the identification of an unknown metal as...

The Identification of an Unknown Metal as Tantalum Using Linear Thermal Expansion and Specific Heat

By: Winston Balmaceda and Jonathon Banick

Hewlett-Packard [Company address]

Balmaceda-Banick

Problem Statement:

To determine if the identity of an unknown metal is tantalum using the intensive

properties of specific heat and linear thermal expansion.

Hypothesis:

t will be determined if the unknown metal is tantalum if the specific heat is

measured to 0.9% error and the linear thermal expansion is measured to 6.2% error.

Data Measured:

In this experiment the independent variables for specific heat were the initial

temperature of the water in which the metal was placed in measured in degrees

Celsius, the final temperature of the water in which the metal was placed in measured in

degrees Celsius, The initial temperature of the metal rod measured in degrees Celsius,

the mass of the metal measured in grams, and the volume of the water measured in ml.

these variables were used to calculate the specific heat of the metal measured in joules

over degrees Celsius. The independent variables for linear thermal expansion were

the initial size of the metal measured in millimeters, the initial and final temperatures of

the metal measured in degrees Celsius. And delta L or the change in length measured

in millimeters. These were used to calculate the alpha coefficient measured in 1/C.

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Introduction

Tantalum, element 73, is a rare, lustrous metal that is highly corrosive resistant.

Due to its corrosive resistant properties it is used often in lab equipment, as a substitute

for Platinum, as well as in electronic components such as resistors. In addition to its

resistance to acid, Tantalum also has an extremely high melting point, high density, and

is rather resistant to oxidation. Its qualities like these that attribute to its ability to be

made into high maintenance products, such as vacuum furnaces.

The purpose of this experiment was to see if a group if unknown metal rods had

the same identity as the known metal rods, Tantalum. In order to determine if the

metals had the same identity, a linear thermal expansion experiment and a specific heat

experiment were conducted. Both LTE and specific heat are intensive properties which

means they are unique values for different elements. Due to this, the data yielded by

known metal rod compared to that of the unknown metal rods would be able to

determine if they were similar materials.

For example the first experiment that the metals would undergo was a specific

heat test, specific heat is an intensive property and each value is unique to a specific

element. In order to determine the specific heat of the metal, the rods were heated to

approximately 100℃ and placed within calorimeters, which had been constructed by

the researchers. The data collected was used to determine the specific heat of the

metals. The specific heats were compared to see if the metal rods were the same or

different.

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Subsequently the second experiment was a test to find the alpha coefficient of

the linear thermal expansion of the metals. To find the alpha coefficient of the known

and unknown metals they were heated to approximately 100℃ and placed in an

apparatus that measured the length of them as they cooled. From there the data

collected from the experiment was used to calculate the alpha coefficients of LTE, which

were compared to determine if the metal rods were the same or different.

As a result, after the collection of data had ended, the properties of the two metal

rods were compared. A 2 sample t-test was conducted and compared along with the

observable physical properties of the rods, as well as the percent error factors from the

specific heat and LTE data. The percent error values in the experiment were used to

compare how the data yielded by the experiment differed from the true value to observe

if there had been any constants between the data sets. The data gathered combined

however was not viewed as more defining than the t-tests conducted, and only helped

enhance the identification process.

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Review of Literature

The purpose of this research was to determine the identity of an unknown metal

rod by comparing its intensive properties with the metal Tantalum. The properties used

in comparison are both chemical and physical properties. For this research, the physical

properties of specific heat and thermal expansion were used to identify the unknown

metal. Both of these properties, LTE (Linear Thermal Expansion) and specific heat are

intensive properties. Intensive properties are properties that are unique for each

element, but the sample size or amount of the element does not affect the properties

and they will remain the same despite so (Helmenstine). In order to measure these

properties heat must be added to the substance. Heat has a direct correlation with the

motion of the molecules within a substance. The motion is caused by the heat in a

substance, heat being energy transferred from one body to another through thermal

processes. This creates a correlation with heat and energy, as heat is lost or gain,

energy is lost or gained as well. Using this relationship between heat and energy the

identity of an unknown metal rod will be found in the course of research.

The intensive property of LTE is the change in length of a substance in response

to a change in temperature. As there is an increase in heat energy in a substance the

motion of the atoms in the substance will increase as well. With the motion of atoms

increased the separation between each atom increases thus causing an expansion of

the substance. Should the change in temperature,ΔT , not cause a change of phase

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within the substance than the substance has undergone LTE, and a change in the

length,ΔL, of the substance will be present. The property, measured in mm/m/K can be

displayed as follows,

α= ΔLLi X ΔT

With Li representing the initial temperature and α representing the linear

expansion coefficient of the object in 10-6/ 1/k. The linear expansion coefficient of

Tantalum being 6.3 x 10-6 1/k. This LTE coefficient is a small expansion but is still

noticeable, unlike some elements which only expand fractions of mm. This effect can

also be measured with contraction, as the change is not limited to heating up the

substance but cooling it as well. This causes a decrease in energy of the substance.

