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The Identification of an Unknown Metal as Tantalum Using Linear Thermal Expansion and Specific Heat
By: Winston Balmaceda and Jonathon Banick
Hewlett-Packard [Company address]
Balmaceda-Banick
Problem Statement:
To determine if the identity of an unknown metal is tantalum using the intensive
properties of specific heat and linear thermal expansion.
Hypothesis:
t will be determined if the unknown metal is tantalum if the specific heat is
measured to 0.9% error and the linear thermal expansion is measured to 6.2% error.
Data Measured:
In this experiment the independent variables for specific heat were the initial
temperature of the water in which the metal was placed in measured in degrees
Celsius, the final temperature of the water in which the metal was placed in measured in
degrees Celsius, The initial temperature of the metal rod measured in degrees Celsius,
the mass of the metal measured in grams, and the volume of the water measured in ml.
these variables were used to calculate the specific heat of the metal measured in joules
over degrees Celsius. The independent variables for linear thermal expansion were
the initial size of the metal measured in millimeters, the initial and final temperatures of
the metal measured in degrees Celsius. And delta L or the change in length measured
in millimeters. These were used to calculate the alpha coefficient measured in 1/C.
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Balmaceda-Banick
Introduction
Tantalum, element 73, is a rare, lustrous metal that is highly corrosive resistant.
Due to its corrosive resistant properties it is used often in lab equipment, as a substitute
for Platinum, as well as in electronic components such as resistors. In addition to its
resistance to acid, Tantalum also has an extremely high melting point, high density, and
is rather resistant to oxidation. Its qualities like these that attribute to its ability to be
made into high maintenance products, such as vacuum furnaces.
The purpose of this experiment was to see if a group if unknown metal rods had
the same identity as the known metal rods, Tantalum. In order to determine if the
metals had the same identity, a linear thermal expansion experiment and a specific heat
experiment were conducted. Both LTE and specific heat are intensive properties which
means they are unique values for different elements. Due to this, the data yielded by
known metal rod compared to that of the unknown metal rods would be able to
determine if they were similar materials.
For example the first experiment that the metals would undergo was a specific
heat test, specific heat is an intensive property and each value is unique to a specific
element. In order to determine the specific heat of the metal, the rods were heated to
approximately 100℃ and placed within calorimeters, which had been constructed by
the researchers. The data collected was used to determine the specific heat of the
metals. The specific heats were compared to see if the metal rods were the same or
different.
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Subsequently the second experiment was a test to find the alpha coefficient of
the linear thermal expansion of the metals. To find the alpha coefficient of the known
and unknown metals they were heated to approximately 100℃ and placed in an
apparatus that measured the length of them as they cooled. From there the data
collected from the experiment was used to calculate the alpha coefficients of LTE, which
were compared to determine if the metal rods were the same or different.
As a result, after the collection of data had ended, the properties of the two metal
rods were compared. A 2 sample t-test was conducted and compared along with the
observable physical properties of the rods, as well as the percent error factors from the
specific heat and LTE data. The percent error values in the experiment were used to
compare how the data yielded by the experiment differed from the true value to observe
if there had been any constants between the data sets. The data gathered combined
however was not viewed as more defining than the t-tests conducted, and only helped
enhance the identification process.
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Balmaceda-Banick
Review of Literature
The purpose of this research was to determine the identity of an unknown metal
rod by comparing its intensive properties with the metal Tantalum. The properties used
in comparison are both chemical and physical properties. For this research, the physical
properties of specific heat and thermal expansion were used to identify the unknown
metal. Both of these properties, LTE (Linear Thermal Expansion) and specific heat are
intensive properties. Intensive properties are properties that are unique for each
element, but the sample size or amount of the element does not affect the properties
and they will remain the same despite so (Helmenstine). In order to measure these
properties heat must be added to the substance. Heat has a direct correlation with the
motion of the molecules within a substance. The motion is caused by the heat in a
substance, heat being energy transferred from one body to another through thermal
processes. This creates a correlation with heat and energy, as heat is lost or gain,
energy is lost or gained as well. Using this relationship between heat and energy the
identity of an unknown metal rod will be found in the course of research.
The intensive property of LTE is the change in length of a substance in response
to a change in temperature. As there is an increase in heat energy in a substance the
motion of the atoms in the substance will increase as well. With the motion of atoms
increased the separation between each atom increases thus causing an expansion of
the substance. Should the change in temperature,ΔT , not cause a change of phase
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Balmaceda-Banick
within the substance than the substance has undergone LTE, and a change in the
length,ΔL, of the substance will be present. The property, measured in mm/m/K can be
displayed as follows,
α= ΔLLi X ΔT
With Li representing the initial temperature and α representing the linear
expansion coefficient of the object in 10-6/ 1/k. The linear expansion coefficient of
Tantalum being 6.3 x 10-6 1/k. This LTE coefficient is a small expansion but is still
noticeable, unlike some elements which only expand fractions of mm. This effect can
also be measured with contraction, as the change is not limited to heating up the
substance but cooling it as well. This causes a decrease in energy of the substance.
