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TRANSCRIPT
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
Evers-Hickey-Maceroni 2
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
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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.
Evers-Hickey-Maceroni 4
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
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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
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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
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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.
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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).
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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.
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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
Evers-Hickey-Maceroni 11
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
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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
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Trial Jig Rod Observations(Known)
16 C 1 Good trans, ran Jig under cold water
17 A 1 Average transfer
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
24 A 1 Average transfer
25 A 2 Average transfer
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
Evers-Hickey-Maceroni 14
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
Evers-Hickey-Maceroni 15
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
16 A 2 Average transfer
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
Evers-Hickey-Maceroni 16
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.
Evers-Hickey-Maceroni 17
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
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Trial Margin of error
(%)
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
Evers-Hickey-Maceroni 19
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
Trial Margin of error
(%)
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
Evers-Hickey-Maceroni 20
Trial Margin of error
(%)
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
Evers-Hickey-Maceroni 21
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.
Evers-Hickey-Maceroni 22
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.
Evers-Hickey-Maceroni 23
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
Evers-Hickey-Maceroni 24
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.
Evers-Hickey-Maceroni 25
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.
Evers-Hickey-Maceroni 26
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.
Evers-Hickey-Maceroni 27
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
Evers-Hickey-Maceroni 28
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.
Evers-Hickey-Maceroni 29
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.
Evers-Hickey-Maceroni 30
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
Evers-Hickey-Maceroni 31
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.
Evers-Hickey-Maceroni 32
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.
Evers-Hickey-Maceroni 33
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.
Evers-Hickey-Maceroni 34
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.
Evers-Hickey-Maceroni 35
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.
Evers-Hickey-Maceroni 36
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.
Evers-Hickey-Maceroni 37
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