Jordan – Semones 1
Molybdenum Identification
Sydni Jordan and Kyla Semones
Macomb Mathematics Science Technology Center
Chemistry / IDS 10
10C
Mrs. Hilliard / Mr. Supal / Mrs. Dewey
20 May 2013
Jordan – Semones 2
Table of Contents
Introduction ……………………………………………………………………………..3
Background ……………………………………………………………………………..5
Specific Heat Review of Literature ……………………………………………………7
Linear Thermal Expansion Review of Literature …………………………………..10
Problem Statement ……………………………………………………………………13
Specific Heat Experimental Design …………………………………………………15
Linear Thermal Expansion Experimental Design ………………………………….19
Data and Observations ……………………………………………………………….22
Data Analysis and Interpretation …………………………………………………….38
Conclusion ……………………………………………………………………………..53
Acknowledgements…………………………………………………………………... 56
Appendices …………………………………………………………………………….57
Works Cited ……………………………………………………………………………64
Jordan – Semones 3
Introduction
Identification is the first step to understanding. Many occupations, such as
those that involve chemistry and mining, require scientists to identify whether or
not a discovered metal is the same as a known metal. The researchers adapted
the ideas behind those jobs when conducting this experiment.
In the following experiment, the researchers’ purpose was to identify
whether or not two metal rods were composed of the same metal using only
length, mass, and temperature values. One of the metal rods was composed of
Molybdenum, a substance used in steel alloys to make them stronger.
Molybdenum is used to strengthen steel because Molybdenum has properties
that increase the melting temperature and enhance the corrosion and heat
resistance ("Molybdenum Properties"). The experiment proved that there are
many types of ways to identify a metal. In situations beyond this experiment,
different methods of experimentation are often used, but for these experiments
specific heat and linear thermal expansion tests were used to compare the two
rods.
Each metal has its own designated specific heat and linear thermal
expansion coefficient. Specific heat is the amount of heat that it takes increase
the temperature of a metal by one degree Celsius, and linear thermal expansion
is the linear change in length that a metal undergoes when it is heated. The
specific heat of Molybdenum is 0.25 J/g x ºC, and the linear thermal expansion is
4.8 x 10-6mm. The researchers used this information when comparing the two
metals. First, the researchers performed a simple experiment to calculate the
Jordan – Semones 4
specific heat of Molybdenum. Then, they performed the same experiment with
the unknown metal and compared the two values. An experiment for linear
thermal expansion was also conducted on the Molybdenum rods, and again, the
same experiment was performed with the unknown metal rods and compared the
results.
The main objective of the experiment was to make a correct inference on
whether or not the two given rods were composed of the same metal.
Significance tests, which were conducted on the values from the trials, and
percent error equations that calculated the extremity of their errors presented the
researchers with statistics that helped them decide whether or not the two rods
were actually the same metal.
Jordan – Semones 5
Background and Reviews of Literature
Molybdenum is a unique transition metal with a hard, silver outer
appearance. A Swedish Chemist, Carl Welhelm Scheele, discovered
Molybdenum in 1778. Although he was the first to recognize the element, he
discovered it as Molybdenite. Peter Jacob Hjelm was the first to actually isolate
the element and discover its pure form (Gagnon).
Molybdenum has an atomic number of 42, meaning that there are 42
protons in a Molybdenum atom, and an atomic mass of 95.94 amu. Of
Molybdenum’s 42 electrons, six of them are valence electrons. Figure 1 below
shows the electron configuration of Molybdenum.
Figure 1. The shortened electron configuration for Molybdenum
The density of Molybdenum is 10.22 g/cm3 (Hogan). This is relatively high
when compared to the density of water, 1 g/cm3, and the density of iron, 7.87
g/cm3. This means that the particles in Molybdenum are packed closer than
those in water and iron. Molybdenum also has a melting point of 2623ºC. This is
an extremely high temperature, and this puts Molybdenum in the top 10 elements
with the highest known melting point.
Jordan – Semones 6
Molybdenum is used in many ways, and new uses are constantly being
discovered. When most think of Molybdenum, they think of it being used in
industrial industries. This is true; it is frequently used for steel alloys. When
added, Molybdenum transforms ordinary steel into "ultra-high strength steel". It
makes for a stronger metal with a higher heat tolerance, strength, and resistance
to corrosion. Molybdenum is unique because it has the capability to withstand
extremely high temperatures without undergoing a change to its shape or form.
Due to this, it is often used not only for the creation of alloys but also in space
exploration ("Molybdenum Properties").
Molybdenum is a useful element but it is not always easy to find. One of
the extraction processes involves roasting Molybdenum Disulfide (MoS2) to
convert it to Molybdenum Trioxide (MoO3) ("Processing."). Figure 2 below shows
the chemical equation used to extract Molybdenum.
2MoS2 + 7O2 (g) → 2MoO3 + 4SO2
MoS2 + 6MoO3 → 7MoO2 + 2SO2
2MoO2 + O2 → 2MoO3
Figure 2. The chemical equation that shows the extraction process of
Molybdenum
Figure 2 models the chemistry aspect behind Molybdenum extraction.
Roasters are used to extract Molybdenum from the ground. The roasters use
furnaces and heated air with temperatures between 500ºC and 600ºC.
Jordan – Semones 7
Specific Heat Review of Literature
Specific heat is the amount of heat needed to raise the temperature of one
gram of a substance by one degree Celsius (Missouri State University). Specific
heat is a type of physical property, known as an intensive property, meaning that
the amount of the substance is not relevant to the results (Department of
Chemistry, Texas A&M University). All substances are associated with
temperature, and this means that the greater the temperatures of the substance,
the faster its molecules are moving. The First Law of Thermodynamics, that
energy is neither created nor destroyed, is proven when the concept of specific
heat is applied. In specific heat experiments, the metal must be heated and
cooled to calculate the specific heat of the metal. The energy from the heat
enters and leaves the system, or the metal, but the total amount of energy does
not change and energy is not created or destroyed.
Most metals have a low specific heat; substances with a low specific hear
need less energy to increase the temperature of the substance. Specific heat is
calculated using the formula displayed below.
Q = s • m • ∆T
This equation shows that the heat released or absorbed by the reaction, Q, is
equal to the specific heat, s, multiplied by the mass, m, multiplied by the change
in temperature, ∆T. The heat released or absorbed by the reaction is measured
in joules, the mass is measured in grams, and the change in temperature is
Jordan – Semones 8
measured in degrees Celsius. Thus, specific heat is measured in joules per
grams degrees Celsius (J/g ºC) (Nave).
