Temperature and cooling curves

September 24, 2021

Week six is all about heat and temperature, starting off with a Eureka science heat and temperature video playlist. This term I used the new rig at the back of A101 to play the series, using the Chromebase to drive the 50" television.

On Wednesday I covered the Celsius temperature scale with a thermometer, but for a second term the coconut oil melted well below 20℃, this term melting closer to 10℃. The oil, many years old now, must be becoming more unsaturated with time. New oil is apparently required to obtain the liquifying/solidifying temperature of the fresh coconut oil.

Laboratory six explored the relationship between time and temperature for water cooling from boiling (100℃). On the board I wrote "How does water cool off with time? When water cools, what mathematical model describes that cooling?"

I also sketched a graph with time on the x-axis, temperature on the y-axis. I laid out a table using times of 1, 2, 3, 5, 8, 13, 21, and 34 minutes. This Fibonacci time sequence proved to be serendipitously optimal for this laboratory. To the right of the table are clock times for the afternoon class. The board above is from the 11:00 section, times are in UTC.

This term the set-up used 100 ml beakers. I had the students wait until the one minute mark to put their thermometers into the water - they melt up at 100℃. I also recommended a first measurement at three minutes to provide two minutes for the thermometer to thermally adjust.

Timeanddate.com has a nice large digital UTC clock that was useful in the class. I filled all of the beakers at one time, with the 8:00 class starting time at 9:20 and the 11:00 class starting at 00:30. the clock times next to the table helped the students immensely. This also meant data arrived in synch and reduced the overall data entry.

This structure of a synchronized start and Fibonacci timing meant that at 34 minutes I had time to cover the mathematical model ahead of the 55 minute measurement - and all students were at the same place at the same time. This worked well for pulling the class together around the exponential decay model. In the above graph the function is the average for the three 11:00 session curves.

As I did last spring, I dropped the current form of Newton's Law of Cooling. The law was always confusing for the students, few of whom have experience with natural logs and e. The start and end temperature are also non-illuminating for the students. I am still unclear whether Newton knew of e and used e is his formulations. The formula above yields a base b of around 0.9, thus the temperature is dropping by powers of about 90% per time period, which was minutes in this case.

Note to that my introduction to the exponential is to review the polynomial forms before showing that now x is in the exponent. Mathematics introduced through data from experiments is a core feature of the course. Never fear the math might be a class motto.

During the 34 to 55 minute gap between measurements, I introduced the exponential decay model as seen above. Then I asked the students to watch and see if the 55 minute temperature measurement didn't agree with the model: the model gave them a predicted temperature. The results at 55 minutes then confirmed that the model was making good predictions of future temperatures. Models provide predictions.

The simultaneous start, the use of the UTC clock with times on the board, and the Fibonacci spacing all proved to be optimal choices for this particular experiment. The sequence naturally spaces out as the rate of temperature change drops, without sacrificing details in the first ten minutes of cooling. Desmos is case sensitive, so the use of lowercase t for time and uppercase T for Temperature, while potentially confusing works.

The small size of the beaker made reading the thermometer easier than when I had used styrofoam and metal cups. This term I did have the 8:00 class test whether dissolved compounds affected the cooling curve, but they did not. The core intent of this laboratory is really to encounter yet another mathematical model that nature obeys: the exponential decay model.

The right side is the Celsius scale, here reading about 62℃.

Derisalyn makes a reading at one of the times set by the Fibonacci sequence.

Marcia reads a temperature while Cynthia records the data.

Having tech in the classroom has changed what we can do during a laboratory. Here Destiny Grace and Faithlyn both have smartphones, allowing direct data entry and math model exploration using the Desmos app.

The simultaneous start is perhaps the hard part - that has to be done by the instructor, with the beakers all over at the pot. Then the hot water has to be moved. Starting at an integer time, preferably a ten minute mark, helps. At one minute the thermometers go in (they melt at 100℃!) and then the first measurement is at three minutes. Again, the Fibonacci spacing works really well when coupled to the clock running on the back screen.

I wrapped up the laboratory with a brief lecture on the relationship of cooling curves, exponential decay, and radioactive decay, and the situation involving the runit dome. Class closed with the following video.