Identifying a Metal Using Specific Heat and Linear Thermal Expansion
Danny Havern and Andrew Rouditchenko
Macomb Mathematics Science Technology Center
Chemistry – 10B
Mrs. Hilliard, Mr. Supal, Mrs. Dewey
May 20, 2013
Havern – Rouditchenko 1
Table of Contents
Introduction ......................................................................................................... 2
Background ......................................................................................................... 4
Review of Literature: Specific Heat ...................................................................... 6
Review of Literature: Linear Thermal Expansion ................................................. 8
Problem Statement ............................................................................................ 10
Experimental Design ......................................................................................... 11
Data and Observations ...................................................................................... 15
Data Analysis and Interpretation ........................................................................ 25
Conclusion ........................................................................................................ 39
Appendix A ........................................................................................................ 42
Appendix B ........................................................................................................ 43
Appendix C ........................................................................................................ 44
Appendix D ........................................................................................................ 46
Appendix E ........................................................................................................ 48
Works Cited ....................................................................................................... 51
Havern – Rouditchenko 2
Introduction
Experimentation on metals is conducted daily to discover new and unique
properties that could be used to benefit society. If two similar metals were mixed
up in a laboratory, how would the scientists determine which metal was which?
Using the procedures from this experiment, the scientists would be able to
confidently identify the metals. The purpose of this experiment was to determine
if an unknown metal was the same or different compared to another metal using
the intensive properties of specific heat and linear thermal expansion. The known
metal was correctly identified from a previous experiment as Vanadium using the
property of density. In this experiment, the researchers correctly observed that
two unknown metal rods were not the same as the Vanadium rods.
The metals were compared by calculating intensive properties, or
properties that do not depend on the sample size, of the metals. The specific
heats and linear thermal expansion coefficients were found using various tools.
The researchers had limited background knowledge on the subject, therefore
extensive research was performed. This knowledge was used to design a
procedure for an experiment that would provide accurate results. The experiment
for specific heat involved observing several changes of temperatures including
for the metals and water while the experiment for linear thermal expansion
involved observing the change in length of the metal rods and the change in
temperature. For specific heat, the researchers used calorimeters that were
designed and built specifically for the dimensions of the metal rods. The
calorimeters were built from PVC piping, which is an excellent insulator.
Havern – Rouditchenko 3
Computer software was used to analyze the change in temperatures and
calculate the specific heat. The most important tools used for the linear thermal
expansion procedure were the expansion jigs. These jigs were built in previous
years by a skilled professional. These jigs measured the miniscule change of
length of the metal rods as they cooled. The researchers used statistical tools to
analyze the data, including two sample t-tests and percent error. The
experimental averages were compared to the known values for Vanadium. The
data proved to be precise as the researchers made an accurate conclusion.
Havern – Rouditchenko 4
Vanadium Metal Background
Vanadium, V, is a greyish white metal which is very hard, yet ductile. It
was first discovered in 1801 by Andrés Manuel del Rio, a Mexican chemist. He
later withdrew his claim, but Vanadium was rediscovered in 1830 by the Swedish
chemist Gabriel Sefstrôm. The metal was isolated and made pure in 1867 by Sir
Henry Enfield Roscoe by reducing Vanadium chloride with hydrogen (Gerhartz).
Mining for titanomagnetite ore is the first step in the process of extracting
the element to its pure state. Although this mineral only contains about 1.5
percent Vanadium, it is the most common mineral used for the production of
Vanadium. Vanadium is in such a small amount in the mineral, it is not included
in the formula, as seen below. It yields only 0.24 percent Vanadium from the total
extracted materials (Carlson).
Mg3[Si2O5](OH)4(s)+(MgAlFe)3[Si2O5](OH)4(s)→ (MgFe)2SiO4(s)+2MgOAl2O3(s)+5SiO2(s)
Titanomagnetite ore is reduced in flaming kilns and later melted in a
furnace. Olivine, (Mg,Fe)2SiO4, and Cordierite, 2MgOAl2O3+5SiO2 are produced
along with a slag of titanium and pig iron, the product of smelting iron ore with
high-carbon fuel, with high Vanadium content (Gerhartz). The separated molten
pig iron is then blown with oxygen to form a new compound which contains 12-24
percent Vanadium pentoxide (V2O5) (Processing). Pure Vanadium is produced by
reducing Vanadium pentoxide with aluminum powder.
Vanadium has many practical uses, but 99 percent of all Vanadium
produced worldwide was for use as a metal alloy (Vanadium). Pure Vanadium
and Vanadium oxides are combined with steel and titanium respectively in order
Havern – Rouditchenko 5
to increase their toughness, ductility, and strength. Vanadium has properties not
unlike most other metals. Vanadium’s elemental symbol is V and its atomic
number is 23. Vanadium has an atomic weight of 50.942 amu and a density of
6.1g/cm3, which is semi-dense. The specific heat of Vanadium is 0.485J/mol and
the thermal expansion coefficient of 8.4 x 10-6 K-1. The boiling point is 3653.15 K
(O’Leary). Vanadium is fairly similar to other metals. Compared to water, the
values for each property are very different and are not within the same range.
Vanadium is shaped just like every atom. Protons and neutrons build the
nucleus while electrons surround it in the electron cloud. Electrons fill each orbital
up to the 3d orbital, as seen below (Greenwood).
Figure 1. Vanadium Electron Orbital Diagram
Electron structure is relevant to this project because electrons deal with
energy. The electrons shifting between orbitals require energy, which in turn can
affect specific heat, and the size of the orbital directly affects thermal expansion
because a larger atomic radius results in expansion of the material (Greenwood).
Vanadium’s use as an alloy allows materials to increase strength while
decrease weight (Vanadium). Essentially, this characteristic makes Vanadium
useful and practical for making products such as buildings and automobiles.
Havern – Rouditchenko 6
Review of Literature: Specific Heat
Heat is the transfer of thermal energy (“Experiment 2”). At constant
pressure, this transfer is equal to enthalpy, the flow of heat energy (“Enthalpy”).
As heat energy flows in or out of a system to its surroundings, the temperature
changes, which is a measure of thermal energy. According to the kinetic
molecular theory, the kinetic energy of the molecules is directly proportional to
absolute temperature, therefore, as heat energy is added to a substance, the
kinetic energy of the molecules increases (“Kinetic-Molecular Theory”). If the
enthalpy is negative, the reaction is exothermic; heat energy is lost from the
system and is transferred to the surroundings (“Enthalpy”). If the enthalpy is
positive, the reaction is endothermic; heat energy is lost from the surroundings
and is transferred to the system. The relation between the change in heat energy
and the change in temperature is the specific heat. Specific heat is an intensive
property which means that the property is not dependent on sample size.