(Eyland)
In manufacturing, LTE is important when creating fitted objects, the element used
may affect the product if it had low heat tolerance. For example, should screws within
an engine block have a high thermal expansion value than the parts will expand when
they heat up from engine, they may cause breakage when they expand, or become too
loose when they cool down and allow movement that would cause damage to an
engine. This is not only a loss for the company as well as a safety hazard for those who
bought the product.
The intensive property of Specific heat will be used to identify the unknown
metal, the value of specific heat is independent of the sample size while heat capacity
varies with sample size. The specific heat of Tantalum is 0.140 J/g˚K and the specific
heat for water (H20) is 4.184 J/g˚K. Knowing these values on can compare the specific
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Balmaceda-Banick
heats calculated during the course of experimentation and use them to help identify the
unknown metal sample. Specific heat is expressed as the following formula,
S= Qm x∆T
With specific heat, S, in J/g˚K. Q is the heat capacity measured in Joules, m the
mass measured in grams, and Delta T the change in temperature measured in kelvin.
(Eyland)
The Appalachian State University Undergraduate Physics Laboratory has
conducted an experiment on Thermal Expansion. In this experiment boiling water was
used to discover how heat affects the dimensions of different metals. The experiment
was conducted with a micrometer which would be pushed against by the metal rod
when it expands and would be used to measure Delta L. The experiment was
conducted with various substances to view the differences between elements. This is
applicable to our experiment as we will need to use thermal expansion to determine a
single element from unknown metal rods.
In manufacturing specific heat is important in making cookware, all cookware
must be able to endure high temperatures in order to be of use. It’s a given that some
metals will melt quickly, but can endure the heat of a stove top. However, many people
use ovens to store cookware and could accidently destroy their appliances should they
leave their cookware in the oven during baking.
David N. Blauch of Dadvison College created a setup in which one would explore
heat capacity and specific heat through Calorimetry. Calorimetry is the measuring of
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Balmaceda-Banick
changes in chemical reactions, physical changes and shifting of phases. A Calorimeter
is an insulated device in which on can boil water and add substances too. Since the
system is closed there is no off effect on the heat capacity or specific heat of the
material, as the set up explains. Using the Calorimeter the heat capacity of a metal was
found, as well as the specific heat of copper (Blauch).
Identifying metals through simply physical properties can be inconclusive alone.
Using thermal properties, such as LTE, which is intensive, one can identify an unknown
metal and state with confidence that they know the true identity of the metal. This aids
an industry where quality means everything, should the material not be up to the
standards of the company or the product not meet the standards of a community, there
would be backlash of immense proportions. Being able to surely test the identity of
metals using thermal properties is a great aide.
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Experimental Design
Linear Thermal Expansion
Materials:
(2) Unknown metal rods (C,D)(2) Tantalum metal rods (A,B)3 x 5 loaf pan Caliper (0.01 mm)(2) Linear thermal expansion apparatus
Thermometer (0. 1 C)⁰Hot plate Tongs TI n-spire cx graphing calculatorFan
Procedure:
1. Randomize the trials between rod A and B for the known, and rod C and D for the unknown rods between the linear thermal expansion apparatus to eliminate bias (see appendix A.)
2. Measure enough water to thoroughly cover the metal rod into a 3x5 loaf pan.
3. Place the loaf pan on a hotplate set on level 6, let sit until water is boiling at approximately 100 C (record using thermometer.)⁰
4. Use the caliper to record the initial length of the metal rods.
5. While keeping the water at a constant, boiling temperature, place the metal rod in the water and allow to sit for at least 7 minutes.
6. Using tongs take the metal out of the boiling water and place it in the linear thermal expansion apparatus.
7. Immediately mark the starting point on the linear thermal expansion apparatus.
8. Allow to sit in the apparatus for approximately 12 minutes or until the dial stops moving (record as delta T.)
9. Use the data collected to calculate the Alfa coefficient.
10.Repeat steps 4 – 9 for 15 trials with the unknown metal rods, and 15 trials with the tantalum rods.
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Diagram:
Figure 1. LTE Material Diagram
Above is a labeled image of the materials used in the LTE experimentation, the
most important being the LTE apparatus and the fan.
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Balmaceda-Banick
Specific Heat
Materials:
(2) Unknown metal rods (C,D)(2) Tantalum metal rods (A,B)50ml graduated cylinder LabQuest LabQuest temperature probe Electronic scale (0.0001g precision)
Thermometer (0.01g precision)3x5 Hot plate Hot plate Tongs TI n-spire cx graphing calculator
Procedure:1. Randomize the trials between rod A and B for the known, and rod C and D for
the unknown rods between the linear thermal expansion apparatus to eliminate bias (see appendix A.)
