Verifying Newton’s Law of Cooling

Newton’s Law of cooling states that, under certain assumptions, the change in temperature of an object is directly proportional to the difference between its temperature and the temperature of its surroundings. This can be written as:

dT / dt = c ( T - K )

where T is temperature, t is time, K is the constant temperature of the surroundings, and c is a constant. This differential equation has solution:

T(t) - K = A ect or ln |T(t) - K| = ct + B

We will collect data and plot it against time to verify this law.

Experiment 1: Hot water

We heat water to the boiling point. Then we use the temperature probe to measure the temperature in time. It takes a couple of minutes for the EA-100 to settle down to an initial temperature. After it settles down take readings every minute (every 60 seconds) for an hour (60 readings). We use an outdoor thermometer to measure the temperature of the surrounding air. If the surrounding air is say 28 degrees centigrade. Newton’s law of cooling says that ln | T - 28 | should be linear.

Using the program EZEXPER, the results are stored in List 1 and List 2 of the CASIO CFX-9850G. To see them press MENU, use the cursor arrows to highlight the LIST icon, and press EXE. To compute ln | T - 28 | and store the result into List 3, use the cursor arrows to highlight the List 3 title and enter the following:

ln OPT F4 (NUM) F1 (Abs) ( OPTN F1 (LIST) F1 (List) 2 - 28 )

Press EXE.

To draw the scatterplot press MENU, use the cursor arrows to highlight the STAT icon, and press EXE.

Press F1 (GRPH) F6 (SET) and set the parameters as follows:

StatGraph1

Graph Type     :Scatter

XList             :List1 (This is time.)

Ylist                      :List3 (This is the natural-log transformed data.)

Frequency     :1

Mark Type     : (This is your choice)

Graph Color     : (This is your choice)

Press EXIT F1 (GPH1) F1 ( X ) to get the least squares line through the data. The r value should be very close to -1.0.

Experiment 2: Cold water

Repeat the experiment outlined above except this time with very cold water. Also this time take a reading every 120 seconds for two hours.