Meli - Stroshein
Zirconium vs. an Unknown Metal
Angelica Meli and David Stroshein
Macomb Mathematics Science Technology Center
Chemistry
10C
Jamie Hilliard, Mark Supal, Christine Dewey
May 23, 2011
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Table Contents
Introduction ................................................................................................ 1
Background ................................................................................................ 2
Review of Literature: Specific Heat ............................................................ 4
Review of Literature: Linear Thermal Expansion ....................................... 6
Problem Statement .................................................................................... 8
Experimental Design: Specific Heat ........................................................... 9
Experimental Design: Linear Thermal Expansion .................................... 12
Data and Observations: Specific Heat ..................................................... 14
Data and Observations: Linear Thermal Expansion................................. 18
Data Analysis and Interpretation: Specific Heat ....................................... 22
Data Analysis and Interpretation: Linear Thermal Expansion .................. 26
Conclusion ............................................................................................... 31
Application ............................................................................................... 33
Appendix A: Labquest .............................................................................. 34
Appendix B: Constructing a Calorimeter .................................................. 35
Appendix C: Jig ........................................................................................ 36
Works Cited ............................................................................................. 37
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Introduction:
Determining whether an unknown metal is Zirconium may seem like an
impossible feat, but a research project achieved this utilizing chemistry. This was
accomplished by comparing two intensive properties of the metals. Intensive
properties are used to identify a substance, regardless of the quantity.
The objective of this experiment was to determine if the unknown metal was
Zirconium.
First, an experiment was performed to calculate the specific heat of the
Zirconium metal sample. Next, the specific heat of the unknown metal sample
was experimentally determined. These values were then compared to each
other using a two-sample t -test to see if the metals were the same or different.
Afterward, an experiment was executed to calculate the linear thermal expansion
coefficient of the Zirconium sample. Subsequently, the linear thermal expansion
coefficient of the unknown sample was experimentally ascertained. Again, a twosample t -test was implemented to interpret the data.
The scientists lastly had to decide whether the metals were the same. They did
this using percent error and a two-sample t -test. Since the data were of a fairly
normal distribution, the p-value was reliable. The small p-value obtained allowed
the scientists to conclude that the unknown metal was not Zirconium. As shown
by this project, chemistry can be applied to a multitude of real-world problems,
including the accurate determination of whether an unknown metal was
Zirconium.
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Background:
The element Zirconium was discovered in 1787 by Martin Heinrich
Klaproth while analyzing jargon. In 1824, Jons Jacob Berzelius isolated it.
Finally, in 1914, it was made into a pure form (Stwertka 117). Zirconium is
obtained from mined baddeleyite (ZrO2). This goes through Sand Chlorination
and the Kroll Process to isolate the Zirconium, as shown in Figures 1 and 2
(“Zirconium | Essential Information”).
ZrO2 (s)  2Cl2 ( g )  2C( g )  ZrCl4 (s)  2ClO( g )
Figure 1. Extraction of Zirconium Tetrachloride Starting With Baddeleyite
ZrCl 4 (s)  2Mg (aq) 
 2MgCl 2 (aq)  Zr (s)
Figure 2. Extraction of Zirconium
In Figure 1, zirconium dioxide reacts with two molecules of dichloride and two
molecules of carbon to produce zirconium tetrachloride and two molecules of
hypochlorite. Then, through the Kroll process in Figure 2, the Zirconium
tetrachloride reacts with two molecules of magnesium, producing zirconium in its
pure form and magnesium chloride (“Zirconium | Essential Information”).
Intensive properties can be used to determine Zirconium’s unique characteristics.
The density of Zirconium is 6.511 g/cm3, higher than that of water, which is 1
g/cm3 (“Density Chart of the Elements"). The specific heat of Zirconium is 0.27
J/g*ºC and is much lower than water’s being 0.39 J/g*ºC ("Specific Heat Capacity
Table"). Zirconium’s linear thermal expansion coefficient is 5.7 m/(m*k)
("Coefficients of Linear Expansion"). Water’s is 51 m/(m*k), and thus its bonding
energy is stronger than that of Zirconium ("Thermal Expansion"). Other intensive
properties can be used to identify Zirconium than just these three. The melting
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point of Zirconium is
1852 ºC, which is higher than 32 ºC for water ("List of
Elements of the Periodic Table”). Also, the boiling point of Zirconium is 4377 ºC
and water’s is 100 ºC ("Chemical Elements.com”). All of these values mean that
Zirconium’s atoms are held a lot tighter than water’s. This is because Zirconium
is a solid and water is a liquid. The molecules in Zirconium have less room to
move due to tighter bonds. Since these bonds are so strong, more energy is
required to them, unlike water’s who are free moving.
[Kr] 4d2 5s2.
Figure 3. Electron configuration of Zirconium.
