Linear Thermal Expansion and Specific Heat of Nickel

BLAH BL

Determining Whether an Unknown Metal is Nickel based on the Intensive

Properties of Specific Heat and Linear Thermal Expansion

Kishwar Basith and Brent Bulgrelli

Macomb Mathematics Science Technology Center

Honors Chemistry

– 10A

Mrs. Hilliard / Mr. Supal / Mrs. Dewey

May 20, 2014

Table of Contents

Introduction .................................................................................................1

Review of Literature ....................................................................................3

Problem Statement .....................................................................................9

Experimental Design for Specific Heat ..................................................... 10

Experimental Design for Linear Thermal Expansion ................................. 13

Data and Observations for Specific Heat .................................................. 16

Data and Observations for Linear Thermal Expansion ............................. 20

Data Analysis and Interpretation ............................................................... 24

Conclusion ................................................................................................ 38

Application ................................................................................................ 41

Appendix A: Calorimeter Instructions ....................................................... 43

Appendix B: Formulas and Sample Calculations ...................................... 45

Works Cited .............................................................................................. 51

Basith - Bulgarelli 1

Introduction

Nickel is the second most abundant element in the Earth’s core after iron

(Lenntech). Due to the depth of which it is found in the Earth, nickel is very hard to access. This makes nickel historically one of the most expensive base metals

(Selby). Chemically, nickel is reactive with other metals, which is why it is usually found bound to copper or iron. Nickel is thought to have arrived on Earth on meteorites (Winter). In industry, nickel is valuable due to the fact that it has the ability to bond with almost any other base metal to create strong and durable alloys, such as stainless steel (Selby).

The purpose of this experiment was to determine whether or not a pair of unknown rods was composed of nickel. The specific heat and linear thermal expansion coefficient of the unknown rods were compared to the specific heat and linear thermal expansion coefficient of the nickel rods. If the values were similar, the unknown rods would be confirmed as nickel. The reason specific heat and linear thermal expansion were chosen as the properties of choice was because they are intensive properties. If a property is intensive, it remains constant no matter what the sample size is.

To determine the specific heat for both sets of rods, an isolated system known as a calorimeter was constructed (see Appendix A). The rods were massed and then heated using boiling water. The rods were then placed into a calorimeter and the initial and final temperatures of the water were recorded.

With the resources provided, it was not possible to take the actual temperatures of the rods. The heat exchanges were found by observing the changes in the

Basith - Bulgarelli 2 temperature of the water. The specific heat was then calculated. The specific heats determined during experimentation were compared to the true specific heat of nickel, which was 0.440 J/g°C.

To find the coefficient of linear thermal expansion for both sets of rods, the initial length of the rods was recorded. The initial temperature of the rods was taken. Boiling water was used to heat the rods. After a set amount of time, the rods were taken out of the boiling water and placed into a linear thermal expansion jig, which calculated the change in length. The alpha coefficient of linear thermal expansion was then calculated. The alpha coefficients determined during experimentation were compared to the true alpha coefficient of linear thermal expansion of nickel, which is 13.3 10 -6 /°C.

To determine whether or not the experiment yielded valid results, percent error was calculated. In order to acquire a more developed conclusion, a two sample t test was conducted on both the specific heat data and the linear thermal expansion data. The purpose of the t test was to see whether or not there was a significant mathematical difference between the two pairs of rods.


Review of Literature

Specific heat and linear thermal expansion are two very helpful ways to identify metals. Specific heat is an intensive physical property of matter (De

Leon). Every substance has a unique specific heat. The specific heat of a substance is the amount of heat required to change the temperature of 1 g of a substance by 1°C (Chang 239). Linear thermal expansion deals with the length of an object or substance after it has been heated. When a substance is heated or cooled, its length changes by an amount proportional to the original length and the change in temperature (Duffy). The alpha coefficient of linear thermal expansion is the ratio of the change in length per degree Kelvin to the length of the object at 273 K (Winter). Every substance also has a unique alpha coefficient of linear thermal expansion. Both specific heat and linear thermal expansion are intensive properties, meaning they do not change with sample size (Helmenstine).