2. Record the mass of each of the four rods using a 0.0001g precision electronic scale, record.
3. Measure enough water to thoroughly cover the metal rod into a 3x5 loaf pan.
4. Set up the Lab Quest to measure temperature every second for eight minutes.
5. Place the water on a hot plate at a heat level 6 and leave it until it is boiling at approximately 100 C. (Measure using a thermometer.)⁰
6. Once the water is approximately 100 C gently place the metal rod into the water.⁰
7. Let the metal rod sit in the boiling water for 5 minutes. After the time is up, measure the temperature of the water once more, record the data. This serves as the internal temperature of the metal.
8. In the time you are waiting, measure 45 ml of water using a 50 ml graduated cylinder. Pour the water into the calorimeter that will be used. (Appendix a.)
9. 90 seconds before taking the metal rod out of the boiling water, start the Lab Quest in the calorimeter. This is allows the temperature probe to reach equilibrium with the water.
10.After the metal has been in the water for five minutes, take the, now hot, metal rod out of the boiling water and immediately place it into the calorimeter. (The temperature should be recorded.)
11.The temperature will be recorded for 5 minutes after placement into the calorimeter.
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Balmaceda-Banick
11.Repeat steps 3 – 11 for 15 trials with the unknown metal rods, and 15 trials with the tantalum rods.
Diagram:
Figure 2. Specific Heat Material Diagram
Above is a labeled image of the materials used in the specific heat
experimentation, the most important being the calorimeter and the labquest. The
electronic scale is not displayed in this image.
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Balmaceda-Banick
Data and Observations:
Table 1Known LTE Trials
Trial Rod ΔL(mm)
Initial Length
Initial Temp.
(Cº)
Final Temp.
(Cº)ΔT (Cº)
Alpha Coefficient
(mm) (°C -1 ) 1 B 0.06 127.97 98.2 21.7 76.5 6.333E-062 B 0.06 127.89 99.0 25.1 73.9 6.348E-063 A 0.06 127.66 99.0 25.1 73.9 5.830E-064 A 0.06 127.71 99.3 21.2 78.1 6.316E-065 B 0.06 127.93 99.3 25.5 73.8 6.673E-066 A 0.06 127.68 99.3 25.5 73.8 6.474E-067 A 0.05 127.64 99.1 23.1 76.0 5.567E-068 B 0.04 127.91 99.1 23.1 76.0 4.115E-069 A 0.05 127.67 99.1 24.3 74.8 5.236E-0610 B 0.05 127.96 99.1 24.3 74.8 5.224E-0611 A 0.04 127.89 99.2 21.3 77.9 4.015E-0612 B 0.05 127.34 99.2 21.3 77.9 5.040E-0613 A 0.05 127.89 98.4 23.2 75.2 5.511E-0614 A 0.03 127.89 98.4 24.4 74.0 3.593E-0615 B 0.04 127.34 98.4 24.4 74.0 3.820E-06
Table 1 above shows the data found during the experimentation for linear
thermal expansion. Each value except for ΔT, which is change in temperature, and the
Alpha coefficient, were found using experimentation. The final result was then recorded
with three significant figures after the decimal place.
Table 2Unknown LTE Trials
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Trial Rod ΔL (mm)
Initial Length
Initial Temp.
(Cº)
Final Temp.
(Cº)ΔT (Cº)
Alpha Coefficient
(mm) (°C -1 ) 1 C 0.04 127.35 98.6 25.3 73.3 4.283E-062 D 0.05 127.88 98.6 25.3 73.3 4.802E-063 C 0.05 127.40 99.0 23.8 75.2 5.532E-064 D 0.05 127.85 99.0 23.8 75.2 4.681E-065 C 0.05 127.31 99.3 23.9 75.4 5.209E-066 D 0.04 127.87 99.3 23.9 75.4 4.045E-067 C 0.05 127.31 99.0 24.5 74.5 5.272E-068 D 0.04 127.94 99.0 24.5 74.5 4.616E-069 C 0.04 127.31 99.2 24.5 74.7 4.416E-0610 D 0.05 127.94 99.2 24.5 74.7 5.232E-0611 C 0.05 127.31 99.3 24.6 74.7 5.468E-0612 D 0.04 127.94 99.3 24.6 74.7 4.499E-0613 C 0.04 127.31 98.4 24.7 73.7 4.370E-0614 D 0.04 127.76 98.4 24.6 73.8 4.136E-0615 C 0.03 127.34 98.4 24.6 73.8 3.618E-06
Table 2 above shows the data found during the experimentation for linear
thermal expansion for the unknown values. Each value except for ΔT, which is change
in temperature, and the Alpha Coefficient, were found using experimentation. The LTE,
or linear thermal expansion, for the unknown value is similar to the known value.
Table 3.Known Specific Heat Trials
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TrialInitial Temp.
(Cº)Equilibrium
Temp.(C°)
Change in Temp.