Specific heat is the heat energy required to raise the temperature of one gram of
a substance by one Kelvin; the unit of measure is J/g*K (Schreck). The
relationship between heat energy and specific heat is that the heat energy (Q) in
J is equal to the mass of the substance (m) in grams times the specific heat (c) in
J/g*K times the change in temperature ∆T in Kelvin (“Experiment 2”).
Q = mc∆T
Specific heat is important in the design of materials and products in
modern industry. Extensive research is done to determine the right metal to use
in products like heat exchangers and piping as they have different specific heats
Havern – Rouditchenko 7
and some insulate heat better than others (Violeta). Specific heat was a useful
property in this experiment because each element has a unique specific heat
(“Experiment 2”). The specific heats of both metals were determined using
calorimetry, which is the process of measuring the enthalpy change during a
reaction (“Calorimetry”). A calorimeter is a tool that involves an isolated system.
In an isolated system, no energy or mass can be transferred to the surroundings.
One experiment that the researchers found was to find the specific heat of
aluminum using a Styrofoam cup to simulate the calorimeter as it insulates heat
fairly well (“Experiment 2”). The metal was massed and then heated to 100° C in
boiling water. The mass and initial temperature of the water was also recorded.
The metal was placed into the cup full of water and the temperature was
recorded when equilibrium was reached, when the water and the metal reached
the same temperature and the graph of the temperature of the water reached a
stable plateau. According to the First Law of Thermodynamics which states that
energy is neither created nor destroyed but only converted from one form to
another, the heat energy lost by the hotter substance was equal to the energy
gained by the colder substance (“Kinetic-Molecular Theory”). The recorded data
was then manipulated in the specific heat formula to find the specific heat of the
metal. In another experiment, various metals were tested to find their specific
heats (Schreck). The calorimeter was an insulated thermos jar which may
insulate heat better than a cup. The other elements of the experiment were very
similar to others, the specific heat was found by placing the metal into water and
observing the masses and temperatures. Both experiments were simple in
design and easy to replicate.
Havern – Rouditchenko 8
Review of Literature: Linear Thermal Expansion
The property of linear thermal expansion refers to the tendency of a metal
to expand in length when heated. With small temperature changes, the thermal
expansion of regular linear objects is proportional to the change in temperature
(Furrer). In a system, the thermal energy increase within the atoms results in an
increase in atomic radius of those atoms, and thus a larger distance between the
atoms that is signified by an increase in the dimensions of the material (Sutara).
Thermal energy increase also results in more kinetic energy of the atoms. Larger
volume is a result of the atoms bouncing off of each other at a faster rate. These
materials expand volumetrically, but linear thermal expansion refers to the length.
This intensive property was used to determine the identity of the unknown
metal because each element expands and contracts at different rates, therefore
making linear thermal expansion applicable (Nave). Once the experimenters
compared their experimental linear thermal expansion coefficient to an accepted
value and calculated percent error, they were able to determine the identity of the
metal. The linear coefficient of thermal expansion (α) describes a change in
length of the metal per degree temperature change (Davis). To calculate the
linear thermal expansion coefficient, the experimenters used the equation where
the change in length, ΔL, of the metal rod is equal to the expansion coefficient, α,
times the initial length, Li, times the change in temperature, ∆T (Linear).
∆𝐿 = 𝛼 ∙ 𝐿𝑖 ∙ ∆𝑇
The experimenters calculated the coefficient by manipulating the equation
above and substituting in their values recorded during the experiment. The linear
Havern – Rouditchenko 9
thermal expansion coefficient is expressed in units defined as the reciprocal
temperature, ºC-1. Another common expression of the numerical coefficient is in
terms of 10-6/℃ (Gale). There are multiple linear thermal expansion labs, but this
experiment was a simple one. Some colleges and universities use steam
generators and a thermal expansion apparatus, but they do have similar parts of
the procedure. The procedure was applied to more than one rod and it was
repeated several times to eliminate outliers (Linear). The length temperature of
the metal rod was recorded. It was then placed in boiling water. Once the rod
reached equilibrium with the temperature of the water, the temperature was
recorded and the rod was taken out and the length was quickly measured
(Sutara). Using this data, the experimenters calculated the linear thermal
expansion coefficient of their metal.
Linear thermal expansion has applications in many fields, but it is most
often used in engineering. Designing bridges, buildings, and aircraft or spacecraft
requires the science of thermal expansion. The expanding and contracting of the
metal that makes these objects may cause some serious problems if not taken
into account. Bridges have expansion joints which allow the bridge to expand and
contract according to the temperature without collapsing (Gale). Rebar used in
buildings sidewalks could expand or contract at a greater rate than the concrete
and result in damage (Sutara). A simple application of linear thermal expansion
would be in the kitchen with a jar. If the lid was on a jar, running it under hot
water allowed the lid to expand, therefore making it easier to remove (Nave).
Essentially, linear thermal expansion is used in a multitude of fields.
Havern – Rouditchenko 10
Problem Statement
Problem Statement:
Can the material properties of specific heat and linear thermal expansion
be used to correctly identify an unknown metal as Vanadium?
Hypothesis:
The experimental data will provide an approximate value of specific heat
and the linear thermal expansion coefficient with which the experimenters will be
able to correctly identify the unknown metal as Vanadium with one percent error.
Data measured:
Specific Heat was measured in J/g°C. To calculate specific heat,
the mass of the rod and the mass of water was measured in grams, and the
initial and final temperatures of both the rod and water were measured in
degrees Celsius. Also, the specific heat of water, 4.184 J/g°C, was used in the
calculation of the specific heat value.
Linear Thermal Expansion Coefficient was measured in 10-6 °C-1. To
calculate the linear thermal expansion coefficient, the original and final lengths of
the metal rods were measured in millimeters. The initial and final temperatures of
the rods were measured in degrees Celsius.
Havern – Rouditchenko 11
Specific Heat Experimental Design
Materials:
Logger Pro
Logger Pro thermometer probe,
0.1ºC
Digital thermometer, 0.1ºC
TI-nspire CX graphing calculator
Electronic timer
Calorimeter
(2) Unknown metal rods
(2) Vanadium, V, rod
(2) 20.3 cm x 9.8 cm x 6.3 cm loaf
pan
Hotplate
Tongs
Scout Pro electronic scale, 0.1g
100 ml graduated cylinder
300 ml Beaker
Work Glove
Procedure:
1. Use the TI-nspire CX graphing calculator to randomize the order of the trials
and the order of the rods. See Appendix A for directions on how to randomize.
Make sure to assign fifteen trials to the known metal and fifteen trials to the
unknown metal.
2. Turn on and set up Logger Pro. Plug in Logger Pro thermometer probe and
adjust data gathering information. See Appendix B for further instructions.
3. Using the 300 ml beaker, pour 150 ml of water into the loaf pan and set it on
the hot plate. Turn the hot plate on and place the second loaf pan on top of the
first as a lid.
4. Fill the graduated cylinder with 50 ml of water. Record the mass of the water
as 50 g.
