effectiveness of linear thermal expansion on determining ...€¦ · understanding linear thermal...

Effectiveness of Linear Thermal Expansion on Determining if Two Metals are the

Same

Elizabeth Evers, Justine Hickey, Jillian Maceroni

Mrs. Dewey, Mrs. HIlliard, Mr. Supal

May 24, 2016

Evers-Hickey-Maceroni 1

Table of Contents

Introduction ……………………………………………………………………..............2

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

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

Experimental Design..............................................................................................9

Data and Observations........................................................................................11

Data Analysis and Interpretation..........................................................................17

Conclusion...........................................................................................................26

Application...........................................................................................................30

Acknowledgments................................................................................................32

Appendix A: Randomization.................................................................................33

Appendix B: Coefficient of Linear Thermal Expansion.........................................34

Appendix C: Percent Error...................................................................................35

Appendix D: t-test................................................................................................36

Works Cited.........................................................................................................37


Introduction

In 2012, interstate 275, near the Ford Road off ramp in Canton Township,

buckled due to an extreme heat wave. Michigan Department of Transportation

spokesman Rob Morosi reported that, “In the heat wave that we're having the

pavement is expanding. Those joints that we put in there have reached their

maximum. When that occurs the pavement has nowhere to go, but to go up”

(“Interstate 275”). Over the course of the extremely hot days, the heat caused the

metal joints to expand and break causing the road the jut upward. By better

understanding Linear Thermal Expansion and how it pertains to infrastructure,

this incident could have been avoided and a better metal could have been placed

in the pavement.

The purpose of this experiment was to identify whether or not the known

and unknown sample were the same type of metal, using the intensive property

Linear Thermal Expansion. In this experiment, each sample had two metal rods

used to test. A metal rod was first measured lengthwise, then set in boiling, hot

water. After one minute the metal was taken out and quickly placed in the LTE

Jig. The metal was measured it condensed back to the original size. By recording

this data and calculating the coefficients, the hope was to solve the identity if the

metals are the same.

The property, Linear Thermal Expansion can be used to in a scientific

community and in everyday life. An industry or a manufacturing company, can

use LTE to test whether the material used is the correct one for the jobs it will be

fulfilling. Depending on the use, a product might need to withstand certain


amounts of heat that it could experience when in use. For example, an engine

would need a material that when heated, it will not expand and break. So a car

company, like Ford, would use Linear Thermal Expansion to test if the material

used in the engines can tolerate this amount of heat.


Review of Literature

Linear Thermal Expansion, also known as LTE, is the material property of

expansion when a substance is heated. As the temperature of the material rises,

the atoms inside the material are excited meaning that the vibration of the atoms

increase. This increase in vibration causes the atoms to bump into one another

more frequently causing an increase of separation between the atoms. This

increase separation between the atoms causes the material to expand. Once the

material returns to room temperature, the vibrations decrease and the material

Contracts shown in figure 1. This change in length is caused by Linear Thermal

Expansion (“Thermal Expansion”).

http://www.bbc.co.uk/bitesize/ks3/science/chemical_material_behaviour/behaviour_of_matter/revision/2/

Figure 1. Cooled vs. Heated Atoms

Figure 1, shows the cooled atoms vibrating at slow speeds. They are

contracted compared to the heated atoms because there are less collisions

among them. When heat is added to the atoms they vibrate at higher speeds

creating more collisions among the atoms expanding the space needed to

contain them ("Expansion and Contraction").

Linear Thermal Expansion is an intensive property, meaning that the


measured value, for instance LTE, melting point, and boiling point, does not

depend on the amount of the substance being measured (Senese). When heated

to the same temperature, the atoms in every element react uniquely. Atoms in

some elements may act more excited than others meaning that all elements

would expand to different lengths. Every element having its own Coefficient of

Linear Thermal Expansion means that LTE will help identify what element you

are working with to make sure you are not using the wrong element, or in the

case of this research, if two metals are the same or different to find out if two that

look the same really are the same.

𝛼 = 𝛥𝐿

𝐿0(𝛥𝑇)

Above, shows the formula to calculate the Coefficient of Linear Thermal

Expansion of a heated substance. For LTE to be measured, an original length

(𝐿0) measured in millimeters (mm) using a caliper; temperature change (𝛥𝑇);

measured in degrees celsius, oC; and the change in length (𝛥𝐿) measured in

millimeters, mm, using the LTE Jig, are required. The units of 𝛼 are measured in

oC-1 *10-6 (Licudine). It will be assumed when the metal stops condensing that it

has reached room temperature. This is because of equilibrium which says that

once something has reached room temperature it will cease cooling and remain

equal to room temperature.

The known metal used as the comparison in this experiment is Copper.