(Cº)Mass
(g)Specific
Heat(J/g x Cº)
Water Metal Water Metal Metal Water 1 24.7 98.3 27.4 2.70 70.9 67.8181 45 0.1062 24.3 98.3 27.7 3.40 70.6 67.6637 45 0.1343 22.9 98.5 26.1 3.20 72.4 67.8181 45 0.1234 26.0 98.8 28.7 2.70 70.1 67.6637 45 0.1075 21.0 98.8 24.2 3.20 74.6 67.8181 45 0.1196 23.3 98.5 26.6 3.30 71.9 67.6637 45 0.1287 24.2 98.6 27.1 2.90 71.5 67.8181 45 0.1138 27.0 98.6 29.8 2.80 68.8 67.6637 45 0.1139 22.7 97.5 25.6 2.90 71.9 67.8181 45 0.112
10 20.7 98.8 23.7 3.00 75.1 67.6637 45 0.11111 21.0 98.8 24.2 3.20 74.6 67.8181 45 0.11912 23.1 99.1 26.3 3.20 72.8 67.6637 45 0.12213 23.1 99.1 26.2 3.10 72.9 67.8181 45 0.11814 22.6 98.6 25.5 2.90 73.1 67.6637 45 0.11015 22.3 98.6 25.5 3.20 73.1 67.8181 45 0.122
The table above shows the values found during experimentation for specific heat
for the known metal rods. All values regarding the temperature, as well as the specific
heat values, were recorded to 3 significant figures.
Table 4. Unknown Specific Heat Trials
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Balmaceda-Banick
TrialInitial Temp.
(Cº)Equilibrium
Temp.(C°)
Change in Temp.
(Cº)Mass
(g)Specific
Heat(J/g x Cº)
Water Metal Water Metal Metal Water 1 29.7 98.1 31.8 2.10 66.3 67.8566 45 0.0882 25.5 98.1 28.7 3.20 69.4 67.9513 45 0.1283 27.8 98.8 30.1 2.30 68.7 67.8566 45 0.0934 28.7 98.8 31.0 2.30 67.8 67.9513 45 0.0945 25.0 98.7 28.0 3.00 70.7 67.8566 45 0.1186 27.1 98.7 29.8 2.70 68.9 67.9513 45 0.1097 25.6 98.6 28.6 3.00 70.0 67.8566 45 0.1198 25.8 98.6 28.4 2.60 70.2 67.9513 45 0.1039 22.4 98.5 25.9 3.50 72.6 67.8566 45 0.13410 22.4 98.5 25.6 3.20 72.9 67.9513 45 0.12211 22.5 98.5 25.5 3.00 73.0 67.8566 45 0.11412 22.6 98.5 25.6 3.00 72.9 67.9513 45 0.11413 22.4 98.7 25.4 3.00 73.3 67.8566 45 0.11414 22.8 98.7 25.9 3.10 72.8 67.9513 45 0.11815 23.6 98.0 26.6 3.00 71.4 67.8566 45 0.117
Table 4 above shows the values found during experimentation for specific heat
for the unknown metal rods.
Table 5Observations of Known Specific Heat Trials
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Balmaceda-Banick
Trial Calorimeter used Observations
1 3Ran along with the first trial of the day, trial 2,
shaken every 50 seconds by researcher 2
2 1Ran along with the first trial of the day, trial 1,
shaken every 50 seconds by researcher 2
3 2Ran along with trial 4, shaken every 50 seconds by
researcher 1
4 3Ran along with trial 3, shaken every 50 seconds by
researcher 1
5 1Ran along with trial 6, shaken every 50 seconds by
researcher 2
6 2 Ran along with trial 5, shaken every 50 seconds by researcher 2
7 3 Ran along with trial 8, shaken every 50 seconds by researcher 1
8 1 Ran along with trial 7 shaken every 50 seconds by researcher 1
9 3 Ran along with trial 10, shaken every 50 seconds by researcher 2
10 1 Ran along with trial 9, shaken every 50 seconds by researcher 2
11 2 Ran along with trial 12, shaken every 50 seconds by researcher 1
12 1 Ran along with trial 11, shaken every 50 seconds by researcher 1
13 3 Ran along with trial 14, shaken every 50 seconds by researcher 2
14 1 Ran along with trial 13, shaken every 50 seconds by researcher 2
15 1 shaken every 50 seconds by researcher 1
Table 5 above displays the observations for all 15 known trials for Specific heat
as well as the Calorimeter used.
Table 6
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Balmaceda-Banick
Observations of Unknown Specific Heat Trials
Trial Calorimeter used Observations
1 2Ran along with trial 2, shaken
every 50 seconds by researcher 2
2 3Ran along with trial 1, shaken
every 50 seconds by researcher 2
3 3Ran along with trial 4, shaken
every 50 seconds by researcher 1
4 1Ran along with trial 3, shaken
every 50 seconds by researcher 1
5 1Ran along with trial 6, shaken
every 50 seconds by researcher 2
6 3 Ran along with trial 5, shaken every 50 seconds by researcher 2
7 1 Ran along with trial 8, shaken every 50 seconds by researcher 1
8 2 Ran along with trial 7 shaken every 50 seconds by researcher 1
9 1 Ran along with trial 10, shaken every 50 seconds by researcher 2
10 3 Ran along with trial 9, shaken every 50 seconds by researcher 2
11 1 Ran along with trial 12, shaken every 50 seconds by researcher 1
12 3Ran along with trial 11, shaken
every 50 seconds by researcher 1
13 1Ran along with trial 14, shaken
every 50 seconds by researcher 2
14 3Ran along with trial 13, shaken
every 50 seconds by researcher 2
15 2 shaken every 50 seconds by researcher 1
The table above displays the observations for all 15 unknown trials for
Specific heat as well as the Calorimeter used.