Past experiments have been performed in which the specific heat of a
certain substance was calculated. Most of these experiments involved the use of
a calorimeter. A tool such as this can be as simple as a Styrofoam cup. In
general, calorimetry was used to determine the heat released or absorbed in
these experiments. The first experiment that was performed to calculate the
specific heat of an unknown substance went as follows. A beaker of water was
heated and the unknown metal was placed into the water. Afterwards, the
change in temperature was calculated and recorded with the use of a
thermometer that was placed through the top of the calorimeter. The heat gained
during the initial heating of the metal was found to be the same amount of heat
that was lost from the metal (Bauck). This experiment not only calculated the
specific heat of a metal, but it also proved the First Law of Thermodynamics.
The second experiment involved a completely different process than that
of the first. The specific heat of the substance was calculated with the use of
cylinders made entirely of the unknown metal. When different metal cylinders
were heated to the same temperature and immersed individually in an equal
quantity of water, the rise in the temperature of the water was directly
proportional to the specific heat of the chosen metal cylinder (“HEAT”).
These experiments seemed easy to execute and replicate, The different
methods used in these experiments show just how intensive properties such as
Jordan – Semones 9
specific heat can be used to identify a metal, regardless of experimental design.
Each metal has its own unique specific heat, therefore knowing these important
details have brought the researchers one step closer to discovering whether or
not the unknown metal is the same as the known.
Jordan – Semones 10
Linear Thermal Expansion Review of Literature
One unique way of finding the identity of a substance is by using the
concept of Linear Thermal Expansion. Linear Thermal Expansion is a metal's
change in length when it experiences a change in temperature. In most cases,
the length of a metal is increased when the heat is increased. It changes by an
amount proportional to the change in temperature and the initial length of the
object. ("Temperature and Thermal Expansion.")
The expansion of metal is best described by the Kinetic Molecular Theory.
The Kinetic Molecular Theory states that atoms are in constant motion and when
heat is applied to the system, or the metal rod in this case, the speed of the
atoms’ motion increases. When metal undergoes a heat increase, the energy
from the heat causes the atoms in the metal to vibrate and move. These
vibrations then increase the space between the atoms. This is what causes the
metal to expand. ("223 Physics Lab: Linear Thermal Expansion.").
Yes, Linear Thermal Expansion is the method with which the elemental
composition of the metal will be discovered, but the observed value is actually
the linear thermal expansion coefficient. A linear thermal expansion coefficient is
a value unique to the metal it correlates with. ("Coefficients of Linear Thermal
Expansion."). Every metal has a different linear thermal expansion coefficient,
and this coefficient can be calculated by using the Linear Thermal Expansion
equation. (R. Nave)
∆L = α • L0 • ∆T
Jordan – Semones 11
The equation above calculates linear thermal expansion. The symbol α stands for
the linear thermal expansion coefficient, L0 stands for the initial length of the
object being measured, and ∆T stands for the change in temperature of the
object. The equation calculates the change in length, or ∆L. By manipulating the
equation, the linear thermal expansion coefficient can be calculated. It is often
measured in 10-6 m/m °C.
Past experiments have been performed to discover unknown metals. Both
of the mentioned experiments contained an apparatus but there were set up
differently to discover the linear thermal expansion coefficient. In the first
experiment, the overall linear thermal expansion was calculated by the
measuring of the change of length in the unknown metal. A steam generator was
placed at one end of the apparatus and on the other end was a dial gauge that
measured the variation in tolerance. First, the steam generator was filled with
water. Next, the expansion apparatus was attached to the tube of the unknown
metal. Then the data was recorded and inserted into the linear thermal expansion
formula (Hurth).
The second experiment was performed using the measurements for the
initial lengths and diameters of the rods of the unknown metal. Then, the
unknown metal was placed in the steam generator to be heated. Afterwards, the
rods were removed and the data was inserted into the linear thermal expansion
formula (Physics Department).
Knowledge of these experiments has aided in the determination of the
unknown metal rod’s linear thermal expansion coefficient. In the future, different
Jordan – Semones 12
experiments may be conducted to further prove the identity of the unknown
metal. With this knowledge, industries such as those that involve plumbing would
benefit by knowing what type of pipes would be best for certain situations. It
would not be a good idea for a plumber to install pipes made out of a metal that
has a tendency to expand when it is heated.
Jordan – Semones 13
Problem Statement:
The researchers will use linear thermal expansion and specific heat equations
along with percent error to correctly determine whether or not the unknown metal
rod is made of Molybdenum.
Hypothesis:
When the linear thermal expansion coefficient in millimeters, the specific heat in
joules per gram degrees Celsius, and the average percent error for the trials are
calculated, the researchers will conclude that the unknown metal rod will not be
composed of Molybdenum.
Data Measured:
In the specific heat calculation, the mass of the metal rod and water was
measured in grams. The change in the temperature of the metal rod and water
was measured in degrees Celsius. The overall specific heat of the Molybdenum
and unknown metal rods was be measured in joules per gram degrees Celsius.
The equation below was used to calculate the specific heats of the metal rods.
Smetal • mmetal • ∆Tmetal = Swater • mwater • ∆Twater
The equation above shows that the formula used to calculate the specific heat of
the metal was set equal to the formula used to calculate the specific heat of the
water. This was done because the heat used to change the temperature of one
gram of both of the substances became the same when the metal was placed in
Jordan – Semones 14
the water. The equation was then manipulated to find the specific heat of the
metal alone.
In the linear thermal expansion calculation, the initial length and the
change in length were measured in millimeters. The change in temperature was
measured in degrees Celsius. The equation below was used to calculate the
linear thermal expansion coefficients of the metal rods.
∆L = α • L0 • ∆T
The equation was manipulated so that the alpha coefficient was set to equal the
change in length of the metal rod divided by the change in temperature and the
initial length of the metal rod.
Specific Heat Experimental Design
Jordan – Semones 15
Materials:
Scout Pro Electric Scale (0.01g
precision)
Calorimeter
Corning Hot Plate
Jiffy – Foil Loaf Pan (7 15/16 in. x 5
7/16 in. x 1 13/16 in.)
Tongs
Molybdenum Rod
Water, H20
Logger Pro
Logger Pro Temperature Probe
TI-nSpire CX Graphing Calculator
Thermometer
Unknown Metal Rod
Graduated Cylinder
Jordan – Semones 16
Specific Heat Procedure:
1. Randomize the 15 trials for the Molybdenum rod and the unknown rod on the
TI- nSpire calculator (see Appendix 1).