(Eyland)

In manufacturing, LTE is important when creating fitted objects, the element used

may affect the product if it had low heat tolerance. For example, should screws within

an engine block have a high thermal expansion value than the parts will expand when

they heat up from engine, they may cause breakage when they expand, or become too

loose when they cool down and allow movement that would cause damage to an

engine. This is not only a loss for the company as well as a safety hazard for those who

bought the product.

The intensive property of Specific heat will be used to identify the unknown

metal, the value of specific heat is independent of the sample size while heat capacity

varies with sample size. The specific heat of Tantalum is 0.140 J/g˚K and the specific

heat for water (H20) is 4.184 J/g˚K. Knowing these values on can compare the specific

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heats calculated during the course of experimentation and use them to help identify the

unknown metal sample. Specific heat is expressed as the following formula,

S= Qm x∆T

With specific heat, S, in J/g˚K. Q is the heat capacity measured in Joules, m the

mass measured in grams, and Delta T the change in temperature measured in kelvin.

(Eyland)

The Appalachian State University Undergraduate Physics Laboratory has

conducted an experiment on Thermal Expansion. In this experiment boiling water was

used to discover how heat affects the dimensions of different metals. The experiment

was conducted with a micrometer which would be pushed against by the metal rod

when it expands and would be used to measure Delta L. The experiment was

conducted with various substances to view the differences between elements. This is

applicable to our experiment as we will need to use thermal expansion to determine a

single element from unknown metal rods.

In manufacturing specific heat is important in making cookware, all cookware

must be able to endure high temperatures in order to be of use. It’s a given that some

metals will melt quickly, but can endure the heat of a stove top. However, many people

use ovens to store cookware and could accidently destroy their appliances should they

leave their cookware in the oven during baking.

David N. Blauch of Dadvison College created a setup in which one would explore

heat capacity and specific heat through Calorimetry. Calorimetry is the measuring of

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changes in chemical reactions, physical changes and shifting of phases. A Calorimeter

is an insulated device in which on can boil water and add substances too. Since the

system is closed there is no off effect on the heat capacity or specific heat of the

material, as the set up explains. Using the Calorimeter the heat capacity of a metal was

found, as well as the specific heat of copper (Blauch).

Identifying metals through simply physical properties can be inconclusive alone.

Using thermal properties, such as LTE, which is intensive, one can identify an unknown

metal and state with confidence that they know the true identity of the metal. This aids

an industry where quality means everything, should the material not be up to the

standards of the company or the product not meet the standards of a community, there

would be backlash of immense proportions. Being able to surely test the identity of

metals using thermal properties is a great aide.

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Experimental Design

Linear Thermal Expansion

Materials:

(2) Unknown metal rods (C,D)(2) Tantalum metal rods (A,B)3 x 5 loaf pan Caliper (0.01 mm)(2) Linear thermal expansion apparatus

Thermometer (0. 1 C)⁰Hot plate Tongs TI n-spire cx graphing calculatorFan

Procedure:

1. Randomize the trials between rod A and B for the known, and rod C and D for the unknown rods between the linear thermal expansion apparatus to eliminate bias (see appendix A.)

2. Measure enough water to thoroughly cover the metal rod into a 3x5 loaf pan.

3. Place the loaf pan on a hotplate set on level 6, let sit until water is boiling at approximately 100 C (record using thermometer.)⁰

4. Use the caliper to record the initial length of the metal rods.

5. While keeping the water at a constant, boiling temperature, place the metal rod in the water and allow to sit for at least 7 minutes.

6. Using tongs take the metal out of the boiling water and place it in the linear thermal expansion apparatus.

7. Immediately mark the starting point on the linear thermal expansion apparatus.

8. Allow to sit in the apparatus for approximately 12 minutes or until the dial stops moving (record as delta T.)

9. Use the data collected to calculate the Alfa coefficient.

10.Repeat steps 4 – 9 for 15 trials with the unknown metal rods, and 15 trials with the tantalum rods.

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Diagram:

Figure 1. LTE Material Diagram

Above is a labeled image of the materials used in the LTE experimentation, the

most important being the LTE apparatus and the fan.

