droplet splashing.

The teaching simulation models an ideal gas under strict isothermal conditions at constant room temperature T_0 = 20.00 °C (293.15 K) and standard barometric pressure B = 101.325 kPa (760.0 mmHg). In physical laboratory practice, barometric pressure drifts over hours as atmospheric weather fronts pass (±0.2 to 0.5 mmHg), capillary depression differences at the mercury-glass menisci introduce minor hydrostatic offsets (±0.2 to 0.4 mm) if tube diameters are not perfectly matched, room temperature fluctuations alter the gas temperature by ±0.2 °C, slight non-uniformity in the bore cross-section of the closed glass tube produces volume scale calibration deviations, and the presence of trace water vapor in non-dehydrated air causes slight departures from ideal gas behavior due to variable saturated vapor pressure contributions. The simulation evaluates absolute pressure using P = k / V, where the reference state constant is k = P_0 V_0 = 101,325 Pa × 30.0 × 10⁻⁶ m³ = 3.03975 J (confined volume V in cm³, with 1 cm³ = 10⁻⁶ m³). The differential mercury head is computed from Δh = (P − B) / (ρ_Hg g), with ρ_Hg = 13,546 kg m⁻³, g = 9.80665 m s⁻², and standard atmosphere B = 101.325 kPa. The calculated pressure in kPa is rounded to 0.01 kPa to simulate precision digital manometry. Ordinary least-squares regression of P (Pa) against 1/V (m⁻³) across five volume settings (20.0, 25.0, 30.0, 35.0, 40.0 cm³) yields the reported isothermal state constant k and statistical uncertainty.

12. Conclusion

The worked example illustrates how to obtain Boyle isothermal state constant k from measurements, graph variables and a fitted coefficient. The estimate is 3.0399 J. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.