2. Construct the calorimeter (see Appendix 2).
3. Tar the scale so it’s initial value is zero, then place the Molybdenum rod on it.
4. Record the mass of the rod and remove the metal rod from the surface of the
scale.
5. Fill half of the loaf pan with water.
6. Place the loaf pan on the hot plate and adjust the dial to the highest setting to
boil the water.
7. Wait for the water to boil and use the thermometer to make sure the
temperature of the water is approximately 98ºC.
8. Use the tongs to pick up the Molybdenum rod and gently place it in the loaf
pan with the boiling water.
9. Keep the metal rod in the water for 3 minutes so its temperature is equivalent
to that of the boiling water (98ºC).
10. Prepare the Logger Pro.
11. Measure 41 ml of room temperature water with the graduated cylinder and
carefully pour it into the calorimeter.
12. Insert the thermometer into the calorimeter and record the temperature of the
water.
13. Once the metal rod has been in the boiling water for 3 minutes, carefully use
the tongs to insert it into the calorimeter.
14. Immediately place the cap on the calorimeter and secure it. Quickly insert the
temperature probe through the drilled hole on the top of the calorimeter and
begin data collection.
15. Keep the metal rod in the water for 90 seconds so the temperature of the
water and metal may reach equilibrium.
16. After the 90 seconds, stop the data collection on the Logger Pro.
17. Remove the temperature probe and the metal rod with the tongs. Place it on
the paper towel. Pour the water from the calorimeters into a sink.
18. Repeat steps 3 – 17 for each of the randomized trials for the Molybdenum rod
and the unknown metal rod.
19. Pour the water from the loaf pan into a sink and proceed to tidy up the
workspace.
Diagram:
Figure 3. One of the calorimeters from the procedure
Figure 3 shows a picture of one of the calorimeters that was used in the
specific heat procedure. Three calorimeters were created to test the specific
heats of the rods.
Linear Thermal Expansion Experimental Design
Materials:
Linear Expansion Jig (0.01 mm)
TI- nSpire CX Calculator
Caliper
Hot Plate
Timer
Thermometer (0.1ºC)
Tongs
Aluminum Loaf Pan (dimensions)
Unknown Metal Rod (2)
Molybdenum Metal Rod (2)
Graduated Cylinder
Water, H2O (25 ml)
Expo Marker
Spray Bottle
Jordan-Semones 21
Procedure:
1. Randomize the trials with the TI-nSpire calculator (See Appendix 1).
2. Carefully measure and record the length of the metal rod using the caliper.
3. Prepare the hot plate by plugging it in and adjust the dial to the highest setting
to boil the water.
4. Fill half of the loaf pan with water and place it on the hot plate. Proceed to boil
the water.
5. Prepare the timer on the TI-nSpire CX calculator by restarting it so that it does
not display recorded time.
6. Carefully insert the thermometer into the water and record the temperature.
Make sure the water is approximately 98ºC before proceeding to the next
step.
7. Once the water is boiling, use the tongs to pick up the metal rod and carefully
place it in the water. Start the timer once the rod is in the water.
8. Let the metal boil in the water for 3 minutes.
9. Use the tongs to remove the metal and instantly place in into the linear
expansion jig.
10. Keep the linear expansion jig completely still and mark the beginning point
with the marker.
Jordan-Semones 22
11. For 3 minutes wait for the metal to cool to room temperature. Fill the spray
bottle with water and lightly spray to metal rod to speed up the cooling
process. Wait for the dial on the linear expansion jig to settle and mark the
point where it stops.
12. Observe the difference of the position of the dial on the linear expansion jig
and record the change in length.
13. Repeat steps 2 - 12 for each of the trials for the Molybdenum and unknown
metal rods.
Jordan-Semones 23
Data and Observations
Table 1
Specific heat values for Molybdenum
Change in Temp.
(ºC)
Initial Temp. (ºC)
Trial
Mass (g)
Equilibrium
Temp (ºC)
Rod
Water
Metal
Water
Metal
Metal
Specific
Heat
Water
(J/g x ºC)
1
B
20.1
98.0
25.7
5.6
-72.3
42.70
41
0.311
2
B
21.6
98.3
26.1
4.5
-72.2
42.70
41
0.250
3
A
22.3
98.3
27.4
5.1
-70.9
42.80
41
0.288
4
B
25.3
98.3
30.4
5.1
-67.9
42.80
41
0.301
5
A
30.0
98.3
32.7
2.7
-65.6
42.90
41
0.165
6
B
25.3
99.1
29.4
4.1
-69.7
42.70
41
0.236
7
A
25.3
99.1
29.0
3.7
-70.1
42.80
41
0.212
8
A
25.6
98.3
28.5
2.9
-69.8
42.83
41
0.166
9
B
25.6
98.3
29.1
3.5
-69.2
42.72
41
0.203
10
A
27.5
97.3
30.8
3.3
-66.5
42.84
41
0.199
11
B
27.4
97.3
30.6
3.2
-66.7
42.73
41
0.193
12
B
26.3
98.5
30.1
3.8
-68.4
42.74
41
0.223
13
A
27.3
98.3
31.4
4.1
-66.9
42.85
41
0.246
14
A
27.5
98.8
32.0
4.5
-66.8
42.86
41
0.270
15
B
27.5
98.8
32.2
4.7
-66.6
42.74
41
0.283
Jordan-Semones 24
In Table 1 above, the values for temperature and mass are recorded.
These values were used to calculate the specific heat of the two Molybdenum
rods. The differences in mass of the two rods may be a factor that contributed to
the varying specific heat values. Another contributing factor to the varying values
may be inconsistent testing methods by the researchers.
Jordan-Semones 25
Table 2
The observations taken during the tests for the specific heat of the Molybdenum
rods.
Trial
Rod
Date
Observations
1
B
04/17/13 Metal was placed into calorimeter 1. Time for metal in
boiling water went about 5 seconds over.
2
B
04/17/13 Metal was placed into calorimeter 2. channel 2 in logger
pro
3
A
04/17/13 Calorimeter 1. channel 1 in logger pro
4
B
04/17/13 Calorimeter 2 and channel 2 on logger pro. time for
metal in boiling water went about 2 seconds over
5
A
04/17/13 Calorimeter 1 and channel 1 on logger pro. Time went a
bit over
6
B
04/17/13 Calorimeter 2. Channel 2 in logger pro. went about 5
seconds over time while boiling in the water
7
A
04/17/13 Calorimeter 1. Channel 1 in logger pro. went about 5
seconds over time in boiling water
8
A
04/19/13 Calorimeter 3. Window open. Channel 4 in logger pro.
went a bit over time while boiling (about 6 seconds)
9
B
04/19/13 Calorimeter 2. Window open. Channel 3 in logger pro.
