Linear Thermal Expansion of Copper lab report

1. Title and aim

Determination of the Coefficient of Linear Thermal Expansion of Copper by Steam Dilatometry. Determine the coefficient of linear thermal expansion (α) of a solid copper specimen using a steam-jacketed dilatometer with precision micrometer screw gauge and electrical contact indicator, measuring the longitudinal elongation (ΔL) resulting from controlled thermal elevation (ΔT) and evaluating the linear dilatation slope.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Linear expansion coefficient alpha of 16.80 × 10⁻⁶ K⁻¹, differing by 0.000% 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

• Steam Dilatometer Enclosure with Rigid End Abutments: A double-walled brass jacket 1.1 m in length mounted upon a rigid heavy cast-iron bed plate, equipped with low-friction internal guide rollers to support the specimen rod horizontally, an internal thermometer socket, a steam inlet nozzle, a bottom condensate drain, and a hardened steel fixed abutment block at the left end.
• Precision Micrometer Screw Gauge with Spherometer Carriage: A hardened stainless steel micrometer head having a calibrated lead screw pitch of 0.500 mm and 50 circular thimble subdivisions, rigidly clamped to the right end frame to measure longitudinal specimen displacements with an instrument resolution of 0.001 mm.
• Low-Voltage Electrical Contact Indicator Circuit: A 1.5 V dry cell battery connected in series with a 10 kΩ current-limiting resistor and sensitive center-zero galvanometer between the insulated fixed frame and the micrometer spindle, providing instantaneous, zero-force visual indication when the advancing micrometer tip touches the expanding copper rod.
• Regulated Steam Boiler and High-Precision Thermometer: An electric steam generator supplying saturated water vapor at 100 °C through flexible insulated tubing to elevate the specimen temperature uniformly, monitored by a 0.1 °C immersion thermometer inserted into the central dilatometer chamber.
• Electrolytic copper rod specimen (OFHC, 99.95% Cu); initial length L_0 = 1.0000 m (1000.0 mm) at ambient baseline temperature T_0 = 20.00 °C.
• Insulated brass steam dilatometer jacket with central mercury thermometer port, steam inlet tube, and condensation drainage nipple.
• Precision micrometer head with pitch p = 0.500 mm and 50 thimble divisions (least count 0.010 mm; optical vernier estimation to ±0.001 mm).
• Low-voltage DC contact indicator circuit (1.5 V cell, current-limiting 10 kΩ resistor, and high-sensitivity microammeter / galvanometer).
• Reference coefficient of linear thermal expansion of pure copper: α = 16.80 × 10⁻⁶ K⁻¹ (16.80 × 10⁻⁶ °C⁻¹ at 20 °C).

4. Theory and graph choice

Thermal expansion in crystalline condensed matter is a direct microscopic manifestation of the anharmonicity of interatomic binding potential energy wells. At absolute zero, lattice ions reside at the bottom of an asymmetric potential energy minimum U(r). In a purely harmonic potential well where U(r) = (1/2)k(r − r_0)², thermal agitation increases vibrational amplitude symmetrically about the equilibrium spacing r_0, resulting in zero net shift in mean atomic separation regardless of temperature. In real solids, however, the short-range repulsive force arising from Pauli exclusion overlap between electron clouds is significantly steeper than the longer-range attractive electrostatic or metallic bonding forces. Consequently, as thermal energy kT increases the mean vibrational energy of phonons in the lattice, atoms oscillate across an asymmetric potential curve, vibrating more widely toward larger interatomic separations than toward compressed ones. The time-averaged atomic position ⟨r⟩ shifts outward with rising temperature, macroscopic evidence of which is observed as the thermal expansion of the solid body. For an isotropic solid bar of initial length L_0 at baseline temperature T_0, a uniform temperature increase to T = T_0 + ΔT produces a longitudinal elongation ΔL proportional to both L_0 and ΔT: ΔL = L_0 · α · ΔT, where α is the coefficient of linear thermal expansion (expressed in reciprocal kelvins, K⁻¹ or °C⁻¹). In steam dilatometry, a copper tube or rod is enclosed within an insulated metal steam jacket supported between rigid end frames. One end of the rod bears firmly against a fixed abutment stop, while the opposite end projects through a low-friction packing gland against a precision micrometer screw gauge. To eliminate mechanical deformation and anvil pressure errors, an electrical circuit comprising a dry cell and galvanometer detects the exact physical contact between the micrometer spindle and rod tip. Passing dry steam through the jacket heats the specimen uniformly, and the resulting elongation ΔL is measured across multiple steady-state temperature elevations. Plotting ΔL against the product L_0 · ΔT generates a straight line whose slope directly yields the linear expansion coefficient α.

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 K⁻¹.

