portfolioae.weebly.com · Web viewThermal expansion causes significant stress on a component if its design does not allow expansion and components (Linear Coefficient of Thermal

Comparing an Unknown Metal with Hafnium Using Specific Heat and Linear Thermal Expansion
Emily Chapman – Aaron Espere
Honors Chemistry – 10C
May 20, 2014
Table of Contents
Problem Statement………………………………………………………………….12
Experimental Design: Linear Thermal Expansion………………………………17
Data and Observations………………………………………………………………20
Conclusion……………………………………………………………………………46
Introduction
All metals have their own specific properties. No two metals are completely alike. Different metal properties are used to determine which metal is the best fit for the job it needs to do. Not just physical properties or chemical properties, but intensive properties are looked into too. Some of the intensive properties observed in different metals to determine which one will fit the job the best are its melting point, specific heat, and linear thermal expansion. These properties can be important when deciding which metal will build the highest quality and most stable building or structure. One example of this is when builders need to determine which metal to use to build the most stable house. Due to the expansion of metal when exposed to heat, knowing what metal expands the least when exposed to heat will enable the best quality house. Thermal expansion causes significant stress on a component if its design does not allow expansion and components (Linear Coefficient of Thermal Expansion), so choosing the best metal that allows expansion for that design allows a stable house, one that can reside under heat and take abuse. Intensive properties may not important, but great problems around the world would be caused if intensive properties were not observed.
The purpose of this experiment was to identify if two sets of metal rods (one known hafnium, one unknown) were the same using the intensive properties of specific heat and linear thermal expansion. Various sources of evidence were used to support the theory if the metals were the same. Different procedures were used to collect the specific heat and linear thermal expansion of the metals. In the specific heat trial, the metal rods were heated up. At the same time, water was put into a calorimeter at room temperature. Then the metal was put in the calorimeter, and the equilibrium temperature was taken. In the linear thermal expansion experiment, a rod was heated up to near-boiling temperature, and then put on a jig and then cooled down. The change in length from when the rod entered the jig from when it was cooled down was taken. Percent error tables were used to compare the values gotten for specific heat and linear thermal expansion for the unknown rods to the true value of hafnium. Also, a two sample t-test was used to tell if the mean of the values were the same.
Hafnium has many uses out in the real world. Hafnium is an excellent neutron absorber. Hafnium also has a very high melting point, making it able to withstand extreme amounts of heat without melting. Hafnium is also corrosion resistant, which makes it very good to work in nuclear environments. Finally, hafnium has an affinity to nitrogen and oxygen, making it very good at scavenging nitrogen and oxygen (Avalon Rare Metals – Hafnium). All these traits give hafnium a nice place in the real world. For example, being corrosion resistant and a good attractor of neutrons, hafnium is able to be used for reactor control rods. These reactor control rods are put into nuclear plants and nuclear submarines (Avalon Rare Metals – Hafnium). With its ability to scavenge oxygen and nitrogen from its affinity, and its high melting point, it can be used in gas-filled, and incandescent lamps. Hafnium is also used in alloys, which are used in liquid rocket thruster nozzles (Periodic Table of Elements: Hafnium).
Background and Review of Literature
The purpose of this experiment was to determine the identity of an unknown sample and comparing it with an already known sample using intensive properties and seeing if they were the same. The previously known sample was found through the intensive property of density, and was discovered to be hafnium. The unknown metal will be identified using specific heat and linear thermal expansion, and compared to the specific heat and linear thermal expansion of the known metal, hafnium.
The metal element known as hafnium was first discovered by Danish chemist, Dirk Coster, and Hungarian chemist, Charles de Hevesy in 1923. Hafnium was discovered when these chemists used a method known as X-ray spectroscopy to analyze the arrangement of the outer electrons of atoms zirconium ore samples. The name hafnium originates from the Latin word for the city of Copenhagen, Hafnia. Hafnium has a density of 13.3 g/cm³. Hafnium also withstands extreme heat with a melting point of 2233°C (Gagnon). The boiling point of hafnium is also extremely high at 4603°C. Therefore, hafnium can withstand very high temperatures and can be used in extreme conditions. This heat resistance along with the density and strength of hafnium makes this metal a vital and very useful material in the production of many products.
Specific heat is one commonly used way to identify what a substance is. This is because it is very accurate and easy to find when the right tools are used. Specific heat is such a good technique to identify a substance because it is an intensive property. When something is intensive, it means that the property will remain the same, and the size of the sample does not matter.
The kinetic molecular theory states that when molecules have energy added to them, they move around faster (Brucat). When they move faster, they collide with each other more often, producing heat. Specific heat is the amount of energy that must be added to vibrate the molecules quickly enough for one gram of a substance to rise one degree Celsius. Even if the sample is big, the ratio between the weight of the sample and the temperature will always be the same, which is why specific heat is intensive. The specific heat of the same material is the same no matter what because the energy used to raise the material one degree Celsius is divided by the weight.
The equation above shows how to calculate the specific heat of any object and the units that go along with it. Specific heat, S, is the change in heat in joules per gram Celsius, Q measured in joules, divided by the mass in grams, m, times the change in temperature in degrees Celsius, t (Nave). The units used when discussing specific heat is J/g*C.
Specific heat is important to industry because industries need to know how much heat an object is able to hold, as well as how long it will be able to hold that heat. Companies that make products such as thermoses and coolers need to know what material will hold the heat or cold in their container for the most amount of time. The company with the container retaining internal heat the most effectively will have the most customers. This is an example as to why it is important for industries that have to deal with retaining heat to know about the importance of specific heat.
