Comparing Zirconium and an Unknown Metal Using Specific...

Comparing Zirconium and an Unknown Metal Using Specific Heat and Linear Thermal

Expansion

Crystal Jacobs and Brendan Kelley

Macomb Mathematics Science Technology Center

Chemistry

10C

Mrs. Hilliard, Mr. Supal, Mrs. Dewey

May 20, 2010

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Table of Contents

Introduction ..............................................................................................................1

Review of Literature .................................................................................................4

Problem Statement ................................................................................................. 8

Specific Heat Experimental Design ........................................................................ 9

Linear Thermal Expansion Experimental Design ................................................. 11

Data and Observations ......................................................................................... 13

Data Analysis and Interpretation .......................................................................... 20

Conclusion ............................................................................................................ 34

Application............................................................................................................. 39

Acknowledgements ............................................................................................... 20

Appendix A ............................................................................................................ 20

Appendix B ............................................................................................................ 20

Works Cited........................................................................................................... 20

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Introduction

Remember that expensive looking diamond ring in the commercial on

television? It might not be a diamond after all; in fact, it may be something

completely different, such as zirconium. Everything in the universe has its own

unique intensive properties, properties of a substance that are independent of

sample size, and there are different ways to identify an object apart from

something else, even if two objects share similar properties. All matter has

different characteristics that set itself apart from all other matter. For example,

the element zirconium is a lightweight metal with a light gray appearance. While

other metals may have this same appearance, there are other properties that set

zirconium off from other similar metals (Properties of Matter).

Zirconium has several uses. For example, zirconium is used in nuclear

power plants because of its low absorption cross section for neutrons. This

means that there is a small chance of a neutron interacting with a nucleus

(Rinard). Because of this, there is a low chance of the zirconium element

becoming unstable due to the addition of a neutron to the nucleus. The zirconium

rods in these power plants prevent any radioactive particles from escaping.

Zirconium is also very resistant to corrosion by most agents such as sea water,

acids, and alkali metals. As a result, zirconium is widely used in the chemical

industry where corrosive agents are used. There are more common, everyday

uses for this metal. Zircon, a common compound of zirconium, can take many

forms; one of these forms being transparent, which makes it create great

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diamond-like jewelry (Thomas Jefferson). Zirconium also creates stronger glass

with a better optical quality (Kawano).

The purpose of the experiment was to determine whether an unknown

metal shared the same identity as the metal zirconium. Specific heat and linear

thermal expansion (LTE) experiments were conducted using the unknown metal

and zirconium to conclude whether or not they were the same. Physical

properties, as well other observations such as weight and length, aided in coming

to a conclusion about the identity of the unknown metal.

To conduct the experiment for specific heat, the metal rods were heated

by being placed in boiling water for a pre-determined time. After the metal was

heated, the rod was taken out of the water and then put into a homemade

calorimeter, which was used to help measure the change in temperature. For

recording the data, a temperature prop was placed in the calorimeters with the

rod while the rod cooled down.

For the LTE experiment, the metal rods were first measured to find the

length of the rods. Once measured, the rods were again placed into boiling water

so that they could heat up. Once heated, the rods were placed into a LTE jig to

find the change in length. Each experiment had a set number of 15 trials for

each metal.

To analyze the data that the researchers collected, a two-sample t test

was used. What a two-sample t test does is it compares the means of two

samples, or data sets in this case, and it help determine whether the two

samples are from the same population by using a t value, which assess whether

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the means of the data are statistically different or not. After the t values were

calculated, the results of the t tests were then used to accept or reject the overall

hypothesis and determine whether or not the metal rods were the same or

different.

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Review of Literature

In this experiment, the specific heat of an unknown metal rod is compared

to the specific heat of a zirconium rod. Specific heat is an intensive property of

matter. It is the amount of heat per unit that is required to raise the temperature

by one degree Celsius (Specific Heat, 2004). The unit that is commonly used is

joules. All matter, whether it is water, Cheetos, or one of the elements on the

periodic table, has a unique specific heat (Flinn Science). Calculating the specific

heat can help to find out what an unidentified material may be. Not all matter

reacts the same way to a flame. For example, a piece of metal will get hotter a lot

quicker than a piece of plastic. This is because metal and plastic have different

specific heats.

The specific heat of an object can be found via experimentation. One tool

that is commonly used in finding the specific heat of a material is a calorimeter. A

calorimeter is an isolated system, which means that no heat nor matter can

escape. Since energy cannot be created nor destroyed, as stated in the first law

of thermodynamics, this type of system is useful for specific heat experimentation

because any energy that is transferred from an object changes the temperature

of the water surrounding it (First Law). This change in temperature can then be

measured using a thermometer that is placed in the water alongside the object.

The equation to find the specific heat is shown below. To find the specific

heat, the added heat, Q, is equal to the specific heat, c, of the material, times the

mass, m, of the material, and times the change in temperature, T. (Specific Heat,

2004).

𝑄 = 𝑐 ∗ 𝑚 ∗ ∆𝑇

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Adding energy to a known mass, one can observe the change in

temperature. The heat that is being found is measured in joules, a unit of energy.

There are many similar experiments that have been conducted to find the

specific heat of an object. For example, one such experiment that was conducted

was finding the specific heat of brass. To do this, the brass was first measured,

and then heated up. After that, the brass was put into some water and the

temperature change of the water was recorded. Once all the measurements were

recorded, the specific heat was then calculated (Specific Heat Experiment,

2003).

Another experiment that was thoroughly described discusses taking a

chunk of copper and placing it in boiling water. Once the metal was thoroughly

heated until it was the same temperature as the boiling water, it was quickly

transferred into a beaker with cooler water that had been previously measured to

be 25°C. As the hot metal copper cools down, the water surrounding it rises until

they reach the same temperature (ChemTeam). This state of equalization, where

the reactants and products of a reaction remain constant, is known as equilibrium

(Chemical Equilibria). In the case of this experiment, the reactant would be

considered the temperature of the copper and the product is the temperature of

the water.

