Physics Laboratory 3

Newton's law of cooling

Objectives:

In this experiment students will study Newton's law of cooling. If an object with temperature T_B is submerged in an environment with a large heat capacity and temperature T_E, then Newton’s law of cooling states that the temperature of the object as a function of time is given by

T_B= T_E+ (T_B- T_E)₀e^-st.

Here s is the rate constant. The difference between the body temperature and the environmental temperature approaches zero exponentially as e^-st. In this lab students will determine s for three calorimeter cans filled with hot water.

The surface of one can is unpainted, the surface of the second can is painted black, and the surface of the third can is painted white. The cans are exposed to air in the laboratory environment. The cans will transfer heat to the environment via conduction, convection and radiation. The surfaces of the three cans have different emissivities, so we may expect different s's for the three cans. Student will determine s for each can and compare the measured rate constants for the different cans. To determine s, students will record each can's temperature as a function of time for approximately 1 hour. Newton's law of cooling states that (T_B- T_E)_t = (T_B- T_E)₀e^-st, or that

ln(T_B- T_E)_t = ln(T_B- T_E)₀- st.

This equation is of the form y = ax + b, with y = ln(T_B- T_E)_t, x = t, a = -s and b = ln(T_B-T_E)₀. Students will determine the slope a = -s of a straight line fit of ln(T_B- T_E)_t versus t and thus determine the rate constant.

Procedure:

With the temperature sensor in air, determine the temperature of the environment, i.e. the room temperature T_E.

Open a Microsoft Excel spreadsheet.

Record the temperature T_E in the spreadsheet.

Monitor the calorimeter cans filled with with hot water for approximately 1 hour. The thermometer inserted into the water reads the temperature of the water. For each can record the temperature as a function of time in your spreadsheet. (Click on each of the thumbnails above to view an enlarged picture.)

	Clock	Can 1(unpainted)	Can 2 (black)	Can 3 (white)	Elapsed	Can 1	Can 2	Can 3
TE (deg C)	Time	Temp (deg C)	Temp (deg C)	Temp (deg C)	Time (s)	ln(T_B-T_E)	ln(T_B-T_E)	ln(T_B-T_E)

Data Analysis:

For each measurement find the elapsed time since the start of the experiment and record it in your spreadsheet.

In your spreadsheet, subtract T_E from the measured temperature T_B of each can and find the natural logarithm of this difference. Use the regression function to find the slope of a straight line fit of ln(T_B- T_E)_t versus time t. (Click Tools, Data Analysis, Regression. Choose appropriate input ranges for x and y. Choose a data range for the output of the function. The x-variable coefficient is the slope and the intercept coefficient is the intercept. The next column gives the standard error in these quantities.) Set the rate constant s equal to the magnitude of the slope.

Open Microsoft Word and prepare a report using the template shown below.

Name:
E-mail address:

Laboratory 3 Report

	In a few sentences summarize the experiment.
	Show your spreadsheet entries.

Answer the following questions:

	What are the rate constants you found for the different cans?
	How does these rate constant compare? Are the different surfaces of the calorimeter cans associated with different rate constants?
	Did you expect a difference?