Copper’s density is 8.96 g/cm3 which is relatively average compared to other

pure metals. Copper’s LTE is 16.6 oC-1 *10-6 ("Thermal Expansion"). This average


density and low LTE makes Copper good for types of metal products that need to

be tolerant of heat because the metal will not expand as much and break. On the

periodic table, Copper is in close proximity as Zinc. Zinc’s LTE is 29.7 oC-1 *10-6,

which is a high heat tolerance ("Coefficients of Linear Thermal

Expansion").Having a high heat tolerance means, the atoms in the metal are

excited and will expand and contract more than other metals with lower

coefficients. This could possibly cause possible cracks for breakage in the

material because it is unable to deal with the strain put on it during the expanding

and contracting. These two LTE’s are very different in value, which means Zinc

would not be the best metal to use, due to its high heat tolerance, compared to

Copper.

There have been numerous experiments previously conducted on Linear

Thermal Expansion quite similar to the current experiment. The first experiment

was a more general study; it was proving that all metals expand when heated,

not testing for comparison. The experiment also used steam to heat the metal

instead boiling water. This seems to be a common thread among the various

prior experiments. There is no particular reason for using steam because it has

been found to be just as effective as using boiling water. This experiment also

had a very detailed step by step procedure of how to find the coefficient, which

helped in understanding how to use the equation and an explanation on what

happens at an atomic level during the heating and cooling process (Licudine).

Another similar experiment calculated and compared the coefficient of

linear thermal expansion of various nickel alloys, adding different percentages of


Copper, cobalt, and iron to nickel-chromium alloys at various temperature

ranges, rather than the comparing LTE of two metals at one temperature. This

gives an example of how similar Coefficients of Linear Thermal Expansion can

be and showing how careful and quick the trials must be executed. It also

explains some of the science behind the experiment including equilibrium. The

samples in the experiment were heated either by an oil bath or an air furnace

which is slightly different than boiling the metal in water which is what will be

used to conduct the current experiment. Although these methods are different

they both are effective (Hidnert). Overall, these experiments had similar designs

to the experiment being conducted and will be used to help conduct this

experiment.


Problem Statement

Problem:

To determine if the unknown metal is Copper, compare the intensive

property, Coefficient of Linear Thermal Expansion of the two metals.

Hypothesis:

Using Linear Thermal Expansion to compare the two metals, it will be

found that there is a ±15.0568% or less error, which was calculated using two

metals with the closest Linear Thermal Expansion Coefficient and the equation

shown in Appendix C, and a α level more than 0.1; meaning the two metals are

the same.

Data Measured:

Linear Thermal Expansion measures how much a metal expands when it

is heated measured in oC-1 *10-6. For this given experiment the change in

temperature (ΔT) will found using degrees celsius (oC). It will be assumed that

the starting temperature (oC) of the metal rod is the temperature of the boiling

water and the end temperature (oC) of the rod will be room temperature (oC). The

change in length (ΔL) of the metal rod after boiling will be measured in

millimeters (mm) with a LTE Jig. The initial length (L0 ) will be measured using

calipers in millimeters (mm).


Experimental Design

Materials:

(2) Samples of the Known Metal Tongs

(2) Samples of the Unknown Metal Loaf pan

(2) LTE Jig (0.01 mm) 100 mL Graduated Cylinder Thermometer (0.1°C) Hot plate

TI-nspire CX graphing calculator Stopwatch

Caliper (0.01 mm) Hot mitt

Procedures:

Randomization

1. Use the TI-nspire CX Graphing Calculator to randomize which metal rod is being tested for each individual trial of known and unknown and which

LTE Jig being used for each trial. See Appendix A for instructions on how to randomize using the calculator.

Testing for Known Metal

1. Using a graduated cylinder measure 100 mL of water and pour it in the loaf pan.

2. Using a hot plate bring the water to a boil (95-105°C).

3. Record the temperature of the water under initial temperature using the 0.1 thermometer.

4. While the water is boiling, use the calipers. to measure the original length of the metal rod being tested. Assume the metal has reached equilibrium with the water and is the same temperature as the water .

5. Using the tongs, submerge metal for that trial in the water for one minute. Measure time with a stopwatch.

6. Once the minute is up, have one researcher hold the LTE Jig and raise the pin.

7. Have a second researcher quickly remove the metal from the boiling water with tongs and slide the metal into the LTE Jig.


8. Release the pin on the LTE Jig so it may begin recording the length change of the metal. Quickly record the initial length of the metal given by the LTE Jig.

9. Let the metal sit in the LTE Jig and cool for three minutes or cool to touch. If needed use a fan to help the cool metal .