Figure 3 shows Zirconium has four valence electrons. Zirconium has 40 protons,
40 electrons, and 51 neutrons (Watt 4-5). Thus, Zirconium’s atomic mass is 91
g/mol, which is slightly less than the median value (Chang A-6). The quantity of
protons is what makes Zirconium unique among all elements, giving Zirconium its
intensive properties for its identification. The 40 electrons show that it is in its
ground state of no net charge. The 51 neutrons increase its atomic mass (Watt
5).
This metal is used for the production of products because it is durable. One main
use for zirconium is to coat a dental crown, allowing it to last long (“Zirconium
Basic Uses”). On a much bigger scale, 90% of used zirconium is in nuclear
reactors because zirconium has a low neutron-capture cross-section (“Neutron
Capture”). By having a low neutron-capture cross-section, neutrons will not be
lost as much (Stwertka 118).
Review of Literature for Specific Heat:
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The specific heat of an element was an intensive property that was unique
to every element. It can be used to figure out if the unknown metal was
Zirconium because the specific heat of the unknown sample must match the
specific heat of Zirconium. If there was no match, they were more than likely
different metals, but that could be difficult to determine because the values of
specific heat were close. For example, the specific heat of Zirconium was 0.27
J/g*ºC. Another metal, Niobium, has a specific heat of 0.26 J/g*ºC which was
very close to Zirconium ("Specific Heat Capacity Table"). Other elements were
even closer in value and it can be difficult to distinguish between elements.
Specific heat was basically “the amount of heat required to change the heat
content of exactly one gram of a material by exactly one degree Celsius
(“Specific Heat”). On an atomic level, heat was absorbed by the atoms in the
metal and they began to vibrate faster and faster as they heated up. This then
raised the temperature of the metal and allowed for the determination of how
much heat was needed to raise the temperature of one gram of the metal one
degree Celsius (Pauling 326). To calculate specific heat, specific heat, s, was
equal to the amount of heat released in Joules, q, divided by the quantity of the
mass in grams, m, multiplied by the change in temperature, ∆T.
s
q
m  T
In the experiments, the addition of the metal to the water changed the amount of
heat in the system, which was enthalpy, and allowed for the calculating of
specific heat. One experiment to test for specific heat was to heat up a metal to
a desired temperature and to note what constant temperature it has reached.
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Then the metal was placed into a beaker filled with water and the temperature
the water changed to was recorded. This was called the equilibrium because a
balanced temperature was achieved. Then the specific heat was calculated and
compared to known values ("Specific Heat Experiment" 1-12).
Another experiment to identify the specific heat of a metal involved the use of a
calorimeter. This isolated system prevented the loss of mass and heat so
accurate results were obtained. First, a needed amount of water was weighed
and then put into the calorimeter. After that, the metal was put in a tube
containing boiling water to absorb heat. Finally the metal was moved quickly into
the calorimeter and the equilibrium temperature of the water (“Calorimetry
Specific Heat").
This experiment was used to identify if the unknown metal matched the known
value of Zirconium because they tested an intensive property, which was specific
heat, of the metal. The methods of the experiments were applicable to the
design of the project because they tested the specific heat of the metal and
allowed for the comparison to Zirconium’s specific heat. Therefore, allowing for
the determination if the metals could possibly be the same or different. In these
specific heat experiments, the metal exothermically released heat into the water.
This was what caused it to have an endothermic absorption of heat and reach an
equilibrium temperature. These changes in temperature and the masses of the
metal and the water were what allowed the scientists to find out if the metal
matched Zirconium.
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Review of Literature: Linear Thermal Expansion:
Linear thermal expansion was the increase of an object’s length due to heating.
This was explained by the kinetic molecular theory. When an object is heated, its
atoms gain energy. Thus, they move faster and farther apart, which
consequently increases the size of the object ("223 Physics Lab: Linear Thermal
Expansion").
The equation to calculate the linear thermal expansion coefficient was the
quantity of the object’s final length, L final , minus its initial length in meters, Linitial .
This was then divided by the initial length, Linitial , multiplied by the quantity of the
final temperature in Kelvin, T final , minus the initial temperature, Tinitial ("Thermal
Expansion Equations Formulas Calculator - Linear Coefficient").
a
L final  Linitial
Linitial (T final  Tinitial )
Linear thermal expansion coefficient was an effective way to decide if the second
metal rod was Zirconium ("Study of Heat Capacity Enhancement in Some
Nanostructured Materials"). Since it was an intensive property, each metal had
its own unique expansion coefficient. It was 5.7 for Zirconium ("Coefficients of
Linear Expansion"). The linear thermal expansion coefficient of silver was 19.5
(10-6 m/m K), and 22.2 (10-6 m/m K) for aluminum ("Coefficients of Linear
Expansion."). Thus, the bonds between the atoms of zirconium are weaker and
less strongly attracted to each other than, with higher electronegativity and than
that of silver or aluminum. On an atomic level, this was because zirconium has
more shielding than aluminum. Zirconium has more energy levels. Also,
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Zirconium has fewer protons than Silver, so surrounding atoms are less attracted
to it.
