# DETERMINATION OF JOULE'S MECHANICAL EQUIVALENT OF HEAT by pptfiles

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DETERMINATION OF JOULE'S MECHANICAL EQUIVALENT OF HEAT AN ACTIVITY ON HEAT BY: Jim Roberts, Professor of Physics and Material Science UNT OBJECTIVE: This exercise is designed to show how mechanical and electrical energy are related by using data collectors and graphing calculator technology. INTRODUCTION In electrical circuits the power generated by a current flowing through a resistance is given by a relationship between voltage, current and resistance. These relationships are given by: Power = Rate of doing work = Energy/time. Electrical energy is given by the product: VoltageX(Charge moved) = electrical energy (2) (1)

or in abbreviated form Energy = Vq, with V the voltage difference and q the amount of charge moved across the gap. These two relationships can be combined to relate electrical energy to mechanical energy. P = E/t = (Vq)/t = V(q/t). (3)

But q/t is electrical current of (Coulomb/sec). Then power is given by VI, with V voltage and I current (Amperes). Another form of the power is: P = (IR)I = I2R. (4)

Here Ohm's law has been used to replace voltage V with IR. The experiment can now be designed to allow us to measure the amount of power over a specific time and obtain the energy expended into a known quantity of water. (Power)X(Time) = (Vq/t)X(t) = Electrical energy. (5) We can use our knowledge of heat energy absorption in a known reference reservoir, water, and the amount of energy expended by the electrical circuit in the form of heat to obtain the mechanical equivalent of heat. The heat exchange arrangement is represented in figure 1.

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Figure 1. A schematic representation of electrical energy generated in one region (A) and transferred to another region (B) to be measured. The energy produced in region A is given by: Electrical Energy = I2Rt. Energy "dumped" into region B is measured by: Q(heat energy) = cmΔT. (7)

(6)

In equation (7), c is the specific heat, for water 1 cal/gm-deg C, m is the mass in grams and ΔT is the change in temperature of the water. The experiment now simply becomes one of operating the heat generator for a given time to determine how much electrical energy is produced, to measure this energy in region B and to compare the values to each other. The heater may be one that is calibrated in Watts or one that needs to have the current and voltage related to it measured in time. By suitable changes, the use of a diode and a calibrating resistor, the power input to the heater element can be monitored and the time measured simultaneously by using the EA-200. This part of the experiment may require experienced students who understand basic laws of electricity. PROCEDURE Collect all of the materials needed. A list is given below for simplicity. MATERIALS: 1. Immersion Heater, Ekco Model 24J2 of 120 volts ac, 200 Watts power and 60 Hz operation. These are available at hardware stores or from Ekco Housewares, Inc, Franklin Park, IL 60131. 2. Two large styrofoam cups for the thermal container. These need to be doubled to hold the heat in. 3. EA-200 Data Collector/Analyzer to measure temperature and time. (Voltage and current can be measured, as well, with suitable modification of the voltage probe.) 2 2

