Newton's Law of Cooling lab report

1. Title and aim

Verification of Newton's Law of Cooling and Determination of the Cooling Constant. Investigate the rate of loss of heat by radiation and convection for a body cooling in a constant-temperature enclosure, verifying Newton's law of cooling for small temperature differences, and determining the cooling constant k from the linear logarithmic temperature excess decay.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Cooling rate constant k of 0.0960 min⁻¹, differing by 0.016% 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

• Double-Walled Water-Jacketed Cooling Enclosure: A heavy cylindrical double-walled copper vessel containing a large thermal reservoir of circulating cold water maintained at a steady ambient temperature T_0 = 20.0 °C, establishing an isothermal, draft-free boundary condition surrounding the radiating test calorimeter.
• Polished Thin-Walled Copper Calorimeter Vessel with Insulated Lid: A cylindrical container (mass 75.0 g, volume 120 mL) having a specularly polished outer surface to suppress radiation losses, suspended centrally within the water jacket on cork supports and sealed with a varnished wooden lid featuring apertures for the thermometer and stirrer.
• Calibrated Precision Calorimetric Thermometer (0.05°C Subdivisions): A sensitive mercury-in-glass thermometer graduated from 10°C to 90°C in 0.1°C intervals (readable to 0.02°C with a cathetometer), immersed to a fixed depth in the calorimeter water bath to record the transient temperature decay.
• Digital Precision Bench Chronometer and Mechanical Loop Stirrer: An electronic quartz digital chronometer resolving elapsed time to 0.1 s, paired with a low-thermal-mass brass wire helical stirrer designed to produce gentle vertical circulation without introducing kinetic frictional dissipation.
• Initial hot water temperature T_i = 80.0 °C; initial temperature excess θ_0 = T_i − T_0 = 60.0 °C.
• Constant temperature water jacket bath T_0 = 20.0 °C maintained by large thermal mass circulation.
• Polished copper calorimeter containing 100.0 g pure water; aggregate system thermal capacity C_sys = 456.8 J K⁻¹.
• Reference Newton cooling constant k = 0.0960 min⁻¹ (thermal cooling time constant τ = 1/k = 10.42 min = 625.0 s).

4. Theory and graph choice

The quantitative formulation of how a heated body sheds thermal energy to its surroundings was first articulated by Sir Isaac Newton in his 1701 treatise Scala Graduum Caloris. Newton posited that whenever the temperature excess between a warm body and its ambient enclosure remains moderate (typically less than 30°C to 40°C), the instantaneous rate of heat loss dQ/dt is directly proportional to the temperature difference (T − T_0). Because the sensible heat content of an incompressible fluid or solid relates to its temperature through the heat capacity C_sys via dQ = C_sys dT, Newton's cooling law manifests as a first-order linear differential equation: dT/dt = −k (T − T_0), where k is the characteristic cooling constant of the system. While heat dissipation in ambient air is a complex composite of blackbody radiation governed by the Stefan-Boltzmann T⁴ law and buoyant natural convection governed by fluid boundary layers, the binomial Taylor expansion of Stefan's law reveals that for small fractional temperature excesses, radiative emission linearizes identically into Newton's proportional format. Integrating this governing rate equation yields an exponential decay of temperature excess θ(t) = T(t) − T_0 = θ_0 exp(−k t). Taking the natural logarithm transforms the nonlinear thermal decay into a linear relationship ln[θ_0 / θ(t)] = k t, allowing the cooling constant k (and the thermal time constant τ = 1/k) to be precisely evaluated from the gradient of an ordinary least-squares fit across serial time observations.

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 min⁻¹.

