The purpose of this lab experiment is to measure the linear expansion coefficients of up to three different metals by observing how these materials expand and contract with changes in temperature. In addition, we will incorporate simple mathematical modeling techniques to fit curves to empirical data.

When heat is added to most materials, the average amplitude of the atoms' vibration within the material increases. This, in turn, increases the separation between the atoms causing the material to expand. If the temperature change, , is such that the material does not go through a phase change, then it can be shown that the change in the object's length, , is given by the equation

(1)

where , is the initial length of the object before heat is added, and , is the linear expansion coefficient of the material. Accepted values of several common materials are given below in Table 1.

This effect, however, is not simply limited to materials whose temperature has increased. If energy is removed from the material then the object's temperature will decrease causing the object to contract. The temperature change, , from Equation 1 is always found by subtracting the initial temperature of the object from the final temperature, or . Therefore, if < 0, will also be negative, indicating a length contraction.

From Equation 1, we see that , is not only dependent on , but also on the initial length of the object, . So, the longer the object, the greater change in its length. Although the phenomena of linear thermal expansion can be problematic when designing bridges, buildings, aircraft and spacecraft, it can be put to beneficial uses. For instance, household thermostats and bi-metallic strips make use of the property of linear expansion.

Table 1

Accepted Linear Expansion Values of Common Materials
Material	a (x10-5 °C-1)
Glass (ordinary)	0.09
Glass (Pyrex)	0.32
Concrete	1.20
Steel	1.24
Copper	1.76
Aluminum	2.34
Lead	2.90

In our experiment today we will use a thermistor to measure the change in the rod's temperature. A thermistor is a small, inexpensive electronic device, which is commonly used to measure temperature. Since the thermistor is essentially a resistor made of a semiconducting material, an increase in temperature rapidly decreases the resistance of the device. Unfortunately the relationship between the thermistor's temperature and resistance is not linear, but rather logarithmic, making it somewhat inconvenient to use. The graph to the left shows this logarithmic behavior. The temperature versus resistance plot of a typical thermistor is given. Notice that the equation that fits this data is T = -24 Ln(R) + 139.48, where R is in kW.

N.B.: this is a plot of a typical thermistor and does not represent the thermistor that you will be using today!

1. Determine the equation that best fits the tabular data as shown on the PASCO Thermal Expansion apparatus.

2. Determine the linear expansion coefficient of Aluminum, Copper, and/or Steel.

(Figure 1.) Thermal expansion apparatus (Figure 2.) Thermistor (Figure 3.) Steam generator (Figure 4.) The meter sticks and vertical stands are located in the window-well at the front of the classroom (Figure 5.) Two squirt bottles (Figure 6.) Funnel and holder (Figure 7.) Water runoff collection pan (Figure 8.) Dial caliper (Figure 9.) Digital multi-meter (DMM) (Figure 10.) Hand protectors Three metal rods (Al, Cu, and steel) Two riser blocks Banana wires Hose clamps Ice Hot water from sink Paper towels

[Click on images to enlarge.] 1 2 3 4 5 6 7 8 9 10

1. Caution!!! Rods may be very hot when using the steam generator! Use the hand protectors to remove hot rods.

2. Get your TA to check your steam generator between uses. If the water level is not at the proper level, you will experience time consuming problems.

3. Think before you start! You will be using a large amount of water throughout this lab; make sure your setup allows the water to flow smoothly and that you have a continuous supply of water available!

4. You will be given a limited amount of ice. Your experiment should be designed so that the ice lasts for the duration of that day's experiment.

5. Note that the thermistor takes longer to reach thermal equilibrium than does the rod, so you must allow a fair amount of time (1-2 minutes) for the temperature measurement to stabilize. Since the rod's length will reach a maximum deviation much more quickly than the thermistor, you should continually watch the dial caliper and the ohmmeter, and record the maximum deviations shown by both devices at whatever time they occur.

6. The zero-mark on the dial caliper may be adjusted to mark the initial position of the caliper needle. Take care to loosen the locking screw before rotating the zero-mark.