10. Once the metal has cooled, record the the finial length of the metal given by the LTE Jig.

11. Record the temperature of the room using the 0.1 thermometer. Assume the metal has reached equilibrium and is the same temperature as the room.

12. Repeat steps 3-11 for the rest of the trials for the known and unknown metals.

Diagrams:

Figure 2. Materials

Figure 2 shows the materials needed to perform the experiment. Items not

pictured include the unknown and known metal rods, TI-nspire CX graphing

calculator, and stopwatch.

TongThermomete

LTE Jigs

Hot Graduate

Hot

Caliper


Data and Observations

Table 1

Data Results for Copper Linear Thermal Expansion

Trial ΔL

(mm)

Initial Length

(mm)

Initial Temp.

(ºC)

Final Temp.

(ºC)

ΔT

(ºC)

Alpha

Coefficient

(°C-1 x 10-6)

1 0.14 129.49 99.6 23.1 76.5 14.133

2 0.14 129.15 97.0 23.5 73.5 14.748

3 0.11 129.20 96.8 23.6 73.2 11.631

4 0.11 129.15 97.4 22.8 74.6 11.417

5 0.14 129.49 98.7 22.7 76.0 14.226

6 0.10 129.49 96.1 22.5 73.6 10.493

7 0.14 129.13 97.5 22.7 74.8 14.494

8 0.13 129.51 97.2 23.2 74.0 13.565

9 0.10 129.49 97.2 22.4 74.8 10.324

10 0.13 129.45 97.7 22.4 75.3 13.337

11 0.13 129.52 98.4 24.0 74.4 13.491

12 0.14 129.22 96.8 24.0 72.8 14.882

13 0.16 129.49 98.3 23.9 74.4 16.608

14 0.13 129.47 96.5 22.3 74.2 13.532

15 0.12 129.18 98.2 23.4 74.8 12.419

16 0.13 129.49 96.7 22.2 74.5 13.476

17 0.12 129.50 97.8 23.8 74.0 12.522

18 0.11 129.50 97.6 22.5 75.1 11.311

19 0.11 129.26 96.4 22.5 73.9 11.516

20 0.15 129.27 97.2 22.3 74.9 15.492

21 0.15 129.49 95.3 23.7 71.6 16.179

22 0.10 129.23 95.4 23.9 71.5 10.823

23 0.12 129.14 98.6 23.8 74.8 12.423

24 0.13 129.23 99.1 24.5 74.6 13.485

25 0.12 129.49 99.0 23.8 75.2 12.323

26 0.12 129.47 95.3 24.1 71.2 13.018

27 0.13 129.42 97.8 23.7 74.1 13.556

28 0.13 129.53 97.7 24.3 73.4 13.673


Trial ΔL

(mm)

Initial Length

(mm)

Initial Temp.

(ºC)

Final Temp.

(ºC)

ΔT

(ºC)

Alpha

Coefficient

(°C-1 x 10-6)

29 0.11 129.53 99.3 23.7 75.6 11.233

30 0.12 129.50 98.4 23.6 74.8 12.388

Avg. 0.13 129.38 97.5 23.3 74.2 13.091

Table 1, above, shows the recorded data for the Copper rods. The change

in length, initial length of the metal, initial (room temperature) and final

temperature (boiling water temperature) and the change in temperature were

recorded. The alpha coefficient, for each trial, was then calculated, a sample

calculation is shown in Appendix B. At the bottom of the table, the averages for

each column were calculated.

Table 2

Observations for Copper Linear Thermal Expansion

Trial Jig Rod Observations (Known)

1 C 1 Strong Boil, Clean landing but Slow and more water after

2 C 2 Stayed in a little extra, little off

3 A 2 Transfer was slow

4 B 2 Slow Transfer, water still in Jig

5 B 1 Good Boil, good transfer

6 C 1 Poor transfer

7 A 2 Moderate boil, good transfer

8 B 1 Good Transfer, low boil

9 C 1 Great/good transfer, Jig was noticeably hot, barely moving

10 A 1 Low boil, good transfer

11 C 1 Fast moving good transfer

12 C 2 Moving fast good transfer

13 A 1 Good trans, low boil

14 B 2 Jig cold, good boil average transfer

15 A 2 Average transfer


Trial Jig Rod Observations(Known)

16 C 1 Good trans, ran Jig under cold water


18 A 1 Ran Jig under cold water

19 B 2 Slow transfer

20 C 2 Good transfer

21 C 2 Water took forever to boil, bad transfer

22 A 1 bad transfer

23 A 1 Bad transfer, water at low temp with low boil



26 C 2 Shakey but fast

27 A 2 Good Transfer

28 B 2 Good Transfer

29 A 2 Bad transfer

30 C 2 Average Transfer

Table 2 above, shows all observations recorded before, during and after

each trial for the Copper rods. How the transfer of the metal from the boiling

water to the LTE Jig went, the strength of the boil during the trial, and if the LTE

Jig was unusually cold or warm before the metal was transferred were recorded

for each trial. Which LTE Jig and Copper sample used in each trial was also

recorded.