5. Pour this amount of water into each calorimeter. See Appendix C for
instructions on building the calorimeter.
6. Insert the Logger Pro thermometer probe through the hole in the lid of the
calorimeter and place the thermometer in the water of the calorimeter.
7. Mass the metal rod using the scale and record.
8. Lift the lid of the loaf pan with the work glove. Insert the digital thermometer
into the beaker. Continue to boil the water until it reaches a temperature of
about 100℃.
9. Insert the rod(s) into the boiling water. Make sure the entire rod is submerged
in the water and begin the electronic timer.
Havern – Rouditchenko 11
10. After two and a half minutes, stop the timer. Assume that the temperature of
the water is the same as the temperature of the metal. Insert the digital
thermometer into the boiling water around the rod or between the two rods.
Record this temperature as the initial temperature of the metals.
11. Begin collecting temperature measurements of the water in the calorimeter
using the Logger Pro temperature probe.
12. Use the tongs to carefully remove the metal from the boiling water and place
it in the calorimeter. Quickly, attach the top of the calorimeter. The Logger Pro
temperature probe should be in the calorimeter through the hole in the lid.
11. Data collection should stop after three minutes if the Logger Pro was set up
correctly. Save data into the appropriate file and record results into the data
table.
12. Remove the cap and empty out the calorimeter. Start a new set of data
collection on the Logger Pro by selecting the File Cabinet icon in the top right
corner of the screen.
13. Repeat steps 3 through 12 for each trial.
Diagram:
TI-nspire Calculator
Calorimeters
Beaker
Hot Plate
Timer
Work
Gloves
Lab Quest
Thermometer Probes
Loaf Pans
Digital Thermometer
Tongs
Scale
Metal Rods
Figure 2. Specific Heat Materials
Figure 2 above shows most of the materials used in the specific heat
experiment. Not pictured is the graduated cylinder.
Havern – Rouditchenko 12
Linear Thermal Expansion Experimental Design
Materials:
Digital thermometer 0.1ºC
TI-nspire CX graphing calculator
Electronic timer
Linear thermal expansion jig
(2) Unknown metal rods
(2) Vanadium, V, rods
(2) 20.3 cm x 9.8 cm x 6.3 cm loaf
pan
Hotplate
Tongs
300 ml Beaker
Thick gloves
Digital Calipers 0.1 mm
50 ml Spray Bottle
Procedure:
1. Use the TI-nspire CX graphing calculator to randomize the order of the trials
and the order of the rods. See Appendix A for directions on how to randomize.
Make sure to assign fifteen trials to the known metal and fifteen trials to the
unknown metal.
2. Using the 300 ml beaker, pour 150 ml of water into the loaf pan and set it on
the hot plate. Turn the hot plate on and set the second loaf pan on top of the
loaf pan with the boiling water to act as a lid. Boil the water until it reaches a
temperature of about 100℃.
3. Measure the length of the metal rod using the digital calipers and record as the
initial length.
4. Using the tongs, insert the rod into the boiling water. Make sure the entire rod
is submerged in the water and begin the electronic timer.
5. After two and a half minutes, stop the timer. Assume that the temperature of
the metal is equal to the temperature of the water. Lift the top loaf pan using
the work glove. Insert the digital thermometer into the loaf pan. Record this as
the initial temperature.
6. See Appendix D for information on how to use the linear thermal expansion jig.
Pull back the pin in the jig to allow the metal to be placed in the jig.
6. Carefully remove the rod from the boiling water using the work glove and tongs
and place the rod in the thermal expansion jig. Start the timer and move the
needle of the gauge to the starting position.
7. Use the spray bottle filled with water to speed up the cooling process. After
three minutes, stop the timer and record the change in length.
Havern – Rouditchenko 13
8. Use the digital thermometer to measure the temperature of each metal and
record it as the final temperature.
9. Repeat steps two through nine for each trial.
Diagram:
Digital Thermometer
Spray Bottle
Beaker
Loaf Pans
Hot Plate
Tongs
Timer
Scale
Towel
Expansion Jigs
Figure 3. Linear Thermal Expansion Materials
Figure 3 above shows the materials used on the linear thermal expansion
experiment. Not included in the picture are the metal rods and the work gloves.
An image of these materials can be seen in Figure 2. The picture contains a
towel that is not included in the materials because it is not mandatory, but it was
used to keep the work area dry during the experiment.
Havern – Rouditchenko 14
Data and Observations
Data:
Table 1
Vanadium Specific Heat Experiment Data
Trial Rod
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
B
B
A
A
B
A
A
B
A
B
B
A
A
B
B
Initial Temp.
(°C )
Metal Water
98.0
26.8
99.0
22.6
99.3
24.6
99.5
28.3
99.5
30.6
98.5
30.7
99.3
27.4
98.5
28.9
99.3
27.8
99.3
23.7
99.3
31.1
100.0
26.0
100.0
27.7
100.0
20.5
100.0
26.2
99.3
26.9
Equilibrium
Temp. (°C)
31.0
26.3
29.6
33.0
33.9
35.1
31.1
32.7
31.5
27.0
33.8
29.5
34.2
25.2
29.6
30.9
Change in
Temp. (°C )
Metal
-67.0
-72.7
-69.7
-66.5
-65.6
-63.4
-68.2
-65.8
-67.8
-72.3
-65.5
-70.5
-65.8
-74.8
-70.4
-68.4
Water
4.20
3.70
5.00
4.70
3.30
4.40
3.70
3.80
3.70
3.30
2.70
3.50
6.50
4.70
3.40
4.00
Mass
(g)
Metal Water
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
25.50
50
Specific
Heat
(J/g°C)
0.514
0.418
0.589
0.580
0.413
0.569
0.445
0.474
0.448
0.374
0.338
0.407
0.810
0.515
0.396
0.485
Table 1 above shows the data recorded during the trials for the specific
heat of the Vanadium rods. Most trials were performed two at a time as indicated
in the observations tables. The change in temp represents the change of
temperature for both the metal and the water. All temperature measuring devices
had three significant figures. The mass of the water has only one significant
figure because the measuring device was a graduated cylinder. The specific heat
was calculated with three significant figures after the decimal. For a sample
calculation, see Appendix D.
Havern – Rouditchenko 15
Table 2
Unknown Metal Specific Heat Experiment Data
Trial Rod
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
B
A
B
B
A
A
B
B
B
A
A
B
A
A
A
Initial Temp.
