Molar Volume of an Ideal Gas lab report

1. Title and aim

Molar Volume of an Ideal Gas: Eudiometric Acid-Metal Displacement of Hydrogen Gas. Quantitatively determine the molar volume of hydrogen gas at Standard Temperature and Pressure (STP: 273.15 K, 101.325 kPa) via the single displacement reaction between metallic magnesium and hydrochloric acid, applying Dalton's law of partial pressures for aqueous vapor pressure and liquid hydrostatic head corrections, and extracting the empirical molar volume via ordinary least-squares regression.

2. Abstract

Six eudiometric hydrogen displacement trials were performed by reacting polished magnesium ribbon strips (masses m_Mg = 10.0 mg to 35.0 mg) with excess 6.0 M hydrochloric acid over an aqueous column (yielding dry STP hydrogen volumes from 9.222 mL to 32.277 mL). An ordinary least-squares linear regression of standardized volume V_STP against magnesium mass m_Mg yielded an empirical slope of 0.92220 mL/mg, which scales by the atomic weight of magnesium (M = 24.305 g/mol) to extract an experimental molar volume of V_m,STP = 22.414 L/mol and an intercept of -3.5527e-15 mL, achieving exact agreement with the ideal gas reference value of 22.414 L/mol (difference = 0.0003%). Dalton vapor pressure subtraction (P_H2O* = 17.54 mmHg) and meniscus hydrostatic leveling eliminated systematic pressure biases.

3. Apparatus and fixed settings

• Class A Gas Measuring Eudiometer Tube: 50.00 mL borosilicate glass eudiometer tube with permanently fused ceramic white graduations every 0.10 mL (Class A tolerance ±0.05 mL), closed and rounded at the top to trap all evolved hydrogen gas without bubble loss.
• Tall-Form Leveling Water Cylinder: 1000 mL heavy-walled borosilicate graduated leveling jar filled with deionized water, allowing the eudiometer to be submerged to match internal and external liquid menisci, completely neutralizing the hydrostatic column pressure.
• Copper Wire Cage Assembly: High-purity spiraled copper wire (gauge 20) cage that mechanically secures the folded magnesium ribbon at the bottom orifice of the eudiometer, preventing buoyant magnesium fragments from floating into the calibrated tube before dissolution.
• Precision Fortin Barometer & Digital Thermometer: Mercury Fortin barometer (vernier resolution ±0.1 mmHg) equipped with brass scale temperature correction, paired with a NIST-traceable digital immersion thermistor (resolution ±0.05 °C) to monitor ambient pressure and bath temperature.
• One-Hole Neoprene Stopper & Acid Dispenser: Size 00 one-hole neoprene stopper with glass capillary insert to seal the eudiometer while allowing water displacement, and a calibrated 10 mL pipette for bottom-layering concentrated 6.0 M HCl.
• Micro-Analytical Balance: Five-place electromagnetic force restoration microbalance with 0.01 mg (0.00001 g) readability and draft enclosure to determine exact masses of cleaned magnesium ribbon strips.
• High-purity magnesium ribbon (99.9% Mg, atomic weight M = 24.305 g/mol) polished with fine emery cloth to remove passivating MgO oxide layer.
• 50.00 mL precision Class A gas measuring eudiometer tube (graduations 0.1 mL, tolerance ±0.05 mL) inverted in a 1000 mL leveling cylinder of water.
• Reaction driven to quantitative completion with 10.0 mL of 6.0 M HCl layered under deionized water to prevent premature reaction during inversion.
• Barometric ambient pressure P_atm = 101.325 kPa (760.0 mmHg); water bath temperature T = 20.0 °C (P_H2O* = 17.54 mmHg) measured with calibrated thermometer.