Table 3

Data Results for Unknown Metal Linear Thermal Expansion

Trial ΔL

(mm)

Initial Length

(mm)

Initial Temp.

(ºC)

Final Temp.

(ºC)

ΔT

(ºC)

Alpha Coefficient

(°C-1 x 10-6)

1 0.11 122.42 96.2 23.5 72.7 12.360

2 0.10 122.46 98.6 23.5 75.1 10.873

3 0.11 118.43 96.3 23.7 72.6 12.794

4 0.08 122.53 96.4 22.7 73.7 8.859

5 0.10 118.47 96.4 22.5 73.9 11.422


Trial ΔL

(mm)

Initial Length

(mm)

Initial Temp.

(ºC)

Final Temp.

(ºC)

ΔT

(ºC)

Alpha Coefficient

(°C-1 x 10-6)

6 0.10 122.48 95.2 22.6 72.6 11.246

7 0.10 118.39 99.2 22.6 76.6 11.027

8 0.11 122.53 96.6 22.3 74.3 12.083

9 0.09 122.44 95.5 22.3 73.2 10.042

10 0.09 122.57 95.4 22.5 72.9 10.072

11 0.09 118.41 95.8 22.3 73.5 10.341

12 0.10 122.48 97.5 23.8 73.7 11.078

13 0.10 118.39 96.8 23.9 72.9 11.587

14 0.09 122.56 96.8 23.6 73.2 10.032

15 0.12 122.59 96.9 23.7 73.2 13.373

16 0.11 118.37 97.6 22.3 75.3 12.341

17 0.10 122.52 96.1 22.1 74.0 11.030

18 0.09 118.49 96.5 22.5 74.0 10.264

19 0.08 118.43 96.9 23.6 73.3 9.216

20 0.11 118.39 96.1 22.4 73.7 12.607

21 0.10 122.56 95.3 23.2 72.1 11.317

22 0.08 122.53 97.3 24.1 73.2 8.919

23 0.11 122.15 97.4 23.8 73.6 12.235

24 0.09 118.41 95.1 23.7 71.4 10.645

25 0.08 118.36 97.3 23.6 73.7 9.171

26 0.08 118.41 95.3 23.8 71.5 9.449

27 0.09 118.56 99.7 24.3 75.4 10.068

28 0.11 122.57 97.8 23.6 74.2 12.095

29 0.09 118.57 99.3 23.8 75.5 10.054

30 0.09 122.55 99.1 24.6 74.5 9.858

Avg. 0.10 120.6 96.88 23.2 73.7 10.882

Table 3, above, shows the recorded data for the unknown rods. The

change in length, initial length of the metal using calipers, initial (room

temperature) and final temperature (boiling water temperature) and the change in

temperature were all recorded. The alpha coefficient was then calculated for


each trial, a sample calculation is shown in Appendix B. At the bottom of the

table, the averages for each column were calculated.

Table 4

Observations for Unknown Metal Rods

Trial Jig Rod Observations (Unknown)

1 A 1 Weaker boil, fast and clean landing

2 C 1 All round good

3 B 2 Good boil, sloppy transfer

4 A 1 Slow boil, transfer terrible, added water after

5 C 2 Low Boil, Ok Transfer

6 C 1 Slow Boil, good transfer, moving slowly

7 A 2 Slow transfer

8 A 1 Started to boil, good transfer

9 C 1 Slow boil, delay pin drop

10 C 2 Average transfer

11 A 2 Low water, wobbly transfer

12 A 1 Ok transfer

13 C 2 Metal slid a little

14 C 1 Bad transfer

15 B 1 Decent trans, added water after


17 A 1 Slow transfer drop pin late, add water after

18 C 2 Bad transfer, Jig is really hot

19 A 2 Jig was warm good transfer

20 B 2 Not good transfer

21 A 1 Bad transfer

22 A 1 Terrible transfer, water low boil

23 C 1 Good transfer

24 B 2 Decent transfer, low water temp

25 B 1 Average transfer

26 B 2 Average transfer

27 A 2 Good boil, average transfer


Trial Jig Rod Observations (Unknown)

28 A 1 Good boil, good transfer

29 B 2 Bad transfer, good boil

30 C 1 Average transfer, good boil

Table 4, above, shows the observations recorded before, during and after

each trial for the unknown metal rods. the strength of the boil, transfer quality,

and if the LTE Jig was unusually warm or cold before the metal was transferred

were recorded. Which LTE Jig and unknown metal sample used in trail were also

recorded.


