Thermal conductivity

After the effective failure of a thermal expansion laboratory this past summer, I returned to time versus temperature for a thermal conductivity laboratory. This laboratory used to pursue only the maximum temperature and led to a group discussion of possible chart types. This almost always led to a column chart and never led to a consideration of possible mathematical models. Shifting to time versus temperature usually yields a curve well modeled by a logistic function.

Laslyn Siden and Jessica Reyes monitor the temperature rise

The intent is not to teach the students the mathematics of the logistic function, one does not have to have to understand a volcano in order to learn what a volcano looks like and enjoy the beauty of one such as Mount Fuji.

Lodonna Osawa and V-Ann Nakamura

The covered side of the apparatus has 100 Celsius water inside, a metal cylinder is glued into the styrofoam cups near their bottom. A small amount of room temperature water is in the cup with the thermometer. While the apparatus is described in the course text, this specific variation of the laboratory is not covered in the text.

By the eleven o'clock laboratory I had a more specific set of times to record. I learned belatedly that the students did not understand how the times on the stopwatch convert to the seconds in the table. The seconds are necessary for graphing the data in a spreadsheet. The rightmost set of minute markers was part of a post hoc effort to bring the students up to speed on the relation between what they saw on their stopwatch and the recommended data table.

In the eight o'clock laboratory I could not stumble on coefficients that would mimic the data. I did finally find some that worked at eleven. A spreadsheet function that well matched the data for one group was:

Temperature = 6/(1+0.15*exp(-0.08*(time-250)))+26

Note that few of the students have taken MS 101 Algebra and Trigonometry, a course which includes exponential and logarithmic functions. The laboratory just provides a chance for students to glimpse mathematics that is presently beyond their reach. Seeing things one does not fully understand does not make the view any less worthy. In class I note that while the students might not "understand" the function, the system is obeying a mathematical model. Like the soap, rolling balls, and marbles before, the system "knows" what to do and then does that. Nature is mathematical, even if some of the mathematics is beyond one's own capacities.

A developing tropical system on Thursday morning led to low room temperatures of 25 Celsius and 23 Celsius outside in the rain.

Data to generate the chart seen above:

time (s) temp (°C) theory (°C)
0 26 26
15 26 26
30 26 26
45 26 26
60 26 26
90 26 26
120 26 26
150 26 26
180 26 26
210 28 27
240 29 30
270 31 32
300 31 32
330 32 32
360 32 32
390 32 32
420 32 32

Raw spreadsheet function: =6/(1+0.15*exp(-0.08*(A2-250)))+26