One experiment done that was a good experimental design for specific heat capacity is Experiment VIII: Specific Heat and Calorimetry. This experimental design fits well with the definition of what calorimetry, as well as what a calorimeter is. Calorimetry is the science associated with figuring out the changes in energy of a system by measuring the heat exchanged with the surroundings. A calorimeter is a container to hold a sample in which the heat measured causes a change of state. A calorimeter being used as an isolated system is talked about in this paper, and can be used for determining the specific heat of various types of elements, such as metals. This experiment goes through the process of measuring specific heat of different metals including aluminum, brass, and steel using a calorimeter. (Experiment VIII: Specific Heat and Calorimetry).
The experimental design for a lab using specific heat capacity is also found in the article ‘H-2 Specific Heat Capacity’. Different theories are listed and described, including heat, Newton’s Law of Cooling, Heat Units, Water Equivalent, Calorimetry, and Specific Heat Capacity. The specific heat section uses the formula to determine specific heat, and the process that takes place during it.
These experiments are both relevant to this project because they provide information the researchers need to know about specific heat, as well as calorimetry. One experiment gives step by step instructions on how to set up and run an experiment using a calorimeter. These instructions work for any metal, not just the ones used in this specific experiment. The second experiment gives an explanation of the formula used for calculating specific heat. With this information, the researchers can have a better understanding of how to use a calorimeter, and a better understanding of how to calculate specific heat.
Another way to identify a metal is through linear thermal expansion. In linear thermal expansion is an intensive property, like how specific heat is. Thus, by definition, the size of the sample will not affect the expansion coefficient (Chang 15).
Kinetic molecular theory explains why a metal expands when heat is added. As the temperature increases, so does its kinetic energy. In a solid, the molecules are packed close together. While these molecules keep pushing farther apart, the substance expands. This expansion is what is measured as the linear expansion coefficient. The coefficient is alpha, which is 1/ºC. This makes linear thermal expansion intensive because the alpha is based off of the amount of energy added to the substance and the amount it expanded. The coefficient is important to engineers so that they know what materials to use for structures such as bridges and tunnels so they do not break or expand.
The equation above shows how to calculate any substance’s linear thermal expansion coefficient. The coefficient alpha,, is measured in 1/°C and is the change in length in millimeters, , divided by the change in temperature in degrees Celsius, , times the initial length in millimeters, L.
The experiment Thermal Expansion Experiment by David Harrison Joe Vise, and Claude Plante explains linear thermal expansion well. It shows what it should visually look like when a metal rod is heated, and how it should look when it expands. This will help with the experiment when linear thermal expansion is being tested because it can visually show how it may look when the metal rods are being heated.
Another good example of linear thermal expansion was found in the paper written by Tong Wa Chao. This experiment goes through how to find linear thermal expansion of printed wiring components. The conclusion of this paper talks about using a change in temperature to determine the significant changes in the thermal expansion coefficient.
Both of these linear thermal expansion experiments are relevant because they go through the processes of finding a thermal expansion coefficient. The first experiment helps create a visualization of how linear thermal expansion works, and what occurs during its process. The second experiment goes through temperature change and how that has an effect on the changing thermal expansion coefficient.
Problem Statement
Problem Statement:
To determine the identity an unknown metal using the intensive properties of specific heat and linear thermal expansion and comparing it to the known values of Hafnium.
Hypothesis:
Using the intensive properties of specific heat and linear thermal expansion, the identity of the unknown rod will be determined to be Hafnium within a 3 percent error of the true values of linear thermal expansion and specific heat.
Data Measured:
In the specific heat experiment, the specific heat of a metal was computed in joules/grams*Celsius (J/g°C). In order to find the specific heat, the temperature of the metal rods when heated up and the water when it was put into the calorimeter was taken in Celsius. The heated metal rod was put into the room temperature water, and left until equilibrium was reached, which was taken in Celsius. The mass of the rod and the water were both taken in grams. Once all variables were found, they were all put into an equation (appendix A) to compute the specific heat.
In the linear thermal expansion experiment, the coefficient of linear thermal expansion was computed in millimeters per degree Celsius (mm/C). The original length of the metal rods was taken in millimeters, and the difference in length from when the metal rod was near-boiling point, and when it was cooled down was taken as the change in length in millimeters also. The temperature of the rod was taken in Celsius. After all the variables were collected, the coefficient of linear thermal expansion was computed through an equation (appendix A).
Specific Heat Experimental Design
Tongs
Labquest
Procedure:
Be aware of safety precautions. Wear gloves, goggles, and appropriate attire.
1. Using the TI-Nspire calculator, randomize 15 trials for both of the Hafnium rods, and both of the unknown metal rods, also randomize the calorimeter to use with each trial.
2. Set up Labquest to collect data. Set the samples/seconds to 1 and leave it to run for 180 seconds.
3. Using the scale, mass the rod being tested. Record the results in the table.
4. Fill the loaf pan with 50 ml of water.
5. Using tongs put the metal being tested in the loaf pan.
6. Place the loaf pan on the hot plate set at 9 until boiling point is reached. After it has been heated, take the temperature of water using the thermometer to make sure the temperature is near boiling point (100°C – 2°).
7. Put metal rod into the loaf pan using tongs and leave it for four minutes. After four minutes, take temperature of the water, and record as the initial temperature (in °C) of the metal.
8. Put 65 mL of room temperature water into the calorimeter and put the temperature probe inside. Record temperature of water that temperature probe gives as initial temperature of water.
9. Take out temperature probe. Using tongs, place metal inside calorimeter, afterwards put temperature probe inside calorimeter through the cap.