Specific Heat

Specific heat is an intensive property of matter. Specific heats of substances are determined using calorimeters. A calorimeter creates an isolated system, or a system in which neither heat nor mass escape. The assumption behind the science of calorimetry is that the heat released by the substance is absorbed by the water (The Physics Classroom).

Specific heat is measured in Joules per gram Kelvin (J/g•K) or Joules per gram Celsius (J/g•°C) (Chang 239). Specific heat measures the amount of heat

Basith - Bulgarelli 4 that is required to increase the vibrations of molecules by 1°C or 1 Kelvin (Kent).

When the heat is transferred to the water, the water’s molecules begin to vibrate as well, signaling a transfer of energy due to the First Law of Thermodynamics.

This law states that energy is not created or destroyed, merely transferred

(Woodward). When the temperature of the water and the temperature of the metal sample is equal, this is called the point of equilibrium. This signals that heat transfer is complete (Chang 240). The specific heat of the substance can be found using the final and initial temperatures of the water and then inserting the values into the following formula (Nave). The variable s w represents specific heat of water, measured in J/g•°C, m w

represents the mass of the water, measured in mL, ΔT w

represents the change in temperature of the water in the calorimeter, measured in °C, s m

represents the specific heat of the metal sample, measured in J/g•°C, m m represents the mass of the metal sample, measured in g, a nd Δ T m represents the change in temperature of the metal sample, measured in °C.

(Chang 240). 𝑠 𝑤 𝑚 𝑤

∆𝑇 𝑤

= 𝑠 𝑚 𝑚 𝑚

∆𝑇 𝑚

Table 1. The Specific Heats of Common Materials

Material

Aluminum

Specific Heat

(J/g•°C)

0.900

Glass

Brass

0.880

0.380

Table 1 shows the specific heat of three common materials (Nave). The researchers compared these values to the specific heat of their known metal, nickel. Nickel has a specific heat of 0.440 J/g•°C (Nave). The values for specific heat are important because the larger the value, the more heat the substance

Basith - Bulgarelli 5 absorbs. This is especially helpful when designing products so that the material does not get too hot during high temperatures to cause scalding or burning

(Nave).

The researchers have looked at 2 past labs concerning specific heat. The first was conducted at Palm Harbor University High School. First, students added water to a calorimeter. The temperature of the water in the calorimeter was measured. The students then placed a metal rod into a test tube. The test tube was boiled in a beaker for ten minutes. Afterwards, the test tube was removed from the beaker and added to the calorimeter. The final temperature was measured. The data collected to calculate the specific heat was the initial and final temperatures of the water in the calorimeter and the metal, as well as their masses. The calorimeter was constructed using Styrofoam cups (Bauck).

The second experiment looked at by the researchers concerning specific heat was conducted at San Juan High School. The students measured the mass of the metal and temperature of the water in the calorimeter. The metal was heated in a test tube and then placed in the calorimeter. The final temperature was measured. The values collected to calculate specific heat were the initial and final temperatures of the water in the calorimeter, the initial and final temperatures of the metal sample, and the masses of both. The calorimeter was constructed using Styrofoam cups and pipe insulation (Montbriand).


Linear Thermal Expansion

Linear thermal expansion is another intensive property of matter. Like specific heat, every substance has a unique alpha (α) coefficient. With a few exceptions, every substance expands when heated (Cassel). Linear thermal expansion requires the use of a device that measures lengths, such as a ruler, or, to receive more precise measurements, a caliper. Depending on the chosen unit, the lengths of linear thermal expansion can be measured with any unit, from inches to centimeters. Temper ature is measured in °C (Raymond). The alpha coefficient is measured in 1/°C or °C -1 (Ellert).

Linear thermal expansion is the expansion of atoms. When heat is added to the atoms of most substances, the atoms will become more energized. This will cause the particles to expand. This is due to Kinetic Molecular Theory, which states that as more energy is added to atoms, they become more energized, causing them to vibrate more and expand. There is also area and volume thermal expansion. However, with longer, slender, and generally smaller objects, such as metal rods, only linear expansion can be measured accurately (Nave).