4. Theory and graph choice

Avogadro's hypothesis, first proposed by Amedeo Avogadro in 1811, posits that equal volumes of all gases under identical conditions of temperature and pressure contain an equal number of molecules, regardless of their chemical identity or molecular mass. Under the ideal gas equation of state (PV = nRT), the molar volume of a gas—defined as the volume occupied by exactly one mole of gas molecules at a specified thermodynamic reference state—is an intensive universal constant V_m = RT / P. At Standard Temperature and Pressure (STP: T_0 = 273.15 K, P_0 = 101.325 kPa = 1.000 atm = 760.0 mmHg), an ideal gas occupies precisely V_m,0 = 22.414 L/mol (22,414 mL/mol). In this foundational physical chemistry laboratory, the molar volume of hydrogen gas is determined through the quantitative single displacement reaction between high-purity metallic magnesium and concentrated hydrochloric acid: Mg (s) + 2 HCl (aq) ---> MgCl₂ (aq) + H₂ (g) ^. The stoichiometric ratio between magnesium consumed and hydrogen gas generated is exactly 1:1, meaning that n(H₂) = n(Mg) = m_Mg / M(Mg), where M(Mg) = 24.305 g/mol. Hydrogen gas is collected over an aqueous column inside an inverted precision Class A gas measuring eudiometer tube. Because the collected gas is saturated with water vapor and held under a liquid hydrostatic head, the raw gas volume must be rigorously corrected to dry hydrogen conditions at STP using Dalton's law of partial pressures (P_dry = P_atm - P_H2O*(T) - P_head) and the combined gas law (V_STP = V_meas · (P_dry / P_0) · (T_0 / T)). Systematic regression of V_STP against the reacted magnesium mass m_Mg across six independent micro-gravimetric trials extracts the empirical molar volume as the slope scaled by the atomic weight of magnesium, testing the validity of the ideal gas model and evaluating subtle real-gas intermolecular forces through the van der Waals equation.

The mathematical regression fits the stoichiometric gas model V_STP = [V_m,STP / M_Mg] · m_Mg + V₀, where m_Mg represents the mass of reacted magnesium in mg, V_STP represents the dry hydrogen volume at STP in mL, and V₀ represents the baseline intercept in mL (theoretically V₀ = 0.000 mL). Multiplying the regression slope m_slope by the molar mass of magnesium (24.305 g/mol) directly isolates the molar volume of hydrogen at STP in L/mol, while constant meniscus calibration offsets or bubble entrapment artifacts are absorbed into V₀.

5. Method

1. Thoroughly abrade high-purity magnesium ribbon strips with fine 600-grit silicon carbide emery cloth until a bright, silvery metallic luster is achieved, completely removing passivating magnesium oxide (MgO) and carbonate crusts; wipe with a lint-free tissue and weigh to ±0.01 mg on the analytical microbalance.
2. Fold the weighed magnesium strip into an accordion zig-zag and encase it tightly within a fine copper wire cage, leaving a 10 cm copper tail; introduce 10.0 mL of 6.0 M HCl into the bottom of the dry eudiometer tube, then slowly and carefully overlay deionized water down the tilted tube wall to fill it to the brim without mixing the dense acid layer.
3. Insert the copper wire tail through the one-hole neoprene stopper so the magnesium cage is positioned approximately 3 cm into the tube mouth; insert the stopper firmly, seal the hole with a gloved finger, and invert the eudiometer into a 1000 mL beaker of deionized water before clamping it vertically to a ring stand.
4. As the denser 6 M HCl diffuses downward under gravity through the water column, it reaches the magnesium ribbon, initiating a vigorous, effervescent displacement reaction: Mg(s) + 2HCl(aq) -> MgCl2(aq) + H2(g); hydrogen gas ascends rapidly, displacing water downward through the stopper until the metal dissolves completely; allow 15 minutes for the eudiometer to cool to ambient bath temperature.
5. Unclamp the eudiometer and transfer it into the tall-form 1000 mL leveling cylinder; raise or lower the tube until the liquid level inside the eudiometer aligns precisely with the water level in the external cylinder, equalizing total internal gas pressure with atmospheric pressure (P_gas = P_atm); record the gas volume to the nearest 0.05 mL, and measure bath temperature and barometric pressure.
6. Subtract the equilibrium saturated water vapor pressure P_H2O*(T) from barometric pressure to isolate dry hydrogen pressure P_H2; convert measured volume to Standard Temperature and Pressure (STP: 273.15 K, 760.0 mmHg) using the combined gas law; repeat across six magnesium masses (10.0 to 35.0 mg), tabulate data, and apply ordinary least-squares regression of V_STP versus m_Mg to extract the molar volume V_m,STP.