Linear Thermal Expansion, or LTE, is an intensive property that is very

useful in determining an element’s identity. LTE occurs in all areas of matter,

especially in metals. LTE is a property of thermodynamics. Thermodynamics is

the study of heat and its relationships with other forms of energy. LTE is the

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change in distance of a material when heated (Thermal Expansion, 2001). There

is a change in distance because as heat is added, there is more motion in the

molecules of the material. More motion in the molecules causes the material to

expand, causing a linear change in length. This is an intensive property because

every element has a different coefficient of LTE. An intensive property is a

property that does not change based on the amount of the material there is. The

coefficient of LTE is a constant number found using the formula below. The

coefficient is represented by the letter α in the formula below (Thermal Expansion

– Linear, 2014).

∆𝐿

𝐿0

= 𝛼∆𝑇

The coefficient of LTE has a unit of K-1 (Coefficient of Thermal Expansion

Data, 2014). This formula also uses the characters L and T. The letter L refers to

length, and the letter T refers to the temperature. In particular, 𝐿0 means the

original length while ∆L means the change in length. The length measurements

can be in any unit since they are being divided by each other. Temperature is

typically measured in Kelvin, although sometimes in Celsius.

One experiment in this research used LTE to see if a known metal shared

its identity with an unknown metal. Since the coefficient of LTE is intensive, if one

finds the coefficient of LTE of the unknown metal and compare it to the coefficient

of the known metal, they can determine whether or not the metals share the

same identity. The known metal in this experiment was zirconium. Zirconium has

a coefficient of linear thermal expansion of 5.7 ∗ 10−6 °𝐶−1. In comparison,

tungsten has a coefficient of 4.5 ∗ 10−6 °𝐶−1. Tungsten has the smallest

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coefficient of LTE of pure metals. Also in comparison, Cadmium has a coefficient

of 30.8 ∗ 10−6 °𝐶−1. Cadmium has the largest coefficient of LTE of pure metals

(Coefficient of Thermal Expansion Data, 2014).

In one experiment, a thermistor and a thermal expansion apparatus were

used to determine a metal’s coefficient of LTE. It used steam to heat the metal

while in the thermal expansion apparatus. Steam would not be very useful for this

research, but putting it in boiling water and measuring the temperature of the

water would be just as accurate (223 Physics Lab, 2014). Another experiment

was very similar except for the use of a micrometer to measure the change in

length of the rods. The use of a micrometer is an efficient and precise way of

measuring the change in length of both rods (Thermal Expansion, 2012). This

research will also take advantage of a thermal expansion apparatus called a

linear thermal expansion jig. With these tools, precise measurements can be

found and used in later calculations.

One industry that benefits from the knowledge of LTE is the transportation

office in every city. Road maintenance would not have to fix as many potholes

per year. One strategy in making the roads is to put spaces in areas where a lot

of expansion takes place. This would reduce costs and save money for more

important things like public education. It would also reduce taxes.

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Problem Statement

Problem:

To determine if an unknown metal rod is made of zirconium using the

intensive properties of specific heat and linear thermal expansion and comparing

these values to the known values of zirconium.

Hypothesis:

The unknown metal will be compared to zirconium by using specific heat

and linear thermal expansion with a percent error of less than 1% for both values.

Data Measured:

The data measured for the specific heat experiment, which will be found

for both the known and unknown metal, was the mass, m, in grams, and the

temperature, T, in Celsius in the surroundings of the calorimeter. The data

measured for the linear thermal expansion was the length in millimeters, L, and

the temperature in Celsius, T.

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Specific Heat Experimental Design

Materials:

(2) Zirconium metal rods (2) Unknown metal rods TI-nspire CX calculator

Loaf pan (24.13 cm long x 13.97 cm wide x 6.35 cm deep)

Tongs (2) Calorimeters

LabQuest OHAUS GA200 (0.0001 g precision) 50 mL graduated cylinder

Water Temperature probe (0.1° Celsius

precision)

Procedure:

1. See Appendix B for how to build the calorimeter.

2. Randomize 15 trials for the known and unknown metals using the TI-

nspire calculator. Make sure that the both metals have an equal number of

trials.

3. Randomize the calorimeter that will be used for each trial.

4. Calibrate the calorimeters. See Appendix B.

5. Mass the rod and record.

6. Measure out 100 mL of water and put in the loaf pan.

7. Heat up the water in the loaf pan until it is boiling at a temperature of

100°C.

8. Place the metal rod into the boiling water and let boil for four minutes.

9. Record the heat of the boiling water. Assume that the heat of the boiling

water is equal to the temperature of the metal rod.

10. Turn on LabQuest and connect the temperature probe to the LabQuest.

11. Go to mode and change the duration for five minutes.

12. For the rate, change it to record a data point every two seconds.


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13. Measure out 40 mL of room temperature water and put in the calorimeter.

Record this as the mass of the water in the calorimeter.

14. Put the temperature probe into the water in the calorimeter.

15. Begin recording the temperature of water for 30 seconds before placing

the metal rod in the calorimeter by clicking on the “play” button.

16. Place the heated metal rod into the calorimeter using the tongs.

17. After the LabQuest has stopped recording data, take the metal rod out of

the calorimeter and dump out the water.

18. Repeat steps 5-17 until all 30 trials are completed.

Diagram:

Figure 1. Specific Heat Experimental Design

Figure 1 above shows the necessary components of the specific het

experiment. Some objects to note are the calorimeters and the loaf pan.


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Linear Thermal Expansion Experimental Design

Materials:

(2) Zirconium metal rods

(2) Unknown metal rods TI-nspire CX calculator Loaf pan (24.13 cm long x 13.97 cm

wide x 6.35 cm deep) Tongs

Linear Thermal Expansion Jig

50 mL graduated cylinder Water Thermometer (0.1° Celsius

precision) Caliper (0.01 mm precision)

Procedure:

1. Randomize 15 trials for both the known and unknown metals for a total of 30 trials. Make sure that the both metals have an equal number of trials.

2. Measure the length of the sample using the caliper and record.

3. Place the rod into the jig and mark the reading of the dial. This will be used for a later calculation.

4. Measure out 40 mL of water and put in the loaf pan.

5. Heat up the water in the loaf pan until it is boiling at a temperature of 100°C.

6. Place the metal rod into the boiling water and let boil for four minutes.

7. Measure the temperature of the water and record. Assume that the temperature of the water is equal to the temperature of the metal rod.