The alpha coefficient of linear thermal expansion can be found using the following formula. The variable ∆𝐿 stands for the change in length of the metal measured in mm, while 𝐿 𝑖

represents the initial length of the metal measured in mm. The variable ∆𝑇 represents the change in tempe rature of the metal in °C.

The alpha coefficient is measured in 1/°C (Ellert).

𝛥𝐿 = ∝ 𝐿 𝑖

∆𝑇


Table 2. The Alpha Coefficient of Thermal Expansion for Common Materials

Material

Alpha Coefficient

(10 -6 /K)

Aluminum

Glass

0.900

0.880

Brass 0.380

Table 2 shows the alpha coefficient for three common materials (Ellert).

The researchers compared these values to the alpha coefficient of their known metal, nickel. The alpha coefficient of nickel is 13.3 10 -6 /K (Ellert). The larger the coefficient, the more the substance will expand when exposed to heat. This is important to producers and industrialists in design and construction because the materials used to construct buildings, bridges, and other works do not collapse in intense heat (Gibbs).

The researchers have looked at 2 past labs concerning linear thermal expansion. The first lab was conducted at St. Louis Community College. The students took the initial temperature of the water that was used. They also measured the initial length of the rod. After letting the rod sit in the boiling water, the students removed it and measured the final temperature of the water, and the final length of the rod. The rods were measured with micrometers, or measuring calipers (Buckhardt).

The second lab looked at by the researchers was conducted at Lawrence

Tech University. The process used was similar to the process used by the students at St. Louis Community College. The difference was that these students used steam from a boiler to heat the metal. To do this, they connected the boiler to an airtight container holding the metal using plastic pipes. The

Basith - Bulgarelli 8 lengths of the rod was measured using a micrometer, or measuring caliper

(Lawrence Tech University Department of Natural Sciences).


Problem Statement

Problem Statement:

To determine whether an unknown metal is nickel using the intensive properties of specific heat and linear thermal expansion.

Hypothesis:

If the percent error of specific heat and linear thermal expansion of the unknown metal sample is less than or equal to 1%, then the metal rod will be identified as nickel.

Data Measured:

The data will be analyzed using percent error. The experimental values of specific heat and linear thermal expansion will be compared to the true values of specific heat and linear thermal expansion of nickel. The measurements required to calculate specific heat include the mass of the water from the calorimeter in g, the change in temperature of the water in the calorimeter in °C, the specific heat of the water in the calorimeter in J/g•°C, the mass of the unknown metal sample in g and the chang e in temperature of the metal in °C. The measurements required to calculate the α coefficient of linear thermal expansion include the initial length of the unknown metal in mm, the final length of the unknown metal in mm, and the change in temperature of the metal in °C. To further analyze the data and acquire a more developed conclusion, a two sample t test and box plots of the data will be used.


Experimental Design for Specific Heat

Materials:

(2) Calorimeters

500 ml Loaf Pan

Thermometer (0.1 °C)

500 mL Graduated Cylinder

Tongs

(2) Unknown Metal Rods

(2) Nickel Rods

Vernier LabQuest

Temperature Probe (0.01 °C)

Scale (0.01g Precision)

TI - Nspire

Hot Mitt

Hot plate

Procedures:

Follow all safety precautions. Wear goggles, lab coats, and appropriate lab attire.

Specific Heat

1. Construct two calorimeters. Place them in the constructed calorimeter stand. Label the calorimeters 1 and 2. Designate the two unknown metal rods as Rod A and Rod B. Designate the known rods as Rod A and Rod

B. See Appendix A for instructions on calorimeters.

2. Using the random integer function of the TI-Nspire, randomize the trials.

Allocate 15 trials for the two unknown rods and 15 for the two nickel rods.

Also randomize which calorimeter will be used for each rod. This is to expose both rods to the same error and eliminate bias.

3.

Turn on the Vernier LabQuest equipment and choose the “collect data over time” setting. Set the time lapses to “collect 1 sample per second for

300 seconds.”