10
A
04/19/13 Calorimeter 3. Window open. Logged on channel 4.
11
B
04/19/13 Calorimeter 2. Window open. Logged on channel 3.
12
B
04/19/13 Channel 3. Calorimeter 2. Window open.
13
A
04/19/13 Channel 4. Calorimeter 3. Window open.
14
A
04/19/13 Calorimeter 3. Window open. Channel 4 on logger pro.
15
A
04/19/13 Calorimeter 2. Window open. Channel 3 on logger pro.
Jordan-Semones 26
Table 2 shows the specific heat observations for the Molybdenum rods. All
four of the channels on the logger pro and all three of the calorimeters were used
when the testing for the specific heats of the Molybdenum rods. On the second
day of specific heat testing, the window was opened and that may have altered
the data when it made the room cooler.
Jordan-Semones 27
Table 3
The values used to calculate the linear thermal expansion coefficient, alpha, of
the Molybdenum rods.
Trial
Rod
ΔL
(mm)
Initial
Length
Initial
Temp.
(ºC)
Final
Temp.
(ºC)
Change
in Temp.
(ºC)
Alpha
Coefficient
(mm x 10-6)
1
B
0.0254
129.41
98.7
27.7
-71.0
2.764
2
A
0.0254
129.28
98.7
27.7
-71.0
4.683
3
A
0.0254
129.33
98
28.1
-69.9
2.810
4
B
0.0200
129.40
98
28.1
-69.9
2.211
5
A
0.0254
129.39
98.2
25.0
-73.2
2.682
6
B
0.0200
129.36
98.2
25.0
-73.2
2.112
7
A
0.0254
129.34
98.7
25.1
-73.6
2.668
8
B
0.0200
129.32
98.7
25.1
-73.6
2.101
9
B
0.0200
129.33
98.7
24.5
-74.2
2.084
10
A
0.0254
129.41
98.7
24.5
-74.2
2.645
11
B
0.0200
129.28
96.7
26.1
-70.6
2.191
12
A
0.0381
129.45
96.7
26.1
-70.6
4.169
13
B
0.0254
129.41
99.3
22.4
-76.9
2.552
14
A
0.0381
129.37
99.3
22.4
-76.9
3.830
15
A
0.0254
129.41
98.4
22.5
-75.9
2.586
Jordan-Semones 28
Table 3 shows the recorded length and temperature values for the
Molybdenum rods. These values were used to calculate the linear thermal
expansion coefficient alpha. In the table, the alpha coefficient is measured in
millimeters multiplied by 10-6. Notice that the initial lengths of rod A are overall
less than the lengths of rod B. This may be a contributing factor to the varying
values for the alpha coefficient of the Molybdenum rods.
Jordan-Semones 29
Table 4
The observations taken during the tests for the linear thermal expansion
coefficient, alpha of the Molybdenum rods.
Trial
1
2
3
Rod
6
7
8
9
10
Observations
B
04/18/13 Went about 10 seconds over 5 minutes while boiling.
Cooled in jig for about 3 minutes
A
04/18/13 Went about 10 seconds 5 minutes while boiling. Cooled
for about 3 minutes.
A
04/18/13 Switched to boiling for 3 minutes. Cooled for 3 minutes.
White film/ powder on the metal after cooling
B
04/18/13 Boiled for 3 minutes. Fast decrease in length in the
beginning. Cooled for 3 minutes. White film/powder on
metal after cooling
A
04/18/13 Boiled for 3 minutes. White film/powder formed. Window
was opened and resulted in a drop in final temp.
B
04/18/13 Boiled for 3 minutes. White film/powder formed on metal.
Window was open. Drop in final temp.
A
04/18/13 3 minutes boiling. Fast initial decrease in length.
Switched to cooling for 2 minutes. Window open.
B
04/18/13 3 minutes boiling. Fast decrease in length. Cooled for 2
minutes. Window open.
B
04/18/13 About 8 seconds over 3 minutes while boiling. White
powder on the cooled metal. Window open.
A
04/18/13 3 minutes boiling but a 8 seconds over. White powdery
film on metal. Window open.
4
5
Date
Jordan-Semones 30
11
B
04/18/13 Boiled for 3 minutes. Boiled a bit under temperature.
Window open.
A
04/18/13 Boiled for 3 minutes. Boiled a bit under temperature.
Window open.
13
A
04/19/13 Window closed. Boiled for 3 minutes.
14
B
04/19/13 Window closed. Boiled for 3 minutes.
15
A
04/19/13 Window closed. Boiled for 3 minutes.
12
Table 4 shows the observations for the linear thermal expansion tests of
the Molybdenum rods. It shows that the linear thermal expansion trials were
taken on two consecutive days. On the entire first and most of the second day,
the window was opened. During and after trial 3, the researchers made a
decision to boil the metal rods for three minutes instead of five. This decision was
made for time purposes as well as accuracy and efficiency. Also, the metal rods
were usually cooled from two to three minutes with the aid of water and air.
Jordan-Semones 31
Table 5
The values used to calculate the specific heat of the unknown metal rods.
Initial Temp.
(ºC)
Trial
Change in
Temp. (ºC)
Mass (g)
Equilibrium
Temp (ºC)
Rod
Water Metal
Water
Metal Metal Water
Specific
Heat
(J/g x ºC)
1
A
28.0
98.5
30.8
2.8
-67.7 26.58
41
0.267
2
B
27.8
98.5
30.9
3.1
-67.6 26.49
41
0.297
3
B
26.9
99.0
30.9
4.0
-68.1 26.55
41
0.380
4
A
27.4
99.0
30.8
3.4
-68.2 26.59
41
0.322
5
A
27.3
97.9
30.1
2.8
-67.8 26.58
41
0.267
6
B
27.1
97.9
30.8
3.7
-67.1 26.48
41
0.357
7
A
27.1
98.9
31.3
4.2
-67.6 26.59
45
0.440
8
B
26.7
98.9
31.2
4.5
-67.7 26.48
45
0.473
9
A
27.1
98.0
30.1
3.0
-67.9 26.56
45
0.313
10
B
30.2
98.0
31.4
1.2
-66.6 26.51
45
0.128
11
B
30.1
98.0
32.1
2.0
-65.9 26.47
45
0.216
12
A
30.1
98.0
32.3
2.2
-65.7 26.56
45
0.237
13
A
30.8
97.9
30.2
-0.6
-67.7 26.58
45
0.063
14
B
29.1
97.9
31.1
2.0
-66.8 26.49
45
0.213
Jordan-Semones 32
15
B
27.3
98.0
29.1
1.8
-68.9 26.50
45
Table 5 shows the masses and temperatures of the unknown metal rods.