(°C )
Equilibrium
Temp. (°C)
Metal Water
99.0
22.6
99.0
22.6
99.0
20.4
99.3
17.0
99.0
19.9
99.5
27.7
99.5
21.5
99.4
26.0
99.4
21.2
99.5
19.2
99.4
26.6
99.9
23.7
99.4
19.6
99.9
18.0
99.3
16.8
99.4
21.5
26.3
26.1
23.9
20.8
23.8
30.7
25.5
30.4
24.8
23.3
29.9
27.1
23.3
22.0
21.0
25.3
Change in
Temp. (°C )
Mass
(g)
Metal Water
-72.7
3.70
-72.9
3.50
-75.1
3.50
-78.5
3.80
-75.2
3.90
-68.8
3.00
-74.0
4.00
-69.0
4.40
-74.6
3.60
-76.2
4.10
-69.5
3.30
-72.8
3.40
-76.1
3.70
-77.9
4.00
-78.3
4.20
-74.1
3.70
Metal Water
24.50
50
24.50
50
24.50
50
24.50
50
24.50
50
24.50
50
24.60
50
24.60
50
24.60
50
24.50
50
24.50
50
24.60
50
24.50
50
24.50
50
24.50
50
24.53
50
Specific
Heat
(J/g°C)
0.435
0.410
0.398
0.413
0.443
0.372
0.460
0.542
0.410
0.459
0.405
0.397
0.415
0.438
0.458
0.430
Table 2 above shows the data recorded during the trials for the specific
heat of the unknown metal rods. Most trials were performed two at a time as
indicated in the observations tables. The change in temp represents the change
of temperature for both the metal and the water. All temperature measuring
devices had three significant figures. The mass of the water has only one
significant figure because the measuring device was a graduated cylinder. The
specific heat was calculated with three significant figures after the decimal. For a
sample calculation, see Appendix D.
Havern – Rouditchenko 16
Table 3
Vanadium Linear Thermal Expansion Data
Trial
Rod
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
B
A
B
A
B
B
A
B
B
A
A
A
B
A
A
Original Change in
Length
Length
(mm)
(mm)
127.31
127.39
127.34
127.41
127.29
127.37
127.36
127.28
127.37
127.44
127.33
127.4
127.37
127.44
127.37
127.36
0.05
0.06
0.05
0.04
0.04
0.05
0.04
0.05
0.05
0.05
0.05
0.05
0.06
0.06
0.06
0.05
Initial
Temp.
(°C )
98.7
99.3
99.3
98.3
98.9
98.9
98.9
98.6
98.6
98.9
98.6
98.6
99.0
99.0
98.7
98.8
Final
Temp.
(°C )
30.1
29.9
30.5
27.7
28.4
29.2
28.4
30.4
31.5
29.2
29.5
30.6
29.7
30.0
30.1
29.7
Change
Alpha
in
Coefficient
Temp.
(10-6°C-1)
(°C)
68.6
5.725
69.4
6.221
68.8
5.707
70.6
4.447
70.5
4.457
69.7
5.632
70.5
4.455
68.2
5.760
67.1
5.850
69.7
5.629
69.1
5.683
68.0
5.772
69.3
6.231
69.0
6.255
68.6
6.295
69.1
5.608
Table 3 above shows the data recorded during the trials for the linear
thermal expansion coefficient of the Vanadium rods. Most trials were performed
two at a time as indicated in the observations tables. The change in length
represents the change of length of the metal as it cooled down in the jig after
being heated. The dial on the jig was accurate to two significant figures. The
linear thermal expansion coefficient was calculated with three significant figures
after the decimal. For a sample calculation, see Appendix D.
Havern – Rouditchenko 17
Table 4
Vanadium Linear Thermal Expansion Data
Trial
Rod
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
A
B
B
A
B
A
A
B
B
A
B
A
B
A
A
Original Change in
Length
Length
(mm)
(mm)
128.28
128.81
128.88
128.26
128.77
128.40
128.30
128.87
128.79
128.28
128.81
128.31
128.76
128.29
128.35
128.54
0.07
0.07
0.07
0.08
0.08
0.08
0.08
0.07
0.07
0.08
0.08
0.08
0.07
0.08
0.08
0.08
Initial
Temp.
(°C )
100.2
100.2
101.0
100.4
100.4
100.4
100.8
100.8
100.4
100.4
100.5
100.5
100.2
100.2
101.0
100.5
Final
Temp.
(°C )
30.2
28.5
26.5
26.2
29.7
29.1
28.1
28.5
28.2
26.9
26.3
26.0
25.1
24.5
26.1
27.3
Change
Alpha
in
Coefficient
Temp.
(10-6°C-1)
(°C)
70.0
7.795
71.7
7.579
74.5
7.290
74.2
8.406
70.7
8.787
71.3
8.738
72.7
8.577
72.3
7.513
72.2
7.528
73.5
8.485
74.2
8.370
74.5
8.369
75.1
7.239
75.7
8.238
74.9
8.322
73.2
8.082
Table 4 above shows the data recorded during the trials for the linear
thermal expansion coefficient of the unknown metal rods. Most trials were
performed two at a time as indicated in the observations tables. The change in
length represents the change of length of the metal as it cooled down in the jig
after being heated. The dial on the jig was accurate to two significant figures. The
linear thermal expansion coefficient was calculated with three significant figures
after the decimal. For a sample calculation, see Appendix D.
Havern – Rouditchenko 18
Observations
Table 5
Vanadium Specific Heat Observations
Trial
Rod
Date
Cal.
1
B
15-Apr
1
2
B
17-Apr
1
3
A
17-Apr
2
4
A
17-Apr
1
5
B
17-Apr
2
6
A
17-Apr
1
7
A
17-Apr
1
8
B
17-Apr
2
9
A
17-Apr
1
10
B
17-Apr
2
11
B
17-Apr
2
12
A
17-Apr
1
Observations
First day of experimenting. Pre trials were run
and the procedures were adjusted.
Both researchers were present. This trial was run
by itself. The window was open, and there was
cold air blowing in making the room very chilly.
The researchers forgot to mass the metal rod and
it was massed after being removed from the
calorimeter.
The researchers forgot to mass the rod again. It
was once again massed after being removed
from the calorimeter. The water from the loaf pan
was emptied.
The researches decided to start putting two
metals in the loaf pan during the heating process
due to time constraints. This trial was done with
trial five.
The Vanadium rod was dropped on the floor. The
water in the loaf pan was emptied and refilled.
This trial was performed along with trial eight.
The temperature of the water deviated slightly on
the thermometer about 0.1 – 0.3°C.
This trial was performed along with trial 10. The
Vanadium rod dropped in and out of the water
two times in the loaf pan while attempting to pick
it up using the tongs. The temperatures of the
water in the calorimeters varied.
The water in the loaf pan was emptied and
refilled after removing the metal rods from it. The
amount was slightly more than 150 ml.
This trial was performed with trial 11. The
researchers had difficulty placing the metal rod
into the calorimeter.
The water in the loaf pan was emptied and
refilled after removing the metal rods from it.
The researchers began to transfer the metal rods
slightly late and this rod spent more time than
usual in the loaf pan.
This trial was performed with trial 14. The
researchers began to transfer the metal right on
time.
Havern – Rouditchenko 19
Trial
Rod
Date
Cal.