10. Once calorimeter is closed, begin temperature recording on Labquest.
11. Wait until water has reached equilibrium.
12. Record temperature when equilibrium is reached as equilibrium temperature (in °C).
13. Repeat steps 3-12 for each trial.
14. After all trials have had measurements taken, refer to Appendix A to calculate the specific heat for all trials.
Diagram:
Calorimeters
Figure 1. Specific heat materials diagram
Figure 1 shows the materials used during the trials of this experiment. The materials labeled in the diagram were the most important materials used in the experiment.
Linear Thermal Expansion Experimental Design
Materials:
Loaf Pan
Hot Plate
Procedure:
Be aware of safety precautions. Wear gloves, goggles, and appropriate attire.
1. Using the TI-Nspire calculator, randomize 15 trials for both of the Hafnium rods, and both of the unknown metal rods.
2. Set aside thermometer where it will record the room temperature.
3. Pour 35 mL of room temperature water into the loaf pan.
4. Using the Caliper, measure the length of the metal rod being tested. Record this as initial length (in mm).
5. Place the loaf pan on the hot plate set at 9 until the water boils. After it has been heated, take the temperature of the water using a thermometer to make sure the temperature is near boiling point (at least 1° off).
6. Put metal rod being tested into the loaf pan using tongs and leave it for four minutes. After four minutes, take temperature of the water, and record as the initial temperature (in °C) of the metal.
7. Using tongs quickly move the metal rod to the LTE jig and mark the point where the dial starts using a marker.
8. Turn on fan and wait for the metal to cool down to room temperature. Once it has cooled down, mark where the dial stops, this is the final point.
9. Find the difference in length between the final point and the initial point (in mm). Record this as the change in length, or L.
10. Once the metal has cooled down, it should be room temperature, using the thermometer set aside in beginning, record as final temperature (°C).
11. Repeat steps 3-10 for all other trials.
12. Once trials have been done, solve for alpha coefficient for all trials (appendix A).
Diagram:
Figure 2. Linear Thermal Expansion Materials.
Figure 2 shows a diagram of the materials used in the linear thermal expansion trials. The most important materials used in this experiment are labeled in the diagram.
Data and Observations
Trial
Rod
Water
Metal
Water
Metal
Water
Metal
1
B
23.6
99.4
25.7
2.1
73.7
65
51.94
0.149
2
A
22.6
99.6
24.7
2.1
74.9
65
51.91
0.147
3
B
24.3
99.1
26.3
2.0
72.8
65
51.94
0.144
4
A
24.2
99.5
26.3
2.1
73.2
65
51.91
0.150
5
B
23.9
99.5
26.0
2.1
73.5
65
51.95
0.150
6
B
26.4
99.3
28.4
2.0
70.9
65
51.97
0.148
7
A
24.3
99.4
26.3
2.0
73.1
65
51.91
0.143
8
A
25.3
99.8
27.3
2.0
72.5
65
51.94
0.144
9
B
25.1
99.4
26.4
1.3
73.0
65
51.95
0.093
10
B
23.5
99.8
25.5
2.0
74.3
65
51.97
0.141
11
B
24.9
99.6
26.9
2.0
72.7
65
51.97
0.144
12
A
24.4
99.5
26.4
2.0
73.1
65
51.92
0.143
13
B
28.4
99.3
30.3
1.9
69.0
65
51.95
0.144
14
A
24.3
99.5
26.3
2.0
73.2
65
51.91
0.143
15
A
24.7
99.6
26.7
2.0
72.9
65
51.94
0.144
Average
24.7
99.5
26.6
2.0
72.9
65
51.94
0.142
Table 1 shows the results gotten for the known hafnium metals A and B when the specific heat trial was conducted. The data is widely consistent throughout the trials, except for trial nine, where it was lower. The explanation for this is that the rod was dropped during the transfer to the calorimeter, possibly lowering the temperature of the rod before it was put into the calorimeter.
Table 2
Trial
Observations
1
Rod B, calorimeter B, the metal boiled for three minutes, rod placed in calorimeter by experimenter 1
2
Rod A, calorimeter A, the metal boiled for three minutes, rod placed in calorimeter by experimenter 1
3
Rod B, calorimeter A, the metal boiled for three minutes, rod placed in calorimeter by experimenter 1
4
Rod A, calorimeter B, the metal boiled for three minutes, rod placed in calorimeter by experimenter 1
5
6
7
8
9
Rod B, calorimeter B, the metal boiled for three minutes, rod placed in calorimeter by experimenter 1, experimenter 1 dropped metal rod while moving it from the loaf pan to the calorimeter
10
11
12
Rod A, calorimeter B, the metal boiled for three minutes, the amount of water being boiled in the loaf pan was low, rod placed in calorimeter by experimenter 1
13
14
15
Table 2 shows the observations made during the trials testing the specific heat of the known metal rods. The rod used, the calorimeter used, how long the metal was boiled for, and which experimenter placed the rod in the calorimeter was observed. Possible errors in the experiment were also observed in Table 2.