Data Analysis and Interpretation

In this experiment there were two sets of metals that were being tested.

One metal was known to be Copper and the other metal was unknown; two

samples for each type of metal were tested. The purpose was to test Linear

Thermal Expansion on both of these metals, calculate their coefficient of Linear

Thermal Expansion and determine if they are the same metal or not.

This experiment used a control, the known metal Copper, which was

compared to the unknown metal. Doing this helped reduce confounding and gave

a basis to compare the unknown to. This experiment also contained repetition; 30

trials were conducted for each set of metal. Repeating many trials reduced

variability helping to produce more accurate results. Lastly this experiment

contained randomness; which metal sample and which LTE Jig used was

randomized for each trial. Randomizing the metal sample and LTE Jig being

used for each trial helped reduced any bias that could have occurred. The data

being measured was quantitative continuous.

Table 5 Margin of Error of LTE for Copper Metal Rods

Trial Margin of error

(%)

1 -14.86

2 -11.15

3 -29.93

4 -31.22

5 -14.30

6 -36.79

7 -12.68

8 -18.29



(%)

9 -37.81

10 -19.66

11 -18.73

12 -10.35

13 0.05

14 -18.48

15 -25.19

16 -18.82

17 -24.57

18 -31.86

19 -30.63

20 -6.67

21 -2.54

22 -34.80

23 -25.16

24 -18.77

25 -25.76

26 -21.58

27 -18.34

28 -17.63

29 -32.33

30 -25.37

Avg. -21.14

Table 5 shows the margin of error for the Copper trials along with an

average calculated at the bottom of the table. Calculating the margin of error

while executing the trials helped gage if the trial went smoothly or if it was

executed inaccurately. If the margin of error was abnormally high, what was done

wrong and how can the next trial be improved was looked at. Such as, if the

transfer of the metal from the water to the LTE Jig was too slow and if enough


time was allotted for the metal to completely cool. Overall the margin of error was

on the high side for the known metal.

Table 6

Margin of Error of LTE for Unknown Metal Rods


(%)

1 -25.54

2 -34.50

3 -22.93

4 -46.63

5 -31.19

6 -32.25

7 -33.57

8 -27.21

9 -39.51

10 -39.32

11 -37.70

12 -33.26

13 -30.20

14 -39.57

15 -19.44

16 -25.66

17 -33.56

18 -38.17

19 -44.48

20 -24.05

21 -31.83

22 -46.27

23 -26.29

24 -35.87

25 -44.75

26 -43.08



(%)

27 -39.35

28 -27.14

29 -39.44

30 -40.62

Avg. -34.45

Table 6 shows the margin of error for the unknown trials and the average

percent error calculated at the bottom of the table. Calculating the margin of error

helped determine if something abnormal occurred while executing the trials. If the

margin of error was unusually high, it was examined further to see if the human

error could have been the result of this. For example, if the transfer of the metal

from the water to the LTE Jig was too slow or not properly positioned in the LTE

Jig, if the metal was allowed to cool down totally before recording final length or if

the water was a strong or weak boil. Even though the margin of error for Copper

was high, the margin of error for the unknown metal was larger than Copper’s

which means that the two metals may be different.

Figure 3. Normal Probability Plot for Copper’s Coefficients of Linear Thermal Expansion


Figure 3, shows the normal probability for Copper’s LTE coefficients.

There does not appear to be any extreme skewness or nonlinear trends which

mens that this data appears to be normal. Based on this Normal probability chart

and the Central Limits Theorem which states that any sample that contains 30

trials or more is most likely normal, it is safe to say that this data comes from a

normal population.

Figure 4. Normal Probability Plot for the Unknown Coefficients of Linear thermal Expansion

Figure 4, shows the normal probability for the unknown metals LTE

coefficients. There does not appear to be any extreme skewness or nonlinear

trends which mens that this data appears to be normal. Based on this Normal

probability chart and the law of large numbers which states that any sample that

contains 30 trials or more is most likely normal, it is safe to say that this data

comes from a normal population.


Figure 5. Copper and the Unknown Metal’s Coefficients of Linear Thermal

Expansion

Figure 5 shows box plots for both Copper and the unknown metal’s

coefficients of LTE, the unknown metals distribution is on top the Coppers

distribution is on bottom. The distribution for the unknown metal is fairly normal

and contains no outliers while the distribution of Copper is slightly skewed to the

right but still containing no outliers. The graph also shows that the range of

Copper’s coefficients is greater than the range of the unknown metal’s

coefficients, meaning that there was more variability in the Copper’s data. There

is also a fair amount of overlap between the two distributions but their medians

are fairly far apart with 100% of the unknown metal’s coefficients smaller than

50% of Copper’s coefficients. This shows that Copper and the unknown metal

may be different.