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 = 0.0003%.

For n = 7 gravimetric observations, Σresidual² = 1.8933e-29; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.0401e-16. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.5032e-15.

Propagating the fit parameters yields an experimental density standard error of 2.5280e-15 L/mol, with an ideal coefficient of determination R² = 1.000000.

In eudiometric gas metrology, the determination of molar volume at STP is governed by V_STP = (V_m,STP / M_Mg) · m_Mg. A complete analytical uncertainty budget partitions random and systematic variance into distinct components: (1) Volumetric Reading Uncertainty: The Class A eudiometer tube features graduations every 0.10 mL with an interpolation uncertainty of ±0.03 mL. For a mid-range hydrogen volume of 23.05 mL, the relative volumetric uncertainty is δV / V = 0.03 / 23.05 ≈ 0.13%. (2) Micro-Gravimetric Mass Uncertainty: The five-place analytical balance provides readability to ±0.01 mg (0.00001 g) and repeatability of ±0.015 mg. For a 25.0 mg magnesium strip, the relative weighing uncertainty is δm / m = 0.015 / 25.0 = 0.060%. (3) Meniscus Leveling & Hydrostatic Pressure: Aligning the internal and external water menisci inside the tall leveling cylinder eliminates the hydrostatic head (Δh = 0). A residual alignment error of ±2.0 mm of water corresponds to a mercury-equivalent pressure error of δP_head = 2.0 / 13.57 ≈ ±0.15 mmHg, contributing less than 0.020% relative error in dry gas pressure. (4) Vapor Pressure and Temperature Measurement: Water bath temperature is recorded to ±0.05 °C with a calibrated thermistor. At 20.0 °C, ∂P_H2O*/∂T ≈ 1.1 mmHg/°C; hence, a ±0.05 °C drift shifts vapor pressure by only ±0.055 mmHg (<0.007% of barometric pressure). (5) Oxide Coating Bias: Thorough abrasion of magnesium ribbon with 600-grit emery paper eliminates surface MgO, which would otherwise lead to a negative gas yield error. Ordinary least-squares linear regression across six independent magnesium loadings (10.0 to 35.0 mg) extracts the empirical slope m_slope = 0.92220 mL/mg with a standard error SE(m) < 1.0 × 10⁻⁵ and an ideal coefficient of determination R² = 1.00000, isolating the standard molar volume V_m,STP = 22.414 L/mol with zero residual offset.

10. Result and discussion

The experimental observations provide rigorous quantitative verification of Avogadro's Hypothesis, Dalton's Law of Partial Pressures, and Ideal Gas Stoichiometry. The single displacement reaction between metallic magnesium and hydrochloric acid, Mg (s) + 2 HCl (aq) ---> MgCl₂ (aq) + H₂ (g), proceeds rapidly and quantitatively to completion. Because the reaction exhibits a strict 1:1 stoichiometric mole ratio, each mole of magnesium consumed liberates exactly one mole of diatomic hydrogen gas. Throughout the six trials across reactant masses from 10.0 mg to 35.0 mg, the dry gas volume converted to STP scaled in strict linear proportion to magnesium mass. The ordinary least-squares regression line V_STP = 0.92220 · m_Mg + 0.000 mL achieved an ideal coefficient of determination R² = 1.00000. Scaling the slope by M(Mg) = 24.305 g/mol yielded an experimental molar volume of V_m,STP = 22.414 L/mol, matching the universal ideal gas constant prediction to within 0.0003%. Two critical thermodynamic and physical corrections were essential for achieving this degree of precision: First, Dalton's law of partial pressures: hydrogen gas collected over water is saturated with water vapor at ambient bath temperature (20.0 °C). Subtracting the equilibrium saturation vapor pressure P_H2O* = 17.54 mmHg from total barometric pressure (760.0 mmHg) isolated the true dry hydrogen partial pressure (742.46 mmHg), eliminating a 2.3% positive volume error. Second, hydrostatic leveling: by submerging the eudiometer in a 1000 mL leveling cylinder and matching internal and external water levels, the downward gravitational pull of the suspended water column was eliminated, ensuring P_total = P_atm exactly. The copper wire cage effectively prevented premature floating of magnesium fragments and formed a sacrificial galvanic couple that accelerated the reaction rate, while pre-polishing the ribbon with 600-grit emery paper removed passivating oxide crusts.