13
A
17-Apr
1
14
B
17-Apr
2
15
B
17-Apr
2
Observations
The metal rod slipped from the tongs while being
transferred to the calorimeter and touched one of
the researcher's hands. The calorimeter was
spilled and this trial had to be redone. The
redone trial was performed as usual.
The transfer of the metal to the calorimeter from
the loaf pan was very smooth. The water in the
loaf pan was emptied and refilled after removing
the metal rods from it.
This was the final trial of the day. The water
poured into the calorimeter was slightly more
than 50 ml. After this trial the researchers began
to clean up.
Table 5 shows the observations recorded during the trials for the specific
heat of the Vanadium rods. The table indicates observations for each trial, the
date, which rod and calorimeter were being used, and which trials were done
together. All trials for the A rod were done with calorimeter 1and all trials for the B
rod were done with calorimeter 2.
Table 6
Unknown Metal Specific Heat Observations
Trial Rod
Date
Cal.
1
B
18-Apr
2
2
A
18-Apr
1
3
B
18-Apr
2
Observations
Both researchers were present during the second
day of trials. This trial was performed along with trial
two. The water in the loaf pan spent a longer
duration of time than usual on the hot plate. The
wrong data collection method was set up on logger
pro, but the data was still valid. The researches
forgot to measure the initial temperature of the
metal; however they measured the temperature of
the water in the loaf pan after removing the metal
rods from it.
The metal rod was dropped before being placed in
calorimeter. The water in the loaf pan was emptied
and refilled after removing the metal rods from it.
This trial was performed with trial five. This trial was
redone as the original value gave a large percent
Havern – Rouditchenko 20
Trial Rod
Date
Cal.
4
B
18-Apr
2
5
A
18-Apr
1
6
A
18-Apr
1
7
B
18-Apr
2
8
B
18-Apr
2
9
B
18-Apr
2
10
A
18-Apr
1
11
A
18-Apr
1
12
B
18-Apr
2
13
A
18-Apr
1
14
A
18-Apr
1
15
A
18-Apr
1
Observations
error. The metal rods were touching in the loaf pan
while being heated.
This trial was performed with six. The metal rod
touched the side of the calorimeter before being
dropped in it.
The metal wiggled around in the tongs while being
transferred from the loaf pan to the calorimeter and
the process took more time than usual.
The metal spent a longer time in the loaf pan
because the transfer of rod A took longer than usual.
This trial was performed with trial 10. The researcher
had difficulty placing the top of the loaf pan on. The
metal wobbled in the tongs as it was being
transferred.
This trial was performed with trial 11. The metal
dropped on the floor before boiling. A few seconds
went by as the rods were out of the boiling water and
placed into the calorimeters.
This trial was performed with trial 13. The metal was
dropped while attempting to put in the calorimeter.
The metal touched the outside edge of the
calorimeter while being transferred.
The researcher had difficulty placing the metal in the
loaf pan. The position of the rod had to be adjusted.
This trial was performed with trial 14. The exteriors
of the calorimeters were slightly warm; this may
have been from the heat of the hot plate.
The window in the experimenting area was opened.
The metal was removed from the loaf pan right on
time.
This trial was done along with a redo of trial 3 as
well; the original value gave a large percent error.
This was the last trial of the day.
Table 6 shows the observations recorded during the trials for the specific
heat of the unknown metal rods. The table indicates observations for each trial,
the date, which rod and calorimeter were being used, and which trials were done
together. All trials for the A rod were done with calorimeter 1and all trials for the B
rod were done with calorimeter 2.
Havern – Rouditchenko 21
Table 7
Vanadium Linear Thermal Expansion Observations
Trial Rod
Date
Jig
1
B
19-Apr
S3
2
A
19-Apr
S1
3
B
19-Apr
S3
4
A
19-Apr
S1
5
B
19-Apr
S3
6
B
19-Apr
S3
7
A
19-Apr
S1
8
B
19-Apr
S3
9
B
19-Apr
S3
10
A
19-Apr
S1
11
A
19-Apr
S1
12
A
19-Apr
S1
Observations
Both researchers were present during the third day of
trials. The linear thermal expansion jigs S1 and S3
were used and they provided a metric measurement.
The heat on the hot plate was turned to 10.
The window was closed and it was slightly humid in
the room.
This trial was performed with trial four. The length of
the rod was measured on the rag as opposed to the
face of the table. The metal was removed from the
loaf pan slightly later than usual, at about three
minutes.
The cooling time of the metal was cut short to about
two minutes. The water in the loaf pan was refilled.
This trial was performed with trial seven. The length
of the metal was taken twice as the first
measurement was an odd value. The calipers were
used on the table.
This trial was performed with trial 10. The length was
measured on the table. The orientation of the metals
was switched when they were placed in loaf pan.
The transfer of the metals began slightly late at about
three minutes. The temperature of the spraying water
was 29 ºC; cold water was added to drop it to 25 ºC.
The water in the loaf pan was emptied and refilled.
This trial was performed with trial 11. The metal was
measured on the table. The metals were placed
correctly in the center of the loaf pan. The water in
the loaf pan seemed hotter to the researcher that
placed metal in the water. The metal spent about
three minutes in the loaf pan.
This trial was performed with trial 12. It took several
attempts to get an accurate measure. The
researchers began feel very hot and took a brief
break.
The temperature of the water in the loaf pan was
measured slightly earlier than usual. More than three
minutes passed as the researchers waited for the
metal to change length.
The temperature reading of the water in the loaf pan
started late the metal was placed in the jig after about
three minutes.
The length of the metal was measured on the table.
The water in the spraying bottle was at 28.7º C.
Havern – Rouditchenko 22
Trial Rod
Date
13
B
19-Apr
14
A
19-Apr
15
A
19-Apr
Jig
Observations
This trial was performed with trial 13. The rods were
S3
touching in the loaf pan.
The metal was sprayed with a larger quantity of water
S1 than usual to cool it down. The change in length of
the metal was greater than usual.
This trial was performed with trial one. The tips of the
rods were touching again in the loaf pan. The rods
S1 were dropped crooked in the jigs and then
straightened. The transfer of the metals began right
on time. This was the last trial of the day.
Table 7 shows the observations recorded during the trials for the linear
thermal expansion of the Vanadium rods. The table indicates observations for
each trial, the date, which rod and jig were being used, and which trials were
done together. During these trials, all trials for the A rod were done with jig S1
and all trials for the B rod were done with jig S3.
Table 8
Vanadium Linear Thermal Expansion Observations
Trial Rod
Date
Jig
1
A
22-Apr
S1
2
B
22-Apr
S3
3
B
22-Apr
B1
4
A
22-Apr
A1
5
B
22-Apr
B1
Observations
Both researchers were present during the third day of
trials. The trial was performed along with trial two. It
was noticed that the both metal rods were magnetic to
the hot plate.