In an experiment to measure linear thermal expansion of a metal rod, the
scientist first measured the length of the rod at room temperature. Then, the
scientist pumped heated water through it, an endothermic process for the metal.
Each minute, the scientist recorded the length. From there, the coefficient was
found. ("Thermal Expansion Expt").
Another experiment measured the linear thermal expansion coefficient with an
apparatus. First, the length of the metal rod was measured. The rod and
thermometer were placed into an apparatus, a tool for measuring the
temperature and length. The hose from the boiler was attached to the metal
jacket. Steam was passed through for five minutes. Meanwhile, the scientist
measured the length and temperature every two minutes. The temperature was
converted from Celsius to Kelvin. Afterward, the equation was used to find the
linear thermal expansion coefficient ("Coefficient of Linear Thermal Expansion").
Additionally, linear thermal expansion was significant in industrial projects.
Sidewalks and roads are built accounting for linear thermal expansion. It was
important to see how much the object will expand, and whether this will cause
cracks or potholes. An object of a high linear thermal expansion coefficient must
be allowed to expand; otherwise, the object can get damaged. Thus, bridges
have expansion joints ( "Temperature and Thermal Expansion."). In this way,
linear thermal expansion has a role in the industrial world.
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Problem Statement:
Is it possible to compare a known metal sample’s intensive properties to an
unknown metal sample’s intensive properties to determine if they were the
same?
Hypothesis:
If the values of specific heat and linear thermal expansion in of an
unknown metal sample are compared against those of a known sample, then the
unknown sample can be determined to be the same or different.
Data Measured:
To compute specific heat in J/g*˚C, the water in the calorimeter’s and the
metal’s temperature was measured in degrees Celsius at the beginning and end
of the experiment, the mass of the water in the calorimeter was measured in
milliliters, and the metal’s mass was measured in grams. To calculate the linear
thermal expansion coefficient, the length of the rod was measured in inches
before it was heated, the change in length was measured in inches while it
cooled down, and the metal’s temperature was measured in degrees Celsius
before and after heating.
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Experimental Design: Specific Heat:
Materials:
Calorimeter
Zirconium samples A and B
Unknown metal samples A and B
Tap water
Metal tongs
Labquest
Labquest Temperature Probe
(0.01 ºC precision)
Corning PC-35 Hot Plate
Loaf pan (5.5” x 3” x 2”)
OHAUS GA200 Scale
(0.0001 g precision)
50 mL Graduated cylinder
Thermometer (0.1 ºC precision)
TI-Nspire calculator
Procedures:
*Safety Precaution: Wear gloves, lab coat and goggles. Also, be extra careful
when dealing with heated equipment to avoid injury.
1.
Construct the calorimeter (See Appendix A: Calorimeter Construction). An
example of a calorimeter can be seen in Figure 4.
2.
Put the temperature probe into the calorimeter through the hole in the
calorimeter’s cap.
3.
Fill the calorimeter with 40 mL of tap water using the graduated cylinder.
4.
For each trial, randomly select Metal Sample A or B to use using the
TI-Nspire calculator’s random integer function.
5.
Record the weight of the metal sample before using it.
6.
Fill the loaf pan with enough tap water to submerge the metal and place it
on the hot plate.
7.
Allow the water to heat up until it boils for three minutes. It should reach
100 ºC.
8.
Place the metal sample into the water and let it stay in the loaf pan for five
minutes.
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9.
Start recording the temperature of the calorimeter’s water with the
Labquest and its temperature probe (See Appendix B) thirty seconds before
adding the metal.
10.
Continue recording the temperature for the rest of the experiment. At the
beginning, this will be the initial temperature and at the end it will be the
equilibrium temperature, which is where the line representing temperature stays
constant on the Labquest’s graph.
11.
After the five minutes, record the temperature of the water the metal is
boiling in as the metal’s initial temperature. Assume it is the same as the water’s
after the five minutes.
12.
Once five minutes have elapsed, remove the metal from the loaf pan with
the metal tongs and put it into the calorimeter. Be sure to do this quickly and
close the cap right away to prevent heat loss.
13.
Continue to record the equilibrium temperature inside the calorimeter for
three minutes.
14.
Uncap the calorimeter and remove the metal carefully and remove the
Labquest temperature probe.
15.
Repeat Steps 1-14 for each trial of the experiment using the unknown
metals. The beaker with the heated water in Step 6 does not have to be emptied
and can be reused for each trial performed.