Table 7
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Balmaceda-Banick
Observations of Known LTE Trials
Trial Apparatus used Observations
13
Ran by Researcher 1, left in hot water for 10 minutes, cooled for 12 minutes
23
Ran by Researcher 1, left in hot water for 10 minutes, cooled for 12 minutes. First trial of the day.
32
Ran by Researcher 2, left in hot water for 10 minutes, cooled for 12 minutes. First trial of the day.
42
Ran by Researcher 2, left in hot water for 10 minutes, Fanned constantly for 11 minutes.
52
Ran by Researcher 1 along with trial 6, Cooled for 12 minutes, and boiled for 11.
63
Ran by Researcher 1 along with trial 5, Cooled for 12 minutes, and boiled for 11.
72
Ran by Researcher 2, along with trial 8. Fanned within 5 minute intervals for 12 minutes.
86
Ran by Researcher 2, along with trial 9. Fanned within 5 minute intervals for 12 minutes.
91
Fanned by researcher 2, run by researcher 1. Ran with 10 and boiled for 10 minutes.
105
Fanned by researcher 2, run by researcher 1. Ran with 9 and boiled for 10 minutes.
113
Fanned by researcher 1, run by researcher 2. Ran with 12 and boiled for 8 minutes.
Trial Apparatus used Observations
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Balmaceda-Banick
122
Fanned by researcher 1, run by researcher 2. Ran with 11 and boiled for 8 minutes.
136
Ran by Researcher 2, along with trial 14. Fanned within 5 minute intervals for 12 minutes.
141
Ran by Researcher 2, along with trial 13. Fanned within 5 minute intervals for 12 minutes.
152
Ran by Researcher 1, fanned for 10 minutes, boiled for 8
The table above displays the observations for all 15 known trials for LTE
as well as the Apparatus used. Observations were recorded so it can be seen how the
different factors effected the data collected for each trial.
Table 8
Observations of Unknown LTE Trials
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Trial Apparatus used Observations
16
Ran by Researcher 1, left in hot water for 10 minutes, cooled for 12 minutes
21
Ran by Researcher 1, left in hot water for 10 minutes, cooled for 12 minutes
33
Fanned by researcher 2, run by researcher 1. Ran with 4 and boiled for 10 minutes.
42
Fanned by researcher 2, run by researcher 1. Ran with 3 and boiled for 10 minutes.
52
Fanned by researcher 2, run by researcher 1. Ran with 6 and boiled for 11 minutes.
66
Fanned by researcher 2, run by researcher 1. Ran with 5 and boiled for 11 minutes.
72
Fanned by researcher 2, run by researcher 1. Ran with 8 and boiled for 10 minutes.
86
Fanned by researcher 2, run by researcher 1. Ran with 7 and boiled for 10 minutes.
9
2
Ran by Researcher 2, left in hot water for 10 minutes, Fanned constantly for 11 minutes. Ran with 10
10
2
Ran by Researcher 2, left in hot water for 10 minutes, Fanned constantly for 11 minutes. Ran with 9
112
Ran by Researcher 2, along with trial 12. Fanned every 3 minute interval of 12 minutes
Trial Apparatus used Observations
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Balmaceda-Banick
123
Ran by Researcher 2, along with trial 11. Fanned every 3 minute intervals in 12 minutes
132
Ran by Researcher 1, along with trial 14. Fanned every 3 minute intervals in 12 minutes
146
Ran by Researcher 1, along with trial 13. Fanned every 3 minute intervals in 12 minutes
152
Ran by Researcher 1, left in hot water for 10 minutes, fanned for 12 minutes
The table above displays the observations for all 15 unknown trials for
LTE as well as the Apparatus used. Observations were recorded so it can be seen how
the different factors effected the data collected for each trial.
Data Analysis
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Balmaceda-Banick
Specific Heat Data Analysis and Interpretation
In this experiment, raw data was collected using 2 known and 2 unknown metal
rods. From there on they were tested upon using three calorimeters. In order to help
eliminate any bias in the experimental design, a random integer function was used to
randomize the calorimeter used in each trial. Following the randomization process, the
data was then collected using the lab quest unit and a temperature probe. The data
collection unit measured the heat of the water surrounding the metal, therefore it
measures the heat exothermically released from the metal. The heat of the water was
taken twice every second in order to monitor any sudden drops or spikes within the data
that could be caused by any interference by the system or its surroundings.
