Mechanical Equivalent of Heat   

(last edited January 16, 2013)

Dr Larry Bortner

 

 Purpose

 

 

To verify the relationship between mechanical energy and heat; that is, to find the relationship between energy associated with motion and the energy transfer between objects at different temperatures.

 

 

 Background

 

 

The parlance of thermodynamics in physics uses familiar words like work, energy, and heat in describing physical phenomena. The definitions of these terms lead to more precision than those used in the everyday lexicon. Briefly, the concepts of these terms are as follows:

 

 

*        Work results from a force moving a mass over a distance.

 

 

*        Energy can become work.

 

 

*        Heat equates to the transfer of energy from a hotter object to a cooler object.

 

 

 

 

 

These quantities combine in the law of the Conservation of Energy (equivalently, the First Law of Thermodynamics) to state that in an isolated system, the total energy remains the same. Though obvious and logical to those of us trained in science today, this concept did not reach full acceptance until outlined by Helmholtz in 1847, based on the experimental work of Mayer and Joule. Their experiments showed that a given amount of work always produced a particular amount of heat.

 

 

 

 

 

In this experiment, we do work on an aluminum drum calorimeter. This work calculated in Joules increases the internal energy of the cylinder, thereby increasing its temperature. We measure this temperature change; using the specific heat of aluminum and assuming that no heat escapes to the environment, we calculate the amount of heat in calories absorbed by the cylinder. The ratio of these two quantities, the work done to the heat absorbed, gives the mechanical equivalent of heat.

 

 

 

 

 

apparatus setup

 

 

Figure 1 Experimental set up.

 

 

 

 

 

 Experimental Specifics

 

 

Attach a nylon cord to a mass M then wrap it several times around the drum.  Rotate the aluminum drum with a crank so that the friction between the cord and the cylinder keeps the mass a steady few inches off the ground. The equilibrium situation means that the cranking torque equals the frictional torque. For a moment arm of r,

 

 

 

 

 

 (1)                    

 

 

 

 

 

with g = 9.80 m/s2. Define dbare as the diameter of the cylinder without any rope and dwrapped as the diameter of the cylinder wrapped with a single layer of rope under tension. Then define the moment arm as

 

 

 

 

 

(2)                     

 

 

                                                                       

 

 

The total angular displacement of the cylinder after N turns is

 

 

 

 

 

(3)                     

 

 

                                                                       

 

 

This gives an expression for the total work done on the calorimeter:

 

 

 

 

 

(4)                     

 

 

                                                                       

 

 

This work done on the aluminum drum of mass m raises its temperature from Ti to Tf, resulting in an absorbed heat Q in calories of

 

 

 

 

 

(5)                     

 

 

                                                                       

 

 

where c equals the specific heat of aluminum.

 

 

 

 

 

The mechanical equivalent of heat J calculates as

 

 

 

 

 

(6)                     

 

 

                                                                       

 

 

Note that this is also the units conversion of calories to joules, 4.186 J/cal.

 

 

 

 

 

We measure the temperature of the drum with an embedded thermistor, composed of a semiconductor material with a resistance that varies nonlinearly with temperature. Thermistors made of the same materials and processed in the same way have the same R vs. T curve. Calibrated properly, any thermistor can be used as a sensitive, accurate thermometer. The thermistor used in the Mechanical Equivalent of Heat apparatus is standardized and the temperature to resistance conversion is done automatically with the Pasco Thermistor Sensor and DataStudio software.

 

 

 

 

 

The Second Law of Thermodynamics states that the direction of the spontaneous energy flow between two objects at different temperatures that are in thermal contact with each other is from the warmer object to the cooler one. The derivation above assumes that the drum is thermally isolated. In our experiment, it is definitely in contact with the lab environment which is at a different temperature. In order to compensate for heat exchanged with this environment, the average of Ti and Tf  in this experiment should be Troom.

 

 

 Procedure

 

 

You need the following items:

 

 

*        Pasco TD-8551 Mechanical Equivalent of Heat apparatus

 

 

*        nylon cord

 

 

*        digital calipers

 

 

*        1-kg weight hanger with 8 1-kg weights

 

 

*        Pasco CI-6725A Thermistor Sensor (u{T} = 0.5 °C)

 

 

*        Pasco CI-6505A Temperature Sensor (u{T} = 1 °C)

 

 

*        BNC- dual banana plug lead

 

 

*        beaker of ice water

 

 

*        silicone spray

 

 

*        paper towels, vinyl gloves

 

 

 

 

 

Please read all the steps thoroughly before attempting the experiment, so that you can anticipate subtle points of the procedure and plan for them accordingly.

 

 

 

 

 

1.     Fill the beaker halfway with ice and add enough water to make it  full.

 

 

2.     On the computer desktop, click on Start> DataStudio Experiments> Third Quarter> Mechanical Equivalent of Heat.

 

 

a.     Click Start at the top of DataStudio to start monitoring the room temperature and the drum temperature. Do not click Stop until the end of the experiment. (Don’t save the activity when you do.)

 

 

b.     Record the room temperature TR to the nearest tenth of a degree.

 

 

3.     Record a mass m of 200.0 ± 0.2 g for the calorimeter.

 

 

4.     Before handling the drum or string, put on a pair of gloves.

 

 

a.     If the nylon string is wrapped around the cylinder, unwrap it.

 

 

b.     Unscrew the black knob and remove the aluminum cylinder.

 

 

c.     Measure and record the drum diameter dbare and your estimated uncertainty.

 

 

5.     Prepare the calorimeter drum for the experiment.

 

 

a.     Take the drum to the front of the room and spray it with silicone.