16.
Compute the specific heat for each trial after all necessary values have
been obtained.
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Diagram:
Graduated
Metal Tongs
Bread Pan
Cylinder
Calorimeters
TI N-Spire
Metal Samples
Temperature Probe
Thermometer
Labquest
Figure 4. Specific Heat Materials
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Figure 4.
Specific Heat
Materials
Meli - Stroshein
Experimental Design: Linear Thermal Expansion:
Materials:
Thermometer (0.1 ˚C precision)
TI-Nspire
Calipers (0.0001 in precision)
Metal Sample A or B
Jig (0.001 in precision)
Bread pan (5.5” x 3” x 2”)
Corning PC-35 Hot Plate
Metal Tongs
Procedures:
*Safety Precaution: Wear gloves, goggles, lab coat, and be careful when working
with heated materials.
1. Randomize if Metal A or B is used using the TI-Nspire.
2. Measure the metal’s length using the caliper.
3. Record the metal sample’s length.
4. Record the temperature with the temperature probe. Assume it is the same
as the water.
5. Place the metal in the boiling water for three minutes. Assume its
temperature reaches that of the heated water.
6. Measure the temperature of the water using the thermometer.
7. Record the water’s temperature.
8. Remove the metal with the metal tongs and place it into the jig.
9. Use the jig, pictured in Figure 5, to measure the change in rod length using
the dial indicator for 4 minutes. (See Appendix C: Jig).
10. Substitute the values obtained into the equation for linear thermal expansion.
11. Repeat Steps 1-10 for all 30 trials.
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Diagram:
Jig
Spray Bottle
Calipers
Graduated
Cylinder
TI-Nspire
Bread Pan
Metal Tongs
Metal Samples
Figure 5. Linear Thermal Expansion Main Materials
Thermometer
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Data and Observations: Specific Heat:
Table 1
Specific Heat Data Zirconium
Initial Temp. (ºC)
Trial
Water
Metal
Equilibri
um
Temp.
(°C)
Change in
Temp. (Cº)
Water
Metal
Specifi
c Heat
Mass
Metal (g)
Water
(mL)
(J/g°C)
1
26.0
97.5
28.5
2.5
-69.0
26.7819
39
0.221
2
26.0
97.5
28.5
2.5
-69.0
26.7647
40
0.227
3
25.6
98.7
29.7
4.1
-69.0
26.7808
41
0.381
4
26.2
98.7
29.6
3.4
-69.1
26.7945
40
0.307
5
24.0
98.7
26.1
2.1
-72.6
26.7306
39
0.177
6
25.4
98.7
28.8
3.4
-69.9
26.6980
40
0.305
7
22.8
97.9
24.1
1.3
-73.8
26.6991
40
0.110
8
27.1
98.3
29.4
2.3
-68.9
26.7763
41
0.214
9
22.2
97.7
25.0
2.8
-72.7
26.7107
40
0.241
10
23.0
97.7
24.7
1.7
-73.0
26.7782
40
0.146
11
22.4
98.0
25.0
2.6
-73.0
26.6982
41
0.229
12
23.3
98.0
25.7
2.4
-72.3
26.7805
40
0.207
13
22.8
98.1
24.9
2.1
-73.2
26.7814
39
0.175
14
22.7
98.1
25.8
3.1
-72.3
26.6984
40
0.269
15
22.3
97.9
24.2
1.9
-73.7
26.7801
40
0.161
Aver
age
0.223
In Table 1 were the data collected from the specific heat experiment for
Zirconium.
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Table 2
Specific Heat Observations Zirconium
Trial
Observations
1
Zirconium A, Scale 1, Labquest 6
2
Zirconium B, Scale 1, Labquest 6
3
Zirconium B, Scale 2, Labquest 6
4
Zirconium A, Scale 4, Labquest 6
5
Zirconium B, Scale 4, Labquest 6
6
Zirconium A, Scale 2, Labquest 6
7
Zirconium A, Scale 2, Labquest 6
8
Zirconium B, Scale 2, Labquest 6
9
Zirconium A, Scale1, Labquest 6
10
Zirconium B, Scale 4, Labquest 6
Zirconium A, Scale 1, Labquest 6, Calorimeter
11
shaken
12
Zirconium B, Scale 2, Labquest 6
13
Zirconium B, Scale 3, Labquest 6
14
Zirconium A, Scale 1, Labquest 6
15
Zirconium B, Scale 2, Labquest 6
In Table 2 were the observations of the specific heat experiment for
Zirconium.
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Table 3
Unknown Metal Specific Heat
Initial Temp.
(ºC)
Trial
Water
Metal
Equilib
rium
Temp.