8. Place the heated metal rod into the linear thermal expansion jig quickly

using the tongs. Record the reading on the dial of the jig.

9. After the metal has cooled to room temperature, take the metal out of the

jig and compute the change in length and record the room temperature.

10. Repeat steps 2-9 until all 30 trials are completed.


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Diagram:

Figure 2. Linear Thermal Experimental Design

Figure 2 above shows a diagram of the materials necessary for the linear

thermal expansion experiment. Some tools to notice are the linear thermal

expansion jigs, the loaf pans and hot plates, and the caliper.


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Data and Observations

Table 1 Specific Heat Data of Zirconium

Trial Rod Calorimeter

Mass of

Sample

(g)

Initial Temp

of Sample

(°C)

Equilibrium

(°C)

Mass of

Water

(g)

Initial Temp

(°C)

Change in

Temp

(Water)

Change in

Temp

(Metal)

Specific Heat of Sample

(J / g

* °C)

1 A B 26.7597 98.5 24.20 40.00 21.70 2.50 74.30 0.210

2 B A 26.6930 98.5 26.10 45.00 23.70 2.40 72.40 0.234

3 B A 26.6936 98.9 23.70 43.00 21.00 2.70 75.20 0.242

4 A B 26.7956 98.8 22.30 40.00 19.60 2.70 76.50 0.220

5 A B 26.7599 98.8 22.50 40.00 20.10 2.40 76.30 0.197

6 A B 26.7598 98.7 22.20 40.00 19.80 2.40 76.50 0.196

7 B A 26.6930 98.7 22.80 45.00 20.50 2.30 75.90 0.214

8 A A 26.7596 98.7 23.50 43.00 21.50 2.00 75.20 0.179

9 B B 26.6933 98.7 22.40 41.00 20.10 2.30 76.30 0.194

10 B A 26.6931 98.8 23.10 44.00 20.90 2.20 75.70 0.200

11 A B 26.7596 98.8 23.10 41.00 20.50 2.60 75.70 0.220

12 B B 26.6929 98.6 23.00 41.00 20.80 2.20 75.60 0.187

13 B A 26.6931 99.0 23.60 44.00 21.30 2.30 75.40 0.210

14 A A 26.7598 98.6 22.90 44.00 20.70 2.20 75.70 0.200

15 B B 26.6934 98.9 23.60 41.00 20.80 2.80 75.30 0.239

Table 1 shows the raw data from the trials conducted for the specific heat

experiment. These trials were done using the known metal rods of zirconium.

The order of the rods and the calorimeter that was used for each trial were both

found randomly. The table includes all measurements found during

experimentation in order to calculate the specific heat of the sample, as well as

the average for each of the measurements.


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Table 2 Specific Heat Data of Unknown Metal

Trial Rod Calorimeter

Mass of Sample

(g)

Initial

Temp of

Sample

(°C)

Equilibrium

(°C)

Mass of

Water (g)

Initial Temp

(°C)

Specific

Heat of Sample

(J / g

* °C)

1 A B 123.6748 98.4 39.80 29.00 22.20 0.29

2 B B 123.8259 98.6 40.80 29.00 21.80 0.32

3 B A 123.8253 98.4 41.90 30.00 24.00 0.32

4 A A 123.6734 98.6 43.40 30.00 23.20 0.37

5 A B 123.6738 98.5 39.80 29.00 21.20 0.31

6 A A 123.6740 98.8 42.40 29.00 22.10 0.35

7 B A 123.8251 98.5 42.60 29.00 22.20 0.36

8 A A 123.6725 98.7 41.40 30.00 21.90 0.35

9 B B 123.8248 98.8 39.50 29.00 21.60 0.30

10 B B 123.8260 98.7 39.40 29.00 20.50 0.31

11 A A 123.6740 98.9 41.00 29.00 20.40 0.35

12 B B 123.8264 98.9 39.40 29.00 21.60 0.29

13 B A 123.8245 98.6 39.40 30.00 21.60 0.30

14 A A 123.6739 98.7 43.10 30.00 24.60 0.34

15 A B 123.6711 98.6 42.00 29.00 21.60 0.35

Table 2 shows the raw data from the trials conducted for the specific heat

experiment. These trials were done using the unknown metal rods. The order of

the rods and the calorimeter that was used for each trial were both found

randomly. The table includes all measurements found during experimentation in

order to calculate the specific heat of the sample, as well as the average for each

of the measurements.

Table 3 Linear Thermal Expansion Data of Zirconium

Trial Rod Jig

Initial Length

of Sample

(mm)

Initial

Temp of

Sample

(°C)

Final Temperature of Sample

(°C)

Change in

Temp

(°C)

Change in

Length (in)

Change in

Length (mm)

Coefficient

(C-1)

1 A B 129.37 99.1 26.0 73.1 0.002 0.046 4.8E-06

2 A B 129.52 98.9 25.0 73.9 0.002 0.043 4.5E-06


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Trial Rod Jig

Initial Length

of Sample (mm)

Initial Temp

of

Sample

(°C)

Final Temperature

of Sample

(°C)

Change in

Temp

(°C)

Change in

Length (in)

Change in

Length (mm)

Coefficient

(C-1)

3 A A 129.42 98.8 25.3 73.5 0.002 0.048 5.1E-06

4 B B 129.20 98.8 25.2 73.6 0.002 0.043 4.5E-06

5 A A 129.37 98.3 24.1 74.2 0.002 0.043 4.5E-06

6 A A 129.39 98.7 24.3 74.4 0.002 0.041 4.2E-06

7 B B 129.25 98.3 24.1 74.2 0.002 0.046 4.8E-06

8 B A 129.26 99.3 25.0 74.3 0.002 0.046 4.8E-06

9 B A 129.21 99.0 25.8 73.2 0.002 0.046 4.8E-06

10 A B 129.39 98.9 25.2 73.7 0.002 0.041 4.3E-06

11 A B 129.39 98.7 23.5 75.2 0.002 0.046 4.7E-06

12 B B 129.24 98.7 24.3 74.4 0.002 0.046 4.8E-06

13 B A 129.23 98.7 23.5 75.2 0.002 0.041 4.2E-06

14 A A 129.47 98.6 23.5 75.1 0.002 0.051 5.2E-06

15 B B 129.22 98.6 23.5 75.1 0.002 0.043 4.4E-06

Table 3 shows the raw data from the trials conducted for the linear thermal

expansion experiment. These trials were done using the known metal rods. The

order of the rods and the jig that was used for each trial were both found


order to calculate the alpha coefficient of the sample, as well as the average for

each of the measurements.