In order to analyze the data found for specific heat trials, a two sample t-test was
performed. In order to use the statistical test for our given data, the data must fit within
three given assumptions, which are SRS, independent trials and normality. This means
that each group is considered to be a distinct sample from a population, that the
responses in each group do not rely upon another group’s data, and that the
distributions of the variable of interests are normal under the null hypothesis. The first
assumption is met by the metals being only a portion of all of the metal tantalum. The
second assumption is met by the data being collected from more than one metal rod
keeping the two populations independent. The data also yields a normal distribution as
can be seen in figure 4 and figure 5.
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Table 9.
Percent Error Values of Specific Heat
Trial Numbers Known Specific Heat Percent Error Unknown Specific Heat Percent Error
2 24.482 37.2252 4.282 8.7423 12.352 33.6484 23.446 32.8615 14.937 15.9026 8.777 22.4437 19.569 15.0618 19.111 26.6989 20.017 4.453
10 20.603 13.12411 14.937 18.55212 12.635 18.55413 15.673 18.88514 21.15 15.72315 13.191 16.726
Averages: 16.344 19.906
The data table above displays the specific heat percent errors for the data. The
table seems to show that there is a vast difference between percent error values in the
two different variables and this might suggest that they are fairly different, but that
cannot be determined from raw data alone.
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Balmaceda-Banick
Table 10.
The Averages of the Specific Heat
Average Known Specific Heat Unknown Specific Heat
0.117 0.112The above table displays the two averages for specific heat. The difference in
these two though seemingly little can make a significant impact on whether the metals
will be identified as the same or different. The averages can be later referred to
depending on what the statistical tests suggest.
Figure 3. Box plots of Specific Heat Trials
The box plot above displays the values for the Specific Heat trials. The above
box plots slightly overlap and have close means, which suggests that the specific heat
of the metals may be the same.
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Figure 4. Unknown Specific Heat Normal Distribution Plot
Figure 5. Known Specific Heat Normal Distribution Plot
The two figures above show the normality of the raw data from the specific heat
trials for both known and unknown metal rods. The plots both show that the data is
normal as there aren’t any significant outliers from the linear function within the plot.
This means that our data is more reliable.
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Figure 6. Histogram of Data for Known Specific Heat Values
Figure 7. Histogram of Data for Unknown Specific Heat Values
The histogram above displays the frequency of values within the data. The
histogram in figure 6 is left skewed while the figure 7 histogram is more right skewed.
This shows that there is a difference between the two, but does not show the
significance of the difference.
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Balmaceda-Banick
Figure 8. Probability Graph of the Specific Heat Values
We are unable to reject the null hypothesis (H ⁰) because the p-value of 0.232 is
greater than the alpha level of 0.1. There is evidence suggesting that the unknown
metal has the same mean as the known metal, tantalum. There is a 23% chance that
the metals have a difference in the x̄ value of 0.0048 by chance alone.
The P value of 0.232 matches the expectations and predictions of the
researchers as it shows that our null hypothesis is true. Our Null hypothesis (H ⁰) of (H ⁰):
Mks = Mus, where Mks is the known values of specific heat and Mus is the unknown values,
suggests that the metals are the same. With the alternate hypothesis (Ha): Mks =/= Mus,
not being accepted, it is safe to assume that the metals are the same.
27
2-Sample t-testAlternate M1=/=M2
T 0.PVal 1.Df 28.x̄1 0.112333x̄2 0.112333sx1 0.012949sx2 0.012949
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Linear Thermal Expansion Data Analysis and Interpretation
In testing the Linear Thermal Expansion of the given metals, raw data was
collected using two known and two unknown metal rods. From there the rods were
placed into a linear thermal expansion apparatus. The apparatus used was randomized
using a random integer function to each trial in order to help eliminate bias. The rods
were then placed in the randomized apparatus in order to measure the change in
length, in millimeters, that the metal underwent during the process of cooling. The
temperatures in which the rods were exposed to varied day to day due to the constant
boiling of pots and different methods cooling used.
In order to analyze the data found during the linear thermal expansion trials, a
two sample t-test was performed. In order to use the statistical test for the trials data,
the data needed to fit within three given assumptions, which are SRS, independent trials
and normality. This means that each group is considered to be a distinct sample from a
population, that the responses in each group do not rely upon another group’s data, and
that the distributions of the variable of interests are normal under the null hypothesis.
The first assumption is met from the metals being only a portion of all of the metal
tantalum. The second assumption is met from the data being collected from more than
one metal rod keeping the two populations independent. The data also yields a fairly
normal distribution as can be seen in figure 10 and figure 11.
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Table 11.Percent Error for LTE
Trial Number Known LTE Percent Error Unknown LTE Percent Error
1 -0.527 32.012 -0.77 23.783 7.461 12.1894 -0.259 25.7065 -5.918 17.3216 -2.757 35.7937 11.641 16.3228 34.687 26.7269 16.893 29.899
10 17.081 16.95711 36.27 13.20812 19.993 28.58313 12.526 30.63914 42.974 34.34415 39.359 42.573
Averages: 15.244 25.737
The data table above displays the percent error for the Linear Thermal
Expansion trials; it can be displayed simply from the table that the values greatly differ.
It can be noted that despite the increasing difference of percent error, no precautions
taken by the researchers caused a positive change in the shape of the data.