(°C)
Change in
Temp. (Cº)
Water
Metal
Specific
Heat
Mass
Metal (g)
Water
(mL)
(J/g°C)
1
22.2
97.5
40.7
18.5
-56.8
123.8445
40
0.440
2
22.9
97.5
41.5
18.6
-56.0
120.8949
41
0.471
3
23.3
99.4
39.2
15.9
-60.2
120.8216
39
0.357
4
24.0
99.4
38.5
14.5
-60.9
123.8443
40
0.322
5
28.5
97.9
41.1
12.6
-56.8
123.8485
41
0.307
6
27.6
97.9
41.4
13.8
-56.5
120.8773
40
0.338
7
32.8
97.8
42.2
9.4
-55.6
123.8640
39
0.223
8
32.0
97.8
41.8
9.8
-56.0
120.8958
39
0.236
9
32.8
98.2
44.1
11.3
-54.1
120.8250
41
0.297
10
32.3
98.2
43.4
11.1
-54.8
123.8430
39
0.267
11
31.4
98.4
42.9
11.5
-55.5
120.8236
40
0.287
12
33.3
98.4
41.9
8.6
-56.5
123.8617
40
0.206
13
31.9
98.2
42.3
10.4
-55.9
123.8545
41
0.258
14
31.6
98.2
42.3
10.7
-55.9
120.8954
40
0.265
15
23.8
98.6
42.7
18.9
-55.9
120.8243
40
0.468
Average
0.316
In Table 3 were the data collected from the specific heat experiment for the
unknown metal.
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Table 4
Specific Heat Observations for Unknown Metal
Trial
Observations
1
Metal B, Scale 4, Labquest 6
2
Metal A, Scale 1, Labquest 6
3
Metal A, Scale 2, Labquest 6
4
Metal B, Scale 3, Labquest 6
5
Metal A, Scale2, Labquest 6
6
Metal B, Scale 4, Labquest 6
Metal A, Scale 2, Labquest 6, Issues grabbing
7
metal
Metal B, Scale 1, Labquest 6, Issues grabbing
8
metal
9
Metal A, Scale 2, Labquest 6
10
Metal A, Scale 4, Labquest 6
11
Metal B, Scale 6, Labquest 6
12
Metal B, Scale 4, Labquest 6
13
Metal A, Scale 5, Labquest 6
14
Metal B, Scale 1, Labquest 6
15
Metal A, Scale 2, Labquest 6
In Table 4 were the observations from the specific heat experiment for the
unknown metal.
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Data and Observations: Linear Thermal Expansion:
Table 5
Linear Thermal Expansion Data Zirconium
Trial
Rod
ΔL
(in)
Initial
Initial
Length Temp.
(in)
(ºC)
Thermal
Final Expansion
Temp. Coefficient
(ºC)
(1/ºC x 106)
5.0870
98.5
18.6
2.460
2
A 0.0010
B 0.0010
5.0910
98.5
18.6
2.458
3
B 0.0011
5.0985
98.1
19.7
2.752
4
A 0.0007
5.0885
98.1
19.7
1.755
5
A 0.0010
5.0865
96.7
19.5
2.547
6
B 0.0011
5.0980
96.7
19.5
2.795
7
A 0.0010
5.0875
98.1
20.5
2.533
8
B 0.0010
5.0965
98.1
20.5
2.529
9
B 0.0010
5.0970
98.6
20.7
2.519
10
A 0.0010
5.0855
98.6
20.7
2.524
11
B 0.0009
5.0990
98.5
21.9
2.304
12
A 0.0010
5.0860
98.5
21.9
2.567
13
B 0.0010
5.0965
98.6
22.5
2.578
14
A 0.0010
5.0845
98.6
22.5
2.584
15
A 0.0010
5.0850
98.7
22.5
2.581
1
Average
2.499
In Table 5 were the data from the linear thermal expansion experiment for
Zirconium.
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Table 6
Observations Thermal Expansion Zirconium
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Observations
Jig 7, Caliper 7
Jig 11, Caliper 7
Jig 11, Caliper 7
Jig 7, Caliper 7
Jig 7, Caliper 7
Jig 11, Caliper 7
Jig 11, Caliper 7
Jig 7, Caliper 7
Jig 11, Caliper 7
Jig 7, Caliper 7
Jig 7, Jig shaken, Caliper 7
Jig 11, Caliper 7
Jig 7, Caliper 7
Jig 11, Caliper 7
Jig 11, Caliper 7
In Table 6 were the observations of the linear thermal expansion experiment for
Zirconium.
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Table 7.
Linear Thermal Expansion Data Unknown Metal
Alpha
Coefficient
Initial
Lengh
(in)
Initial
Temp.
(ºC)
Final
Temp.