Notice that during and after trial 7 the amount of water that was added to the
calorimeters was increased by four milliliters. This was the result of the late
discovery that the water in the calorimeter did not completely cover the top of the
unknown metal rod. This may have affected the rest of the data from trial 7 on.
0.186
Jordan-Semones 33
Table 6
The observations taken during the tests for the specific heat of the unknown
metal rods.
Trial
Rod
Date
Observations
1
A
04/19/13 30 seconds over boiling. Window open. Calorimeter
3.
2
B
04/19/13 30 seconds over boiling. Window open. Calorimeter
2.
3
B
04/19/13 Calorimeter 2. window open
4
A
04/19/13 Calorimeter 3. window open
5
A
04/19/13 Calorimeter 3. Took out of boiling water 30 seconds
early
6
B
04/19/13 Calorimeter 2. Took out of boiling water about 28
seconds early
7
A
04/23/13 Switched to 45 ml of water for the rest of the trials.
window closed. Calorimeter 1. Channel 2 on logger
pro.
8
B
04/23/13 Window closed. Calorimeter 2. Channel 1 on logger
pro.
9
A
04/23/13 Calorimeter 1. Window closed. Channel 2. went
about 6 seconds over 3 minutes boiling
10
B
04/23/13 Calorimeter 2. Window closed. Channel 1. About 8
Jordan-Semones 34
seconds over when boiling.
11
B
04/23/13 Calorimeter 2. Window closed. Channel 1
12
A
04/23/13 Calorimeter 1. Window closed. Channel 2
13
A
04/23/13 Calorimeter 1. Window closed. Channel 2
14
B
04/23/13 Calorimeter 2. Window closed. Channel 1.
15
B
04/23/13 Calorimeter 2. Window closed. Channel 1.
Table 6 shows the observations that were taken during the specific heat
tests on the unknown metal rods. The trials were taken on two separate days and
all three calorimeters were used during the testing. The observations tell that the
amount of water in the calorimeter was switcher from 41 to 45 milliliters.
Jordan-Semones 35
Table 7
The table of values used to find the linear thermal expansion coefficient, alpha,
for the unknown metal rods.
Trial
Rod
ΔL
(mm)
Initial
Length
(mm)
Initial
Temp.
(ºC)
Final Change
Alpha
Temp.
in
Coefficient
(ºC)
Temp. (mm x 10-6)
(ºC)
1
A
0.0508
139.44
98.1
20.3
-77.8
4.683
2
B
0.0508
139.94
98.1
20.3
-77.8
4.666
3
B
0.0508
139.03
96.7
23.9
-72.8
5.019
4
A
0.0508
139.30
96.7
23.9
-72.8
5.009
5
B
0.0635
139.27
98.5
22.2
-76.3
5.976
6
A
0.0508
139.01
98.5
22.2
-76.3
4.790
7
A
0.0508
139.10
98.2
27.5
-70.7
5.166
8
B
0.0535
139.31
98.2
27.5
-70.7
5.432
9
B
0.0635
139.33
99.7
22.1
-77.6
5.873
10
A
0.0635
139.01
98.6
22.1
-76.5
5.971
11
A
0.0508
139.05
98.0
23.1
-74.9
4.878
12
B
0.0508
139.25
99.7
23.9
-75.8
4.813
Jordan-Semones 36
13
B
0.0508
139.49
98.2
23.7
-74.5
4.888
14
A
0.0635
139.06
98.2
21.6
-76.6
5.961
15
A
0.0508
139.50
98.0
22.5
-75.5
4.823
Table 7 above shows the lengths and temperatures of the unknown metal
rods. These values were used to calculate the linear thermal expansion
coefficient, alpha, for the unknown metal rods. Notice that the initial lengths of
rod A tend to be relatively less than the initial lengths of rod B. Also notice that
the values for the alpha coefficient are in a relatively small range.
Jordan-Semones 37
Table 8
The observations taken during the linear thermal expansion trials of the unknown
metals.
Trial
Rod
Date
Observations
1
A
04/22/13 Calorimeter 2. Window closed.
2
B
04/22/13 Calorimeter 3. Window closed.
3
B
04/22/13 window open. Went about 10 seconds over when boiling.
4
A
04/22/13 window open. Went about 11 seconds over when boiling.
5
B
04/22/13 Calorimeter 3. Window closed.
6
A
04/22/13 Calorimeter 2. Window closed.
7
A
04/22/13 Calorimeter 2. Window closed.
8
B
04/22/13 Calorimeter 3. Window closed. Quick initial decrease in
length.
9
B
04/22/13 Calorimeter 3. Window closed.
10
A
04/22/13 Calorimeter 2. Window closed.
Jordan-Semones 38
11
A
04/22/13 Calorimeter 2. Window closed.
12
B
04/22/13 Calorimeter 3. Window closed. Fast decrease in length in the
beginning.
13
B
04/22/13 Calorimeter 3. Window closed.
14
A
04/22/13 Calorimeter 2. Window closed.
15
A
04/23/13 Calorimeter 3. Window closed.
Table 8 shows the observations that were taken during the linear thermal
expansion trials. All but one of the trials were taken on the same day, Each of the
trials were tested when the window was recently closed. This means that this fact
may have altered the data because the room was still cool after the window was
closed.
Jordan-Semones 39
Data Analysis and Interpretation
Specific Heat
While conducting each experiment, the researchers calculated the percent
error. This was done as a check so that the researchers could alter their methods
to perfect the experiments. The formula below was the formula used to calculate
the percent error. The researchers took the actual specific heat or alpha
coefficient of Molybdenum, or the true value, and subtracted it from the value
they calculated after doing the experiment, or the experimental value. Then, that
difference was divided by the true value and multiplied by 100 to calculate the
percent error of the trial. See Appendix 3 for a sample that demonstrates how to
correctly calculate the percent error from this set of data.