5. Method

1. Measure the overall length of the clean, polished copper rod at ambient room temperature using a precision steel meter scale and vernier calipers to ±0.5 mm, recording L_0 = 1.0000 m. Insert the rod into the steam jacket, seating its left end firmly against the hardened steel fixed abutment stop. Insert the calibrated thermometer into the central socket and record the steady initial room temperature T_0 = 20.00 °C.
2. Connect the electrical contact indicator leads to the rod and micrometer head. Slowly advance the micrometer thimble until the galvanometer needle deflects, indicating that the spindle has touched the cold rod tip without exerting compressive mechanical strain. Read the micrometer drum to ±0.001 mm and record this initial baseline reading R_0. Back the micrometer screw off by 2 to 3 turns to allow room for free expansion.
3. Connect the steam generator to the dilatometer inlet and place a beaker beneath the drain nipple to collect condensed water. Activate the boiler and admit steam steadily into the jacket. Observe the rapid temperature rise on the central thermometer and allow steam to circulate until condensate flows smoothly and the thermometer reading stabilizes.
4. Once steady-state equilibrium is attained at the target temperature (e.g., T = 65.0 °C, giving ΔT = 45.0 °C), carefully advance the micrometer thimble until the galvanometer needle just twitches. Record the hot micrometer reading R_hot to ±0.001 mm. The longitudinal elongation is evaluated as ΔL = R_hot − R_0. Immediately retract the screw to prevent jamming.
5. Repeat the measurement procedure across five distinct temperature elevations (ΔT = 15.0 °C, 30.0 °C, 45.0 °C, 60.0 °C, and 75.0 °C) by adjusting the steam-air mixing ratio or tracking the cooling curve of the jacket. For each setting, record the stable temperature elevation ΔT and corresponding micrometer elongation ΔL.
6. For each observation, calculate the independent variable X = L_0 · ΔT in m K (or m °C) and the dependent displacement Y = ΔL in meters. Plot Y against X, perform ordinary least-squares linear regression to determine the gradient S, and evaluate the linear thermal expansion coefficient α = S. Compare α with the standard reference value for electrolytic copper.

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 = 0.0000%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.8204e-38; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 2.3790e-21. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.1836e-19.

Propagating only the relevant fit coefficient gives a fit-only standard error of 2.3790e-15 × 10⁻⁶ 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.

Micrometer screw gauge zero-point indexing and backlash error (±0.002 mm), electrical contact sensitivity threshold and contact point oxidation, temperature non-uniformity along the longitudinal steam jacket (temperature gradient between steam inlet and outlet), heat conduction loss at the end supports where the rod protrudes, thermal response lag of the enclosed thermometer, and scale uncertainty in measuring initial rod length L0 (±0.5 mm) contribute experimental uncertainty. The OLS regression standard error reflects random scatter across temperature stations, but does not capture shared length measurement offsets or longitudinal thermal gradients.

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 advance the micrometer spindle in the forward direction when approaching the rod tip to eliminate mechanical screw backlash.
• Ensure the electrical contact circuit is operational and watch for the earliest galvanometer deflection to avoid compressive strain on the specimen.
• Allow saturated steam to flow through the jacket until condensation drains steadily and the thermometer reading is completely stationary before recording hot measurements.
• Ensure that the fixed end stop remains completely rigid and immovably clamped to the bed plate throughout the thermal cycle.

The teaching simulation models linear thermal dilatation under steady-state isothermal conditions with an initial rod length L_0 = 1.0000 m at T_0 = 20.00 °C and an invariant reference expansion coefficient α = 16.80 × 10⁻⁶ K⁻¹. In physical experiments, minor temperature gradients exist along the length of the steam jacket (especially near the protruding ends), thermometer thermal lag and stem emergent column errors can cause small temperature reading offsets, mechanical micrometer backlash or pitch errors contribute displacement uncertainty (±0.002 mm), electrical contact threshold sensitivity can vary with surface oxide film formation on the copper tip, and slight frame flexion during thermal elevation can introduce small structural offsets. The simulation calculates longitudinal elongation using the linear dilatation relationship ΔL = L_0 · α · ΔT, where L_0 = 1.0000 m is the baseline length, α = 16.80 × 10⁻⁶ K⁻¹ is the reference linear expansion coefficient for pure copper, and ΔT = T − T_0 is the temperature elevation above 20.00 °C. Elongation readings are rounded to 0.001 mm to match high-precision micrometer screw resolutions, and ordinary least-squares regression of ΔL against L_0 · ΔT across five temperature stations yields the reported linear expansion coefficient.

12. Conclusion

The worked example illustrates how to obtain Linear expansion coefficient alpha from measurements, graph variables and a fitted coefficient. The estimate is 16.80 × 10⁻⁶ K⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.