In order to carry out any statistical test of significance certain conditions

must be met. Each sample must be a simple random sample, or SRS, both data

sets must come from a normal population and the entire population must be

larger than ten times the sample size. Both sets of data are simple random

samples; the LTE Jig and sample of metal used was randomized. As shown if

figure 1 and 2 both data sets appear to be fairly normal and shown in figure 3

both distributions are fairly normal and their means are not being pulled by any

outliers. All conditions are met so a statistical test can be carried out and the

results can be trusted.

The statistical test of significance that will be used for this data is a two

sample t-test. This test is used to test if two averages from two different

populations are significantly different or occurred by chance. A two sample t-test

is appropriate for this data because it contains two SRSs from two distinct,

independent populations and both populations are normally distributed as

previously explained.

Ho: μc = μu

Ha: μc ≠ μu

Figure 6. Null and Alternative Hypotheses

Figure 6 shows the null and alternative hypotheses for the Two Sample t-

test being carried out. The null hypothesis, Ho, states that the mean LTE

Coefficient for Copper, μc, and the mean LTE Coefficient for the unknown metal,

μu, are the same. The alternative hypothesis, Ha, states that the mean LTE


Coefficient for Copper and the mean LTE Coefficient for the unknown metal are

different.

Figure 7. One Variable Statistics for Copper and the Unknown Metal

Figure 7 shows the one variable statistics for Copper, on the left, and the

unknown metal, on the right. This function on the TI-Nspire calculator computes

the mean, 𝑥, standard deviation, 𝜎𝑥, and many other important values.


Figure 8. Probability Graph of the Two Metal Samples

Being the Same

Figure 8 shows the t-test results and the probability graph of the two metal

samples being the same. From the results of the t-test, the null hypothesis is

rejected because the p-value is 1.92658*10-7 or about 0 which is less than the

alpha level of 0.1. There is convincing evidence that these two metal samples

are not the same. If the null is true there would be about a 0% chance that the

mean coefficient of Linear Thermal Expansion for the two metals are the same.

Since this is so unlikely it can be said that the mean coefficient of Linear Thermal

Expansion for the two metal samples are not the same. Sample calculations on

how to compute the t and p values are shown in Appendix D.


Conclusion

In this experiment, the objective was to figure out, by calculating the

Linear Thermal Expansion Coefficient, if two metal samples (the known sample

being Copper and the other unknown) are the same metals with a ±15% margin

of error and a 0.1 alpha level. The hypothesis of this experiment stated that the

two metal samples would be found to be the same type, based on how they had

similar appearances. This hypothesis, however, was rejected; the two metals

were not the same.

Many factors were taken into consideration upon reaching this conclusion.

Even though the percent error for Copper was high at an average of 21%, which

could have been caused by the atoms instantly cooling down and contracting

before the metal was placed in the LTE Jig; the the average percent error for the

unknown metal was even higher at 34% which was outside the range of 15% to -

15%. The distribution graphs of Copper and the unknown metal’s LTE

Coefficients were also strikingly different; 100% of the unknown metal’s LTE

Coefficients were smaller than 50% of Copper’s LTE coefficients. Their mean

LTE Coefficients were also far apart with Copper’s at 13.091 oC-1 *10-6 and the

unknown metal’s at 10.882 oC-1 *10-6. The last piece of information that was

looked at was the two sample t-test results. The p-value for this this test was

found to be 1.927*10-7 , or approximately 0, which is far less than the 0.1 alpha

level. This means the mean of Copper’s LTE Coefficients and the mean of the

unknown metal’s coefficients were not the same. By knowing this information, it

can be said that the two metals were different.


When metal is heated their atoms are excited and start to vibrate at

increasing speeds. This causes the atoms to have more collisions with each

other and requiring more space to accommodate these accelerated atoms, thus

causing the metal itself to expand. When cooled, the atom’s vibrations slow down

and have less collisions, allowing for the metal to contract. This expansion and

contraction is known as Linear Thermal Expansion. However, the atoms of every

metal expand and contract uniquely which allows LTE to be used to identify

which metal is being tested or if two metals are the same or different.

Having two different metals means that the two metals atoms’ reacted

differently when heated. The unknown metal had a much lower mean LTE

Coefficient than Copper which means that the atoms in Copper were more

excited, when heated, and expanded and contracted (had a greater length

change) than the atoms in the unknown metal.

Current research on the Linear Thermal Expansion, that was found, was

used for the basis of the experimental design. While some were useful in giving

insight into how to calculate the Coefficient using other methods beside an LTE

Jig, others showed how different metal alloys compare to each other. Even

though all methods and metals were different, they all accomplished the same

objective: successfully testing and calculating a Linear Thermal Expansion

Coefficient.