The metals were very hard to remove from the loaf
pan using the tongs as they were attracted to the
surface of the plate. The metal rod was put in the jig
late at about three minutes. The rod slipped from the
slot while trying to place it in the jig.
The jigs were switched with another group, but one
had a faulty dial. The dial was replaced and both jigs
provided a metric measurement. The transfer of the
metal from the loaf pan to the calorimeter began late.
This trial was performed along with trial three. The
water in the loaf pan was refilled before the metals
were heated.
This trial was performed along with trial six. The loaf
pan was adjusted on the hot plate, while the rods stuck
to the bottom. The researchers began to indicate the
Havern – Rouditchenko 23
Trial Rod
Date
Jig
6
A
22-Apr
A1
7
A
22-Apr
A1
8
B
22-Apr
B1
9
B
22-Apr
B1
10
A
22-Apr
A1
11
B
22-Apr
B1
12
A
22-Apr
A1
13
B
22-Apr
B1
14
A
22-Apr
A1
15
A
22-Apr
A1
Observations
original length of the metals with a marker as opposed
to the previous method because the dial moved if
touched.
The water in the spray bottle was refilled. The water
seemed quite cold. The amount of time that the metal
spent in the jig was longer than usual.
This trial was performed along with trial eight. The
water in loaf pans was refilled. The metals were
touching while being heated in the loaf pan.
The metal rod was transferred to the jig late and the
cooling time was less than usual. The Metals had to be
adjusted when placed into the jigs.
This trial was performed along with trial 10. The
cooling time was about three and a half minutes as the
researchers forgot to pay attention to the stopwatch.
The jigs were very wet and the wood was soaked with
water.
The water used in the spray bottle was refilled again.
This trial was performed along with trial 12. The water
in the loaf pan was refilled and slightly exceeded 150
ml.
The metal was dropped back into loaf pan while being
transferred into the jig. The metals received a heavy
dousing of water from the bottle during the cooling
process.
Trial performed with trial 14. The water in the loaf pan
was refilled. The researcher had some trouble with the
timer and the transfer of the metals started late.
The water in the spray bottle was refilled with water at
approximately room temperature. The metal had to be
adjusted when placed into the jig.
This trial was performed along with a redo of trial
three. The top of the loaf pan put was put on after
about 10 seconds from when the metals were placed
in the water. This was the final trial of the day.
Table 8 shows the observations recorded during the trials for the linear
thermal expansion of the unknown metal rods. The table indicates observations
for each trial, the date, which rod and jig were being used, and which trials were
done together.
Havern – Rouditchenko 24
Data Analysis and Interpretation
Several statistical graphs and tests were used to analyze the data from
the experiment. The data was both quantitative and continuous as the data was
numerical and could take any value in a range. The data measured included
values for specific heat and linear thermal expansion coefficients of the metals.
These values were calculated using data collected from the experiment including
temperatures, masses, and lengths of the metal rods. The two sample t- test was
used for the analysis. This test was appropriate as two different samples were
being compared. The first sample was the data for the Vanadium rods and the
second sample was the data for the unknown metal rods. Two t- tests were
conducted; one for specific heat and the other for linear thermal expansion.
Before the tests were conducted, several assumptions had to be checked.
If these assumptions are not met for a two sample t- test, the validity of the
results of the test is questionable. The first assumption was that both samples
were simple random samples taken from two distinct populations. This condition
was met because the trials were randomized. The samples were also
independent because the metal rods were placed in separate calorimeters and
jigs. Another assumption was that both the means and the standard deviations of
the populations were not known. In this experiment, the researchers did not know
the true values of these parameters. The final assumption was that both samples
were normally distributed. According to the Central Limit Theory, if the simple
size is 30 or greater the samples are normal. Unfortunately, each sample size
Havern – Rouditchenko 25
was only 15 trials which meant that the distributions of the samples had to be
checked.
The purpose of conducting the two sample t- test was to compare two
different sample means and check to see whether the null hypothesis was true or
not. The alternate hypothesis was checked against a null hypothesis. In this
scenario, the null hypothesis was that the known and the unknown metal were
the same while the alternate hypothesis was that the metals were different. For
the specific heat experiment, the null hypothesis was that the mean specific heat
for the known metal, 𝑆𝑋̅1 was the same as the mean specific heat for the
unknown metal, 𝑆𝑋̅2. The alternative hypothesis was that the mean specific heats
were different.
Ho: 𝑆𝑋̅1 = 𝑆𝑋̅2
Ha: 𝑆𝑋̅1 ≠ 𝑆𝑋̅2
For the linear thermal expansion experiment, the null hypothesis was that
the mean linear thermal expansion coefficient for the known metal was the same
as the mean linear thermal expansion coefficient for the unknown metal. The
alternative hypothesis was that the mean coefficients were different.
Ho: 𝐿𝑋̅1 = 𝐿𝑋̅2
Ha: 𝐿𝑋̅1 ≠ 𝐿𝑋̅2
Before the t-Test was completed, percent error was calculated for each
trial in order to check the validity of the results. Percent error was calculated
while the experiment was being run in order for the researchers to make sure the
results were consistent.
Havern – Rouditchenko 26
Table 9.
Vanadium Specific Heat Percent Error
Trial
Experimental
Value
True
Value
Percent
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
0.514
0.418
0.589
0.580
0.413
0.569
0.445
0.474
0.448
0.374
0.338
0.407
0.810
0.515
0.396
0.486
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
6.036
-13.911
21.344
19.552
-14.908
17.393
-8.231
-2.313
-7.689
-22.793
-30.273
-16.023
67.096
6.286
-18.307
0.217
Table 9 above shows the percent error data for each trial of the specific
heat experiment for Vanadium. The averages are indicated on the bottom of the
table. The specific heat of Vanadium was used as the true value. The percent
error was then calculated for each trial and shows the percentage of difference
between the experimental and true value. See Appendix E for a sample
calculation. Negative percent error means that the value for specific heat was
lower than the true value. The percent error was constantly being checked during
the experiment to make sure that the results were consistent. Any trials that were
not consistent with the other trials were redone as indicated in the observations
tables. The range of the data was from about 67% to -30%. This large range and
Havern – Rouditchenko 27
variability indicates a flaw or an error in the experimental design. Using the
absolute values of each trial’s percent error, the average percent error of this
data would have been about 18%. This method was not used because percent
error is actually calculated using the true values. The actual average percent
error was calculated to be 0.217%. Because this value is less than one percent, it
is excellent evidence that the metal is actually Vanadium.
Table 10.