4. CFX-9850GB graphing calculator to produce graphed data and to analyze slope of heat energy versus time.

Figure 2. A diagram of the heater, container and temperature probe to measure the mechanical equivalent of heat. The schematic is shown on the left and the pictorial set up is shown at the right. The insulation is simple news paper. Paper makes an excellent insulator of heat. The top will need to be covered during the experiment to insure low loss of heat from the container. The graphing calculator is available to receive the data from the EA-200 for a visual graphic display of the heat curve. 4. Styrofoam sheet or other insulating material to place over the top of the styrofoam cups to reduce the heat loss during the experiment. 5. You will need a watch or timer that will keep time to the nearest second unless you program the EA-200 for time intervals. Place the thermometer into the styrofoam cups with the heater placed away from the thermometer. Fill the cup with 500 cc of water. Insert the temperature probe in channel 1 of the EA-200 data collector. The internal clock of the unit will mark the passage of time as the temperature data are being taken. When everything is in place, start the experiment by plugging in the heater and starting the watch and EA-200 at the same time. Be sure that you record the temperature as you begin. Watch the temperature as it moves up a degree at a time. Read the time for each degree and record the data in the data sheet given below. If you program the data collector to take time intervals of one minute, the temperature and time will be correlated by the unit. It may be good to use both procedures to satisfy the requirement that “students know how to observe and record data”. The data are also plotted in figure 3. Part of the Texas TEKS requirement is that students how to read and graph data. This procedure will help the kids learn this requirement. A set of sample data is given in table 2 for you to compare your results with. The data shown in the tables is taken for a container of 500 cc (500 gms) of water heated by a commercial coffee heater of 200 Watts rating. The data are as follows: Column 1 is elapsed time in seconds, column 2 is the minute reading, column 3 is the second reading, column 4 is the observed temperature and column 5 is the calculated temperature, assuming linear response for the system. 3 3

The total amount of heat energy “dumped” into the water is given by the equation: Power(Watts)Xtime(sec) = Energy(Joules) 200 Watts X 367sec = 73,400 Joules The heat energy in the water is given by: Q(calories) = mass(gms)X(specific heat)X(ΔTemperature) Q(Calories) = 500gmX1(cal/gm-deg C)X32 = 16,000 Cal. The ratio of the two energies is given by: (73,000J)/(16,000Cal) = 4.56J/C. The accepted standard value is 4.18J/C giving a percent error for these data of: %Error = [(accepted value)-(measured value)]X100 (accepted value) %Error = 9.15%. When you have completed the first cycle of the experiment, repeat the procedure for two more times. Take fresh water each time from the faucet to start under the same conditions. When you have three sets of data, calculate the average of the mechanical equivalent of heat and see how the numbers vary from an average. If all of the values are above the accepted value, one procedure is dictated. If all of the values are below the accepted value, another procedure may be dictated. Discuss any differences and how these can be accounted for in your measurements. Tell how you would improve the experiment to guarantee a good value of the mechanical equivalent of heat. Table 1. Data for determining the mechanical equivalent of heat. ΔTIME(S) TIME(H) TIME(M) TIME(S) TEMP(C)

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TABLE 2. Sample data for mechanical equivalent of heat constant. 1 ΔTIME(S) 0 23 34 45 55 65 80 97 122 TIME(MIN) 10 10 10 10 11 11 11 11 12 2 TIME(SEC) 10 33 44 55 5 15 30 47 12 3 TEMP(C) 23 24 26 28 29 30 32 33 35 4 TEM(CAL) 24.05551 26.04432 26.99549 27.94666 28.81136 29.67607 30.97312 32.44311 34.60486 5

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137 146 164 177 192 202 212 227 238 259 275 289 301 314 328 343 356 367

12 12 12 13 13 13 13 13 14 14 14 14 15 15 15 15 16 16

27 36 54 7 22 32 42 57 8 29 45 59 11 24 38 53 6 17

36 37 39 40 41 42 43 44 45 47 48 49 50 51 52 53 54 55

35.90192 36.68015 38.23661 39.36072 40.65778 41.52248 42.38718 43.68423 44.6354 46.45128 47.8348 49.04538 50.08302 51.20713 52.41772 53.71477 54.83888 55.79005

1A

2A Regression Output:

3A

4A

5A

Constant Std Err of Y Est R Squared No. of Observations Degrees of Freedom

24.05551 0.704235 0.994986 27 25

X Coefficient(s) Std Err of Coef.

0.08647 0.001228

Table 2A. Summary data for the fitting routine.

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Figure 3. A plot of the experimental data for determining the mechanical equivalent of heat constant. The bold line is a linear least squares regression fit of the data. The 's are the raw data points. Data were taken with the EA-200 Data Collector input to the CFX 9850GB graphing calculator for observation. The curve above was produced from Excel as the data were input into a computer. All of these activities can be used to teach technology in the class room.

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