Table 4

Linear Thermal Expansion Data of Unknown Metal

Trial Rod Jig

Initial

Length of

Sample

(mm)

Initial

Temp of

Sample

(°C)


(°C)

Change in

Temp

(ºC)

Change in

Length (in)

Change in

Length (mm)

Coefficient

(C-1)

1 A A 123.16 98.2 23.7 74.5 0.004 0.088 9.6E-06


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Trial Rod Jig

Initial Length

of

Sample (mm)

Initial Temp

of Sample

(°C)


(°C)

Change

in Temp

(ºC)

Change in

Length

(in)

Change in

Length

(mm)

Coefficient

(C-1)

2 A A 123.24 98.5 23.9 74.6 0.003 0.083 9.1E-06

3 A B 123.19 98.9 23.8 75.1 n/a 0.075 8.1E-06

4 B B 123.32 97.9 23.7 74.2 0.004 0.086 9.4E-06

5 A A 123.11 98.5 24.0 74.5 0.004 0.096 1.0E-05

6 A B 123.27 98.3 24.4 73.9 n/a 0.075 8.2E-06

7 B A 123.24 98.9 23.8 75.1 0.004 0.098 1.1E-05

8 B B 123.22 98.5 23.9 74.6 n/a 0.080 8.7E-06

9 B B 123.27 98.5 24.0 74.5 n/a 0.073 7.9E-06

10 A A 123.26 98.5 23.1 75.4 0.004 0.091 9.8E-06

11 A A 123.20 98.2 23.9 74.3 0.004 0.098 1.1E-05

12 B B 123.29 98.2 23.9 74.3 n/a 0.080 8.7E-06

13 B B 123.29 98.5 23.1 75.4 n/a 0.074 8.0E-06

14 A A 123.26 98.6 23.7 74.9 0.075 8.1E-06

15 B A 123.33 98.3 24.4 73.9 0.003 0.080 8.7E-06

Table 4 shows the raw data from the trials conducted for the linear thermal

expansion experiment. These trials were done using the unknown metal rods.

The order of the rods and the jig that was used for each trial were both found


order to calculate the alpha coefficient of the sample, as well as the average for

each of the measurements. As seen above, in trials were jig B was used,

excluding trials 1 and 2, the coefficient was larger than the trials were jig A was

used. This was due to jig B measuring in millimeters, while jig A, excluding trials

1 and 2, measured in inches. The inches then had to be converted to millimeters.

This might have given different measurements, thus changing the end results.

Table 5

Specific Heat Observations for Zirconium

Trials Observations

1 Researcher A transferred metal rod into calorimeter and some w ater spilled.

2 Transfer was smooth.

3 Researcher B transferred metal rod into calorimeter.


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Trials Observations

4 Metal rod w as dropped.

5 Metal w as dropped for a few seconds before put in calorimeter

6 Metal rod dropped.

7 Trial w as redone.


9 A little w ater spilled out of calorimeter during transfer.

10 Temperature probe had been touching the metal rod for a couple of minutes.

11 Researcher A transferred metal rod into calorimeter.

12 Metal rod bounced back and some w ater was spilled.



15 Temperature probe had been touching the metal rod for about 30 seconds of minutes.

Shown in Table 5 are the observations for the 15 trials for the specific heat

for the known metal. The metal rods were thin and the researchers had some

minor problems trying to transfer the rod.

Table 6 Specific Heat Observations for Unknown Metal

Trials Observations

1 Water w as spilled out w hen metal rod w as put into calorimeter.

2 Trial redone, metal rod w as dropped.


4 Calorimeter w as not cooled dow n before putting new water into it.

5 Temperature probe w as taken completely out of w ater while the metal rod w as being put into the calorimeter.



8 Metal rod dropped for few seconds before getting into the calorimeter.

9 Trial redone because rods got mixed up.

10 Trial redone because rods got mixed up.






Shown in Table 6 are the observations for the 15 trials for the specific heat

for the unknown metal. The metal rods were thicker than the known metal rods.


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This may have caused the temperature probe to come in contact with the metal

rods rather than just the water in the calorimeters. In a few of the trials, when the

metal rods were transferred into the calorimeter, some water spilled out due to

the metal rod bouncing in the calorimeter. This would cause the mass of the

water to decrease a little bit, and might change the end result slightly.

Table 7 Linear Thermal Expansion Observations for Zirconium

Trials Observations

1 Researcher A conducted experiment.

2 Researcher A conducted experiment.

3 Researcher B placed metal into jig and Researcher A measured.

4 Researcher B conducted experiment. Metal w as out in the air for a few seconds before placed into the jig.

5

Researcher A measured, metal w as out in air for a few seconds before put into jig. The

tick on the jig moved slightly due to the jig being hit.

6 Researcher A measured and Researcher B put the metal into the jig.

7 Researcher A measured and placed metal into the jig.

8 Trial w as redone because the metal had been dropped.

9 Jig got bumped into and the tick moved slightly.

10 Trial w as redone because the tick on the jig moved to a completely new point.

11 Researcher A conducted experiment and Researcher B measured.

12 Researcher A measured.

13 Metal w as out in the air for a few seconds before put into the jig.

14

Researcher B conducted experiment. Metal w as out in the air for a few seconds before

placed into the jig.

15 Researcher A conducted experiment and Researcher B measured.

Shown in Table7 are the observations for the 15 trials for the linear

thermal expansion for the known metal. The metal rods dried quickly. The

change in length was very small, so the researchers had to make an educated

guess on the change of the length.