Table 3
Water
Metal
Water
Metal
Water
Metal
1
B
29.4
99.4
35.7
6.3
63.7
65
72.80
0.369
2
B
27.2
99.5
33.4
6.2
66.1
65
72.85
0.350
3
A
25.9
99.4
33.9
8
65.5
65
71.98
0.462
4
B
20.3
99.6
29.7
9.4
69.9
65
72.82
0.502
5
A
26
99.7
33.6
7.6
66.1
65
71.98
0.434
6
B
27.3
99.9
33.7
6.4
66.2
65
72.86
0.361
7
A
22.1
99.5
29.9
7.8
69.6
65
71.98
0.423
8
A
23.2
99.8
31
7.8
68.8
65
71.98
0.428
9
B
21.4
98.8
30.2
8.8
68.6
65
72.84
0.479
10
A
23.2
99.8
31.5
8.3
68.3
65
71.97
0.459
11
B
22.4
99.8
32.3
9.9
67.5
65
72.95
0.547
12
B
20.6
99.5
29.8
9.2
69.7
65
72.97
0.492
13
A
27
99.9
32.9
5.9
67.0
65
72.82
0.329
14
A
22.2
99.9
29.9
7.7
70.0
65
71.94
0.416
15
A
21.1
99.9
29.7
8.6
70.2
65
71.98
0.463
Average
23.9
99.6
31.8
7.9
67.8
65
72.45
0.434
Table 3 shows the results gotten for the unknown metal rods A and B when the specific heat trial was conducted. The specific heat did not seem to be consistent throughout all 15 trials, but the initial heats of the water wasn’t consistent, and the change in temperature wasn’t consistent, and it may have been affected by that.
Table 4
Trial
Observations
1
Rod B, scratches on side of rod, calorimeter A, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
2
Rod B, calorimeter B, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
3
Rod A, calorimeter A, the metal boiled for four minutes, rod placed in calorimeter by experimenter 2
4
Rod B, calorimeter A, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
5
Rod A, calorimeter B, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
6
Rod B, calorimeter B, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
7
8
9
Rod B, calorimeter B, the metal boiled for four minutes, rod placed in calorimeter by experimenter 2
10
11
Rod B, calorimeter A, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
12
Rod B, Calorimeter A, the metal boiled for five minutes, rod placed in calorimeter by experimenter 2
13
14
15
Rod A, calorimeter B, the metal boiled for four minutes, rod placed in calorimeter by experimenter 2
Table 4 shows the observations made during the trials testing the specific heat of the unknown metal rods. The rod used, the calorimeter used, how long the metal was boiled for, and which experimenter placed the rod in the calorimeter was observed. Possible errors in the experiment were also observed in Table 4.
Table 5
Trial
Rod
T
1
A
125.28
0.010
98.9
25.5
73.4
1.087E-06
2
A
125.28
0.030
98.7
24.7
74.0
3.236E-06
3
B
125.09
0.030
98.7
25.7
73.0
3.285E-06
4
A
127.47
0.060
98.9
26.0
72.9
6.457E-06
5
B
127.35
0.060
98.3
25.5
72.8
6.472E-06
6
A
127.48
0.055
98.7
23.8
74.9
5.760E-06
7
B
127.46
0.050
98.6
23.8
74.8
5.244E-06
8
B
127.35
0.050
98.8
24.8
74.0
5.306E-06
9
A
127.47
0.050
98.5
24.6
73.9
5.308E-06
10
B
127.46
0.049
98.6
24.5
74.1
5.188E-06
11
B
127.46
0.058
98.4
24.4
74.0
6.149E-06
12
A
125.28
0.051
98.7
24.5
74.2
5.486E-06
13
A
125.28
0.049
98.9
24.3
74.6
5.243E-06
14
B
127.46
0.051
98.8
24.6
74.2
5.393E-06
15
A
125.28
0.060
98.5
24.6
73.9
6.481E-06
Table 5 shows the results gotten for the known hafnium rods A and B when the linear thermal expansion trail was conducted. The alpha coefficient was not very consistent for the first three trials, that is because a sufficient cooling method was not found to completely cool down the metals in the set amount of time that was given. As more trials were done for the linear thermal expansion, the more the trials started to go to a more consistent number.
Table 6
Trial
Observations
1
Rod A, the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for five minutes
2
Rod A, the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for six minutes
3
Rod B, the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for five minutes
4
Rod A, boiled for six minutes, placed into jig by experimenter 1, rod left in jig for five minutes
5
Rod B, the metal boiled for four minutes, placed into jig by experimenter 1, rod left in jig for five minutes
6
Rod A, the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for seven minutes
7
8
Rod B, the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for eight minutes
9
Rod A , the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for five minutes
10
Rod B, the metal boiled for five minutes, placed into jig by experimenter 1, rod left in jig for four minutes
11
12
Rod A, the metal boiled for six minutes, placed into jig by experimenter 2, rod left in jig for six minutes
13
Rod A, the metal boiled for five minutes, placed into jig by experimenter 2, rod left in jig for five minutes
14
15
Rod A , the metal boiled for four minutes, placed into jig by experimenter 2, rod left in jig for five minutes
Table 6 shows the observations made during the trials testing the linear thermal expansion of the known metal rods. The rod used, how long the metal was boiled in the water, which experimenter placed the rod in the jig, and how long the rod was in the jig was observed. Varying times while boiling the metal in trials were observed in Table 6.
Table 7
Trial
Rod
T
1
A
128.57
0.120
96.0
26.0
70.0
1.333E-05
2
B
128.61
0.100
99.0
26.0
73.0
1.065E-05
3
A
128.61
0.110
98.0
24.0
74.0
1.156E-05
4
B
128.67
0.090
99.0
24.0
75.0
9.326E-06
5
A
128.67
0.100
99.4
27.0
72.4
1.073E-05
6
B
129.98
0.089
99.8
26.0
73.8
9.278E-06
7
A
128.58
0.100
99.0
29.0
70.0
1.111E-05
8
B
129.87
0.098
98.0
26.0
72.0
1.048E-05
9
A
128.50
0.090
99.0
24.6
74.4
9.414E-06
10
A
128.50
0.110
98.9
23.9
75.0
1.141E-05
11
A
128.50
0.098
99.4
24.6
74.8
1.020E-05
12
B
129.87
0.090
97.9
25.1
72.8
9.519E-06
13
A
128.50
0.090
98.7
23.9
74.8
9.363E-06
14
B
129.87
0.090
99.1
22.9
76.2
9.094E-06
15
A
128.50
0.100
98.6
23.4
75.2
1.035E-05
Averages
128.92
0.098
98.7
25.1
73.6
1.039E-05
Table 7 shows the results gotten for the unknown rods A and B when the Linear Thermal Expansion trials were conducted. The Alpha Coefficient is very inconsistent even though most of the other numbers were consistent. A fan was used in these trials, so the metals could cool faster, but it still didn’t give consistent numbers.
