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One of the fundamental properties of all matter is specific heat. Conceptually we can think of this as being a resistance to changes in temperature, or something like a thermal inertia. Because changes in temperature are closely related to changes in entropy, many interactions in a material have an impact on the specific heat. Further, the conditions associated with the measurement affect the outcome. This is commonly seen when discussing ideal gases. The specific heat of an ideal gas is smaller if the volume if the gas is held constant than if the pressure is held constant. For theoretical work, constant volume is easire to manage. For experimental work, constant pressure is often easier to maintain. Fortunately in most solids and liquids the difference is small. So, we measure specific heat at constant pressure, c_p.

Many methods exist for measuring c_p. The one we're chasing is called "relaxation specific heat." The basic notion is to hold the surroundings at a fixed temperature. Then apply a know amount of heat (heater power multiplied by the time it is applied) to the sample while recording its temperature as a function of time. This gives what we call the warming curve. Shut off the heater and let the sample cool down, approaching the temperature of the surroundings, while recording the sample's temperature. This gives us the cooling curve. By fitting models to T_warm(t) and T_cool(t) we should be able to pull out c_p.

What we need to do

1. Have a functioning temperature controller.
2. Construct a reasonably thermally isolated sample stage with a trusted thermometer attached.
3. Create a Python program that will
   1. Set the temperature of the stage via the temperature controller
   2. Apply a small heat pulse of known power to the stage for a known time while recording the temeprature
   3. Shut off the pulse while continuing to record temperature.
   4. Move to the next temperature and repeat.
4. Using the obtained data, we will fit our stage T(t) curves on the warming and cooling side to our model. From the fit parameters, we can extract the specific heat.

This is really what Richard is chasing. Creating a temperature controller from nothing is a big ask for a student. The progress we're making gives me confidence that we'll be able to take some basic c_p measurements before Spring Break 2025.

Progress

• 22 Nov 2024: We found that the digital output from the LabJack U6 (link: https://labjack.com/products/labjack-u6-pro) is sufficient to supply just under 2 mW of power to our sample heater. I'm cleaning up the code to make sure we are collecting temperature/time data with sufficient density and that the fitting procedure makes sense. Once we return from Thanksgiving break, we're going to try to take our first data. It's exciting because of what we learned while making progress.
• 15 Nov 2024: This week Richard mounted the thermometer and heater so they are floating on the tape. He's soldering the leads to header pins so we can minimize the stress on the heater and thermometer joints. In terms of the programming, we can record data rapidly into a large buffer, then write line by line from a set of Python arrays. We've also worked out how to do the fits to the warming and cooling curves. Once we start collecting data (hopefully by winter break) we'll be able to see if we get a reasonable value for the specific heat of copper and aluminum at room temperature.
• 9 Nov 2024: Richard has begun assembling the parts (platinum thermometer and strain gauge) for the sample stage. Each piece needs four leads in twisted pairs. We'll measure the temperature using a 4-wire technique. In order to monitor the power to the heater, we'll drive a fixed current through the strain gauge while measuring the voltage drop across it. This is just like a 4-wire resistance measurement, but the purpose of the current is to heat. My work is on the Python script to collect the data.
• 1 Nov 2024: Now that we have a reasonable temperature controller, we can move on to the actual heat capacity apparatus. From previous attempts we have a stage made of a square aluminum slab with a 100 Ω heater and thermometer attached. The block has a large-ish hole drilled through it. We will be suspending our sample stage (a 120 Ω strain gauge) with its thermometer (a 1 kΩ Pt resistance element) in this hole. Richard had the brilliant idea to use tape as a first go suspending the stage. Our second iteration will be a fine, non-conducting mesh, perhaps from an aquarium net. So, we're almost at the point where we need to do some significant programming. I do have some circuit design and construction, and we'll need to make a box for the electronics. But we're far closer than I imagined a week ago.

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Updated 24 November 2024.