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Specific Heat

Materials:

(2) Unknown metal rods (C,D)(2) Tantalum metal rods (A,B)50ml graduated cylinder LabQuest LabQuest temperature probe Electronic scale (0.0001g precision)

Thermometer (0.01g precision)3x5 Hot plate Hot plate Tongs TI n-spire cx graphing calculator

Procedure:1. Randomize the trials between rod A and B for the known, and rod C and D for

the unknown rods between the linear thermal expansion apparatus to eliminate bias (see appendix A.)

2. Record the mass of each of the four rods using a 0.0001g precision electronic scale, record.

3. Measure enough water to thoroughly cover the metal rod into a 3x5 loaf pan.

4. Set up the Lab Quest to measure temperature every second for eight minutes.

5. Place the water on a hot plate at a heat level 6 and leave it until it is boiling at approximately 100 C. (Measure using a thermometer.)⁰

6. Once the water is approximately 100 C gently place the metal rod into the water.⁰

7. Let the metal rod sit in the boiling water for 5 minutes. After the time is up, measure the temperature of the water once more, record the data. This serves as the internal temperature of the metal.

8. In the time you are waiting, measure 45 ml of water using a 50 ml graduated cylinder. Pour the water into the calorimeter that will be used. (Appendix a.)

9. 90 seconds before taking the metal rod out of the boiling water, start the Lab Quest in the calorimeter. This is allows the temperature probe to reach equilibrium with the water.

10.After the metal has been in the water for five minutes, take the, now hot, metal rod out of the boiling water and immediately place it into the calorimeter. (The temperature should be recorded.)

11.The temperature will be recorded for 5 minutes after placement into the calorimeter.

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11.Repeat steps 3 – 11 for 15 trials with the unknown metal rods, and 15 trials with the tantalum rods.

Diagram:

Figure 2. Specific Heat Material Diagram

Above is a labeled image of the materials used in the specific heat

experimentation, the most important being the calorimeter and the labquest. The

electronic scale is not displayed in this image.

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Data and Observations:

Table 1Known LTE Trials

Trial Rod ΔL(mm)

Initial Length

Initial Temp.

(Cº)

Final Temp.

(Cº)ΔT (Cº)

Alpha Coefficient

(mm) (°C -1 ) 1 B 0.06 127.97 98.2 21.7 76.5 6.333E-062 B 0.06 127.89 99.0 25.1 73.9 6.348E-063 A 0.06 127.66 99.0 25.1 73.9 5.830E-064 A 0.06 127.71 99.3 21.2 78.1 6.316E-065 B 0.06 127.93 99.3 25.5 73.8 6.673E-066 A 0.06 127.68 99.3 25.5 73.8 6.474E-067 A 0.05 127.64 99.1 23.1 76.0 5.567E-068 B 0.04 127.91 99.1 23.1 76.0 4.115E-069 A 0.05 127.67 99.1 24.3 74.8 5.236E-0610 B 0.05 127.96 99.1 24.3 74.8 5.224E-0611 A 0.04 127.89 99.2 21.3 77.9 4.015E-0612 B 0.05 127.34 99.2 21.3 77.9 5.040E-0613 A 0.05 127.89 98.4 23.2 75.2 5.511E-0614 A 0.03 127.89 98.4 24.4 74.0 3.593E-0615 B 0.04 127.34 98.4 24.4 74.0 3.820E-06

Table 1 above shows the data found during the experimentation for linear

thermal expansion. Each value except for ΔT, which is change in temperature, and the

Alpha coefficient, were found using experimentation. The final result was then recorded

with three significant figures after the decimal place.

Table 2Unknown LTE Trials

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Trial Rod ΔL (mm)

Initial Length

Initial Temp.

(Cº)

Final Temp.

(Cº)ΔT (Cº)

Alpha Coefficient

(mm) (°C -1 ) 1 C 0.04 127.35 98.6 25.3 73.3 4.283E-062 D 0.05 127.88 98.6 25.3 73.3 4.802E-063 C 0.05 127.40 99.0 23.8 75.2 5.532E-064 D 0.05 127.85 99.0 23.8 75.2 4.681E-065 C 0.05 127.31 99.3 23.9 75.4 5.209E-066 D 0.04 127.87 99.3 23.9 75.4 4.045E-067 C 0.05 127.31 99.0 24.5 74.5 5.272E-068 D 0.04 127.94 99.0 24.5 74.5 4.616E-069 C 0.04 127.31 99.2 24.5 74.7 4.416E-0610 D 0.05 127.94 99.2 24.5 74.7 5.232E-0611 C 0.05 127.31 99.3 24.6 74.7 5.468E-0612 D 0.04 127.94 99.3 24.6 74.7 4.499E-0613 C 0.04 127.31 98.4 24.7 73.7 4.370E-0614 D 0.04 127.76 98.4 24.6 73.8 4.136E-0615 C 0.03 127.34 98.4 24.6 73.8 3.618E-06

Table 2 above shows the data found during the experimentation for linear

thermal expansion for the unknown values. Each value except for ΔT, which is change

in temperature, and the Alpha Coefficient, were found using experimentation. The LTE,

or linear thermal expansion, for the unknown value is similar to the known value.