Table 8
Trial
Observations
1
Rod A, the metal boiled for four minutes, placed into the jig by experimenter 1 , fan used to cool metal on jig during all unknown trials, rod left in jig for three minutes
2
Rod B, the metal boiled for six minutes, placed into the jig by experimenter 1, rod left in jig for five minutes
3
Rod A, the metal boiled for four minutes, placed into the jig by experimenter 2, rod left in jig for three minutes
4
Rod B, the metal boiled for six minutes, placed into the jig by experimenter 2, rod left in jig for three minutes
5
Rod A, the metal boiled for four minutes, placed into jig by experimenter 2, rod left in jig for three minutes
6
Rod B, the metal boiled for four minutes, placed into jig by experimenter 2, rod left in jig for two minutes
7
Rod A, the metal boiled for five minutes, placed into jig by experimenter 2, rod left in jig for three minutes
8
Rod B, the metal boiled for four minutes, placed into jig by experimenter 2, rod left in jig for three minutes
9
Rod A, the metal boiled for four minutes, placed into jig by experimenter 2, rod left in jig for four minutes
10
Rod A, the metal boiled for four minutes , placed into jig by experimenter 2, rod left in jig for three minutes
11
12
Rod B, the metal boiled for five minutes, placed into jig by experimenter 2, rod left in jig for four minutes
13
Rod A, the metal boiled for five minutes, placed into jig by experimenter 2, rod left in jig for three minutes
14
Rod B, the metal boiled for four minutes , placed into jig by experimenter 2, rod left in jig for three minutes
15
Table 8 shows the observations made during the trials testing the linear thermal expansion of the known metal rods. The rod used, how long the metal was boiled in the water for, which experimenter placed the rod in the jig, and how long the rod was in the jig was observed. Rod times while in the jig were shorter in these trials because a fan was used.
Data Analysis and Interpretation
In the specific heat trials, the specific heat of the known and unknown rods was being found. First, the mass of the metal and the water was found. The amount of water put into the calorimeter was converted into millimeters, and the metal was massed using a scale. These both were measured in g (grams). The initial temperature of the metal rods and the water were collected in °C (Celsius), and after the metal was put into the calorimeter, where the equilibrium temperature was found (°C). The specific heat then calculated through the use of a formula, which was calculated in J/g°C (Joules/grams per degree Celsius).
In the linear thermal expansion trials, the alpha coefficient of the known and unknown rods was being found. The length of the rod was measured through the use of a calorimeter, and measured in mm (millimeters). The rod was then heated up and the initial temperature of the rod was taken, and then put onto a jig for it to cool down, and the final temperature of the rod was taken too. Both were measured in °C. Once on the jig, the rod would’ve changed length, which was shown on the dial. This change in length was measured in mm.
To deem that the data was valid, percent errors and others methods were used. Percent errors show how close the values gotten in the experiment match up with the true value of the experiment. This allows the experimenters to see how close their data is to the actual value, and be able to change any factors to help make the percent error closer. The experiment was also carried out the same way in every trial, and all the trials were randomized. Each set of rods (known hafnium and unknown metal) were each given 15 trials, and even though this amount of trials is not enough to perform a two sample t-test, normal probability plots could be used to check the validity of the data.
Table 9
Trial
Rod
1
B
2.192
2
A
0.609
3
B
-1.476
4
A
2.948
5
B
2.447
6
B
1.106
7
A
-1.820
8
A
-1.063
9
B
3.568
10
B
-3.525
11
B
-1.402
12
A
-1.832
13
B
-1.266
14
A
-1.962
15
A
-1.615
Average
-0.206
Table 9 shows the percent error of the specific heat calculated for both rods A and B when compared to the true known specific heat of Hafnium. A low percent error means that the specific heat calculated of the rod is very close to the true known value of Hafnium. A positive percent error means the value calculated in the experiment is higher than the true value, and a negative percent means that the value was less than the true value. When the rods were tested, they got very low percent errors, and the average percent error is -0.206%, which is very close to 0, which suggests that the known rods A and B are actually Hafnium.
Table 10
Trial
Rod
1
B
153.049
2
B
139.825
3
A
216.096
4
B
244.019
5
A
197.566
6
B
147.181
7
A
190.039
8
A
193.407
9
B
228.046
10
A
214.510
11
B
274.485
12
B
236.934
13
A
125.273
14
A
184.807
15
A
217.049
Average
197.486
Table 10 shows the percent error of the specific heat calculated for unknown rods A and B when compared to the true specific heat value of Hafnium. The percent errors are somewhat inconsistent, some of it going over 200 percent, and the lowest being 125.273%, which may mean something in the experiment went wrong. Overall percentages are high, but there is a lot of variability in the results. It was not because of performance of the experiment, because the specific heat trials for the unknown metal rod was performed the same as the one for the known hafnium rod. This may be due to little mistakes in the procedure, like inconsistent room temperature water, since all water was taken from tap. The percent error is very off from zero, and the average is 197.486% (which is very far from zero), which provides evidence that unknown metal rods may not be Hafnium.