The experimental design was followed very well in the process of this

experiment. The metal was always left in water to heat up a constant amount of

time and left to cool for a consistent amount of time amount of time. Due to how


well it was executed it is highly unlikely this had any effect on the outcome of the

experiment.

Most of the error in this experiment were caused by human or equipment

error. The transfer of the metal rod from the water to the LTE Jig was often slow

which could have allowed the metal to start contracting before the length change

was being measured. Other human error could have occurred by not letting the

metal to completely cool, which means that the atoms would still be excited and

expanded after the metal was removed from the LTE Jig. The unknown metal

rods were substantially larger than the Copper rods and sometimes the unknown

metal rods were slightly warm but were removed due to time constraints. Even

though the length change of these metals were not moving they could have not

been given enough time to completely cool and finish contracting. Other error in

the experiment was equipment error. On day three of the trials the hot plate was

barely getting up to an appropriate temperature, this could have caused the

metals not to expand fully or as much as previous days. If the metal is not heated

to a consistent heat then the atoms would not become fully excited and would

would not fully expand resulting in a smaller length change and a lower LTE

coefficient. Other equipment errors could have been from the LTE Jigs being

abnormally warm not allowing for the metals to cool all the way so the atoms

would be still be expanded when the metal was removed this could have been

fixed by leaving more time in between trials but due to time constraints that was

not possible. Also the thermometer being inaccurate could have caused error in

the calculations of the Linear Thermal Expansion Coefficient.


To identify this metal, research using other intensive properties such as

specific heat and density can be conducted. Research in this area could be

expanded further by identifying other elements, metal alloys or non-pure metals

with these intensive properties. Further research could also include

experimenting with different methods or devices to calculate the change in

length. Many industries such as, automotive industries, manufacturers of metal

products or products involving heat, and engineers can benefit from this

experiment. These industries can use this research as another method to

determine unknown metals and which metal is most appropriate for a certain job

in which heat is a possible factor. For example, a construction company, creating

a bridge, would need to take into consideration the Linear Thermal Expansion

Coefficient of the metal used so that it will not expand and break on scorching,

hot days.


Application

Due to its conductivity, Copper is a very versatile metal and can be used

for many different products and functions. It is used for anything from piping and

wiring homes to just fun decorations around people's homes.

Figure 9. 3-D Model of Tea Kettle

Figure 9, above, displays a teapot made out of Copper. Copper was

commonly used for teapots because it was much cheaper than iron and other

metals that were available at the time. Also for those who do not know, teapots

are used to boil water for delicious cups of tea.

Figure 10. Drawing of Teapot


Figure 10, above, shows the drawing of the teapot from solidworks. Using

the mass tool in solidworks, the mass of the teapot came out to be 17.47 pounds.

So to get the cost of making this the cost of Copper per pound will be used, the

cost per pound of Copper is $3.38. For the total cost 17.74 and 3.38 are

multiplied together to get the total cost of the teakettle to be $59.96.


Acknowledgements

We would like to thank Mrs. Hilliard for her knowledge, guidance and her

lab that helped us conduct our experiment. Without her, we would not have had

equipment and an area to conduct our experiment. We would also like the thank

Mrs. Cybulski and we knowlegde of statistics. A lot of the statistical tests that

were taught in her class, were used in the Data Analysis and Interpretation to

come up with our conclusion.


Appendix A: Randomization

Randomization for LTE Jigs

1. Assign each of the LTE Jigs the number A, B, and C.

2. Using the TI-nspire randomization function, found by going to a calculator page and clicking menu, random, integer, and inputting the numbers (randInt(lowerbound,upperbound,how many it prints out), or (randInt(1,3,1)) generate and assign a LTE Jig to each trial. LTE Jig A corresponds with 1, LTE Jig B corresponds with 2, and LTE Jig 3 corresponds with 3. Repeat this function for all the trials.

Randomization for Known Metals

1. Assign each metal sample the number 1 and 2.

2. Then using the random integer function, described in Randomization for LTE Jigs step 2, (randInt(1,2,1)), assign each trial a metal by using the numbers generated by the calculator, Metal 1 corresponds with 1 and Metal 2 corresponds with 2. Repeat this function for all the trials.

Randomization for Unknown Metals

1. Repeat the steps listed in the Randomization for Known Metals for all the

trials of the unknown metal.


Appendix B: Coefficient of Linear Thermal Expansion

To find the Coefficient of Linear Thermal Expansion, the formula below is

used, where the alpha coefficient, 𝛼; equals the change in length, 𝛥𝐿; divided by

the original length, 𝐿0; times the change in temperature, 𝛥𝑇.