4. Attach the temperature probe in the appropriate slot. Do not begin data collection.

5. Measure the mass of the metal sample with the scale. Record in the appropriate column of the data table.

Basith

– Bulgarelli 11

6. Fill the loaf pan with water. Don’t fill the loaf pan all the way, as there will be displacement when a metal rod is placed into the pan.

7. Place the loaf pan with the water on the hot plate. Turn the hot plate on and set to the highest setting.

8. When the water has reached a boil, gently place the metal rod into the pan using the tongs.

9. When the water is boiling, assume that the temperatu re is 100 °C.

Assume that the initial temperature of the rod is the initial temperature of the water. Record in the appropriate column in the data table.

10. Measure out 45 mL of tap water into the graduated cylinder. Pour the water into one of the calorimeters.

11. Repeat step 10 for the other calorimeter.

12. Slide the temperature probe of the LabQuest down the hole in the lid of the calorimeter. Begin data collection.

13. Allow the temperature to stabilize on the LabQuest. This will be the initial temperature of the water in the calorimeter. Record in the appropriate column of the data table.

14. Wait until the temperature reaches equilibrium on the screen of the

LabQuest. This will be the final temperature of the metal and the water of the calorimeter. Record in the appropriate column in the data table.

15. To begin a new trial, tap on the file cabinet logo.

16. Repeat steps 4-15 to complete all 15 trials.

17. Repeat steps 1-16 for the unknown rods as well.

Basith

– Bulgarelli 12

Diagram:

Calorimeters

Hot Plate

Loaf

Pan

Graduated

Cylinder

Tongs

Hot

Mitt

Temperature

Probe

Unknown Rods

Thermometer

Nickel Rods

Figure 1. Specific Heat Experiment Materials

Figure 1 shows the materials that were used during the specific heat experiments. Note the two pairs of rods, the calorimeters, the Vernier TM

LabQuest, and the temperature probe.

Vernier TM

LabQuest

Basith

– Bulgarelli 13

Experimental Design for Linear Thermal Expansion

Materials:

(2) Unknown Metal Rods

(2) Nickel Rods

(2) 500 mL Loaf Pan

Tongs

Graduated Cylinder

Linear Thermal Expansion Apparatus (0.001 in Precision)

Thermometer (0.1 °C Precision)

Caliper (0.01 mm Precision)

TI – Nspire

Hot Plate

Hot Mitt

Blow-Off TM Compressed Air

Procedures:

Follow all safety precautions. Wear goggles, lab coats and appropriate lab attire.


1. Designate the unknown rods as Rod A and Rod B. Designate the nickel rods as Rod A and Rod B.

2. Using the TI – Nspire random integer function, randomize the two sets of rods. Allocate 15 trials for the unknown rods and 15 for the nickel rods.

3. Fill a loaf pan with tap water. Do not fill all the way, as there will be displacement when the metal rod is placed into the pan.

4. Place the loaf pan on the hot plate. Turn the hot plate on and set to the highest setting.

5. Fill another loaf pan with room temperature water. Again, do not fill up all the way, as there will be displacement when the metal rod is placed into the pan.

6. Place the metal rod into the loaf pan that contains the room temperature water using tongs.

7. Use the thermometer to measure the temperature of the water in the room temperature pan. Assume the temperature of the water is the initial temperature of the metal. Record in the appropriate column of the data table.

Basith

– Bulgarelli 14

8. Using the tongs, remove the rod from the pan.

9. Using the caliper, measure the initial length of the rod. Record in the appropriate column of the data table.

10. When the water has reached a boil, assume the temperature of the water is 100 °C. Record the temperature of the water as the final temperature of the metal.

Place the metal rod into the water using tongs.

11. Set up the timer to run for five minutes.

12. When the timer reaches five minutes, measure the temperature of the water in the boiling water pan. Assume the temperature of the water is the final temperature of the metal. Record in the appropriate column of the data table.

13. Remove the rod from the pan using tongs. Quickly slide the rod into the linear thermal expansion jig. Wait until the dial stops moving. Quickly rotate the dial so that the zero lines up with the hand and lock into place using the stop. This is done because only the change in temperature can be accurately recorded.