Table 11
Trial
Rod
1
A
8.748
2
A
-10.111
3
B
2.210
4
A
7.613
5
B
7.862
6
A
-3.996
7
B
-12.594
8
B
-11.572
9
A
-11.536
10
B
-13.533
11
B
2.488
12
A
-8.561
13
A
-12.618
14
B
-10.125
15
A
8.012
Averages
-3.847
Table 11 shows the percent error of the alpha coefficient calculated for the known Hafnium rods A and B when compared to the true known alpha coefficient of Hafnium. The percent errors are decently away from zero, and vary a little; this may be due to the metals not being completely cooled. The metals were left on the jig for a set amount of time, and were hand fanned to help assist the cooling. Some metals may have not been completely cooled at the end of the time period given, and may have gotten a specific heat a little but away from the true alpha coefficient of hafnium.
Table 12
Trial
Rod
1
A
122.225
2
B
77.522
3
A
92.635
4
B
55.436
5
A
78.909
6
B
54.634
7
A
85.173
8
B
74.676
9
A
56.897
10
A
90.229
11
A
69.930
12
B
58.654
13
A
56.058
14
B
51.575
15
A
72.476
Averages
73.135
Table 12 shows the percent error of the alpha coefficient calculated for the unknown rods A and B when compared to the true known alpha coefficient of Hafnium. There is a big outlier from the first trial, where the percent error calculated was 122.225%; this is from a brief mishap in the trial. The experimenters did not find a cooling method on the first three trials, and the metal was just left on the jig until the set time was over. Only the second and the third trial were redone, and the first trial could not be redone due to the constriction on time. The percent errors are very far from zero, which supplies evidence that the unknown rods A and B may not be Hafnium.
Another way of looking through the data is with the usage of box plots. With box plots, the overlaps, median, mean, and a five number summary can be found. These can also be used to test the validity of the data and provide evidence suggesting that the two metals are the same.
Hafnium Specific Heat: 0.14
Figure 3. Specific Heat Box Plots
Figure 3 shows the box plots of the specific heat values recorded during both known and unknown metal trials. Figure 3 shows the box plots for the hafnium rod and the unknown metal rod to be significantly spread apart. The true specific heat of hafnium is 0.14 J/g°C. The median specific heat value of the data collected for hafnium was 0.144 J/g°C, which is significantly close to the true specific heat hafnium value. The median specific heat value of the data collected for the unknown metal was 0.434 J/g°C, which is significantly far away from the true value of hafnium shown in figure 3. The top box plot in figure 1 is narrow, meaning that there was not a wide variability in the specific heat recorded during trials. The lower box plot is wide, meaning that there was a wide variability in the specific heat recorded while performing the trials. The hafnium rod box plot data was slightly skewed right, and the unknown metal rod box plot data was slightly skewed right.
Hafnium Alpha Coefficient: 0.000006
Figure 4. Linear Thermal Expansion
Figure 4 shows the box plots for the alpha coefficient values during both the hafnium and unknown metal trials, also that there was a significant separation between the hafnium box plot and the unknown metal box plot. The median alpha coefficient value for hafnium was 0.00000549, which is significantly close to the true alpha coefficient of hafnium shown in figure 4. The median alpha coefficient value for the unknown metal was 0.000001035, which is a tiny bit away from the true alpha coefficient value of hafnium. Even though the difference between these alpha coefficients is very small, it provided very large percent errors. The hafnium box plot was fairly normally distributed, while the unknown metal box plot was skewed to the left.
The statistical test that was suitable for the data in the experiments was a two sample t-test; this test compares the means of two populations of data in which the standard deviation was unknown. The two sample t-test was carried out twice, one for the specific heat experiment and one for the LTE experiment. For a two sample t-test, there are three assumptions that must be met before testing, and if they are all met, and then the t-test can be conducted. Two of the three assumptions were met from the start; the data collected was a simple random sample, randomly selected using the T-Nspire calculator, and that both populations are independent. The last assumption was that the data has more than 30 trials in each population; in the experiment only 15 trials the known rods and unknown rods in each experiment were done. Normal probability plots were used to check the data.
Figure 5. Specific Heat Hafnium Normal Probability Plot
Figure 5 shows the normal probability plot of the specific heat values of hafnium. Most of the data points stay near the line of best fit, but there is a definite trend away from the line of best fit on the 0.144 specific heat value. This may affect the validity, and overall the outcome of the two sample t-test.
Figure 6. Specific Heat Unknown Metal Normal Probability Plot
Figure 6 shows the normal probability plot of the unknown metal specific heat values. The data tends to stay close to the line of best fit, with no outliers shown. No major trends were found in this probability plot.
Figure 7. Linear Thermal Expansion Hafnium Normal Probability Plot
Figure 7 shows the normal probability plot of the linear thermal expansion values of hafnium. Overall, the data is close to the line of best fit. No outliers were shown. There were no significant trends in the data.
Figure 8. Specific Heat Unknown Metal Normal Probability Plot
Figure 6 shows the normal probability plot of the unknown metal linear thermal expansion values. The data is less normal than the one shown in figure 7, but still somewhat stays near the line of best fit. One outlier was shown around 0.0000130 in figure 8, this may affect the outcome of the two sample t-test.
After the normal probability plots were used to check if the data was in fact normal, the following null and alternate equations were used for the two sample t-test.