 

 

b.     Place the drum back on the shaft with the copper contacts toward the crank, making sure that the alignment pins on the shaft fit into the notches on the drum.  Replace the black knob and tighten it until the drum is held securely.

 

 

c.     Attach the loop end of the cord to the mass hanger.  Wrap the other end of the cord around the drum in a clockwise direction (as viewed facing the crank) four to four and a half times, in a single layer with no overlap. This rope end hangs free after it has been wrapped.

 

 

6.     You need to practice your cranking technique that keeps the mass an inch or two off the floor.

 

 

a.     Gently hold on to the free end of the rope and start turning the crank clockwise.

 

 

b.     Gradually increase the tension on the free end of the cord. Ideally, the mass should raise no more than ankle height off the floor and remain there as you turn. The applied tension is minimal; you should not be lifting the 9 kg weight by pulling on the rope.  Most of the lifting torque is provided by the friction between the drum and the cord.    

 

 

c.     If the mass doesn’t rise off the floor or you have to pull tight on the free end of the rope, add another turn of the rope.

 

 

d.     If the mass keeps rising as you crank, either reduce the tension on the free end or remove a turn of the rope.

 

 

e.     The rope must be dry to slide freely over the surface of the drum. If your rope is wet, replace it with a fresh, dry length of rope.

 

 

7.     Measure and record dwrapped. This is larger than dbare by twice the thickness of the cord. Also record your estimated value of u{dwrapped}, which should be larger than u{dbare}.

 

 

8.     Set up a table on your data sheet with the following headings and quantities:

 

 

Trial

M (kg)

ΔT (°C)

Ti ideal (°C)

Tf ideal (°C)

Ti (°C)

Tf (°C)

N

1

9

±7

 

 

 

 

 

2

8

±7

 

 

 

 

 

3

7

±7

 

 

 

 

 

4

6

±7

 

 

 

 

 

5

9

±9

 

 

 

 

 

6

9

±8

 

 

 

 

 

7

9

±6

 

 

 

 

 

8

9

±5

 

 

 

 

 

9

9

-3, +7

 

 

 

 

 

10

9

-7, +3

 

 

 

 

 

9.     Unwrap the rope and remove the cylinder.

 

 

a.     Immerse the drum in ice water for 10 to 15 seconds.

 

 

b.     Remove and dry quickly and thoroughly.

 

 

i.      Be aware that water may condense on the cooler surface.

 

 

ii.     A wet drum means that some of the heat produced would go into evaporating the water. Our theoretical model assumes that all of the heat goes to raising the temperature of the drum.

 

 

c.     Return the cylinder to the shaft, pressing it all the way in to make contact with the leads.

 

 

i.      Check the drum temperature. It should be a few tenths less than the ideal Ti for your particular trial.

 

 

ii.     If the temperature is not less, remove the drum and immerse it in the ice water for a few more seconds, then repeat the above steps from 9b.

 

 

d.     Secure the cylinder with the knob and make sure it is dry.

 

 

e.     Rewrap the drum with the nylon cord.

 

 

10.  Set the counter to zero.

 

 

11.  When the temperature reaches the ideal starting point, start cranking. Record the actual initial temperature in your table, or your best guess, as the temperature may be changing rapidly.

 

 

12.  Crank rapidly until the temperature is a few tenths less than your ideal Tf.

 

 

a.     Crank slowly until you reach this temperature.

 

 

b.     Stop cranking and watch the drum temperature carefully. Record the highest temperature reached as Tf.

 

 

c.     Record the counter reading as N. If you are outside the nominal range of 100-400, something is wrong.

 

 

13.  Repeat the process from Step 9 as needed to fill out the table. You may need to repeat some of the trials.

 

 

14.  Please discard used paper towels.

 

 

 Analysis

 

1.     Create a spreadsheet from Start> Templates> Third Quarter> Mechanical Equivalent of Heat and enter your names.

 

2.     In labeled cells, fill in values for the mass of the calorimeter, the uncertainty of the mass of the calorimeter, the acceleration of gravity, and the specific heat of aluminum, 0.220 cal/g·°C.

 

3.     Enter your values of the drum diameter, with and without string, and the estimated uncertainties. Convert to meters.

 

4.     Enter the room temperature, its uncertainty from the manufacturer’s specifications, the uncertainty of the drum temperature (the thermistor), and the uncertainty in the number of turns as 1.

 

5.     Enter your experimental values of the trial number, the hanging mass, the actual beginning and ending temperatures, and the number of turns of the crank.

 

6.     For each trial:

 

a.     Since we assume that as much heat was gained from the environment as was lost to it, we want to check that the difference between room temperature and the initial temperature (ΔTi) and the difference between the final temperature and room temperature (ΔTf) are roughly the same. Find ΔTi and ΔTf and ΔT = Tf Ti.

 

b.     Calculate the work done by friction W in Joules, the total heat absorbed by the cylinder Q in calories, and the ratio J of these two values, which is the mechanical equivalent of heat.

 

c.     After propagating the errors for Eqs. (4), (5), and (6), calculate the uncertainties of W, Q, and J.

 

i.      For u{W}, assume that that M and g are constant.

 

7.     Find the average of your values of J and the average of u{J).

 

a.     For this experiment, use the average propagated uncertainty as the uncertainty for the average J.

 

8.     Perform a standard equivalency test between your experimental value of J and the defined value of 4.186 J/cal.

 Questions

 

1.     Is it experimentally possible that the heat absorbed by the aluminum cylinder could exceed the work performed on it? Explain.

 

2.     Your last three trials had roughly the same ΔT but probably had values of J that varied more than the average u{J}. Explain these differences.