The idea is just to get the students making an actual measurement that they will get to analyze subsequently.

Estimated Time: 2 hours

The students will set up a stretched rubber band in a configuration that allows them to monitor both its temperature and tension. The students will then adjust both the length and the temperature in order to determine its “equation of state”.

Very little. The students should have some idea of what thermodynamics is, but needn't yet know any details. The analysis will involve the the use of Maxwell relations and the First Law, but that can be developed after the lab.

• Thermometer
• Force meter
• One rubber band per group
• PVC pipe or (ideally) glass cylinder
• Rubber stopper with screw hook
• Boiling water (e.g. coffee urn or tea kettle), cool water and ice
• Various clamps, etc.
• A Lab handout for each student
• A pre-lab handout for each student .pdf version .tex version

The basic idea is to set up the rubber band stretched between the force meter and a hook in the rubber stopper at the bottom of the tube. We used chains to ensure that the entire rubber band was immersed in the tube, which is then filled with either water or acetone to serve as a heat bath. Students can adjust the length of the rubber band either by shifting the force gauge up, or by changing which link of the chain is hooked to the force gauge.

The students pour hot water into the empty tube, and then once it has been stirred and reached equilibrium, they take force measurements at all the lengths they are measuring. They can then either add ice to cool it down, carefully dump the water out and replace it with water of a different temperature. Students should be able to measure around five different lengths at about five different temperatures during a class period, with temperature ranging range from 0 centigrade to close to 90 centigrade.

Getting five lengths at five different temperatures may be impossible based on certain lab setups. The use of chain links made it easy to get consistent length differences at temperatures, but only 3-4 of the lengths actually accrued useful data. As well, the need to keep the water temperature constant while the lengths were measured made it difficult to get the five lengths in a short period of time. It was far easier to do that than to leave a rubber band at one length and vary the temperature though. If I were to do it again as a student, I would ignore the five lengths and simply go with 3-4 that could be measured reliably. - Amanda Abbott

The analysis of the lab requires the introduction of several concepts, and is likely to take several days.

At the first level of analysis, we can relate the slope $\left(\frac{\partial\tau}{\partial T}\right)_L$ to the spring constant $\left(\frac{\partial\tau}{\partial L}\right)_T$ and the coefficient of thermal expansion $\alpha \equiv\left(\frac{\partial L}{\partial T}\right)_\tau$. Since the spring constant is easily measured, this gives us an indirect measurement of the coefficient of thermal expansion—which we see is negative.

We can use the Maxwell's relation generated from the Helmholtz free energy \[ dF = \tau dL -SdT \] which is \[ \left(\frac{\partial\tau}{\partial T}\right)_L = - \left(\frac{\partial S}{\partial L}\right)_T \] to relate the measured variation of tension with temperature with the change of entropy with length under isothermal conditions. By integrating this over length (assuming sufficient measurements are made), we can find measure change in entropy for a finite isothermal stretch. This then allows us to find the energy of heating for a finite isothermal stretch. Since the work is easy to measure (simply by integrating the tension), we have a measurement of the change in both the change in internal energy and entropy for the isothermal stretch!