Ho: μknown = μunknown
Ha: μknown ≠ μunknown
The null hypothesis (Ho) states that the mean of the known Hafnium rod values and the mean of the unknown rod values are the same. The alternate hypothesis (Ha) states that the mean of the known Hafnium rod and the mean of the unknown rod are not the same. After the two sample t-test is used, the experimenters could fail to reject the null hypothesis, or reject it in favor of the alternate hypothesis.
The two sample t- test was calculated using a formula where t, the test statistic equals the quotient of x1, the mean of the first sample, subtracted by x2, the mean of the second sample, divided by the square root of the sum of s12, the first standard deviation squared, divided by n1, the number of trials in the first sample, added to s22, the second standard deviation squared, divided by n2, the number of trials in the second sample.
Figure 9. Specific Heat Density Curve.
Figure 7 shows the density curve for the specific heat. In here, the p value was determined to be 0.0000, basically zero. It is not actually zero; it is just so small the Ti-Nspire software could not fully recognize it. With the p value being so small, it provides significant evidence that the metals are not the same.
Figure 10. Calculator Page for Specific Heat Two Sample T-Test.
Figure 8 shows the results of the two sample t-test that was calculated for specific heat. It also lists all the components in the two sample t-test formula for specific heat (see appendix A).
After all calculations were completed, the null hypothesis was rejected; the p value of 3.75407E-11 was less then alpha level of α = 0.10. This provides significant evidence that the means of specific heats of the known Hafnium rods and the unknown rods are not the same, which suggests that the known rods and unknown rods are not the same metal. If the null hypothesis was true, then it would be almost a 0% chance for it to happen by chance alone.
Figure 11. Linear Thermal Expansion Density Curve.
Figure 9 shows the density curve for linear thermal expansion. The p value here is also 0.0000 on the graph, but is actually not zero. This suggests that the two metals are not the same since their means are not equal.
Figure 12. Calculator Page for LTE Two Sample T-Test.
Figure 10 shows the results of the two sample t-test that was calculated for linear thermal expansion using technology. It also lists all the components for the two sample t-test for LTE.
After all the calculations were made, the null hypothesis was rejected. The p value of 9.446E-12 was less than the alpha level of α = 0.10. This provides significant evidence that the mean alpha coefficients for the known Hafnium rods and the unknown metal rods are not the same (providing evidence that the known and unknown rods are not the same metal). If the null hypothesis was true, then it would be almost a 0% chance for it to happen by chance alone.
Conclusion
The purpose of this experiment was to determine the identity of an unknown metal specific heat and linear thermal expansion, and comparing it to the known values of Hafnium. The hypothesis was that the identity of the unknown rod will be determined to be Hafnium within a 3% error of the true values of linear thermal expansion and specific heat. The hypothesis was rejected. The unknown metal sample given was not the same as the known Hafnium metal sample.
The values for specific heat of the unknown metal rod varied greatly from the known Hafnium value. The true specific heat value for Hafnium is 0.14 J/g°C. The average specific heat recorded from the trials for the known Hafnium rods was 0.142 J/g°C. These values are very close, unlike the average specific heat collected of the unknown metal rod, which was 0.415 J/g°C. The percent error calculated from the known metal specific heat trials ranged from -2% to 3%, while the unknown metal specific heat percent error ranged from 125% to 245%. The percent error calculated from the linear thermal expansion trials also supports the hypothesis being rejected. The known metal rod linear thermal expansion trials had percent errors that ranged from -14% to 9%. This differed greatly from the unknown metal trials, which had percent errors ranging between 51% and 123%.
Throughout trials during this experiment, some procedures were performed out of order. While calculating specific heat, some metal rods may have been left in the loaf pan longer than others. This may have given a smaller increase in the specific heat of the metal being tested. During the linear thermal expansion trials, there were three trials where the experimenters did not know of a cooling method for the metals while they were on the jig, and only two out of the three trials that were done to the best ability were redone. This gave a large outlier in the alpha coefficient for the first trial of the linear thermal expansion, and may have affected the mean of the unknown rods, which may have affected the t-test. Similarly to the specific heat trials, the metal rods were left in the loaf pans for inconsistent times, due to the time it took the metal to be at boiling temperature.
The data collected helped to properly identify that the hypothesis was rejected. For the metal samples to both be the same, the p-value of the two sample t-test conducted needed to be over an alpha level of 0.1. The p-value calculated for specific heat was 3.75×10-11, which is significantly lower than the alpha level. The p-value for linear thermal expansion, 9.45×10-12 was also a significantly lower value than the alpha level used for the test. This gave significant evidence that the means of the known and unknown rods were not the same, suggesting that the metals were not the same either.
During the linear thermal expansion trials, the times the metals were left in the jig varied between each trial. Having an electric fan would have helped speed up the cooling process. Conducting more in-depth background research on the known metal used, hafnium may also had an effect on the experiment.
Errors could have been made throughout this experiment, as in all experiments conducted. One error that could have affected the experiment was the calorimeters used. The calorimeters could have been smaller, making the readings recorded more accurate. During one of the trials, while transferring the metal from the loaf pan to the calorimeter, the rod was dropped on the ground. This happening could have caused readings of the metal in trials conducted after that to be off.
Research could have gone farther and knowing the identity of the unknown rod is not hafnium, the true identity of the rod could be looked into and found. Also, different types of intensive properties could be used to compare hafnium and unknown metals. The metal could be tested for many other things. Other properties that could be tested are the melting point, or boiling point of hafnium to the melting point, or boiling point of an unknown metal. The procedure could be redone again with more precise equipment, and a more efficient way of cooling down the metal to room temperature for both the specific heat and linear thermal expansion trials.