Table 8

Linear Thermal Expansion Observations for Unknown Metal

Trials Observations

1 Metal was not completely cooled off, but the tick on the jig had not moved for a few minutes. No cooling method was used.

2 Data was recorded after 4 minutes of cooling. Different cooling method used.

3 Data was recorded after 4 minutes. No cooling method. Millimeter jig used.


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Trials Observations

4 Metal was not completely cooled off, but the tick on the jig had not moved for a few minutes.

5 Data was recorded after 4 minutes.

6 Data was recorded after 4 minutes. Trial had to be redone because the metal was dropped. Millimeter jig used.


8 Data was recorded after 4 minutes. Metal was cooling for about 10 seconds before put into the jig. Millimeter jig used.

9 Data was recorded after 4 minutes. Millimeter jig used.



12 Data was recorded after 4 minutes. Millimeter jig used.

13 Data was recorded after 3 minutes. New cooling method used. Millimeter jig used.



Shown in Table 8 are the observations for the 15 trials for the linear

thermal expansion for the unknown metal. Due to time constraints, most of the

trials were shortened, and the researchers measured the change in length of the

metal after four minutes of cooling. Trials 1 and 3 were not done in a shortened

amount of time and they were not undergoing any sort of cooling method like the

other thirteen trials. Trials 13, 14, and 15 were shortened even more, down to 3

minutes, and a different cooling method was used to try to cool the rods down

faster. There were two different jigs used for measurement, one in inches and

another in millimeters. The measurements that were in inches were converted to

millimeters in order for all the data to be in the same units. This did affect the

results.


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Data Analysis and Interpretation

To determine if the two metal rods were both composed of the metal

zirconium, two experiments were conducted. The first experiment was used to

find the specific heat of the zirconium and unknown metal rods. The rods were

placed into boiling water for four minutes, and then transferred into a calorimeter

containing cool water. A temperature probe was then used to record the change

in temperature of the water, which was then recorded.

Once all the data was collected, the researchers had to use that data, as

well as the physical properties of the rods, to determine whether they would

accept or reject their hypothesis. To analyze the data that was collected, a two-

sample t test was conducted. A two sample t test is used to compare two means

of two different samples to determine whether or not they come from the same

population. In this case, for the specific heat experiments, the mean specific heat

from both the zirconium and unknown metal rods were compared using the two-

sample t test.

In order for a two-sample t test to be calculated, several assumptions had

to be met. One of the assumptions was that the data was normal. Normally, this

would be achieved by running thirty or more trials. However, because only fifteen

trials were conducted on the zirconium and unknown metals, the data must be

found reliable in another way, such as being graphed on a normal probability

plot. All trials, calorimeters, and rods were randomized by using the random

integer setting in the TI-nspire calculator. This makes the experiment a simple

random sample, or SRS. The trials were also repeated, which eliminates any


22

bias from the researchers or equipment. These trials were also run

independently. This means that one trial did not depend on another trial’s results.

These assumptions allow one to run two-sample t tests and ensures that the data

is unbiased. After the t test is conducted for the specific heat, the researchers

must then determine whether to reject or fail to reject the null hypothesis. The

null hypothesis, or 𝐻𝑜, is 𝑈𝑧 = 𝑈𝑢𝑘 . U is for the mean, meaning the mean of the

sample of the data. z is for zirconium while uk is for unknown metal. The

alternate hypothesis, or 𝐻𝑎, is 𝑈𝑧 ≠ 𝑈𝑢𝑘 .

Table Averages for Specific Heat Experiment

Metal

Mass of

Sample (g)

Initial Temp of

Sample

(°C)

Equilibrium

(°C)

Mass of

Water (g)

Initial

Temp

(°C)

Change

In Temp (Water)

Change

in Temp (Metal)

Specific Heat of

Sample (J/ g * °C)

Zirconium 26.7266 98.7 23.27 42.13 20.87 2.400 75.467 0.21

Unknown 123.7444 98.6 41.06 29.33 22.03 19.03 57.59 0.33

Table 9 above shows the specific heat averages for the zirconium rods

and the unknown rods. As can be seen in the table above, the specific heat of

the two metals are different from one another in terms of specific heats, as these

values are generally very low in the periodic table. A point of interest in the table

above are is the difference in the mass of the samples. The zirconium metal rod

weighed an average of only 26.7266 grams while the unknown metal rod

weighed an average of 123.7444 grams. This led the researchers to suspect a

difference in identity between the two metals, but could not be determined until

further tests were conducted.


23

Table 10

Percent Error for Specific Heat of Zirconium

Trial Percent Error

(%)

1 -25.111

2 -16.790

3 -13.882

4 -21.552

5 -29.992

6 -30.175

7 -23.935

8 -36.366

9 -31.060

10 -28.671

11 -21.645

12 -33.446

13 -25.132

14 -28.849

15 -14.959

Table 10 shows the percent errors for the specific heat experiments

conducted with the metal rods. The range of these values is 22.484. A majority of

the trials are between -20 and - 30% error. This could be because of heat

escaping from the calorimeter. This escape of heat is considered an error in the

experiment.

Table 11

Percent Error for Specific Heat of Unknown Metal

Trial Percent Error (%)

1 4.862

2 14.630

3 14.288

4 32.173

5 10.632

6 25.667

7 27.260

8 22.917

9 5.263


24


10 11.142

11 24.221

12 4.321

13 8.467

14 20.178

15 25.843

Table 11 is the percent error for the unknown metal rods. It shows the

percent error for each trial, along with the average percent error of all the trials.

The range of these percent errors is 27.311. The calculated specific heats from

each trial were compared to the true value to find the percent error. A percent

error measures the difference between the experimental value and the true value

and presents it as a percentage. Percent errors are useful to compare two

different samples to see if they are from the same population. The average

specific heat values of zirconium and the unknown metal can then be compared

using a two sample t test.


25

Figure 3. Specific Heat Percent Error Box Plots

Figure 3 shows box plots that show the minimums (Min), quartile 1’s (Q1),

medians (Med), quartile 3’s (Q3), and maximums (Max) of the percent errors

from the specific heat experiments of both the zirconium and unknown metal

rods. Some points of interest include that all of the trials in each trial do not

overlap. This shows that there is a very large difference in the specific heats of

the metal rods. Another point of interest is that there are no outliers and little

skewness. The normality of this data allows a two sample t test to be run.