𝛼 =𝛥𝐿

𝐿0 × 𝛥𝑇

Below, in figure 1, is a sample calculation of the alpha coefficient of Linear

Thermal Expansion.

𝛼 =(76.5 𝑚𝑚)

129.49 𝑚𝑚 × (23.1 − 99.6)°𝐶

𝛼 = 14.133 °𝐶−1

𝑥 10−6

Figure 1. Sample Calculation of LTE Coefficient

Figure 1, above, shows the substitution of numbers to find the Linear

Thermal Expansion Coefficient. These numbers were taken from the data of first

trial for known metal, Copper.


Appendix C: Percent Error

To calculate the percent error, which is used to show how close or far the

Linear Thermal Expansion Coefficient was to the coefficient of Copper, the

formula below is used. Percent error equals the average of the measured value

minus the accepted value all divided by the accepted value, then multiplied by a

hundred.

𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟

= (𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑣𝑎𝑙𝑢𝑒) − (𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒)

(𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒)

× 100

Below, in figure 1, is the sample calculation to find the percent error of the

alpha coefficient.

𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟

=(13.091 °𝐶−1 × 10−6

) − (16.6 °𝐶−1 × 10−6)

16.6 °𝐶−1 × 10−6× 100

𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 = −21.14%

Figure 1. Sample Calculation for the Percent Error

Figure 1, above, shows the substitution to find the average percent error.

These numbers were taken from the average row of the Copper data table, figure

1, and the Linear Thermal Expansion Coefficient of Copper.


Appendix D: t-test

To calculate the t-value, or how many standard deviations away the two

means are, use the following equation 𝑡 equals the mean of the LTE of Copper

minus the mean of the LTE of the unknown metal divided by the square root of

copper’s standard deviation squared divided by the number of trials plus the

unknown metal’s standard deviation squared divided by the number of trials.

𝑡 =𝑥𝑐 − 𝑥𝑢

√𝑆𝑐2

𝑛𝑐+

𝑆𝑢2

𝑛𝑢

The following sample calculation, in figure 1, is how to find the t-valve for a

two sample t test.

𝑡 =13.0906 °𝐶

−1× 10

−6 − 10.8819 °𝐶−1 × 10−6

√1.61202

30 +1.23182

30

𝑡 = 5.9630

Figure 1. Sample Calculation of t-value

Figure 1, above, shows the substitution to find the t-value used to

determine the p-value, found from a statistical table or a formula. These numbers

were gathered from running a one variable statistical test on both data sets then

plugged into the formula.


Works Cited

Chang, Raymond. Chemistry. Boston: McGrawHill Higher Education, 2007. Print.

"Coefficients of Linear Thermal Expansion." Coefficients of Linear Thermal

Expansion. Web. 15 Apr. 2016.

<http://www.engineeringtoolbox.com/linear-expansion-coefficients-

d_95.html>.

"Expansion and Contraction." BBC, 2014. Web. 11 Apr. 2016.

<http://www.bbc.co.uk/bitesize/ks3/science/chemical_material_behaviour/

behaviour_of_matter/revision/2/>.

Hidnert, Peter. "Thermal Expansion of Some Nickel Alloys." (n.d.): n. pag. 2 Feb.

1957. Web. 10 Apr. 2016.

<http://nvlpubs.nist.gov/nistpubs/jres/58/jresv58n2p89_A1b.pdf>.

"Interstate 275 in Canton Buckles in Extreme Heat." WDIV. 2012. Web. 23 May

2016.

Licudine, Kylie Anne. "Lab Report." Lab Report. De La Salle University, 2016.

Web. 10 Apr. 2016. <http://www.academia.edu/8236887/Lab_report>.

Senese, Fred. "What Are Extensive and Intensive Properties?" General

Chemistry Online: FAQ: Matter:. N.p., 17 May 2015. Web. 15 Apr. 2016.

<http://antoine.frostburg.edu/chem/senese/101/matter/faq/extensive-

intensive.shtml>.

"Thermal Expansion." Thermal Expansion. NDT Resource Center, n.d. Web. 10

Apr. 2016.

http://nvlpubs.nist.gov/nistpubs/jres/58/jresv58n2p89_A1b.pdf

http://www.academia.edu/8236887/Lab_report


<https://www.nde-

ed.org/EducationResources/CommunityCollege/Materials/Physical_Chemi

cal/ThermalExpansion.htm>.

Wiggins, Arthur W., and Sidney Harris. The Joy of Physics. Amherst, NY:

Prometheus, 2011. Print.

https://www.nde-ed.org/EducationResources/CommunityCollege/Materials/Physical_Chemical/ThermalExpansion.htm



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