Table 3.Known Specific Heat Trials

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TrialInitial Temp.

(Cº)Equilibrium

Temp.(C°)

Change in Temp.

(Cº)Mass

(g)Specific

Heat(J/g x Cº)

Water Metal Water Metal Metal Water 1 24.7 98.3 27.4 2.70 70.9 67.8181 45 0.1062 24.3 98.3 27.7 3.40 70.6 67.6637 45 0.1343 22.9 98.5 26.1 3.20 72.4 67.8181 45 0.1234 26.0 98.8 28.7 2.70 70.1 67.6637 45 0.1075 21.0 98.8 24.2 3.20 74.6 67.8181 45 0.1196 23.3 98.5 26.6 3.30 71.9 67.6637 45 0.1287 24.2 98.6 27.1 2.90 71.5 67.8181 45 0.1138 27.0 98.6 29.8 2.80 68.8 67.6637 45 0.1139 22.7 97.5 25.6 2.90 71.9 67.8181 45 0.112

10 20.7 98.8 23.7 3.00 75.1 67.6637 45 0.11111 21.0 98.8 24.2 3.20 74.6 67.8181 45 0.11912 23.1 99.1 26.3 3.20 72.8 67.6637 45 0.12213 23.1 99.1 26.2 3.10 72.9 67.8181 45 0.11814 22.6 98.6 25.5 2.90 73.1 67.6637 45 0.11015 22.3 98.6 25.5 3.20 73.1 67.8181 45 0.122

The table above shows the values found during experimentation for specific heat

for the known metal rods. All values regarding the temperature, as well as the specific

heat values, were recorded to 3 significant figures.

Table 4. Unknown Specific Heat Trials

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TrialInitial Temp.

(Cº)Equilibrium

Temp.(C°)

Change in Temp.

(Cº)Mass

(g)Specific

Heat(J/g x Cº)

Water Metal Water Metal Metal Water 1 29.7 98.1 31.8 2.10 66.3 67.8566 45 0.0882 25.5 98.1 28.7 3.20 69.4 67.9513 45 0.1283 27.8 98.8 30.1 2.30 68.7 67.8566 45 0.0934 28.7 98.8 31.0 2.30 67.8 67.9513 45 0.0945 25.0 98.7 28.0 3.00 70.7 67.8566 45 0.1186 27.1 98.7 29.8 2.70 68.9 67.9513 45 0.1097 25.6 98.6 28.6 3.00 70.0 67.8566 45 0.1198 25.8 98.6 28.4 2.60 70.2 67.9513 45 0.1039 22.4 98.5 25.9 3.50 72.6 67.8566 45 0.13410 22.4 98.5 25.6 3.20 72.9 67.9513 45 0.12211 22.5 98.5 25.5 3.00 73.0 67.8566 45 0.11412 22.6 98.5 25.6 3.00 72.9 67.9513 45 0.11413 22.4 98.7 25.4 3.00 73.3 67.8566 45 0.11414 22.8 98.7 25.9 3.10 72.8 67.9513 45 0.11815 23.6 98.0 26.6 3.00 71.4 67.8566 45 0.117

Table 4 above shows the values found during experimentation for specific heat

for the unknown metal rods.

Table 5Observations of Known Specific Heat Trials

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Trial Calorimeter used Observations

1 3Ran along with the first trial of the day, trial 2,

shaken every 50 seconds by researcher 2

2 1Ran along with the first trial of the day, trial 1,

shaken every 50 seconds by researcher 2

3 2Ran along with trial 4, shaken every 50 seconds by

researcher 1


researcher 1


researcher 2

6 2 Ran along with trial 5, shaken every 50 seconds by researcher 2


8 1 Ran along with trial 7 shaken every 50 seconds by researcher 1







15 1 shaken every 50 seconds by researcher 1

Table 5 above displays the observations for all 15 known trials for Specific heat

as well as the Calorimeter used.