14. Wait until the rod has completely cooled down and the dial has stopped moving. Use the Blow-Off TM to help with the process. Record as the change in temperature once the hand stops moving.

15. Repeat steps 3-14 for all of the trials.

16. Repeat steps 1-15 for the unknown rods.

Diagram:

Basith

– Bulgarelli 15

Loaf Pan

Blow-Off Compressed Air

Hot Plate

Graduated Cylinder

Hot Mitt Tongs Thermometer

LTE Apparatus Nickel Rods Unknown Rods

Figure 2. Linear Thermal Expansion Experiment Materials

Figure 2 shows the materials that were used during the experiments concerning linear thermal expansion. Note the two pairs of rods, the linear thermal expansion apparatus and the Blow-Off TM compressed air.

Basith

– Bulgarelli 16

Data and Observations for Specific Heat

Table 3

Specific Heat of Nickel Data

Trial Rod Cal

Initial Temp.

(°C)

W M

Equil.

Temp.

(°C)

Change in Temp.

(°C)

W M M

Mass

(g)

W

Correction

Factor

(J/g°C)

1 1 A

2 2 A

3 1 B

4 2 B

5 1 A

6 2 A

7 1 B

8 2 B

9 1 A

10 2 A

11 1 B

12 2 B

13 1 A

14 2 A

15 1 B

Average

23.1 100.0 29

22.6 100.0 28.5

22.4 100.0 28.3

22.8 100.0 28.6

22.7 100.0 28.5

22.9 100.0 28.8

23.4 100.0 29.3

23.2 100.0 29.1

23.5 100.0 29.4

23.4 100.0 29.2

23.2 100.0 29.1

22.9 100.0 28.8

23.4 100.0 29.3

23.3 100.0 29.1

22.6 100.0 28.4

26.7 100.0 28.9

5.9 -71.0 36.028 45

5.9 -71.5 36.028 45

5.9 -71.7 36.028 45

5.8 -71.4 36.028 45

5.8 -71.5 36.028 45

5.9 -71.2 36.028 45

5.9 -70.7 36.028 45

5.9 -70.9 36.028 45

5.9 -70.6 36.028 45

5.8 -70.8 36.028 45

5.9 -70.9 36.028 45

5.9 -71.2 36.028 45

5.9 -70.7 36.028 45

5.8 -70.9 36.028 45

5.8 -71.6 36.028 45

5.9 -71.1 36.028 45

0.008

0.010

0.008

0.010

0.008

0.010

0.008

0.010

0.008

0.010

0.008

0.010

0.008

0.010

0.008

0.009

Table 3 shows the data collected during the specific heat experiments on

Specific

Heat

(J/g°C)

0.442

0.441

0.438

0.435

0.432

0.443

0.444

0.445

0.445

0.438

0.443

0.443

0.444

0.438

0.431

0.440 the two nickel rods. The M is an abbreviation for metal, and the W is an abbreviation for water. The Cal is the abbreviation for calorimeter. The average equilibrium temperature was 28.9 °C. The average change in temperature for metal was 71.1 °C. The negative symbol shows that the metal lost heat. This heat was then absorbed by the water. The average change in temperature for the water was 5.9 °C. The average correction factor was 0.009 J/g•°C. The correction factor of calorimeter 1 is 0.008 J/g°C, while the correction factor for calorimeter 2 is 0.010 J/g°C. The average specific heat was 0.440 J/g°C.

Basith

– Bulgarelli 17

Table 4

Observations for Nickel Specific Heat

Trial

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Observations

LabQuest operated by Researcher 1. Tongs operated by Researcher 2.

Trial conducted directly under light.

LabQuest reset by Researcher 1. Tongs operated by Researcher 2.


LabQuest reset by Researcher 2. Small spill cleaned up by Researcher

1. Tongs operated by Researcher 1. Trials conducted directly under light.

Labquest reset by Researcher 1. Tongs operated by Researcher 2.



Trial redone due to rod mix-up. Trial conducted directly under light.





