Appendix A: Sample Calculations
In order to find the specific heat of the metal, the following equation was used. Sm, the specific heat of the metal, was equal to the quotient of Sw, the specific heat of the water multiplied by Mw, the mass of the water multiplied Tw, the temperature change of the water divided by Mm, is the mass of the metal, multiplied by Tm, the temperature change of the water.
Below in Figure 13 is a sample calculation that shows how to find the specific heat using the data recorded in the first trial of the known Hafnium rod.
Figure 13. Specific Heat Calculation.
Next, for the linear thermal expansion trials, the alpha coefficient was looked for. To find the alpha coefficient, the following equation was used, where l, the change in length of the rod is equal to α, the alpha coefficient, multiplied by l0, the original length of the rod, multiplied by T, the change in temperature of the rod.
Below in Figure 14 is a sample calculation that shows how to find the alpha coefficient using the data recorded in the first trial of the known Hafnium rod.
Figure 14. Linear Thermal Expansion Equation.
To check the data gotten in the experiment for accuracy, a percent error formula was used. The closer the percent error is to 0%, the closer it is to the true value. The percent error formula has the percent error, % error, equal to the quotient of the experimental value, experimental value, subtracted by the true value of whatever is being tested, true value, divided by the true value, and multiplied by 100.
Below in Figure 15 is a sample equation for percent error using the specific heat of the first known Hafnium rod trial.
Figure 15. Percent Error Sample Calculation.
To compare the metals and see if they were the same, a two sample t-test was used. To carry out the two sample t-test, the following equation was used, with t being equal to the quotient of x1, the mean of the first sample subtracted by x2, the mean of the second sample, divided by the square root of the sum of S1, the standard deviation of the first sample over n1, the number of trials for the known rods, added to S2, the standard deviation of the second sample, over n2, the number of trials for the unknown rods.
Below in Figure 16 is a sample calculation that shows how to perform the two sample t-test.
Figure 16. Two sample T-Test Sample Calculation.
Appendix B: Calorimeter
Buzz Saw
PVC Primer
PVC Cement
Drill
Procedures:
1. Take a PVC pipe and cut two seven-inch pieces from it.
2. Brush PVC primer on one end of the PVC pipe, as well as one pipe cap.
3. Brush PVC Cement on the end of both PVC pipes, as well as one pipe cap.
4. Place cement-covered pipe cap on the end of one cement-covered pipe.
4. Drill one hole through one cap using the drill.
5. Place the drilled cap on the PVC pipe, opposite of the cemented cap.
6. Place the pipe into the safety cap, with the cemented end in the cap.
7. Wrap insulation foam around PVC pipe.
8. Wrap duct tape fully around insulation.
9. Repeat steps 2-6 with the second seven-inch PVC pipe.
10. Use the expo marker to label one calorimeter ‘A’ and the other calorimeter ‘B’ on the base.
11. Place the temperature probe through the drilled hole. The final products are displayed in Figure 17.
Figure 17. Finished Calorimeters.
Figure 17 shows the appearance of the calorimeters when building was finished.
Appendix C: Calculator Randomization
In order to reduce bias, all the trials were randomized. To help randomize all of the trials, a Ti-Nspire calculator was used. The Ti-Nspire calculator randomization function randint(1,2,1) was used. What this did was randomly generate one number, 1 or 2, one number every time enter was pressed. The metal rods A were designated to the number 1, and the metal rod B was designated to the number 2. The trials were also evenly distributed as best as they could. If any rod had 8 trials designated to it before the 15 trials limit was reached, the rest of the trials would be designated to the other rod.
Figure 18. Ti-Nspire with Random Integer Function.
Figure 18 shows the Ti-Nspire with the random integer function on it.
Works Cited
Brucat, Philip J. "Molecular Motion." Kinetic-Molecular Theory. University of Florida, 2008. Web. 10 May. 2014. <http://www.chem.ufl.edu/~itl/2045/lectures/lec_d.html>.
Chang, Raymond, Ph.D. "Chemistry The Study of Change." Chemistry. 9th ed.
New York: McGraw-Hill, 2007. 15. Print.
Chao, Tong Wa. "Determining the Coefficient of Thermal Expansion of Printed Wiring Board Components." University of Texas, n.d. Web. 27 Mar. 2014. <http://www2.galcit.caltech.edu/~tongc/html/publications/THESIS.pdf>.
Experiment VIII: Specific Heat and Calorimetry. N.p., n.d. Web. 24 Mar. 2014. <http://www.physics.fsu.edu/courses/fall04/phy2053c/labs/heat.pdf>.
Gagnon, Steve. "The Element Hafnium." It's Elemental - The Periodic Table of Elements. Jefferson Science Associates. Web. 19 May 2014. <http://education.jlab.org/itselemental/ele072.html>.
"Hafnium." Avalon Rare Metals. Web. 20 May 2014. <http://www.avalonraremetals.com/projects/target_commodities/rare_metals/hafnium/>.
Harrison, David, Claude Plante, and Joe Vise. "Thermal Expansion Experiment." Faraday Physics. N.p., May 2003. Web. 27 Mar. 2014.
"Periodic Table of Elements: Los Alamos National Laboratory." Periodic Table of Elements: Los Alamos National Laboratory. Los Alamos National Laboratory. Web. 20 May 2014. <http://periodic.lanl.gov/72.shtml>.
Simanek, Donald E. "H-2 Specific Heat Capacity." Lock Haven University. N.p., n.d. Web. 26 Mar. 2014. <http://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm>.

Documents

portfolioae.weebly.com · Web viewThermal expansion causes significant stress on a component if its design does not allow expansion and components (Linear Coefficient of Thermal