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Table 12.Average of LTE
Average Known LTE Unknown LTE
5.340E-06 4.679E-06The table above shows the difference between the averages for the raw data for
Linear Thermal Expansion. It can be noted that while the difference may seem small, for
such a significant value being measured any small change is a significant difference.
Figure 9. Box plots for LTE Values
The above figure is a box plot of the data for Linear Thermal Expansion. The
means are farther away and the plots do not overlap that much, this suggests that the
LTE of the two metal rods are different.
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Figure 10. Unknown LTE Normal Distribution Plot
Figure 11. Known LTE Normal Distribution Plot
The two figures above show the normality of the raw data from the linear thermal
expansion trials for both the known and unknown values. The reliability of the known
LTE values is questionable due to a number of offset points within the plot. It suggests
that the known LTE trials are not as reliable as they could be.
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Figure 12. Histogram of Data for Known Values
Figure 13. Histogram of Data for Unknown Values
The histogram above displays the frequency of values within the data. The
histogram in figure 12 is right skewed while the figure 13 histogram is more left skewed.
This shows that there is a difference between the two, but does not show the
significance of the difference.
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Figure 14. Probability Graph of LTE
With the linear thermal expansion p-values it is possible to reject the null
hypothesis (H ⁰) because the p-value of 0.042 is smaller than the Alpha level of 0.1.
There is evidence suggesting that the unknown metal has a different LTE then the
known metal, tantalum. There is only a 4.2% chance that the metals have a difference
in the x̄ value of 0.661 by chance alone.
The P value of 0.042 contradicts the value received for specific heat. It suggests
that the Null hypothesis (H ⁰) of (H ⁰): MKl = MUl, where MKl is the known values of specific
heat and MUl is the unknown values, is rejected and the metals are different. With the
alternate hypothesis (Ha): MKl =/= MUl, being accepted, it is hard to determine if the metal
is the same as the known or if it differs. It can be considered through any flaws or errors
in the experiment. With specific heat trials it was an uncommon event for water to spill
from the calorimeter, and they were insolated systems that had a fitting hole for the
thermometer. While with the Linear Thermal Expansion trials the equipment used was
unreliable and inaccurate, even measuring the known data it gave off abnormal results,
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2-Sample t-testAlternate M1=/=M2
T 2.16634PVal 0.041577Df 21.6502x̄1 0.000005x̄2 0.000005sx1 0.000001sx2 5.65833E-7
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as displayed in the normality plots. This leans the reliability of the data onto the specific
heat values.
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Conclusion
The purpose of the experiment was to find if the hypothesis, at a percent error of
0.9% for specific heat and 6.2% linear thermal expansion is needed to identify the
unknown metal as tantalum is true. This hypothesis was not accepted, due to the
percent errors of both the specific heat values and the linear thermal expansion values.
The average percent error value for specific heat was 18.1; however the data suggested
that the metals were the same despite the percent error values being much larger than
the suspected value.
Furthermore the linear thermal expansion data does not provide insight to the
hypothesis as the average is larger than the suspected value and does not suggest that
the metals are the same. In the experiment, 2 groups of 2 metal rods were tested upon,
a known group of tantalum, and an unknown group of what was later found to be
tantalum. The two groups were tested using linear thermal expansion and specific heat.
During linear expansion trials the metal rods were heated to around 100℃ and cooled
down to room temperature where the change in length was recorded.
In addition specific heat trials were performed, the metals rods underwent
experimentation using calorimeters, the calorimeters were designed and built to keep
the heat within the system and keep the reaction a closed system. When testing the
metals rods were heated to about 100oc and then were quickly placed into a calorimeter
with 40 mL of water. The data was recorded using a lab quest device and a
temperature probe, the calorimeters were shaken often to ensure good data.
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The methods which data were collected for both conducted experiments may
have been flawed. For the specific heat trials, the volume of water in the calorimeter
varied slightly from trial to trial due to the droplets of water that remained after each trial.
Due to the accuracy of the numbers within specific heat, inaccurate numbers within the
calorimeter would affect the data as a whole. The data collection methods however did
not affect the normality of the specific heat trials and therefore the data was trust
worthy. The linear thermal expansion data methods were flawed as well; the LTE
apparatus were extremely sensitive units in the sense that any contact with them
caused change to the data. In addition tantalum has a low LTE and moves very little.
When calculating the alpha coefficient of linear thermal expansion each fraction of a
millimeter had drastic results on the data resulting in a higher percent error. These
caused an abnormal data resulting in untrustworthy results. Within both experiments the
temperature for the data often changed, the heating was inconsistent despite the values
all being above 90oC and the room temperature differed from day to day. The
temperature of the room may have been the cause for the abnormality of the linear
thermal expansion data, such as the drastic change between the minimum and
maximum values and the amount of outliers accounted for. The inconsistent
temperature of the boiling water and the loss of heat from the calorimeters because of
them not being a truly isolated system may have also effected the expansion and data
values for both experimentations.
