# Expansion of Solids

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. An object's temperature, on the Kelvin [K] scale, is proportional the object's internal kinetic energy. When heat energy (measured in Joules [J]) is transferred to the object, its constituent particles begin moving faster, thus maintaining a greater average separation and therefore expanding. Materials which contract with increasing temperature are uncommon.

If the object undergoing expansion has a predominant dimension (for example, a long metal rod), it is helpful to only observe expansion in that dimension. This is called linear thermal expansion. Each material has its own coefficient of linear thermal expansion (α). This coefficient indicates what percentage of its original length an object will expand per degree Celsius. The coefficient is found by dividing the percent change in length by the change in temperature (ΔT). Percent change in length is equal to the change in length (ΔL) divided by the original length (L_{0}).

ΔL = L_{0}α(ΔT) (1)

WARNING: THIS LAB USES BOILING WATER AND STEAM! PLEASE BE VERY CAREFUL WHEN HANDLING THE STEAM POT OR THE HOT METAL!

### Materials

- Pasco thermal expansion apparatus
- Metal rods
- Steam generator, cap, and rubber hose
- Watch glass
- Water
- Paper towels
- Digital multimeter
- Thermistor or Digital Thermometer

### Objectives

- To discover how heat affects the dimensions of different metals.
- Review how to use a micrometer.
- To learn how determine temperature using a thermistor.
- To practice controlling variables in order to determine a physical property of a material.

### Activities

### Activity 1: Measuring the coefficient of linear thermal expansion

Examine Eq. 1. What are the variables? What are the constants? What type of curve fit would you use to model the relationship between ΔL and ΔT?

Follow the directions below to develop your data set and test this relationship.

As steam is sent through a metal pipe, the pipe expands. This expansion is very small, and must be measured using a metric dial. See the document on error and uncertainty for help using the dial. When the metal rod expands, it will push against the micrometer needle. Examine the metric dial to determine the units and precision. In order to get three significant figures in your measurement of ΔL, you will need to estimate the reading to the nearest fifth of a division.

We will use an Extech EasyView digital thermometer placed into the end of the metal sample to measure temperature. Here is a link to the manufacturer's user manual. Use this user manual (the section titled "Accuracy" in the Specifications table) to determine the uncertainties in your temperature measurements.

One more thing: Although the entire metal sample is expanding you are only measuring the expansion that takes place between the two clips. One clip is anchored in the mounting bracket, and the other presses against the micrometer needle. Therefore when determining L_{0}, you are not concerned with the total length of the pipe, just the effective length between the clips.

The tube from the steam pot attaches to one end of the metal pipe, and a watch glass should be placed at the other end to catch the condensing steam. Make sure your pot is around 2/3 full of water before turning it on. Befoe turning he steam pot on, record the initial temperature and length of your pipe.

It takes a while for steam to enter the pot. When it finally happens, the rod will expand very quickly. You will need to watch carefully for maximum temperature and expansion.

As the pipe then cools, work in conjunction with your partne to get 7-8 data points of ΔL and ΔT. You will be ploting these points.

You will determine he coefficient of linear thermal expansion twice: Once during the heating-up stage, and once during the cooling-down stage.

For the heating-up stage, simply determin ΔL and ΔT, then calcualte the expansion coefficient using Eq. `1. Propagate your error and display the uncertainty in your final result.

For he cooling-down stage, create a graph of ΔL vs. ΔT in Logger Pro or Excel. From the slope of the graph, determine the expansion coefficient.

Locate a table of linear thermal expansion coefficients online or in your textbook and compare your experimental results to the accepted value using percent error. Be sure to cite your reference! Which method (heating-up or cooling-down) was closer to the accepted value? Why do you think this is the case?

### Activity 2: Repeating for a different material

Allow the apparatus to cool sufficiently and repeat the experiment on one other metal pipe.

### Activity 3: Analysis of error

Use the following questions to develop a thorough analysis of error.

- Measurements of three different types were made to determine the coefficient of linear expansion, namely the original length, the change in length, and the temperature. Each measurement has its own associated uncertainty, which should be recorded in your data section. For each of your pipes, which of these measurements contributes the greatest percent uncertainty in the final result? (Hint: Which has the greatest fractional error?) Does this make sense? Explain your reasoning.
- Is uncertainty of measurement alone enough to explain the percent error in your results? If not, identify the sources of experimental error that could explain the difference between your values and the accepted values.