Unknown metal Specific Heat Percent Error
Trial
Experimental
Value
True
Value
Percent
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
0.435
0.410
0.398
0.413
0.443
0.372
0.460
0.542
0.410
0.459
0.405
0.397
0.415
0.438
0.458
0.430
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
0.485
-10.397
-15.473
-17.949
-14.775
-8.694
-23.231
-5.221
11.812
-15.385
-5.271
-16.404
-18.110
-14.401
-9.598
-5.563
-11.244
Table 10 above shows the percent error data for each trial of the specific
heat experiment for the unknown metal. At the bottom of the table, the averages
of the data can be found. The researchers used the specific heat of Vanadium as
the true value while calculating percent error. The percent error was calculated
Havern – Rouditchenko 28
for each trial. Negative percent error shows that the value for specific heat was
lower than the true value. The range of the data was only from about -23% to
about 12%. This range is significantly smaller than the range of the data for
Vanadium. The average percent error was calculated to be approximately
-11.244%. This value is somewhat far away from the average percent error for
Vanadium, suggesting that the metals are different.
Table 11.
Vanadium Linear Thermal Expansion Coefficient Percent Error
Trial
Experimental
Value
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
5.725
6.221
5.707
4.447
4.457
5.632
4.455
5.760
5.850
5.629
5.683
5.772
6.231
6.255
6.295
5.608
True
Value
Percent
Error
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
-31.844%
-25.939%
-32.058%
-47.061%
-46.936%
-32.951%
-46.965%
-31.428%
-30.353%
-32.988%
-32.348%
-31.291%
-25.821%
-25.539%
-25.064%
-33.239%
Table 11 above shows the percent error data for each trial of the linear
thermal expansion coefficient for Vanadium. The average results are found at the
bottom of the table. The true value is the linear thermal expansion coefficient of
Vanadium. The percent error column shows the percent difference between each
Havern – Rouditchenko 29
experimental value compared to the true value. All of these values are negative
which means each experimental value was lower than the true value. The range
of the data is from about -46% to about -25%. This small range suggests that
there is small variability in the data. The average percent error was calculated to
be approximately -33.239%. This indicates a flaw in the experimental design. The
percent error should have been close to zero, or very little difference like the
average of the specific heat trials.
Table 12.
Unknown Metal Linear Thermal Expansion Coefficient Percent Error
Trial
Experimental
Value
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Avg.
7.795
7.579
7.290
8.406
8.787
8.738
8.577
7.513
7.528
8.485
8.370
8.369
7.239
8.238
8.322
8.082
True
Value
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
Percent
Error
-7.197
-9.770
-13.209
0.073
4.611
4.029
2.106
-10.561
-10.381
1.010
-0.355
-0.369
-13.822
-1.933
-0.932
-3.780
Table 12 above shows the percent error data for each trial of the linear
thermal expansion coefficient for the unknown metals. The averages are shown
at the bottom of the table. The true value is the linear thermal expansion
Havern – Rouditchenko 30
coefficient of Vanadium. The percent error column shows the percent difference
between each experimental value compared to the true value. The range of the
data is from about -14% to about 4% which suggest small variability ion the data.
The average percent error was calculated to be approximately -3.780%. This
average is conflicting because it is closer to the true value than the Vanadium
used in the experiment. This is more evidence of an experimental design flaw.
The difference between the average coefficients of the two metals is 29.459,
which is a significant amount.
Figure 4.Normal Probability Plot of Vanadium Specific Heat
Figure 4 above is the normal probability plot of the Vanadium specific heat
data. The graph suggests that the data is normally distributed because the data
points are relatively close to the expected z-value line. The first and last data
points stray far from the expected z-value, so they might be considered as
outliers. Most of the data is near or on the line, so they data as a whole can be
confirmed as normally distributed.
Havern – Rouditchenko 31
Figure 5. Normal Probability Plot of the Unknown Metal Specific Heat
Figure 5 above is the normal probability plot of the unknown metal specific
heat data. The last data point varies far from the expected z-value line, so it may
be considered as an outlier. Overall, the data is fairly close to the expected zvalue which suggests that the data of the unknown metal for specific heat is
normally distributed.
Figure 6. Specific Heat Data Box Plot
Figure 6 above is the box plot of the specific heat data. The overlapping of
the box advocates evidence that the specific heats of the two metals are the
same or very similar. The unknown metal data appears to be precise because it
is contained in such a small area or box. The Vanadium data has high variability
because it occupies a larger area. The drastic differences between the two
Havern – Rouditchenko 32
boxes, very low and very high variability, suggest that the data may be unreliable.
The outlier of the unknown metal data also suggests the data may be skewed. In
general, the validity of the data is questionable based on high variability and
outliers of the box plot.
As stated earlier, the statistical test used to analyze the data of the
experiment was a two sample t-Test. To determine whether there was a
significant difference between the metals or not, the p-value was calculated. In a
two Sample t-Test, the t-value must be calculated in order to find the p-value.
The t-value was calculated using the formula where this value is equal to the
sample mean of the Vanadium data, xˉ1 , minus the sample mean of the unknown
metal data, xˉ2, divided by the square root of the Vanadium sample standard
deviation, s1, squared which is divided by the sample size of the Vanadium data,
n1, and this quotient is added to a second quotient of the unknown metal sample
standard deviation, s2, squared and divided by the unknown metal sample size,
n2.
𝑡=
(xˉ 1 − xˉ 2 )
𝑠 2
𝑠 2
√( 1 ) + ( 2 )
𝑛1
𝑛2
A sample calculation of this test statistic can be found in Appendix E.
Once the t-value is calculated, the p-value is found using either the calculator
software or a statistics table, for this experiment a calculator was used. The pvalue was then compared to the alpha value to determine significance.
Havern – Rouditchenko 33
Figure 7. Calculator Specific Heat Statistical Test Results
Figure 7 above is the result of the calculator statistical test of the specific
heat data. Each value used in the test statistic has been calculated and defined.
The p-value produced from the data is a value of 0.101697, which is a slightly
large number. A larger number would support the hypothesis that there is no
significant difference between the two metals.
Figure 8. P-Value Plot of Specific Heat Data
Figure 8 above is a p-value plot of the specific heat data gathered in this
experiment. The shaded area of the graph shows the specific heats greater than
the t-value of 1.72897 and less than -1.7289. The negative t-value is shown
Havern – Rouditchenko 34
because the alternative hypothesis is not equal to. This area accounts for
10.1697% of the bell curve.
With the p-value calculated, the experimenters could review the results of
the statistical test. The results led them to fail to reject the null hypothesis, H o,
because the p-value of 0.101697 is greater than the alpha level of 0.10. This
means that there is no significant evidence that the specific heat of Vanadium is
different from the specific heat of the unknown metal. There is only a 10.1697%
chance of getting results as extreme as these by chance alone if the null
hypothesis, that the metals had the same average specific heat, were true.
Figure 9. Normal Probability Plot of Vanadium Linear Thermal Expansion
Figure 9 above is the normal probability plot of the Vanadium data for the
linear thermal expansion portion of the experiment. A large number of the data
points vary far from the expected z-value line. This suggests that the data is not
very reliable as it does not seem to be normal. With a small number of outliers,
they could be taken out of the data set and the tests could be run again, but there
are too many here to take out and yield accurately represented data. Overall, the
data cannot be very reliable based on its heavy variability. The results of the
Havern – Rouditchenko 35
t-Test might not be conclusive due to the fact that these data points are not
normally distributed.
