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. Explore a virtual apparatus, collect readings and analyse a graph.
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Adjust the apparatus, then take a reading.
Readings are generated by an ideal model. The diagram is schematic and is not to scale.
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.
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.
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.
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.
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.
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. Suggested independent settings: 2.0, 4.0, 6.0, 8.0, 10.0 min.
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.
Newton's law of cooling assumes an invariant environmental boundary temperature T_0. Any unmodelled drift in the enclosure temperature directly alters the instantaneous driving potential (T − T_0), distorting the apparent exponential decay rate.
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.
Heating the water prior to introducing it into the vessel ensures that the calorimeter metal and water rapidly reach thermal equilibrium near 80 °C without exposing the enclosure jacket to open flames or heating coils.
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.
Central bulb placement avoids boundary-layer edge effects near the metallic walls, ensuring the thermometer samples the bulk core liquid. Synchronizing timer start with a precise initial temperature sets the exact boundary condition θ_0 = 60.00 °C.
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.
Because water has a low thermal conductivity (k ≈ 0.6 W m⁻¹ K⁻¹), thermal stratification occurs rapidly in stagnant cooling liquid, creating cold upper layers and hot lower pools. Gentle stirring maintains spatial isothermality without inducing viscous friction heating.
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.
Transforming the exponential cooling trajectory into natural logarithmic coordinates linearizes the underlying differential equation, mapping elapsed time to a straight line whose slope directly represents the Newton cooling constant k.
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.
Linear regression over multiple serial points distributes instrument reading uncertainty across the full cooling arc, suppressing random parallax errors and rigorously testing the validity of the first-order cooling hypothesis through the goodness-of-fit statistic R².
Points are your observations. The fit uses SI x and y values before display scaling.
| # | Elapsed cooling time t (min) | Water temperature T (°C) | Elapsed cooling time t (min) | Logarithmic temperature excess ln[(T_i − T_0) / (T − T_0)] | Remove |
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