(ºC)
98.8
21.8
4.118
2
B 0.0015 4.7305
A 0.0030 4.8465
98.8
21.8
8.039
3
A 0.0021 4.8485
98.5
22.1
5.669
4
B 0.0022 4.7285
98.5
22.1
6.090
5
B 0.0023 4.7275
98.1
22.0
6.254
6
A 0.0030 4.8475
98.1
22.0
8.132
7
B 0.0030 4.7270
99.3
21.9
8.200
8
A 0.0025 4.8565
99.3
21.9
6.651
9
B 0.0029 4.7280
98.5
22.1
8.028
10
B 0.0040 4.7785
98.2
21.6
10.928
11
A 0.0025 4.8425
98.5
22.1
6.757
12
A 0.0031 4.8420
98.7
23.1
8.469
13
B 0.0030 4.7300
98.7
23.1
8.390
14
A 0.0050 4.7185
97.2
23.3
14.339
15
B 0.0031 4.8410
99.0
22.6
8.382
Trial
Rod
1
ΔL
(in)
Average
(1/ºC x 106)
7.876
In Table 7 were the data from the linear thermal expansion experiment for the
unknown metal.
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Table 8
Linear Thermal Expansion Observations Unknown Metal
Trial
Observations
1
2
Metal A, Jig 7, Caliper 3, Air blew in room
3
Metal A, Jig 7, Caliper 3, Air blew in room
4
Metal B, Jig 10, Caliper 3, Air blew in room
5
Metal A, Jig 7, Caliper 3, Air blew in room
6
Metal B, Jig 10, Caliper 3, Air blew in room
7
Metal A, Jig 7, Caliper 3, Air blew in room
8
Metal B, Jig 10, Caliper 3, Air blew in room
9
Metal A, Jig 7, Caliper 3, Air blew in room
10
11
Metal B, Jig 10, Caliper 3, Air blew in room
12
Metal B, Jig 10, Caliper 3, Air blew in room
13
Metal A, Jig 7, Caliper 3, Air blew in room
14
15
Metal B, Jig 10, Caliper 3, Air blew in room
Metal B, Jig 10, Caliper 3, Air blew in room
Metal A, Jig 7, Caliper 3, Air blew in room
Metal A, Jig 7, Caliper 3, Air blew in room
In Table 8 were the observations of the linear thermal expansion experiment for
the unknown metal.
.
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Data Analysis and Interpretation: Specific Heat:
Throughout the specific heat experiment, percent error was calculated to see the
accuracy of the results (See Appendix D: Sample Calculations). The percent
errors from the experiment were scattered, but for the most part the results were
acceptable. These percent errors were displayed in Table 9.
Table 9 contains percent errors for both Zirconium and unknown metals. These
values were used to check if the data were viable or not. For Zirconium, it was
optimal to be as close to 20% as possible while for the unknown it was best for it
to be a lot more than 20%.
There were some errors during the experiment. During specific heat, the metal
cooled slightly during its transport from the bread pan to the calorimeter. This
may have caused slightly inaccurate results. Also, a design flaw that contributed
to some of the inaccuracy of the results was the insulation tape did not cover the
entire calorimeter. Solving these errors would have possibly lowered the percent
error and produced better results.
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Table 9
Percent Error for Specific Heat
Percent Error (%)
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Average
Zirconium
18.240
16.089
40.969
13.826
34.602
12.930
59.104
20.792
10.623
46.095
15.242
23.168
35.261
0.453
40.329
25.848
Unknown
63.018
74.554
32.114
19.169
13.8
25.249
17.51
12.518
9.8338
1.153
6.3019
23.827
4.562
1.8591
73.454
25.261
To examine the results of the specific heat experiment, a two sample t -test was
used. This was appropriate because there were two independent samples, and
the population means and standard deviations were unknown. The null
hypothesis, H 0 : 1   2 , where 1 was the population mean of Zirconium’s
specific heat and  2 population mean of the unknown metal’s specific heat. This
was the standard hypothesis of the experiment. The alternative hypothesis,
H a : 1   2 , was a hypothesis tested to see if the mean of specific heat for the
unknown sample was not the same as the mean of specific heat for Zirconium. It
had the same variables as the null hypothesis except it was tested to see if the
23
Meli - Stroshein
specific heats were unequal. As shown in Figure 6 and Figure 7, the data was
not skewed enough to really affect the results of the experiment. The data can
then be further used for statistical tests. This was also confirmed in Figure 8
where the box plots were normally distributed and no outliers were apparent.