Table 6

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Observations of Unknown Specific Heat Trials

Trial Calorimeter used Observations

1 2Ran along with trial 2, shaken

every 50 seconds by researcher 2











8 2 Ran along with trial 7 shaken every 50 seconds by researcher 1










15 2 shaken every 50 seconds by researcher 1

The table above displays the observations for all 15 unknown trials for

Specific heat as well as the Calorimeter used.

Table 7

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Observations of Known LTE Trials

Trial Apparatus used Observations

13

Ran by Researcher 1, left in hot water for 10 minutes, cooled for 12 minutes

23

Ran by Researcher 1, left in hot water for 10 minutes, cooled for 12 minutes. First trial of the day.

32

Ran by Researcher 2, left in hot water for 10 minutes, cooled for 12 minutes. First trial of the day.

42

Ran by Researcher 2, left in hot water for 10 minutes, Fanned constantly for 11 minutes.

52

Ran by Researcher 1 along with trial 6, Cooled for 12 minutes, and boiled for 11.

63

Ran by Researcher 1 along with trial 5, Cooled for 12 minutes, and boiled for 11.

72

Ran by Researcher 2, along with trial 8. Fanned within 5 minute intervals for 12 minutes.

86


91

Fanned by researcher 2, run by researcher 1. Ran with 10 and boiled for 10 minutes.

105


113



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122


136


141


152

Ran by Researcher 1, fanned for 10 minutes, boiled for 8

The table above displays the observations for all 15 known trials for LTE

as well as the Apparatus used. Observations were recorded so it can be seen how the

different factors effected the data collected for each trial.

Table 8

Observations of Unknown LTE Trials

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16


21


33


42


52


66


72


86


9

2

Ran by Researcher 2, left in hot water for 10 minutes, Fanned constantly for 11 minutes. Ran with 10

10

2

Ran by Researcher 2, left in hot water for 10 minutes, Fanned constantly for 11 minutes. Ran with 9

112

Ran by Researcher 2, along with trial 12. Fanned every 3 minute interval of 12 minutes


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123

Ran by Researcher 2, along with trial 11. Fanned every 3 minute intervals in 12 minutes

132


146


152

Ran by Researcher 1, left in hot water for 10 minutes, fanned for 12 minutes

The table above displays the observations for all 15 unknown trials for

LTE as well as the Apparatus used. Observations were recorded so it can be seen how

the different factors effected the data collected for each trial.

Data Analysis

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Specific Heat Data Analysis and Interpretation

In this experiment, raw data was collected using 2 known and 2 unknown metal

rods. From there on they were tested upon using three calorimeters. In order to help

eliminate any bias in the experimental design, a random integer function was used to

randomize the calorimeter used in each trial. Following the randomization process, the

data was then collected using the lab quest unit and a temperature probe. The data

collection unit measured the heat of the water surrounding the metal, therefore it

measures the heat exothermically released from the metal. The heat of the water was

taken twice every second in order to monitor any sudden drops or spikes within the data

that could be caused by any interference by the system or its surroundings.

In order to analyze the data found for specific heat trials, a two sample t-test was

performed. In order to use the statistical test for our given data, the data must fit within

three given assumptions, which are SRS, independent trials and normality. This means

that each group is considered to be a distinct sample from a population, that the

responses in each group do not rely upon another group’s data, and that the

distributions of the variable of interests are normal under the null hypothesis. The first

assumption is met by the metals being only a portion of all of the metal tantalum. The

second assumption is met by the data being collected from more than one metal rod

keeping the two populations independent. The data also yields a normal distribution as

can be seen in figure 4 and figure 5.

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Table 9.

Percent Error Values of Specific Heat

Trial Numbers Known Specific Heat Percent Error Unknown Specific Heat Percent Error

2 24.482 37.2252 4.282 8.7423 12.352 33.6484 23.446 32.8615 14.937 15.9026 8.777 22.4437 19.569 15.0618 19.111 26.6989 20.017 4.453

10 20.603 13.12411 14.937 18.55212 12.635 18.55413 15.673 18.88514 21.15 15.72315 13.191 16.726

Averages: 16.344 19.906

The data table above displays the specific heat percent errors for the data. The

table seems to show that there is a vast difference between percent error values in the

two different variables and this might suggest that they are fairly different, but that

cannot be determined from raw data alone.

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Table 10.

The Averages of the Specific Heat

Average Known Specific Heat Unknown Specific Heat

0.117 0.112The above table displays the two averages for specific heat. The difference in

these two though seemingly little can make a significant impact on whether the metals

will be identified as the same or different. The averages can be later referred to

depending on what the statistical tests suggest.