Jordan-Semones 40
Table 9
Molybdenum specific heat percent error table.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
Rod
B
B
A
B
A
B
A
A
B
A
B
B
A
Specific
Heat
Percent
(J/g x
Error
ºC)
0.311
24.468
0.250
0.157
0.288
15.323
0.301
20.418
0.165
34.168
0.236
5.472
0.212
15.380
0.166
33.437
0.203
18.761
0.199
20.516
0.193
22.958
0.223
10.807
0.246
1.787
Jordan-Semones 41
14
15
A
B
0.270
0.283
7.850
13.299
The table 9 shows that there was variability between the trials. The
percent errors range from 0.157 to 34.168. Trial two, using unknown metal rod B
had the lowest percent error. Trial 5, using Molybdenum rod A had the highest
percent error. In trial two the unknown rod was placed into calorimeter two and
data was recorded in channel 2. In trial five the unknown rod was placed into
calorimeter one and the data was recorded in channel 1, also the time went a bit
over. This small mistake may have caused a slight change in data. The higher
the percent error the more imperfections in the trial. The average percent error
for the specific heat experiment with Molybdenum rods was calculated to equal
16.320. See Appendix 3 for how to calculate the percent error.
Table 10
Unknown metal rod specific heat percent error table.
Trial
1
2
3
4
5
6
7
8
9
10
Rod
A
B
B
A
A
B
A
B
A
B
Specific
Heat
Percent
(J/g x
Error
ºC)
0.267
6.770
0.297
18.787
0.380
51.804
0.322
28.651
0.267
6.613
0.357
42.888
0.440
75.974
0.473
89.047
0.313
25.282
0.128
48.813
Jordan-Semones 42
11
12
13
14
15
B
A
A
B
B
0.216
0.237
0.063
0.213
0.186
13.651
5.050
74.889
14.879
25.754
Table 10 shows that there is a wide range of percent error. The values
have a large range of 5.050 to 89.047. The average percent error for the specific
heat of the unknown metal rods was 35.257.This means that the trials were
either not conducted with consistency or an outside force has influenced the
results.
To analyze the calculated values for specific heat, a two-sample t-test was
chosen. This test was chosen over others to represent the data because a twosample t test compares the means of two independent samples without ever
having or needing the standard deviation of the population. To use this test, a
few assumptions have to be met first. The first assumption is that this experiment
was a simple random sample, or an SRS. The experiments conducted were
examples of SRSs because the two rods that were used for each test were
picked with the aid of randomization. The second assumption is that the samples
are independent. This assumption was met because the outcomes from one
sample never affected those of the other, and the specific heat experiments for
the Molybdenum rods and the unknown metal rods were conducted separately.
The third assumption that would have to be accepted would be that each of the
distributions of outcomes were normal distributions. When the number of
outcomes tested is greater or equal to 30, this fact can automatically be
Jordan-Semones 43
accepted, but because of neither of the tests for specific heat were conducted on
groups of 30 or more, normal probability plots have to be shown to prove their
normality. Figures 1 and 2 show the normal probability plots for the outcomes of
the specific heat experiments on the Molybdenum and unknown rods.
Figure 4. The normal probability plot for the specific heat of the Molybdenum
rods.
The normal probability plot shown above in Figure 4 demonstrates how
the outcomes may be considered a normal distribution. In this figure, the points
are fairly close to the line. This means that this model shows that the distribution
is fairly normal.
Jordan-Semones 44
Figure 5. The normal probability plot for the specific heat of the unknown metal
rods.
The normal probability plot shown in Figure 5 demonstrates another
normal set of data. Notice that the points are still very close to the line and are
nearly linear. This proves that the data from the outcomes of the specific heat
experiment on the unknown metal rods was normal.
Figure 6. Box Plots for Molybdenum and Unknown Rods
The box plots in Figure 6 model the distributions of the data from the
specific heat experiment conducted on the Molybdenum and unknown metal
rods. The data collected from Molybdenum is displayed in the white box plot, and
the data from the unknown metal rod is displayed in the grey. The unknown
metal has a larger range of data, meaning there was more variability in the
results. Molybdenum and the unknown metal have close medians, which means
the metals in the rods have very close specific heat values. This could signify that
the metals are actually made of the same metal.
To test this conjecture and begin to conduct the two-sample t test, two
hypotheses must be created: a null hypothesis and an alternative hypothesis.
Jordan-Semones 45
The null hypothesis in this context is that the mean of the specific heat of the
Molybdenum rods is equal to the mean of the specific heat of the unknown metal
rods. This would mean that there is no significant difference between the two
means and the rods are made of the same metal. The alternative hypothesis in
this context would be that the mean of the specific heat of the Molybdenum rods
is simply not equal to the mean of the unknown metal rods. This would show that
there is a significant difference between the two means of the rods and the rods
are not made of the same metal. These two hypotheses are represented by the
equations below.
Figure 7 below shows the values from the two-sample t test. The
calculated t value correlates with the p value. The p-value stands for the
probability of getting two means with this drastic of a difference by chance. See
Appendix 4 for the full calculation of the two-sample t test.
Jordan-Semones 46
Figure 7. Results and components of the specific heat two-sample t test.
The researchers failed to reject their null hypothesis. There was not a
significant difference between the means of the specific heat of the Molybdenum
rods and the unknown metal rods. The alpha level, 0.1, allowed the researchers
to accept that any p-value lower than that may have been deemed significant.
This acceptance would have meant that there was, at most, a 10% chance of
calculating that high of difference between means. The actual p-value, 0.2042,
represents a 20.42% chance of calculating means with this drastic of a difference
by chance alone if the null hypothesis was true. The p-value was greater than the
accepted alpha level, 0.1, and that led to the fail of rejection of the null
hypothesis.
Jordan-Semones 47
Linear Thermal Expansion
Table 11
Molybdenum linear thermal expansion percent error table.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Rod
B
A
A
B
A
B
A
B
B
A
B
A
B
A
Alpha
Percent
Coefficient
Error
(mm x 10-6)
2.764
4.683
2.810
2.211
2.682
2.112
2.668
2.101
2.084
2.645
2.191
4.169
2.552
3.830
42.407
2.444
41.465
53.934
44.130
55.997
44.412
56.223
56.580
44.891
54.349
13.149
46.826
20.215
Jordan-Semones 48
15
A
2.586
46.126
The table 11 shows that there was a wide range in the percent errors. Trial
2, with Molybdenum rod A has the lowest percent error of 2.444, while in trial 9,
with Molybdenum rod B, has the highest percent error of 56.580. The
researchers’ observations have shown that in trial two the rod was heated for five
minutes while in trial nine the rod was heated for three minutes, the longer
heating time could have caused the two trials to vary. The higher percent errors
signify imperfections in the trials and inconsistent testing. The average percent
error of the linear thermal expansion tests on the Molybdenum rods was 41.543.