Figure 6. Zirconium Probability Plot
Figure 7. Unknown Probability Plot
Figure 8. Comparison of Zirconium
and Unknown Data
The equation to test this was composed of the quantity of Zirconium’s average,
x1 , minus the average of the unknown metal sample’s average, x 2 , divided by
the square root of Zirconium’s standard deviation, s1 , divided by Zirconium’s
sample size, n1 , plus the standard deviation of the unknown metal sample, s2 ,
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divided by the unknown metal’s sample size, n2 . This equation determined the tvalue, which was later used to find the p-value.
t
x1  x 2
2
2
s1
s
 2
n1
n2
Figure 9. Two-sample t -test along with Probability Graph
Figure 9 showed the standard deviations and means. The low standard
deviation meant that the data diverged very little from the mean, and it promotes
the accuracy of the data. The mean shows the average value of data. In
conjunction, they greatly increased the potency of the acquired data.
The statistics led to the rejection of the null hypothesis. The metals were not the
same based on their compared average specific heats. If the metals were the
same, the scientists would obtain these results only 0.337% of the time. Figure 4
gave an illustration of just how much that probability value of 0.337% really was.
From the bell curve in Figure 9, it was shown just how much that
p-value really
was. When this p-value was compared to an alpha level of 5.000%, the p-value
was significant. These results would only occur 0.3161% of the, time making
them very unlikely to have happened by chance alone.
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Data Analysis and Interpretation: Linear Thermal Expansion:
Again, for the next experiment, which was linear thermal expansion, percent error
was calculated as shown in Table 10. It was used as before to check and make
sure the trials were done accurately. Zirconium’s results should be around 20%
while the unknown results should be a lot over. The percent errors for the
experiment were not where they should have been, but these percent errors
were just a guide. Different statistical tests could be used to determine if
Zirconium and unknown metals were the same.
Many prominent errors caused the high percent error. The metal was not moved
quickly enough from the bread pan into the jig. Also, it may not have always
cooled to room temperature because not enough time was given to do so. There
were many mistakes when moving the sample into the jig such as stalling and
dropping it back in the water. Together, these lurking variables could have
confounded the results. The dropping of the metal could have been caused by
the design of the metal prongs that were used since it was hard to grab the metal
with them.
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Meli - Stroshein
Table 10
Percent Error for Thermal Expansion
Percent Error (%)
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Average
Zirconium
56.837
56.870
51.721
69.217
55.323
50.966
55.562
55.640
55.815
55.715
59.575
54.968
54.766
54.659
54.723
56.157
Unknown
27.753
41.035
0.541
6.839
9.721
42.674
43.853
16.681
40.849
91.719
18.550
48.573
47.185
151.563
47.048
38.173
For linear thermal expansion experiment, a two-sample t -test was again
performed. The two-sample t -test was appropriate because there were two
samples that were independent of each other and the population means and
standard deviations were unknown. The null hypothesis was H 0 : 1   2 , where
1 was the mean of Zirconium’s linear thermal expansion and  2 was the mean
of the unknown metal’s linear thermal expansion. The alternative hypothesis,
H a : 1   2 , was tested. The variables were the same as the null hypothesis.
As shown in Figure 10 and Figure 11, the data were skewed and located far from
the line in the probability plot. Thus, the results of the two-sample t -test may be
27
Meli - Stroshein
unreliable. Also, in Figure 12, there were two box plots displaying the data.
Figure 12 shows that the box plots for the unknown and known samples had
strikingly different values. The data were distributed differently, and the sample
means were far apart.
Figure 10. Zirconium Probability Plot
Figure 11. Unknown Probability Plot
Figure 12. Comparison of Zirconium
and Unknown Data
The equation to test this was composed of the quantity of Zirconium’s average
coefficient, x1 , minus the average of the unknown metal sample’s average
coefficient, x 2 , divided by the square root of Zirconium’s standard deviation, s1 ,
divided by Zirconium’s sample size, n1 , plus the standard deviation of the
unknown metal sample, s2 , divided by the unknown metal’s sample size, n2 .
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Meli - Stroshein
This equation determined the t-value, t , which allowed the p-value to be
evaluated.
t
x1  x 2
2
2
s1
s
 2
n1
n2
Figure 13. Two-Sample t -Test and Probability Graph
Figure 13 showed the standard deviation. The low values of standard deviation
mean little variance from the mean. The mean shows the average value. All
these promote the validity of the data.
There was a weakness with comparing huge quantities with minuscule quantities.
The coefficient, as shown by the jig, was extremely small. In comparison, the
mass was large. These could have contributed to the weaknesses of the
experimental design because one small change could yield a huge effect.
The statistical test results caused the failing to reject the null hypothesis. The
metals were not the same, according the results from when their thermal
expansion coefficients were compared. If the metals were the same, the
scientists would obtain these results almost 0.000% of the time. This was
displayed in Figure 13. The results were not obtained by chance alone. When
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Meli - Stroshein
this value was compared to an alpha level of 5%, the results of the experiment
were statistically significant. However, due to the abnormality of the data, this
statistical test should not fully be trusted.