26

Figure 4. Specific Heat Normal Probability Plot For Zirconium

Figure 4 above shows the normal probability plot for zirconium in the

specific heat experiment. This graph shows how normal the data is based on

how closely the data follows the line. The data appears to follow the line closely

with only a few data points being a bit further away from the line than the rest,

which means that the data was normally distributed. The few data points that did

not follow as closely to the line could be due to some type of error during that

trial.


27

Figure 5. Specific Heat Normal Probability Plot for Unknown Metal

Figure 5 shows the normal probability plot for the unknown metal in the

specific heat experiment. A majority of the data points on the graph were within a

small distance from the line. Because of this, is can be assumed that the

distribution of this data is somewhat normal, although there would most likely be

some sort of skewness to it.

Figure 6. T Test Results for Specific Heat


28

Figure 6 above shows the results of a two sample t test performed on the

percent errors of the specific heat experiment. As shown above, the t score of

this test is -13.3349. This means that the results of the test is 13.3349 standard

deviations away from the null hypothesis. The P value of the test is 9.86461 ∗

10−14 . This P value is much less than the predetermined alpha level, 𝛼 = 0.10.

Figure 7. P-Value Graph for Specific Heat Experiment

Shown in Figure 7 is the P value graph for the specific heat experiment.

This graph shows a normal bell curve and that the P value is almost 0. There is

no visible area inside the curve because the number of standard deviations is so

high. The area outside of these standard deviations is equivalent to the P value,

which is 9.86461 ∗ 10−14. The researchers rejected the null hypothesis. There is

evidence that the unknown metal rod is not zirconium. There is a 9.86461 ∗

10−12 % chance that the researchers would have gotten the results they had by

chance.


29

A second experiment was then conducted to help aid in the identification

of the unknown metal rods. This second experiment was conducted to find the

alpha coefficient of linear thermal expansion (LTE) of the zirconium and unknown

metal rods. For this experiment, the rods were first measured to find the length,

and then placed into boiling water. After sitting in the boiling water for four

minutes, the rods were quickly transferred into a LTE jig. This jig was used to

measure the change in length of the rods. All of the trials, rods, and jigs were

randomized again using the TI-nspire calculator’s random integer setting, making

the LTE experiment a SRS as well. The researchers ran independent trials for

this experiment, which means that the trials did not depends on another trial’s

results. This allows the researchers to run a two-sample t test on the data found

in order to compare the means of the coefficients of the LTE experiments for the

zirconium and metal rods. After the t test is conducted for the linear thermal

expansion experiment, the researchers must then determine whether to reject or

fail to reject the null hypothesis. The null hypothesis, or 𝐻𝑜, is 𝑈𝑧 = 𝑈𝑢𝑘 . U is for

the mean, meaning the mean of the sample of the data; z is for zirconium while

uk is for unknown metal. The alternate hypothesis, or 𝐻𝑎, is 𝑈𝑧 ≠ 𝑈𝑢𝑘 .

Table 12 Averages for Linear Thermal Expansion Experiment

Metal

Initial Length of Sample

(mm)

Initial Temp of Sample

(°C)


(°C)

Change in Temp

(°C)

Change in Length

(in)

Change in Length

(mm)

Coefficient (C-1)

Zirconium 129.33 98.8 24.6 74.2 0.0018 0.045 4.641E-06

Unknown 123.243 98.1 23.7 74.6 0.0 0.083 9.070E-06

Table 12 above shows the averages of the linear thermal expansion

experiments. The table shows the averages for both the zirconium rods and the


30

unknown metal rods. Looking at the average lengths of the two rods alone, it can

be seen that these metals may be different because of the six millimeter

difference. The coefficients of the rods are also different from each other, as was

seen with the specific heat. This would suggest that the metals are not the same.

Although the differences are not very large, it is large enough for the researchers

to suspect that the metals are not the same. However, this cannot be determined

until a statistical test is conducted on the data.

Table 13

Percent Error for Linear Thermal Expansion of Zirconium


1 -15.18

2 -20.85

3 -10.99

4 -20.34

5 -21.08

6 -25.94

7 -16.36

8 -16.48

9 -15.19

10 -25.23

11 -17.56

12 -16.58

13 -26.63

14 -8.34

15 -21.94

Table 13 above shows the percent errors for the 15 trials as well as the

average percent error for the linear thermal expansion experiment using the

zirconium metal. The range of the percent errors is 18.29.


31

Table 14

Percent Error for Linear Thermal Expansion of Unknown Metal


1 68.643

2 58.957

3 42.223

4 64.408

5 82.771

6 44.439

7 85.763

8 52.685

9 39.455

10 71.119

11 87.824

12 53.214

13 39.655

14 42.522

15 53.272

Table 14 shows the percent errors for the unknown metal rod in the linear

thermal expansion. The range of these percent errors is much larger than the

range of the linear thermal expansion experiment of the zirconium rods with a

range of 48.369%. As with the specific heat experiments, the percent error of

each trial was found by comparing the experimental alpha coefficient value with

the true alpha coefficient value of zirconium. The percent errors are then used to

conduct the two-sample t test.


32

Figure 8. Alpha Coefficient of Linear Thermal Expansion Box Plots.

Shown above in Figure 8 are the boxplots for the coefficients of the linear

thermal expansion experiments for both the zirconium rods and the unknown

rods. The minimums (min), quartile 1’s (Q1), medians (med), quartile 3’s (Q3),

and maximums (max) of the coefficients from the linear thermal expansion

experiments of both metal rods are shown. As can be seen above, none of the

trials overlap. This shows that there is a big enough difference in the alpha

coefficients of the metal rods to assume that they may be different metals. There

is also no outliers in either of the box plots, and there is little skewness. With little

skewness, this shows that the data was normal, and there wasn’t a large

variability in the data. Looking at the figure above, it can be seen that the boxplot

for the zirconium rods has less variability than the boxplot for the unknown metal

rod. This means that there is a smaller range within the data collected for the

zirconium rods during experimentation than the unknown rods.