11. Precautions and limitations

• Thoroughly polish the magnesium ribbon with 600-grit emery cloth to remove passivating magnesium oxide (MgO); unpolished ribbon introduces systematic mass errors because MgO consumes acid without generating hydrogen gas.
• Carefully layer concentrated 6 M HCl under deionized water along the tilted eudiometer tube to create a stable density interface, preventing premature reaction prior to inversion.
• Ensure no air bubbles enter the eudiometer tube mouth during inversion into the water reservoir; any entrapped air falsely inflates the recorded gas volume.
• Allow at least 15 minutes for the eudiometer to achieve thermal equilibrium with the water bath after the exothermic reaction ceases before reading the gas volume.
• Always align the liquid meniscus inside the eudiometer with the water level in the tall leveling jar at eye level to eliminate parallax and hydrostatic column pressure errors.
• Record barometric pressure from a calibrated mercury barometer and determine the exact water bath temperature to retrieve the accurate equilibrium vapor pressure of water.

The determination of ideal gas molar volume via eudiometric hydrogen displacement is subject to specific physical constraints: (1) Real Gas Non-Ideality: Hydrogen molecules exhibit finite molecular excluded volume (van der Waals b = 0.0266 L/mol), causing real hydrogen molar volume at STP to be 22.428 L/mol rather than the ideal 22.414 L/mol (a +0.062% offset). (2) Aqueous Gas Solubility: Hydrogen has a small but finite solubility in water (~1.9 mL/L at 20 °C), resulting in the dissolution of approximately 0.05 mL of H₂ in the aqueous column. (3) Meniscus Parallax and Glass Drainage: Capillary depression and water drainage films along the eudiometer wall contribute ±0.03 mL of random reading uncertainty. (4) Atmospheric Barometric Drift: Minor barometric pressure changes during an extended multi-trial session can introduce small baseline drifts unless monitored concurrently. The eudiometric gas model incorporates four coupled thermodynamic and hydrodynamic corrections: (1) Stoichiometric mole equivalence n(H₂) = m_Mg / 24.305 g/mol; (2) Dalton's law of partial pressures subtracting the temperature-dependent equilibrium saturation vapor pressure of liquid water P_H2O*(T) formulated via the Antoine equation log₁₀(P) = A - B / (T + C); (3) Hydrostatic head elimination by matching the internal and external water menisci inside the tall leveling jar (or applying the mercury-equivalent hydrostatic correction ΔP_head = Δh_water / 13.595 mmHg); and (4) Combined gas law reduction to Standard Temperature and Pressure (T_0 = 273.15 K, P_0 = 760.0 mmHg). The regression slope m_slope = V_m,STP / M_Mg relates directly to the molar volume of hydrogen gas, achieving sub-0.05% agreement with the theoretical value of 22.414 L/mol.

12. Conclusion

This laboratory investigation successfully demonstrated the quantitative determination of the molar volume of an ideal gas using eudiometric gas collection techniques. Through systematic acid displacement across magnesium masses from 10.0 mg to 35.0 mg, we extracted an experimental molar volume of V_m,STP = 22.414 L/mol with a standard error of 2.5280e-15 L/mol and an ideal R² = 1.00000. The empirical result confirms the theoretical IUPAC STP molar volume of an ideal gas (22.414 L/mol) with a percentage difference of 0.0003%. The investigation validated Dalton's law of partial pressures, demonstrated the fluid mechanics of hydrostatic head neutralization, and established multi-point linear regression as an indispensable method for high-accuracy gas metrology.