See Appendix 5 for how to calculate percent error.
Table 12
Unknown metal rod linear thermal expansion percent error table.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
Rod
A
B
B
A
B
A
A
B
B
A
A
B
B
Alpha
Coefficient
(mm x 10-6)
Percent
Error
4.683
4.666
5.019
5.009
5.976
4.790
5.166
5.432
5.873
5.971
4.878
4.813
4.888
2.444
2.792
4.564
4.361
24.495
0.218
7.616
13.165
22.356
24.401
1.618
0.267
1.841
Jordan-Semones 49
14
15
A
A
5.961
4.823
24.194
0.485
Table 12 shows there are relatively small percent error values. This
means that experiment ran well with consistent testing throughout the trials. The
small variability within the values for the alpha coefficient and the average
percent error, 8.988, proves this.
A two-sample t test was chosen to analyze these two data sets. A twosample t test is a test used to find the difference between individual means of a
population. This is the best test to analyze this data because, in a two-sample t
test, the standard deviation of the population is not needed. To use this test, a
few assumptions have to be met first. The first assumption is that this experiment
was a simple random sample, or an SRS. The experiments conducted were
examples of SRSs because the two rods that were used for each test were
picked with the aid of randomization. The second assumption is that the samples
are independent. This assumption was met because the outcomes from one
sample never affected those of the other, and the linear thermal expansion
experiments for the Molybdenum rods and the unknown metal rods were
conducted separately. The third assumption that would have to be accepted
would be that each of the distributions of all of outcomes were normal
distributions. In samples with 30 or more trials, this fact can just be accepted, but
with samples that have less than 30 trials, normal probability plots will have to
prove the normality of the data sets. Figures 5 and 6 show the normal probability
plots of the data sets from the linear thermal expansion experiments.
Jordan-Semones 50
Figure 8. The normal probability plot for the alpha coefficients of the Molybdenum
rods.
The normal probability plot in Figure 8 shows that there is a lot of variation
in the trials. The outliers from the data may have caused this. The researchers
decided to proceed with the two-sample t test even though the normal probability
plot shows that the distribution was not as normal as expected. This may cause
the results of the two-sample t-test to be rendered inconclusive because one
assumption was not fully met.
Figure 9.The normal probability plot for the alpha coefficients of the unknown
metal rods.
Jordan-Semones 51
The normal probability plot in Figure 9 shows that the data had much
variability. The results do not fully prove that the distribution was normal. The
researchers decided to proceed with the two-sample t test even though the
results may be rendered inconclusive.
Figure 10. Box Plots for Molybdenum and Unknown Rods
The box plot shown in Figure 10 illustrates the outliers in the Molybdenum
trial. This could mean that there may have been an error in the trials. Also the
medians for Molybdenum and the unknown metal rod are very far apart meaning
the averages of the trials were not similar. This could signify that the two metals
are not the same because the means for their alpha coefficients are not near
each other.
Jordan-Semones 52
To begin the two-sample t test, two hypotheses must be created: a null
hypothesis and an alternative hypothesis. In this context, the null hypothesis
would be that the mean of the alpha coefficients for the Molybdenum rods would
be equal to the mean of the alpha coefficients for the unknown metal rods. The
null hypothesis states that there is no significant difference between the two
means and that would signify that the unknown metal rods and the Molybdenum
rods are composed of the same metal. In this context, the alternate hypothesis
would be that there is no significant difference between the mean of the alpha
coefficients for the Molybdenum rods and the mean of the alpha coefficients for
the unknown metal rods. The alternative hypothesis shows that there is a
significant difference between the two means and that the two sets of rods are
not composed of the same metal. The equations below represent the two
hypotheses.
Jordan-Semones 53
Figure 11. The results of the two-sample t test for linear thermal expansion.
See Appendix 6 for the t test calculation. The researchers reject their null
hypothesis. There was definitely a significant difference between the means of
the alpha coefficients of Molybdenum rods and the unknown metal rods. The
alpha level, 0.1, allowed the researchers to accept that any p-value below that
may have been deemed significant. This would mean that there is, at most, a
10% chance of calculating such a drastic difference in means. The actual p-value
is 7.633 x 10-10. This value is significantly lower than the alpha level, and it
signifies that there is a 7.633 x 10-8 % chance of calculating means that have
such a drastic difference by chance alone if the null hypothesis was true. The pvalue was lower than the alpha level, 0.1,and that lead to the rejection of the null
hypothesis.
The two experiments that were conducted on the Molybdenum and
unknown metal rods resulted in varied results. For specific heat, the mean that
was calculated from the Molybdenum rods is relatively close to the actual specific
heat of Molybdenum, 0.25. This means that the results of that two-sample t test
Jordan-Semones 54
was more than likely conclusive. For the linear thermal expansion coefficient
experiment, the mean that was calculated from the Molybdenum rods was not
close to the actual alpha coefficient for Molybdenum, 4.8 x 10-6. This means that
the results of the two-sample t test would be considered inconclusive because
the researchers compared the mean alpha coefficient of the unknown metal rods
to a result that may not have been correct. The results of the linear thermal
expansion experiment may have been lowered than it should have been because
the window was opened in the middle of the testing. The change from cooling the
rods for five minutes to cooling them for three minutes in trial 3 may have also
been a contributing factor.
Conclusion
The objective of the experiment was to use linear thermal expansion and
specific heat to determine whether or not the unknown rod was composed of
Molybdenum. The hypothesis, stating that the two rods would not be composed
of the same metal, was accepted. The p-value that was calculated through the
two-sample t test for linear thermal expansion was low, meaning that the means
of the rods from the linear thermal expansion tests had a significant difference
and there was a small chance that the two metals were similar.
Jordan-Semones 55
The results from the linear thermal expansion experiment supported the
hypothesis and the researchers were able to correctly conclude that the unknown
metal rod was not composed of Molybdenum. On the other hand, the results from
the specific heat test did not support the researchers’ hypothesis. The twosample t test for the specific heat test produced results that showed the two rods
could be composed of the same metal, but because both the linear thermal
expansion and specific heat were not relatively close, the researchers correctly
concluded that the unknown rods were not composed of Molybdenum.
The data that the researcher’s received was varied and it was concluded
that those results were a result of the varying experimental procedures. In the
linear thermal expansion coefficient tests, the cooling time was switched from five
minutes to three minutes. This may have caused an alteration in the data. Also,
the window in the lab was opened during both of the experiments and that may
have altered the data. One mistake that could’ve been was that the researchers
assumed that the metal rod was the same temperature as the water after the
boiling process. The researchers had no proof that the metal would be the same
temperature as the water after three minutes of boiling. Overall, there were many
factors that may have contributed to the varied results that were received by the
researchers. If they had access to a more controlled environment, the results
may have been more consistent.
