Specific Heat Capacity lab report

1. Title and aim

Determination of Specific Heat Capacity by the Method of Mixtures. Determine the specific heat capacity c_s of copper using a water-equivalent jacketed calorimeter and the classical method of mixtures, verifying thermal energy conservation across variable specimen masses, and evaluating radiation heat exchange corrections.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Specific heat capacity of copper of 385.0 J kg⁻¹ K⁻¹, differing by 0.004% 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

• Precision Double-Walled Jacketed Calorimeter Assembly: Consists of an inner polished thin-walled copper containment vessel (mass 150.0 g, thermal capacity ~57.8 J K⁻¹), supported on insulating vulcanized cork pegs inside an outer concentric nickel-plated draft-shield jacket with an annular stagnant air gap to suppress convective and conductive heat transfer to the laboratory environment.
• Regnault Steam Boiler and Specimen Heating Chamber: A cylindrical brass double-walled steam generator equipped with a vertical central heating chamber, rubber delivery tubing, and a sliding drop bottom shutter, designed to suspend metallic specimens in dry saturated water vapor at 100.0 °C until thermal equilibrium is verified without wetting the sample with condensation.
• Calibrated Precision Immersion Thermometers (0.01°C / 0.1°C Division): A paired set of mercury-in-glass standards: one graduated from 0°C to 105°C in 0.5°C divisions for monitoring the steam chamber core temperature, and a high-precision calorimetric thermometer graduated from 15°C to 35°C in 0.05°C subdivisions for detecting slight temperature excursions in the calorimeter.
• Mechanical Helical Brass Stirrer and Precision Analytical Balance: A low-thermal-capacity looped brass wire stirrer designed to produce rapid convective mixing throughout the 200.0 g water bath without inducing kinetic frictional heating, alongside a two-pan analytical balance resolving mass to ±0.01 g for gravimetric metrology of water and copper specimens.
• Initial specimen temperature T_s = 100.0 °C (steam boiler at standard atmospheric pressure 101.325 kPa).
• Initial calorimeter water mass m_w = 0.200 kg (200.0 g), initial temperature T_0 = 20.0 °C, specific heat c_w = 4186.0 J kg⁻¹ K⁻¹.
• Calorimeter inner copper vessel (mass 150.0 g) and brass stirrer water equivalent W_cal = 0.020 kg; effective system thermal capacity C_sys = (m_w + W_cal) · c_w = 920.92 J K⁻¹.
• Pure copper reference specific heat capacity c_s = 385.0 J kg⁻¹ K⁻¹.

4. Theory and graph choice

The quantitative measurement of heat as an extensive physical entity forms the bedrock of classical thermodynamics. Pioneered by Joseph Black in Edinburgh during the mid-eighteenth century, the distinction between temperature (an intensive thermodynamic state variable measuring average molecular kinetic energy) and heat (an extensive form of thermal energy transferred across a system boundary due to a macroscopic temperature gradient) transformed calorimetry into a rigorous precision science. The method of mixtures operates upon the foundational postulate of thermal energy conservation: in an adiabatic enclosure isolated from external work and heat exchange, the total enthalpy lost by a hot solid specimen upon immersion into a cooler liquid medium must identically equal the total enthalpy gained by the liquid, the containing calorimetric vessel, the thermometer bulb, and the mechanical stirrer as the combined system equilibrates to a uniform final temperature T_f. Because the container and its accessories participate directly in the thermal exchange, their aggregate thermal capacity is represented through the system water equivalent W_cal, defined as the mass of pure liquid water that would absorb an identical quantity of heat per degree temperature rise. By measuring the initial temperature of the boiling-water-heated specimen T_s, the initial ambient temperature of the calorimeter water T_0, and the asymptotic equilibrium temperature T_f across an incremental sequence of specimen masses, the specific heat capacity c_s of the metallic analyte is directly extracted from the linear slope of heat absorbed versus the thermal mass factor m_s (T_s - T_f).

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 kg⁻¹ K⁻¹.