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Meli - Stroshein
Conclusion:
The hypothesis, an unknown metal can be compared to Zirconium to
decide if the metals were the same by determining specific heat and thermal
expansion coefficient of an unknown sample, was accepted. The samples had
contrasting values for specific heat and linear thermal expansion. This was
because these intensive properties were unique to each metal. Thus, they were
used to determine if the metals were the same or different. The problem
statement was if it was possible to compare two metal’s intensive properties to
determine if they were the same or different. The experiments were conducted
to answer this question.
The hypothesis was accepted because of the data’s comparison to
Zirconium’s specific heat and linear thermal expansion values. It was clear that
the data for the Zirconium and unknown samples were starkly different. Thus,
the samples they were unlikely to have the same identity. The large percent
errors helped decide that the samples were not the same. Finally, the twosample t -test showed that it was incredibly unlikely for the metals to be the same
due to extremely small p-values. These p-values meant that the results proved
unlikely to have occurred by chance alone. In conjunction, these substantial
forms of evidence proved that the unknown metal was not Zirconium.
Although this experiment ran well, it displayed a few minor errors. During
specific heat, the transport of the metal from the bread pan to the calorimeter
was of long duration, and this caused a loss of heat, confounding the data.
During linear thermal expansion, many difficulties arose while placing the sample
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Meli - Stroshein
into the jig. Often, it did not align precisely straight in the ridge of the jig, which
results in inaccurate measurements. Measuring the metal’s change of length
with the jig also was inaccurate because it was sometimes difficult to read the jig
and mark the starting and stopping locations to obtain the change in length. To
increase the validity of the data, the scientists could have transferred the metal
from the bread pan more quickly to prevent the loss of heat from the metal. Also,
it would have been prudent to have practiced placing a metal into the jig more
often. Consequently, the data would have been more accurate.
Additional experiments could have been performed to support the
research. Other intensive properties could have been tested in these
experiments. The scientists could have measured density. They would have
found the volume after measuring diameter and length. Mass would have been
found using a scale. Density would have been the quotient of mass divided by
volume. Densities would have been found for the Zirconium and unknown
metals. Then, these values would have been compared using a two-sample t test. Thus, the comparison of density values would have further revealed
whether the unknown sample was Zirconium.
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Application:
The scientists constructed a ring from Zirconium, represented in
Figure 14. The diameter of the inner circle is 27 mm and the diameter of the
outer circle is 40 mm. The height of the ring is 10 mm. This is just a sample ring,
so other sizes of the ring can be made.
This ring can be useful in many ways, but one of the most important is because
Zirconium is resistant to corrosion when it comes in contact with oxygen, allowing
for it to keep its shiny surface undamaged. Also, it is light weight and will last a
long time. Therefore, it is long lasting for the consumer’s use. The mass of this
ring is 45.15 g with a surface area of 3473.03 mm2. The cost to make this
product would be around $6.77.
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Appendix A: Labquest:
Materials:
Labquest
Temperature Probe
Procedures:
1.
Turn on the Labquest.
2.
Plug the temperature probe into Channel 1 on the top of the Labquest.
More temperature probes can be added if more than one trial is done at once. In
Figure 9, two temperature probes are plugged in to do two trials.
3.
On the far left tab, click rate.
4.
Change collection to 2 samples per second.
5.
Set length of trial to 240 seconds.
6.
Press okay.
7.
Press the play button to begin data collection.
Figure 9. Labquest Set-up
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Appendix B: Constructing a Calorimeter:
Materials:
PVC Pipe (60 in by 3/4 in)
PVC Pipe Cap (3/4 in diameter)
Buzz Saw
Cork (3/4 in diameter)
Insulation tape
Drill
Procedures:
1.
Take a PVC pipe and cut it to the length of seven inches using the buzz
saw.
2.
Cap one end with the PVC cap.
3.
Wrap the pipe and the cap with the insulation tape until it is completely
covered.
4.
Drill a hole through the cork using a drill.
5.
Put the cork in the hole opposite of the cap. The final product is displayed
in Figure 15.
6.
Put the temperature probe through the drilled hole.
7.
Repeat Steps 1-6 for additional calorimeters.
7 in
3/4 in
Figure 15. Constructed Calorimeter
35
Meli - Stroshein
Appendix C: Jig:
During the thermal expansion experiment, a jig was used to measure the change
in length of the metal. The jig, as shown in Figure 1, consisted of a dial that
measured the change in length connected to a wooden board with a ridge to
insert the metal sample. After the metal was heated, the scientist pulled up the
spring in the dial to insert the sample. The metal was inserted, and the dial was
set to zero. While the metal cooled, it was sprayed every two seconds to
expedite the cooling process. Once it cooled to room temperature, the scientists
could read the dial to ascertain the change in length in inches.
18 cm
4 cm
9 cm
Figure 1. Jig Construction
36
2 cm
Meli - Stroshein
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39