Specific Heat

 

The next phase in our study of energy this year concerns thermal energy, the energy stored in random agitation at the atomic/molecular level. Today’s lab investigates thermal energy storage in two common substances, water and lead.

 

The system we study is an insulated cup holding measured masses of water and lead. (We start with just water, and add lead in a second phase of the investigation.) An electric immersion heater adds thermal energy at a known rate, and a computer-linked temperature probe measures the rise in the water’s temperature as it absorbs thermal energy. For these substances, and for many others, the thermal energy absorbed (DU) is proportional to the absorbing mass (M) and the temperature change (DT). The constant of proportionality is called the specific heat of the substance[1]:

 

DU = c M DT

The specific heat is the amount of heat needed to raise the temperature of a unit mass (usually 1 gram) of substance by one degree (usually Celsius).  This is useful because, as you will see in this lab, the energy-temperature relation is often quite linear, and the specific heat is its slope (for unit mass).

 

 

Equipment: § styrofoam cups

§ lab stand and clamps

§ water

§ lead wire

§ top-loading balance

§ stainless steel temperature probe with LabPro interface

§ computer with LoggerPro software

§ hotplate / stirrer with magnetic stir bar

 

 

Part I: Specific Heat of Water

Procedure: Tare the cup, fill it (3/4 or so) with water and determine the mass of water. Set up the cup with immersion heater, temperature probe, and stirrer. Take care that the heater element doesn’t contact the cup (it will melt it) or the temperature probe (you want the temperature of the water, not the heater element). Make sure the stir bar rotates rapidly when the stir control is turned up. Make sure NOT to use the hot plate as a heater (see comment on melting).

 

Set up LoggerPro to record temperature data. This is generally similar to the setup for photogates in the previous lab. Once the interface has power and is connected to the probe and the computer, the software should recognize the probe and bring up a blank data table and graph ready for you to start recording.

 

Calibration: the immersion heater puts out thermal energy at a constant rate (marked on each one in watts). We will have simple wattmeters in the lab, and sometime before doing your calculations you should check the rating of your heater. [2]

 

For each run, take 10 seconds or so of baseline readings before plugging in the immersion heater. Once the heater is on, the temperature should increase steadily after a few seconds. Let it rise for 10-15 degrees before unplugging. Continue recording temperatures for 45-60 seconds.  This is to see how soon the temperature stops rising and how quickly it declines when no heat is being supplied. After each run, store its data. If possible, connect to the lab network and store in your own folder as well as in the copy of LoggerPro running on your machine.

 

Since each run goes fairly quickly, do three or so, just to be sure your results are reproducible. Also, take one run with the stirrer off. Note any differences in the temperature graph, and when the run is over, move the temperature probe gently around within the volume of water to see if there are hotter or cooler places.[3]

 

Analysis: For each run, select the longest convenient linear portion of the temperature graph and determine the units and numerical value of its slope. You can use numbers off the data table, or you can experiment with the built-in slope-finding feature of LoggerPro. Be careful that you are just finding the slope of the linear portion, and not of the whole data set.

 

Your data is temperature versus time. Since the immersion heater puts out thermal energy at a constant rate, a known interval of time corresponds to a definite amount of energy provided.

 

Using the basic specific heat equation, and rating of the heater, and the measured mass of water, calculate the specific heat of water.

 

Part II: Specific Heat of Lead

In this part, you will put some lead wire in the cup along with the water. Heat will go partly into water and partly into lead, so the energy equation is a little more complicated:

 

DU = cW MW DT +  cL ML DT

 

where W and L refer to water and lead, respectively. You will carry out the same basic measurement. The slope of the temperature-time graph is now a function of two masses and two specific heats, but you know three of the four, and can use algebra to solve for the new quantity of interest, cL, the specific heat of lead.

 

It turns out that this is a delicate measurement. The quantity you are looking for depends on the small difference between two large numbers, and if you remember how significant figures work, the difference will have fewer significant figures (i.e. reliable digits) than the two large numbers. This means that precision in the basic measurements is quite important. (We will use the handbook value for the specific heat of water, rather than relying on your value from Part I, as one way to increase the reliability of your result in this part.)

 

Procedure.Procedure: Determine the mass of lead wire you’re working with. Make a coil of the lead wire that fits snugly against the wall of the cup, allowing room for the stir bar to rotate freely at the bottom, and leaving enough space between turns so that water can make good contact with the whole length of wire. Fill the cup, and determine the mass of water. (Taring is quite important here.)

 

Take temperature data as before.

 

Analysis: Determine the needed slope. Solve the two-term energy equation for cL and calculate its value. (The standard value for the specific heat of water is 4.186 joules per gram per degree Celsius.)

 

 

Dessert Course (Optional) Repeat both parts using mineral oil as the liquid. Be observant about stirring: does the stir bar behave the same in this light oil as it does in water. The specific heat here is a good deal smaller than water’s. That may give a better value for the specific heat of lead. Do you see why?

 

Write-up: Give details of anything that caught your attention about the apparatus, the materials, or the measurement process. Give a full account of your calculations.

 


[1] Names and symbols vary somewhat. You’ll see q, Q, and Uth for the energy, dU and dT for DU and DT, and upper and lower case c’s. As always with notation, read any surrounding text carefully for the author’s specific definitions. This is especially important for specific heat, because there are situations where the DU - DT relation depends on the specific conditions in which the heating occurs. This is not an issue in this lab, though. Also, you may well encounter the molar specific heat, the energy it takes to raise one mole (i.e. Avogadro’s number) of the given kind of atom or molecule by one degree.

[2] We do not expect trouble here, but it is important to be aware the calibration is a key link in the chain of analysis, and it is good procedure to check it whenever feasible.

[3] The point is that the basic specific heat equation assumes that all the mass of water is at the same temperature. If that isn’t true, the equation doesn’t apply, at least not to the whole mass at once.