Figure 3. Box plots of Specific Heat Trials

The box plot above displays the values for the Specific Heat trials. The above

box plots slightly overlap and have close means, which suggests that the specific heat

of the metals may be the same.

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Figure 4. Unknown Specific Heat Normal Distribution Plot

Figure 5. Known Specific Heat Normal Distribution Plot

The two figures above show the normality of the raw data from the specific heat

trials for both known and unknown metal rods. The plots both show that the data is

normal as there aren’t any significant outliers from the linear function within the plot.

This means that our data is more reliable.

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Figure 6. Histogram of Data for Known Specific Heat Values

Figure 7. Histogram of Data for Unknown Specific Heat Values

The histogram above displays the frequency of values within the data. The

histogram in figure 6 is left skewed while the figure 7 histogram is more right skewed.

This shows that there is a difference between the two, but does not show the

significance of the difference.

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Figure 8. Probability Graph of the Specific Heat Values

We are unable to reject the null hypothesis (H ⁰) because the p-value of 0.232 is

greater than the alpha level of 0.1. There is evidence suggesting that the unknown

metal has the same mean as the known metal, tantalum. There is a 23% chance that

the metals have a difference in the x̄ value of 0.0048 by chance alone.

The P value of 0.232 matches the expectations and predictions of the

researchers as it shows that our null hypothesis is true. Our Null hypothesis (H ⁰) of (H ⁰):

Mks = Mus, where Mks is the known values of specific heat and Mus is the unknown values,

suggests that the metals are the same. With the alternate hypothesis (Ha): Mks =/= Mus,

not being accepted, it is safe to assume that the metals are the same.

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2-Sample t-testAlternate M1=/=M2

T 0.PVal 1.Df 28.x̄1 0.112333x̄2 0.112333sx1 0.012949sx2 0.012949

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Linear Thermal Expansion Data Analysis and Interpretation

In testing the Linear Thermal Expansion of the given metals, raw data was

collected using two known and two unknown metal rods. From there the rods were

placed into a linear thermal expansion apparatus. The apparatus used was randomized

using a random integer function to each trial in order to help eliminate bias. The rods

were then placed in the randomized apparatus in order to measure the change in

length, in millimeters, that the metal underwent during the process of cooling. The

temperatures in which the rods were exposed to varied day to day due to the constant

boiling of pots and different methods cooling used.

In order to analyze the data found during the linear thermal expansion trials, a

two sample t-test was performed. In order to use the statistical test for the trials data,

the data needed to fit within three given assumptions, which are SRS, independent trials

and normality. This means that each group is considered to be a distinct sample from a

population, that the responses in each group do not rely upon another group’s data, and

that the distributions of the variable of interests are normal under the null hypothesis.

The first assumption is met from the metals being only a portion of all of the metal

tantalum. The second assumption is met from the data being collected from more than

one metal rod keeping the two populations independent. The data also yields a fairly

normal distribution as can be seen in figure 10 and figure 11.

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Table 11.Percent Error for LTE

Trial Number Known LTE Percent Error Unknown LTE Percent Error

1 -0.527 32.012 -0.77 23.783 7.461 12.1894 -0.259 25.7065 -5.918 17.3216 -2.757 35.7937 11.641 16.3228 34.687 26.7269 16.893 29.899

10 17.081 16.95711 36.27 13.20812 19.993 28.58313 12.526 30.63914 42.974 34.34415 39.359 42.573

Averages: 15.244 25.737

The data table above displays the percent error for the Linear Thermal

Expansion trials; it can be displayed simply from the table that the values greatly differ.

It can be noted that despite the increasing difference of percent error, no precautions

taken by the researchers caused a positive change in the shape of the data.

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Table 12.Average of LTE

Average Known LTE Unknown LTE

5.340E-06 4.679E-06The table above shows the difference between the averages for the raw data for

Linear Thermal Expansion. It can be noted that while the difference may seem small, for

such a significant value being measured any small change is a significant difference.

Figure 9. Box plots for LTE Values

The above figure is a box plot of the data for Linear Thermal Expansion. The

means are farther away and the plots do not overlap that much, this suggests that the

LTE of the two metal rods are different.

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Figure 10. Unknown LTE Normal Distribution Plot

Figure 11. Known LTE Normal Distribution Plot

The two figures above show the normality of the raw data from the linear thermal

expansion trials for both the known and unknown values. The reliability of the known

LTE values is questionable due to a number of offset points within the plot. It suggests

that the known LTE trials are not as reliable as they could be.