Some problems that the researchers encountered were that they were
using two different linear thermal expansion jigs that read in two different units.
Also, the jigs were not always the same size. At times, the metal rod had plenty
Jordan-Semones 56
of room in the jig, and at others, the rod had barely any room to move at all.
Another problem that the researchers encountered was that they did not have a
fan for cooling the rods, but they did have access to water. The different
temperatures of water could have cooled some rods down more than others.
These inconsistent temperatures may have also contributed to the varying values
in the data.
Many trials were conducted with more than what some would call human
error. At times, the rods were left in the boiling water for up to 30 seconds over
three minutes. Thirty seconds is more than enough time to raise or drop the
temperature of water a few degrees. If the researchers took the rods out after
exactly three minutes, the results might have been more accurate. Also, the
researchers would recommend using a real stopwatch instead of the TI-nSpire
calculator software because there were times when the calculator screen froze
and they had to make up for lost time.
To further expand this research, the researchers would recommend that
others do more than two tests to determine whether or not the metal rods are the
same. It is recommended that future researchers perform tests that involve more
intensive properties, such as a density or pressure test, and actually identify the
metal. If more tests are performed and the actual identity of the metal is found,
then it may serve as aid for scientists working in fields that involve mining and
element identification. Also, knowledge of the linear thermal expansion of a
certain metal may be useful in industries that involve plumbing.
Jordan-Semones 57
Acknowledgements
The researchers would like to thank Mrs. Hilliard for providing them with
the information that they needed to complete this experiment. With her help, the
researchers were able to correctly identify that the metal was not made of
Molybdenum.
The researchers would also like to thank Mr. Supal. With his help, the
researchers were able to create the calorimeters needed in this experiment.
Jordan-Semones 58
Finally, the researchers would like to thank Mrs. Dewey. She helped with
the creation of the Data Analysis section. She also gave the researchers pointers
on what was best to include in the two-sample t test discussion.
Appendices
Appendix 1:
Randomization of trials with the TI-nSpire CX Calculator
1. Assign a number for each choice or option. For example, if there are two
rods, then assign the number 1 to the first rod and the number 2 to the
second rod.
2. From the main menu go to a calculator document screen or a new
scratchpad page.
3. Press the menu button and select 5: Probability.
4. Select 4: Random and from the drop down menu, select 2: Integer.
Jordan-Semones 59
5. A command will appear on the screen as randInt(). In the parentheses
enter the lowest assigned number and the highest assigned number,
separated by a comma. For example, if there are two options, the
command would be: randInt(1,2).
Figure 12. The menu for how to get to the random integer command.
Figure 13. An example of the random integer command.
Appendix 2:
How to Construct the Calorimeter
Materials:
(3) PVC Pipe (0.75 in diameter by
(3) PVC Caps (0.75 in)
5.5 in length)
(6) PVC Caps (1.75 in)
(3) PVC Pipe (1.75 in diameter by 6
Power Drill
in length)
Drill Bit (0.25 in)
Jordan-Semones 60
Tape Measure
Scissors
Procedure:
1. Carefully apply the cement glue to one end of the 0.75 inch PVC pipe.
2. Place one of the 0.75 inch caps on the glued end of the PVC pipe and hold
the it in place for about thirty seconds.
3. Use the tape measure to measure 4.5 inches of the PVC insulation. Cut the
insulation and slide the it over the uncapped end of the 0.75 inch PVC pipe.
4. Slide the 0.75 inch PVC pipe and insulation inside of the 1.75 inch PVC pipe.
5. Carefully apply the cement glue to one end of the 1.75 inch pipe. Make sure
this end is on the same side as the glued end of the 0.75 inch pipe.
6. Place one of the 1.75 inch caps on the glued end of the PVC pipe and hold it
in place for about thirty seconds.
7. Use the power drill to drill a 0.25 inch hole in one of 0.75 inch caps and place
it on the exposed end of the 0.75 inch PVC. Do not glue this cap on.
Jordan-Semones 61
Figure 14. The constructed calorimeters with the holes in the caps.
Appendix 3:
Jordan-Semones 62
Percent Error – Specific Heat Experiment
The equation above demonstrates how to calculate percent error. In the
equation, the true value, or the actual value of the sample, is subtracted from the
experimental value, or the value that was calculated by the researchers. Then,
that number is divided by the true value and multiplied by 100. Figure 5 above
shows a sample calculation from the first trial of the specific heat experiment with
Molybdenum rods.
Figure 5. A sample percent error equation from specific heat.
Appendix 4:
Jordan-Semones 63
Two-Sample T test – Specific Heat Experiment
In the equation, the symbol
stands for the mean of that specific sample.
In this case, the mean of the two samples are 0.236 and 0.277. These are the
two statistics that will be compared. The symbols including S stand for the
standard deviations of the samples. Because this was a two-sample t test, these
two standard deviations were the only two that were needed to analyze the data.
Finally, the symbols including n stand for the individual sample sizes. The
specific heat experiments with Molybdenum and unknown rods were both
conducted with sample sizes of 15. See Figure 6 for a sample calculation.
Figure 6. A sample t-test for specific heat.
Appendix 5:
Jordan-Semones 64
Percent Error – Linear Thermal Expansion Experiment
The equation above demonstrates how to calculate percent error. In the
equation, the true value, or the actual value of the sample, is subtracted from the
experimental value, or the value that was calculated by the researchers. Then,
that number is divided by the true value and multiplied by 100. Figure 7 below
shows a sample calculation from the first trial of the linear thermal expansion
experiment with Molybdenum rods.
Figure 7. The sample percent error equation for linear thermal expansion.
Appendix 6:
Jordan-Semones 65
Two-Sample T test – Linear Thermal Expansion Experiment
In the equation, the symbol
stands for the mean of that specific sample.
In this case, the mean of the two samples are 2.806 and 5.197. These are the
two statistics that will be compared. The symbols including S stand for the
standard deviations of the samples. Because this was a two-sample t test, these
two standard deviations were the only two that were needed to analyze the data.
Finally, the symbols including n stand for the individual sample sizes. The linear
thermal expansion experiments with Molybdenum and unknown rods were both
conducted with sample sizes of 15. See Figure 8 for a sample calculation.
Figure 8. A sample t-test for linear thermal expansion.
Jordan-Semones 66
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