Table 4 shows the observations for the trials conducted during the specific heat experiment for the nickel rods. Trial 3 featured a spill in the middle of the trial. Trial 5 had to be redone because the rods had been mixed up by the researchers.

Basith

– Bulgarelli 18

Table 5

Specific Heat of Unknown Metal Data

Trial Rod Cal

Initial Temp.

(°C)

W M

Equil.

Temp.

(°C)

Change in

Temp. (°C)

W M M

Mass

(g)

W

Correction

Factor

(J/g°C)

Specific

Heat

(J/g°C)

1 1 A 24.9 100.0 29.3 4.4 -70.7 33.375

2 2 B 23.8 100.0 28.2 4.4 -74.2 33.375

3 2 A 23.9 100.0 28.8 4.9 -71.2 33.375

4 1 B 25.5 100.0 30.2 4.7 -69.8 33.375

5 1 A 25.6 100.0 30.1 4.5 -69.9 33.375

6 2 B 24.5 100.0 29.0 4.5 -71.0 33.375

7 2 A 22.6 100.0 26.9 4.3 -73.1 33.375

8 2 B 22.4 100.0 26.6 4.2 -73.4 33.375

9 1 A 23.0 100.0 27.1 4.1 -72.9 33.375

10 1 B 22.6 100.0 26.9 4.3 -73.1 33.375

11 2 A 24.4 100.0 29.0 4.6 -71.0 33.375

12 2 B 23.8 100.0 28.8 5.0 -71.2 33.375

13 1 A 24.8 100.0 29.3 4.5 -70.7 33.375

14 1 B 23.9 100.0 28.3 4.4 -71.7 33.375

15 2 A 24.5 100.0 29.1 4.6 -70.9 33.375

Average 24.0 100.0 28.5 4.5 -71.7 33.375

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

0.008

0.010

0.010

0.008

0.008

0.010

0.010

0.010

0.008

0.008

0.010

0.010

0.008

0.008

0.010

0.009

0.359

0.345

0.398

0.388

0.371

0.368

0.342

0.333

0.325

0.340

0.375

0.406

0.367

0.354

0.376

0.363

Table 5 shows the data collected during the specific heat experiments on the two unknown metal rods. The M stands for metal, while the W stands for water. The Cal stands for calorimeter. The average equilibrium temperature was

28.5 °C. The average change in temperature of the metal was -71.7 °C. The average change in temperature of the water was 4.5 °C. The average correction factor was 0.009 J/g°C. The correction factor for calorimeter 1 was 0.008 J/g°C, while the correction factor for calorimeter 2 was 0.010 J/g °C. The average specific heat was 0.363 J/g°C.

Basith

– Bulgarelli 19

Table 6

Observations for Unknown Rods Specific Heat

Trial

1

2

3

4

5

6

Observations

LabQuest operated by Researcher 2. Tongs operated by Researcher 1.

Trial conducted directly under light. Rod was not massed, so trial was redone.




Trials conducted directly under light. Redone due to mix-up of the metal rods.




Trial redone due to rod mix-up. Trial conducted directly under light.



7

8





9

10

11

12

13

14

15















Table 6 shows the observations collected during the specific heat experiments on the unknown metal rods. Trial 1 was redone because the rods had not been massed prior to being placed into the calorimeter. Trials 3 and 5 were redone due to rod mix-up.

Basith

– Bulgarelli 20

Data and Observations for Linear Thermal Expansion

Table 7

Linear Thermal Expansion for Nickel Rods Data

Trial Rod ΔL (mm)

Initial

Length

(mm)

Initial

Temp.

(°C)

Final

Temp

(°C)

1

2

3

4

B

B

A

B

0.127

0.127

129.388

0.127

129.388

0.127

129.388

129.388

25.9

100.0

25.7

100.0

25.4

100.0

25.9

100.0

Change in Temp.