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Figure 12. Histogram of Data for Known Values

Figure 13. Histogram of Data for Unknown Values

The histogram above displays the frequency of values within the data. The

histogram in figure 12 is right skewed while the figure 13 histogram is more left skewed.

This shows that there is a difference between the two, but does not show the

significance of the difference.

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Figure 14. Probability Graph of LTE

With the linear thermal expansion p-values it is possible to reject the null

hypothesis (H ⁰) because the p-value of 0.042 is smaller than the Alpha level of 0.1.

There is evidence suggesting that the unknown metal has a different LTE then the

known metal, tantalum. There is only a 4.2% chance that the metals have a difference

in the x̄ value of 0.661 by chance alone.

The P value of 0.042 contradicts the value received for specific heat. It suggests

that the Null hypothesis (H ⁰) of (H ⁰): MKl = MUl, where MKl is the known values of specific

heat and MUl is the unknown values, is rejected and the metals are different. With the

alternate hypothesis (Ha): MKl =/= MUl, being accepted, it is hard to determine if the metal

is the same as the known or if it differs. It can be considered through any flaws or errors

in the experiment. With specific heat trials it was an uncommon event for water to spill

from the calorimeter, and they were insolated systems that had a fitting hole for the

thermometer. While with the Linear Thermal Expansion trials the equipment used was

unreliable and inaccurate, even measuring the known data it gave off abnormal results,

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2-Sample t-testAlternate M1=/=M2

T 2.16634PVal 0.041577Df 21.6502x̄1 0.000005x̄2 0.000005sx1 0.000001sx2 5.65833E-7

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as displayed in the normality plots. This leans the reliability of the data onto the specific

heat values.

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Conclusion

The purpose of the experiment was to find if the hypothesis, at a percent error of

0.9% for specific heat and 6.2% linear thermal expansion is needed to identify the

unknown metal as tantalum is true. This hypothesis was not accepted, due to the

percent errors of both the specific heat values and the linear thermal expansion values.

The average percent error value for specific heat was 18.1; however the data suggested

that the metals were the same despite the percent error values being much larger than

the suspected value.

Furthermore the linear thermal expansion data does not provide insight to the

hypothesis as the average is larger than the suspected value and does not suggest that

the metals are the same. In the experiment, 2 groups of 2 metal rods were tested upon,

a known group of tantalum, and an unknown group of what was later found to be

tantalum. The two groups were tested using linear thermal expansion and specific heat.

During linear expansion trials the metal rods were heated to around 100℃ and cooled

down to room temperature where the change in length was recorded.

In addition specific heat trials were performed, the metals rods underwent

experimentation using calorimeters, the calorimeters were designed and built to keep

the heat within the system and keep the reaction a closed system. When testing the

metals rods were heated to about 100oc and then were quickly placed into a calorimeter

with 40 mL of water. The data was recorded using a lab quest device and a

temperature probe, the calorimeters were shaken often to ensure good data.

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The methods which data were collected for both conducted experiments may

have been flawed. For the specific heat trials, the volume of water in the calorimeter

varied slightly from trial to trial due to the droplets of water that remained after each trial.

Due to the accuracy of the numbers within specific heat, inaccurate numbers within the

calorimeter would affect the data as a whole. The data collection methods however did

not affect the normality of the specific heat trials and therefore the data was trust

worthy. The linear thermal expansion data methods were flawed as well; the LTE

apparatus were extremely sensitive units in the sense that any contact with them

caused change to the data. In addition tantalum has a low LTE and moves very little.

When calculating the alpha coefficient of linear thermal expansion each fraction of a

millimeter had drastic results on the data resulting in a higher percent error. These

caused an abnormal data resulting in untrustworthy results. Within both experiments the

temperature for the data often changed, the heating was inconsistent despite the values

all being above 90oC and the room temperature differed from day to day. The

temperature of the room may have been the cause for the abnormality of the linear

thermal expansion data, such as the drastic change between the minimum and

maximum values and the amount of outliers accounted for. The inconsistent

temperature of the boiling water and the loss of heat from the calorimeters because of

them not being a truly isolated system may have also effected the expansion and data

values for both experimentations.

Furthermore within both experiments temperature was a factor to the expected

outcome, Specific heat being the amount of heat energy a metal can hold, and linear

thermal expansion being the change in length of a metal due to heat. Both

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experimentations dealt with placing the metals into boiling water and allowing them to

cool down. Any inconsistencies of the temperature may have had an effect on the data

yielded within the experiments.

Further research can be conducted to find out the identity of an unknown element

using different intensive properties such as density, or melting point. While density

would be the most efficient, it was outside the parameters of the experiment, testing

melting point, which is intensive, would require much more expensive equipment and is

not as efficient as density.

