Cooling Curve in the Physics Lab
A physics class records the temperature of 500 mL of water in an insulated cup after it is removed from a hot plate. The room is held at 22.0 °C. Temperatures are recorded every 5 minutes: 0 min, 95.0 °C; 5 min, 78.0 °C; 10 min, 65.0 °C; 15 min, 55.0 °C; 20 min, 47.0 °C. A student claims the data are linear because the temperature keeps dropping and proposes using the average rate of change from 0 to 20 minutes to predict the temperature at 30 minutes. Another student says the cooling must be exponential and wants to use the ratio of successive temperature differences above room temperature. The class must choose a model, state its domain, and decide whether either prediction is trustworthy outside the observed interval.
1. Test the linear and exponential models using first differences and ratios of temperature excess above room temperature. Which model fits better, and why?
Read the worked answer
First differences are $-17$, $-13$, $-10$, and $-8$ °C per 5 min. They are not constant, so a linear model is a poor fit. Now subtract the room temperature, 22 °C, to get the excess: 73, 56, 43, 33, and 25 °C. The successive ratios are
$$56/73\approx0.767,\quad 43/56\approx0.768,\quad 33/43\approx0.767,\quad 25/33\approx0.758.$$
These are nearly constant, which supports exponential decay toward the room temperature. A suitable model is
$$T(t)=22+73(0.767)^{t/5},$$
where $t$ is in minutes and 0.767 is the decay factor per 5-minute interval. The exponential model is better because Newton’s law of cooling predicts a rate proportional to the temperature difference from the surroundings, not a constant rate.
2. Predict the temperature at 30 minutes using both the exponential model and the linear average-rate model. Explain which prediction is physically reasonable.
Read the worked answer
Exponential model: at 30 min there are six 5-minute intervals, so
$$T(30)=22+73(0.767)^6.$$
Since $(0.767)^6\approx0.205$,
$$T(30)\approx22+73(0.205)\approx36.97\ ^\circ\mathrm{C}\approx37.0\ ^\circ\mathrm{C}.$$
Linear model: the average rate from 0 to 20 min is
$$\frac{47-95}{20}=-2.4\ ^\circ\mathrm{C/min}.$$
Extrapolating 10 more minutes gives $47-2.4(10)=23.0\ ^\circ\mathrm{C}$.
The linear prediction is physically unreasonable because it nearly reaches room temperature at 30 min and would drop below 22 °C shortly afterward, which would mean the water cools below the room without refrigeration. The exponential prediction approaches 22 °C asymptotically, consistent with the data and with Newton’s law of cooling.
3. State a reasonable domain and range for the model and explain one condition under which the model would fail.
Read the worked answer
A reasonable domain is $t\ge0$ minutes, or more conservatively $0\le t\le 60$ for the planned lab period. The range is $22<T\le95$ °C if the model is used only while the cup remains in the same room. The model would fail if the room temperature changed, if the cup were opened and lost heat by evaporation, if the water were stirred inconsistently, or if the insulation degraded. It would also fail at very large $t$ if the water stratifies or if the sensor’s resolution makes the small excess over 22 °C indistinguishable from noise. The asymptote at 22 °C is a limiting behavior, not a guarantee that the water reaches exactly 22 °C in finite time.