5. Method

1. Fill the annular space of the double-walled enclosure with room-temperature water. Stir the jacket bath thoroughly and record its steady temperature T_0 using a reference thermometer. Confirm that T_0 remains stationary at 20.0 °C throughout the duration of the laboratory session, providing a constant ambient thermal sink.
2. Weigh the clean, dry polished copper calorimeter vessel with its stirrer to ±0.01 g. Add exactly 100.0 g of distilled water preheated in a separate beaker to approximately 85 °C. Mount the calorimeter securely inside the central chamber of the water-jacketed enclosure, seating it firmly upon insulating vulcanized cork pegs to eliminate conductive thermal bridging to the bottom plate.
3. Place the varnished wooden lid firmly over the calorimeter, insert the calorimetric thermometer through the central rubber grommet so its bulb is submerged in the geometric center of the liquid, and insert the loop stirrer. Stir the hot water gently until the thermometer stabilizes at the target initial temperature T_i = 80.00 °C. Start the digital bench chronometer simultaneously as time t = 0.0 min.
4. Maintain continuous, gentle vertical agitation with the loop stirrer at approximately one stroke every two seconds to prevent vertical thermal stratification. At exact half-minute intervals, inspect the thermometer meniscus using an optical magnifier to eliminate parallax, recording the water temperature T at 2.0, 4.0, 6.0, 8.0, and 10.0 minutes of elapsed time.
5. For each recorded observation time t, compute the instantaneous temperature excess θ(t) = T(t) − T_0 above the jacket temperature. Evaluate the logarithmic decay ratio y = ln[θ_0 / θ(t)] = ln[60.00 / (T(t) − 20.00)], where θ_0 = 60.00 °C is the initial temperature excess at t = 0.
6. Plot the logarithmic temperature excess y against elapsed cooling time t (min). Perform an ordinary least-squares linear regression to extract the slope m, standard error, and coefficient of determination R². Confirm that the gradient m equals the system cooling constant k = 0.0960 min⁻¹, and compute the thermal time constant τ = 1/k.

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 = 0.0155%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.5581e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.7219e-05. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.1422e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1.7219e-05 min⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Ambient water jacket temperature drift (±0.1°C), convection drafts around the wooden lid apertures, thermometer thermal response lag and emergent stem temperature gradient, spatial temperature stratification inside the unagitated cooling water, and evaporation mass loss contribute experimental uncertainty. The OLS regression standard error reflects random scatter across time stations, but does not capture non-linear convective cooling departures at larger initial temperature excesses.

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

• Keep the water jacket filled with a large volume of water and confirm that its temperature T_0 remains constant within ±0.1 °C throughout the observation period.
• Stir the water bath gently and continuously with the helical stirrer before each reading to prevent vertical thermal stratification without causing splashing.
• Ensure the outer surface of the calorimeter is cleanly polished and dry before immersion in the jacket to maintain a stable, uniform surface emissivity.
• Read the thermometer with an optical magnifier aligned horizontally with the meniscus to eliminate parallax errors.

The teaching simulation models heat dissipation under ideal linearized Newton cooling kinetics with constant ambient enclosure temperature T_0 = 20.0 °C and an invariant cooling constant k = 0.0960 min⁻¹. In physical experiments, natural convection exhibits an empirical temperature dependence of (T − T_0)^1.25, Stefan-Boltzmann radiation departs from perfect linearity at larger temperature differences (T > 75 °C), evaporative cooling from the water surface introduces non-radiative enthalpy loss, and ambient jacket temperature drift can alter the effective cooling rate. The simulation evaluates water temperatures using the exact analytical solution T(t) = T_0 + (T_i − T_0) exp(−k t) with initial water temperature T_i = 80.0 °C, constant jacket temperature T_0 = 20.0 °C, and cooling constant k = 0.0960 min⁻¹ (time constant τ = 10.417 min). Readings are rounded to 0.01 °C to simulate high-precision calorimetric thermometry, and the linear transformation ln[(T_i − T_0) / (T − T_0)] = k t is evaluated using ordinary least-squares regression.

12. Conclusion

The worked example illustrates how to obtain Cooling rate constant k from measurements, graph variables and a fitted coefficient. The estimate is 0.0960 min⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.