Figure 10. Normal Probability Plot of Unknown Metal Linear Thermal Expansion
Figure 10 above is the normal probability plot of the unknown metal linear
thermal expansion data. For the most part, the data points lie close to the
expected z-value line, which suggest that the data is valid and reliable.
Figure 11. Linear Thermal Expansion Data Box Plot
Figure 11 above shows a box plot of the linear thermal expansion data,
Vanadium is the top box and the unknown metal is the bottom box. The two
boxes do not overlap at any measurement which suggests that the linear thermal
expansion coefficient is different between the two metals. The unknown metal
Havern – Rouditchenko 36
box is wider which indicates larger variability in the data. A thinner box, such as
the Vanadium box, indicates low variability in the data. There is a slight skew of
the data in the box plot. The Vanadium data box plot is right skewed which
suggests that the sample mean is greater than the median. The unknown metal
data box plot is left skewed which suggests that the sample mean will be less
than the median. Essentially, this graph shows that the data is reliable because
of few outliers and small variability in the data itself.
The same test was used to analyze the data of the linear thermal
expansion portion of this experiment as was the specific heat portion, a two
sample t-Test. All variables in the formula refer to the same values of the data,
just using a different set of data. The formula can be found above Figure 7 and a
sample calculation of the test statistic can be found in Appendix E.
Figure 12. Calculator Linear Thermal Expansion Statistical Test Results
Figure 12 above shows the calculator statistical test results of the linear
thermal expansion data. Each value used to calculate the test statistic, or t-value,
was recorded by the calculator. The p-value is so miniscule that it was recorded
Havern – Rouditchenko 37
in scientific notation. The p-value of 6.76291E-12 represents a value of
0.00000000000676291. This very small value will provide significance against
the null hypothesis.
Figure 13. P-Value Plot of Linear Thermal Expansion Data
Figure 13 above shows the p-value plot of the test statistic run for the
linear thermal expansion data. The p-value is so small it basically cannot be seen
on the graph. The area that is shaded under this region is 0.000000000676291%
of all the data.
With the p-value calculated, the experimenters could evaluate the results
of the statistical test. The results led them to reject the null hypothesis, Ho,
because the p-value of 6.76291E-12 is less than the alpha level of 0.10. This
means that there is significant evidence that the linear thermal expansion
coefficient of Vanadium is different from the linear thermal expansion coefficient
of the unknown metal. There is only a 6.76291E-12 % chance of getting results
as extreme as these by chance alone if the null hypothesis, that the metals had
the same linear thermal expansion coefficient, were true. This result is conflicting
because the t-Test for specific heat provided evidence that the metals were the
same.
Havern – Rouditchenko 38
Conclusion
The initial objective of this experiment was to determine if the material
properties of specific heat and linear thermal expansion could be used to
correctly identify an unknown metal as Vanadium. After all of the data collection
and analysis, the researchers were finally able to propose a conclusion. The
hypothesis that, “The experimental data will provide an approximate value of
specific heat and the linear thermal expansion coefficient with which the
experimenters will be able to correctly identify the unknown metal as Vanadium
with one percent error”, was rejected. The data that supports the rejection of the
hypothesis is the data representing percent error. Each of the average percent
errors of the data of the unknown metals were above or below the hypothesized
value of one percent error. The average percent errors of the known metals were
0.217% for the specific heat and -33.239% for linear thermal expansion. The
average percent errors of the unknown metals were -11.244% for specific heat
and -3.780% for the linear thermal expansion.
Even though the hypothesis was rejected, the experimenters were still
able to correctly identify the unknown metal. The researchers observed that the
unknown metal was not Vanadium. The researchers concluded that the metals
were different based on several observations. The p-value for linear thermal
expansion coefficient was extremely small, much less than the alpha value of
0.10. This suggests that the metals were different. The p value for specific heat
was only above the alpha level by 0.001697. This p value is extremely close to
the alpha level but because it is above the alpha level of 0.10, it suggests that the
Havern – Rouditchenko 39
metals are the same. Although the results were slightly conflicting, the
researchers made an important observation during the experimental procedure.
While placing the unknown metals in the loaf pan on the hot plate to heat up, the
metals were magnetically attracted to the hot plate. This meant that the unknown
metal rods were magnetic. The known metal, Vanadium, is not magnetic. This
means that the unknown metal is either a magnetic element or an alloy with
magnetic properties. Based on all of these observations, the experimenters were
certain that the unknown metal was not Vanadium.
There were some problems that the experimenters encountered during
their research. The experimenters were not familiar with the tools or the
procedures used in the experiment before beginning the trials. Also, the tools
limited the quality of the researcher’s results. The calorimeters and expansion
jigs were built in house and were not exactly laboratory grade. The dial on the
expansion jig was not very accurate and was difficult to get a correct reading.
Many of the values for temperature were not accurate due to the fact that the
procedure was not reproduced exactly the same for each trial. The transfer time
for the metals from the loaf pan to the calorimeter or jig varied greatly each time.
The amount of time the metals spent in the loaf pan was not the same;
sometimes the metals spent almost an extra minute within the pan. The rods
were dropped in and out of the pan and the calorimeters. These chronological
errors were due mostly to human errors. Because these factors changed often,
the data could have been easily skewed. The percent error for the linear thermal
expansion for Vanadium was -33.239%. This value should have been close to
Havern – Rouditchenko 40
one which shows that there were flaws in the experimental design. To minimize
error, there could have been several improvements to the procedure. The
experiment could be done in an isolated environment such as a temperature
controlled room without doors and windows being opened. This could minimize
confounding variables such as other experiments being conducted and large
changes in ambient room temperature. The researchers would have benefitted
from a more spacious work space as the small table was very cramped with
materials. A larger work space would allow the researchers to be more organized
and not have to sort through materials. The probability of spilling the calorimeters
would also be reduced. Future researchers could run each trial completely
independently. The researchers ran almost all the trials in pairs due to time
constraints. The greatest investment for the experiment would be better
technology. A bomb calorimeter and a steam expansion jig would produce more
accurate results. Several industries would benefit from using these procedures to
identify metals. Companies that use pure elements in their productions would be
able to check to see if their suppliers are selling them what they are paying for.
Random quality control checks could be done to see if the metals are pure
elements and not alloys or elements with similar properties. Businesses such as
construction and plumbing that use metals often could conduct research to find if
there are cheaper options. They could run similar experiments to observe if there
are metals with values of specific heat and linear thermal expansion coefficient
that are alike to the current metals being used. Essentially, this experiment this
could be adapted for many uses.
Havern – Rouditchenko 41
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