5. Method

1. Carefully clean, dry, and weigh the empty inner copper calorimeter vessel together with the dry brass spiral stirrer to ±0.01 g. Record the dry mass m_cal. Multiply m_cal by the specific heat of copper (385 J kg⁻¹ K⁻¹) and the stirrer mass by that of brass (380 J kg⁻¹ K⁻¹), dividing by the specific heat of water (4186 J kg⁻¹ K⁻¹) to establish the theoretical water equivalent W_cal = 0.020 kg (20.0 g). Pipette or dispense exactly 200.0 g of distilled water into the inner vessel, reweigh the filled vessel to confirm m_w = 0.200 kg, mount the vessel on its cork support within the outer insulating jacket, and insert the precision calorimetric thermometer.
2. Weigh the dry copper specimen pellet charge (selected from 60.0 g to 220.0 g) on the analytical balance to ±0.01 g. Secure the charge inside the perforated copper specimen basket attached to a fine silk suspension thread. Lower the basket into the central heating tube of the Regnault steam generator, plug the top aperture with a loose non-absorbent cotton wool stopper to prevent vertical convective draft losses, and insert the 0–100°C reference thermometer directly into the core of the copper mass. Bring the boiler water to an active, vigorous boil, ensuring steam vents freely through the exhaust port, and soak the specimen until the core thermometer remains perfectly stationary at 100.0 °C for at least five continuous minutes.
3. Position the jacketed calorimeter assembly adjacent to the steam boiler delivery chute, but shielded by a rigid thermal insulating cardboard screen to prevent radiant heat from the boiler flame reaching the cold calorimeter. Stir the calorimeter water bath gently and continuously with the helical stirrer in steady vertical strokes. Read and record the calorimetric thermometer every 30 seconds for three minutes prior to the transfer to establish the preliminary natural background drift rate ΔT / Δt under ambient laboratory conditions. Confirm that the initial water temperature T_0 is steady at approximately 20.0 °C.
4. At an agreed timer signal, smoothly swing the calorimeter under the boiler transfer chute or rapidly extract the copper specimen basket by its suspension thread. Immediately lower the glowing hot specimen cleanly into the water of the inner calorimeter vessel, submerging it completely beneath the water surface without touching the thermometer bulb or inner walls, and close the calorimeter lid within one to two seconds. Ensure that no water droplets splash outward from the vessel during entry.
5. Commence continuous, smooth vertical agitation using the mechanical brass stirrer at approximately two strokes per second, ensuring intimate liquid circulation around the submerged copper pellets while avoiding surface splashing. Continuously observe the calorimetric thermometer meniscus using a magnifying cathetometer or reading lens to avoid parallax. Record the temperature at 15-second intervals as the mercury column climbs rapidly through an initial transient, peaks at an asymptotic maximum, and begins a very slow subsequent decay. Identify and record the peak equilibrium temperature T_f to ±0.01 °C.
6. Continue recording the calorimeter temperature at one-minute intervals for five minutes following the peak to measure the final cooling rate into the room. Apply the Rumelin or Regnault-Pfaundler radiation correction to adjust T_f for convective and radiative dissipation during the mixing interval. Compute the net thermal energy absorbed by the system Q_sys = C_sys (T_f - T_0) and the thermal mass factor X = m_s (T_s - T_f). Repeat the complete operational sequence for all five specimen masses (60.0, 100.0, 140.0, 180.0, 220.0 g) and perform an ordinary least-squares regression of Q_sys against X to extract the specific heat capacity c_s from the gradient.

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 = 0.0043%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 20.47814; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.28901. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 3.24913.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.28901 J kg⁻¹ 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.

Thermometer immersion depth and stem emergent exposure correction, heat lost by radiation and convection during the physical transfer of the hot copper specimen from the steam boiler to the calorimeter vessel (~1–2 s latency), water splashing during immersion, evaporation loss from the warm water surface, non-uniform temperature gradients prior to complete mechanical stirring, and balance mass measurement precision (±0.05 g) contribute experimental uncertainty. The OLS regression standard error reflects random scatter across sample mass settings, but does not capture unmodelled environmental radiation exchanges or systematic calibration offsets in the water equivalent determination.

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

• Shield the calorimeter completely from the steam boiler flame and radiant heating using an insulating wooden or double-layered cardboard partition screen.
• Ensure the metallic specimen is suspended in dry steam for at least five minutes after the internal thermometer stabilizes at 100.0 °C to guarantee isothermal equilibrium without condensed water droplets.
• Execute the transfer of the heated specimen into the calorimeter within two seconds, keeping transit distance minimal to prevent convective cooling in ambient air.
• Maintain smooth, continuous vertical agitation with the stirrer while reading the thermometer to eliminate local thermal stratification without causing splashing.

The teaching simulation assumes zero heat transfer during the rapid 1.5-second transit from the steam boiler to the calorimeter, constant specific heats of water and copper over the 20°C to 100°C interval, and instantaneous thermal conduction throughout the specimen pellets. In actual laboratory calorimetry, sensible cooling during transit, radiative and convective heat exchange with room air during the non-isothermal mixing interval, evaporative losses from the agitated warm water surface, and finite thermometer response time constants must be rigorously quantified using Regnault-Pfaundler radiation corrections. The mathematical model integrates a constant calorimeter system thermal capacity C_sys = 920.92 J K⁻¹ (comprising 200.0 g pure water with c_w = 4186.0 J kg⁻¹ K⁻¹ and a 20.0 g water equivalent calorimeter vessel and brass stirrer) and evaluates final equilibrium temperatures using the exact energy balance equation T_f = (m_s c_s T_s + C_sys T_0) / (m_s c_s + C_sys) with pure copper reference specific heat c_s = 385.0 J kg⁻¹ K⁻¹, initial sample temperature T_s = 100.0°C, and initial water temperature T_0 = 20.0°C.

12. Conclusion

The worked example illustrates how to obtain Specific heat capacity of copper from measurements, graph variables and a fitted coefficient. The estimate is 385.0 J kg⁻¹ K⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.