7. The steam generators take approximately 15 minutes to sufficiently heat the water.

1. Adding a trendline to an Excel plot
2. Adding a non-linear trendline to an Excel plot
3. Create plots of two data series on one graph
4. Fitting multiple curves (trendlines) to one data set
5. Clemson Physics Lab Tutorials
6. Measurement uncertainties
7. Using Excel
8. Graphing data using Excel
9. Using error bars in Excel

Each lab group should download the Lab Report Template and fill in the relevant information as you perform the experiment. Each person in the group should print-out the Questions section and answer them individually. Since each lab group will turn in an electronic copy of the lab report, be sure to rename the lab report template file. The naming convention is as follows:

For example the group at lab table #5 working on the Ideal Gas Law experiment would rename their template file as "5 Gas Law.doc".

These Nudge Questions are to be answered by your group and checked by your TA as you do the lab. They should be answered in your lab notebook.

Objective 1 Nudges

1. Describe the method that you will use to determine the equation that best fits the temperature/resistance data.
2. Do you expect your trendline to be logarithmic, linear, exponential, a power series, etc?
3. Most commonly your data trendlines have been linear. How do you add a trendline that is not linear?
4. Does one trendline accurately fit the temperature/resistance data? If not, try breaking up the single data set into two. We suggest breaking up the data around 35°C and fitting two curves accordingly.
5. What is(are) the equation(s) that best fit the temperature/resistance data?
6. For what temperature range(s) are the equation(s), found in the previous step, valid?

1. How will you measure the initial length of the rod, ?
2. When will you measure the initial length of the rod?
3. In this experiment, you should have at least 3 data points using cold water, hot water, and steam. In what order will you proceed? Why?
4. You have a limited supply of ice (about 0.5L for each day). How do you ensure that this supply will last?
5. What are your initial conditions? What, if anything, will you measure before heating or cooling the metal rods?
6. What exactly do you want the thermistor to measure during the experiment? In reality, what will the thermistor actually measure? How will you reduce any errors in the temperature measurement technique?
7. Using the dial caliper, how can you determine if the rod is contracting or expanding?
8. What is the sign of DeltaL if the rod is contracting? Expanding?
9. When, if ever, is it necessary to zero the dial caliper? When,if ever, should you re-zero it?
10. What is the least count of the dial caliper?
11. What are the uncertainties in the length measurements? ?
12. What are the uncertainties in the temperature measurement? ? The person in charge of Error Analysis should think how to use the raw data along with the data trendlines to quantify this uncertainty. Note the uncertainty may be different for all temperatures.
13. What parameters will you graph in order to measure the linear expansion coefficient, ,
14. Did your best-fit line fall within the data points' error bars?
15. What is your value for ? What are the units for this coefficient?
16. How did your value of , compare to the accepted values?

These Questions are also found in the lab write-up template. They must be answered by each individual of the group. This is not a team activity. Each person should attach their own copy to the lab report just prior to handing in the lab to your TA.

1. How did the linear expansion coefficients for copper, aluminum and steel compare? How can you interpret this?

2. Using the thermistor calibration equations you found, what could be said about the importance of an accurate thermistor reading when dealing with hot and cold temperatures? In other words, in what temperature regime should the experimenter make the most careful thermistor reading? Explain.

3. Explain how linear expansion of metal rods can be used as a thermometer. What would be some of the problems with this method?

4. An aluminum baseball bat, which is 0.8950m in length when the temperature is 34.7°C, is left outside over night. If the temperature that night reaches freezing, what would be the bat's new length?

5. A slab of concrete used in highway construction is set in place when the ambient temperature is 20.0°C. If its length at that time is 20.478m, calculate the minimum gap required between concrete slabs if buckling is to be prevented. Assume the maximum temperature that will be reached is 55°C.

6. From your results, can you approximate the coefficient of volume expansion for aluminum, copper, and steel? That is, . Give your answer in terms of .

• You may wish to assign each lab group only one metal rod with which to experiment. It might be instructive to have two groups analyze copper, two groups analyze steel and two analyze aluminum, and then have the common groups compare results. This would make for some excellent oral presentations.
• Students will have to be briefed on the use of the digital multi-meter.

For this first experiment, it may be helpful for students to see a sample data table. We have them for viewing as a Word document or as a PDF document.

As of now, there are no CUPOL experiments associated with this experiment.