Furthermore within both experiments temperature was a factor to the expected
outcome, Specific heat being the amount of heat energy a metal can hold, and linear
thermal expansion being the change in length of a metal due to heat. Both
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experimentations dealt with placing the metals into boiling water and allowing them to
cool down. Any inconsistencies of the temperature may have had an effect on the data
yielded within the experiments.
Further research can be conducted to find out the identity of an unknown element
using different intensive properties such as density, or melting point. While density
would be the most efficient, it was outside the parameters of the experiment, testing
melting point, which is intensive, would require much more expensive equipment and is
not as efficient as density.
Within industries, cost and quality are everything. Anything sold by a company
has become a liability of the company. Should a company find a metal supplier with
cheap wares, they would be wary but could test it to ensure it was the proper product. It
especially comes into factor when considering what tantalum can be used for, such as
hip replacements. Should the metal hip not be real tantalum it could cause harm to the
owner and lead to monetary loss from the company as well as legal repercussions.
In conclusion, the experimentation led to the result that the hypothesis that
required having a 0.9% error for specific heat and a 6.2% error for linear expansion to
identify an unknown metal as tantalum was rejected. With high percent errors from both
sets of data, the linear thermal expansion data was not used in identification as it had
abnormality in comparison to the specific heat. Using the specific heat the metal rod
was identified as being tantalum, using percent errors of 18% to 35%.
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Appendix A: Sample Calculations
In order to analyze the data, the equation below is used where SH, specific heat, equals
the product of 4.184, the specific heat of water, times ∆ tw, the change in temperature of
the water, times mw, the mass of the water, divided by the product of ∆ tm, the change in
tempature of the metal, times mm, the mass of the water.
SH=4.184 x ∆ tw x mw∆ tm xmm
Shown in figure 15 below is a sample calculation of using the equation for specific heat.
SH=4.184 x ∆ tw x mw∆ tm xmm
SH=4.184 J / g x℃ x 2.7℃ x 45 g70.9℃ x67.8181g
SH=0.106 J / g x℃
Figure 15. Specific heat calculation
Sample calculation of the specific heat from the first known trial.
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In order to analyze the data, the equation below is used where∝, Alfa, equals∆ L, the
change in length, divided by the product ofLi, the initial length, and∆ t , the chance in
temperature.
∝=∆L /(Li x ∆ t)
Shown in figure 16 below is a sample calculation of using the equation for linear thermal
expansion.
∝=∆L /(Li x ∆ t)
∝=0.06mm /(127.97mm x76.5℃)
∝=0.06mm /(9789.705mm x℃)
∝=6.333 x10−6℃−1
Figure 16. Linear thermal expansion
Sample calculation of linear thermal expansion from the first known trial.
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In order to analyze the data, the equation below is used wherePE, the percent error,
equals the summation oftruevalue, the actual value, minusobservedvalue, the results of
the trial, divided bytruevalue, the actual value, times 100.
PE= truevalue−observedvaluetruevalue
x 100
Shown in figure 17 below is a sample calculation of using the equation percent error.
PE= truevalue−observedvaluetruevalue
x 100
PE=6.3 x10−6℃−1−6.333 x10−6℃−1
6.3 x10−6℃−1 x100
PE=−.527%
Figure 17. Percent Error Calculation
Sample calculation of percent error from the first known trial of LTE.
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In order to analyze the data, the equation below, a 2 sample t-test, where T, the t value,
equals x̄1, the mean of the first data set, minus x̄2, the mean of the second data set,
divided by the square root of the summation of s1, the standard deviation of the first data
set squared, divided by n1, the number of sampled in the first data set, plus s2, the
standard deviation of the second data set squared, devided by n2, the number of
sampled in the second data set.
T=x̄1−¿ x̄2
√(s12/n1+s2
2/n2)¿
Shown in figure 18 below is a sample calculation of specific heat using the equation for
a 2 sample t-test?
T=x̄1−¿ x̄2
√(s12/n1+s2
2/n2)¿
T=0.1171J /g x℃−0.1123J / gx℃√¿¿¿
T=0.0048 J /g x℃√¿¿¿
T=1.2267495105146
Figure 18. Specific heat t-test
Sample calculation of the t value of specific heat from the first known trial.
Appendix b
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Materials:
*(7.5”) ¾” PVC pipe *(2) ¾” PVC end cap *(8”) Pool noodle Drill press
Chop saw 1/32” drill bit Ruler Black marker
*The materials listed above are sufficient for only 1 calorimeter in this experiment three were used
Procedure:
1. Using a ruler measure 7.5” of PVS and mark with a marker.
2. Using a ruler measure 8” of pool noodle and mark with a marker.
3. Cut the pipe at the marked line using a chop saw.
4. Cut the pool noodle at the marked line using a chop saw.
5. On one of the end caps use a drill press to drill a 1/32” hole slightly off-center.
6. Assemble the calorimeter by tightly putting the undrilled end cap one end of the PVC and slide the shaft into the pool noodle until there is ¼” of PVC exposed.
7. Gently attach the drilled end cap to the exposed shaft.
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