Within industries, cost and quality are everything. Anything sold by a company

has become a liability of the company. Should a company find a metal supplier with

cheap wares, they would be wary but could test it to ensure it was the proper product. It

especially comes into factor when considering what tantalum can be used for, such as

hip replacements. Should the metal hip not be real tantalum it could cause harm to the

owner and lead to monetary loss from the company as well as legal repercussions.

In conclusion, the experimentation led to the result that the hypothesis that

required having a 0.9% error for specific heat and a 6.2% error for linear expansion to

identify an unknown metal as tantalum was rejected. With high percent errors from both

sets of data, the linear thermal expansion data was not used in identification as it had

abnormality in comparison to the specific heat. Using the specific heat the metal rod

was identified as being tantalum, using percent errors of 18% to 35%.

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Appendix A: Sample Calculations

In order to analyze the data, the equation below is used where SH, specific heat, equals

the product of 4.184, the specific heat of water, times ∆ tw, the change in temperature of

the water, times mw, the mass of the water, divided by the product of ∆ tm, the change in

tempature of the metal, times mm, the mass of the water.

SH=4.184 x ∆ tw x mw∆ tm xmm

Shown in figure 15 below is a sample calculation of using the equation for specific heat.

SH=4.184 x ∆ tw x mw∆ tm xmm

SH=4.184 J / g x℃ x 2.7℃ x 45 g70.9℃ x67.8181g

SH=0.106 J / g x℃

Figure 15. Specific heat calculation

Sample calculation of the specific heat from the first known trial.

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In order to analyze the data, the equation below is used where∝, Alfa, equals∆ L, the

change in length, divided by the product ofLi, the initial length, and∆ t , the chance in

temperature.

∝=∆L /(Li x ∆ t)

Shown in figure 16 below is a sample calculation of using the equation for linear thermal

expansion.

∝=∆L /(Li x ∆ t)

∝=0.06mm /(127.97mm x76.5℃)

∝=0.06mm /(9789.705mm x℃)

∝=6.333 x10−6℃−1

Figure 16. Linear thermal expansion

Sample calculation of linear thermal expansion from the first known trial.

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In order to analyze the data, the equation below is used wherePE, the percent error,

equals the summation oftruevalue, the actual value, minusobservedvalue, the results of

the trial, divided bytruevalue, the actual value, times 100.

PE= truevalue−observedvaluetruevalue

x 100

Shown in figure 17 below is a sample calculation of using the equation percent error.

PE= truevalue−observedvaluetruevalue

x 100

PE=6.3 x10−6℃−1−6.333 x10−6℃−1

6.3 x10−6℃−1 x100

PE=−.527%

Figure 17. Percent Error Calculation

Sample calculation of percent error from the first known trial of LTE.

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In order to analyze the data, the equation below, a 2 sample t-test, where T, the t value,

equals x̄1, the mean of the first data set, minus x̄2, the mean of the second data set,

divided by the square root of the summation of s1, the standard deviation of the first data

set squared, divided by n1, the number of sampled in the first data set, plus s2, the

standard deviation of the second data set squared, devided by n2, the number of

sampled in the second data set.

T=x̄1−¿ x̄2

√(s12/n1+s2

2/n2)¿

Shown in figure 18 below is a sample calculation of specific heat using the equation for

a 2 sample t-test?

T=x̄1−¿ x̄2

√(s12/n1+s2

2/n2)¿

T=0.1171J /g x℃−0.1123J / gx℃√¿¿¿

T=0.0048 J /g x℃√¿¿¿

T=1.2267495105146

Figure 18. Specific heat t-test

Sample calculation of the t value of specific heat from the first known trial.

Appendix b

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Materials:

*(7.5”) ¾” PVC pipe *(2) ¾” PVC end cap *(8”) Pool noodle Drill press

Chop saw 1/32” drill bit Ruler Black marker

*The materials listed above are sufficient for only 1 calorimeter in this experiment three were used

Procedure:

1. Using a ruler measure 7.5” of PVS and mark with a marker.

2. Using a ruler measure 8” of pool noodle and mark with a marker.

3. Cut the pipe at the marked line using a chop saw.

4. Cut the pool noodle at the marked line using a chop saw.

5. On one of the end caps use a drill press to drill a 1/32” hole slightly off-center.

6. Assemble the calorimeter by tightly putting the undrilled end cap one end of the PVC and slide the shaft into the pool noodle until there is ¼” of PVC exposed.

7. Gently attach the drilled end cap to the exposed shaft.

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