33

Figure 9. Linear Thermal Expansion Normal Probability Plot for Zirconium Rod

Figure 9 shows the normality of the data for the linear thermal expansion

experiments of the zirconium rods. The data is fairly normally distributed, and this

can be seen by how close the data points are to the line on the graph. Some

points are a little further away from the line than others, but most are fairly close,

which means that the data collected was fairly consistent and normal.

Figure 10. Linear Thermal Expansion Normal Probability Plot for the Unknown

Rod


34

Figure 10 is the normal probability plot for the unknown metal rods. This

data was not as normally distributed as the data seen in Figure (the one above

this one). This can be seen because most of the data points do not lie closely to

the line that is one the graph; however, because some of the points do lie closely

to the line, there is some normalcy to the data that was collected.

Figure 11. T Test Results for Linear Thermal Expansion

Figure 11 above shows the results of the t test that was performed on the

percent errors of the linear thermal expansion experiments. One point of interest

is that the P-value, 5.80212 ∗ 10−12 , is much smaller than the alpha level, 0.10.


35

Figure 12. P-Value Graph for Linear Thermal Expansion

Figure 12 shows the P-value graph for the linear thermal expansion

experiment. This graph, much like Figure #, shows a bell curve with no area

shaded in it. There is no shaded area because the area outside the t value much

too small. This P value, 5.80212 ∗ 10−12 , is much smaller than the predetermined

alpha level, 0.10.

The researchers rejected the null hypothesis. There is evidence that the

unknown metal rod is not zirconium. There is a 5.80212 ∗ 10−10% chance that

the researchers would have gotten the results they had by chance.


36

Conclusion

This experiment was conducted to determine if the unknown metal the

researchers were given was the same as their known metal rod, which was

zirconium, by finding the specific heat and the alpha coefficient of linear thermal

expansion (LTE). The hypothesis that the unknown metal will be compared to

zirconium with a specific heat and LTE percent error of less than 1% was

rejected. The data that was collected did not support the hypothesis. The

researchers were able to correctly identify whether the unknown metal was the

same; however, the percent errors for both experiments were much higher than

the researchers hypothesized.

Some general observations were made throughout the experiment. One

observation was that the unknown metal and the zirconium were different colors.

Another observation was that the rods were drastically different weights. Another

property noted was that the unknown metal rods stayed much hotter over a long

period of time compared to the zirconium rods.

Some things that were used to analyze the data were boxplots, percent

error tables, and t tests. Boxplots and percent error tables were used to check for

normalcy and suggest differences within the data. Every boxplot and percent

error table showed an immense difference between the values of the zirconium

and unknown metal rods. These large differences suggest that the rods are

different metals. The percent error tables also revealed that the data was normal.

This data could therefore be tested using a two sample t test due to its normalcy.


37

The results of the experiment suggest that the metals are different. The

average percent error of the specific heat experiment for zirconium was found to

be -25.438%. This means the specific heat that was found during

experimentation was less than the published value for specific heat for zirconium.

The average percent error that was found for the specific heat of the unknown

metal was 16.791%. Although this percent error meant that on average, the

specific heats of the unknown metal was close to that of zirconium, it is not the

same metal. These metals are different because of the large difference in the

percent errors. When a two-sample t test was calculated and the means of

specific heat for both the known and unknown metal were compared, it showed

that the two metals were not the same. The results showed that the P value,

9.86461 ∗ 10−14 , was less than the alpha value of 0.10. Since the P value is less

than the established alpha level, the test suggested that the metals were

different.

In the LTE experiment, the average percent error for the zirconium rods

was found to be -18.58%. The average for the unknown metal rods was 59.13%.

This shows that the unknown metal had a larger coefficient than the published

LTE coefficient value that is published for zirconium. There is a large difference

in the percent errors, giving the researchers an indication that the metals are

different. Once the two-sample t test was conducted, as with the specific heat, on

the means of the alpha coefficients of both the zirconium and unknown metal

rods, the p-value was found to be 5.80212 ∗ 10−12 . This P value was then


38

compared to the established alpha level of 0.10. Since the P value is less than

the alpha level, this suggests that the metals are different. Both t tests, the

percent error tables, and the boxplots all suggested that the metals were

different. No tests suggested that they were the same metal.

Although the experimentation went smoothly, some errors occurred. One

error that occurred throughout the experiment was an inconsistent room

temperature. The researchers could not control the temperature of the room that

they were experimenting in, and this would affect the data by creating differences

in the initial temperature. Another error was that the calorimeters were not made

well enough. They did not insulate enough heat, so some heat did escape and

this did affect the data. The loss of heat means that some energy was not

transferred into the water and therefore was not read by the thermometer. The

loss of heat would make both an inconsistency between trials and an

inconsistency with the true values. Another error was inconsistent timekeeping

while doing the LTE experiment. The timing used for this experiment was not

exact and varied between trials. This means that the molecules inside the rod did

not slow down to the speed they were at when the experiment began and

therefore the rod was not at its final length.

The experiment was well designed and successful, but changes could

have been made to make it more reliable. A design flaw when it came to the LTE

experiment was that there was not enough time. The unknown metal did not cool

down very fast, so the researchers had to compromise on accuracy and wait just

five minutes for the metal to cool down to measure the change in length. This


39

made the data inaccurate. Having more time would have made the experiment

much more reliable and would minimize the amount of flaws in the data. Another

experimental flaw was that there was no way for the researchers to be able to

measure the temperature of the center of the metal rods. Instead, it was

assumed that the entire temperature of the rods was the same as the

temperature of the water. This would create some error in the data because the

center of the rods may not have been given enough time to heat up to the

temperature of the water.

However, there were some good aspects to the experiment as well. For

instance, there was enough time during the specific heat experiment for rods to

reach their equilibrium state. This ensured that the temperature readings were

accurate.

Making more accurate calorimeters can improve the results of this

research. Accurate calorimeters should not leak any heat or matter, and this will

create more reliable data. Making the calorimeters a true isolated system would

force the heat energy to transfer to the water instead of escaping. The water

would then have a true reading of the specific heat of the metal. Allowing the

rods to cool completely would also increase the validity of this research. Having

the rods cool completely would give a more accurate reading of the change in

temperature, or delta T, reading. Letting the rods cool completely allows all of the

molecules inside it to slow down and lets the rod fully contract.

A way the experiment could be expanded upon is to conduct a density

experiment on the unknown metal. This would help to identify what the metal is


40

because the densities are different between the unknown metal and zirconium.

Running more trials would be good way to expand the experiment. Running more

trials would make it possible to make the t test more reliable. T tests rely on the

normality of data and the best way to make data normal according to the Central

Limit Theorem is to conduct many trials.


41

Application

Zirconium is a light gray transition metal that has a low neutron absorption

cross section. This means that there is a lower chance that the zirconium will

accept a neutron and become less stable. This allows zirconium to be used in

nuclear reactors as tubes. This tubing is very expensive, but is the perfect

material for nuclear reactors and the reactions that take place there.

Figure 13. Drawing and Isometric View of Zirconium Tubing

Figure 13 above shows a drawing of a zirconium tube. This tube has an

outer radius of 3 inches and an inner radius of 2.5 inches. The length of the tube

is 2 feet long, or 24 inches. The weight of this tube is 49.44 pounds. The cost of

pure zirconium is around $157 per 100 grams. The cost of this tube made of pure

zirconium is around $35,208.24. This tubing will be used inside the nuclear

reactor because it will stay stable during the dangerous reactions.


42

Acknowledgments

The researchers would like to acknowledge a few people for helping them

with their research. They would like to acknowledge Mrs. Hilliard for supplying

some of the necessary materials that were needed. They would also like to

acknowledge Mr. Supal for letting them use his tools to help cut the PVC pipe

that was needed for the calorimeters. They would also like to thank Mrs. Dewey

for help with the data analysis and interpretation.


43

Appendix A: Formulas and Sample Calculations

To find the specific heat of the zirconium metal rods and the unknown metal rods,

the following equation where the specific heat, s, of the water is multiplied by the

mass, m, of the water and the change in temperature, ΔT, of the water. This is

set equal to the equation where the specific heat, s, of the metal rod is multiplied

by the mass, m, of the rod and the change in temperature, ΔT.

Shown in Figure 14 is a sample calculation using the equation for specific heat

by using the data that was collected in trial 1 of the zirconium metal rod.

Figure 14. Specific Heat Equation

To find the linear thermal expansion for the zirconium and unknown metal rods,

the change in length, ΔL, is divided by the original length, Lo, is set equal to the

alpha coefficient, α, multiplied by the change in temperature, ΔT.

Shown below in Figure 15 is a sample calculation of the linear thermal expansion

equation using the data found in trial 1 of the zirconium rods.

Figure 15. Linear Thermal Expansion Sample Calculation

To find the percent error of

the trials that were measured, the experimental value, e, of the metal rods was

subtracted from the true value, t, of the zirconium rods. This was then divided by

the true value, t, of zirconium, and then multiplied by 100.

Figure 16 below shows a sample calculation from the first specific heat trial done

with the zirconium rod. It was found in research that the true specific heat value

of zirconium was 0.281 j/g * °C.

Figure 16. Specific Heat Percent Error Sample Calculation


44

To analyze the data, a two-sample t test was conducted. In order to do this, the t

value was set equal to the difference of the zirconium mean, , and the unknown

metal mean, , divided by the square root of the sample standard deviation of

zirconium, s1, squared divided by the number of trials for that experiment, n1,

added to the sample standard deviation of the unknown rods, s2,squared divided

by the number of trials for that experiment, n2.

Figure 17 below shows a sample calculation for finding the t test value of the first

trial of the linear thermal expansion for zirconium.


45

Appendix B: Calorimeter Procedure

Materials:

PVC pipe

Styrofoam cups Electrical tape

(2) PVC pipe caps

Procedure:

1. Measure out 5 inches on PVC pipe and cut at line.

2. Take one of the caps and cut a hole (x) for the thermometer.

3. Take the Styrofoam cup and make a circle around where the bottom meets the top (Figure 18).

4. Cut off the bottom of the Styrofoam cup.

5. Repeat steps 3 and 4 so that there is two bottoms.

6. With the rest of the Styrofoam cup, cut it up in strips.

7. Curl the strips and put them inside of the bottoms (Figure 19).

8. Put the two caps on either end of the PVC pipe.

9. Wrap the strips around the PVC pipe, except for the cap with the hole on

the top.

10. Use the electrical tape to tape the Styrofoam strips onto the pipe.

11. Repeat steps 9 – 12 to add extra insulation to the calorimeter.

12. Trace a circle onto one of the bottoms of the cut Styrofoam cup, placing the marker on the edge of the insulation (Figure 20).

13. Cut out the circle.

14. Put the insulated PVC pipe through the hole of the one bottom.

15. Place the other end one of the PVC pipe into the other one, making sure

that the insulation is around the bottom of the PVC. Move the bottoms together so that they are touching. Tape around the intersection of the two bottoms.


46

16. Whatever PVC pipe is still showing on the top, repeat steps 8 – 12 to insulate again.

17. Wrap tape around the top part of the calorimeter and around the edge so that no heat can escape.

Note: Depending on what kind of Styrofoam cups are used, the bottoms may not be needed to be cut and there may be more of less insulation needed.

Figure 18. Circle with Where to Cut Figure 18 above shows the cup with the drawn on circle, as stated in step

3, indicating where someone would want to cut the cup if they had a cup this

size. However, due to the variation in cup sizes around the world, this step may

be optional.

Figure 19. Curled Styrofoam


47

Figure 19 shows step 7 of how to make a calorimeter. Because the pieces

needed to fit within another cup, they had to be curled, as seen above, in order

for them to fit.

Figure 20. Circle to Cut Out

Figure 20 shows step 12 of the how to make a calorimeter. In order for the

PVC pipe to be able to pass through the insulation of the calorimeter, a hold had

to be cut through one of the two cups that could be holding all the insulation

around the PVC pipe.

Figure 21. Finished Calorimeter


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Figure 21 above shows a finished version of the calorimeter. As can be

seen, there is tape around the top to ensure that no heat escapes from the

calorimeter during experimentation.

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