(°C)

74.1

74.3

74.6

74.1

Alpha

Coefficient

(10 -6 /°C)

13.246

13.211

13.157

13.246

5

6

A

B

0.127

0.127

129.388

129.388

0.127

129.388

25.6

25.6

25.5

100.0

100.0

100.0

74.4

74.4

74.5

13.193

13.193

13.175

7

8

9

10

A

A

B

A

11

12

13

14

15

B

B

A

A

A

Average

0.127

0.127

0.127

0.127

0.127

0.127

0.127

0.127

0.127

129.388

129.388

129.388

129.388

129.388

129.388

129.388

129.388

129.388

25.4

25.8

25.7

25.8

25.2

25.9

25.8

25.9

25.7

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

74.6

74.2

74.3

74.2

74.8

74.1

74.2

74.1

74.3

13.157

13.228

13.211

13.228

13.122

13.246

13.228

13.246

13.206

Table 7 shows the data that was collected during the linear thermal expansion experiments for the nickel rods. The average change in length of the metals was 0.127 mm. The average change in temperature was 74.3 °C. The average alpha coefficient was 13.206 1/°C.

Table 8

Observations from Linear Thermal Expansion for Nickel Rods

Trial Observations

Tongs used by Researcher 1. Jig operated and metal cooled by

1

Researcher 2. Room unusually warm due to other experiments.

2

Tongs used by Researcher 2. Jig and metal cooled by Researcher

1. Trial redone due to rod mix-up. Room unusually warm due to other experiments.

3

4

5







Basith

– Bulgarelli 21

Trial

6

7

8

9

10

11

12

13

14

15

Observations




Researcher 2. Room unusually warm due to other experiments. Trial redone due to rod mix-up.

















Table 8 shows the observations collected during the linear thermal expansion experiment with the nickel rods. The room was unusually warm due to other experiments that were being conducted. Trials 2 and 7 had to be redone due to mix-up of the rods.

Table 9

Linear Thermal Expansion for Unknown Rods Data

Trial

1

2

3

4

5

6

7

8

9

10

Rod

A

B

A

B

B

B

A

A

A

B

ΔL

Initial

Length

(mm)

(mm)

0.003

129.28

0.003

129.28

0.003

129.28

0.003

129.28

0.003

129.28

0.003

0.003

0.003

0.003

0.003

129.28

129.28

129.28

129.28

129.28

Initial

Temp.

(°C)

25.7

100.0

25.6

10.00

25.5

25.7

25.6

25.7

25.6

25.9

25.5

25.7

Final

Temp

(°C)

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

Change in Temp.

(°C)

74.3

74.4

74.5

74.3

74.4

74.3

74.4

74.1

74.5

74.3

Alpha

Coefficient

(10 -6 /°C)

0.312

0.312

0.311

0.312

0.312

0.312

0.312

0.313

0.311

0.312

Basith

– Bulgarelli 22

Trial Rod

11

12

13

14

15

B

A

B

A

B

Average

ΔL

Initial

Length

(mm)

(mm)

0.003

129.28

0.003

129.28

0.003

129.28

0.003

129.28

0.003

129.28

0.003

129.28

Initial

Temp.

(°C)

Final

Temp

(°C)

25.9

100.0

25.4

100.0

25.1

100.0

25.7

100.0

25.9

100.0

25.6

100.0

Change in Temp.

(°C)

74.1

74.6

Alpha

Coefficient

(10 -6 /°C)

0.313

0.311

74.9

74.3

74.1

74.4

0.310

0.312

0.313

0.312

Table 9 shows the data collected during the linear thermal expansion trials for the unknown rods. The average change in length was 0.003 mm. The average change in temperature was 74.4 °C. The average alpha coefficient was

0.312 1/°C.

Table 10

Observations from Linear Thermal Expansion for Unknown Metal Rods.

Trial Observations

1



2

3

4

5

6

7

8

9

10

11

Tongs used by Researcher 2. Jig and metal cooled by Researcher

1. Room unusually warm due to other experiments.










Researcher 2. Room unusually warm due to other experiments. Trial redone due to rod mix-up.








Researcher 2. Room unusually warm due to other experiments. Trial redone due to jig malfunction.

Basith

– Bulgarelli 23

Trial

12

13

14

15

Observations









Table 10 shows the observations from the linear thermal expansion trials for the unknown rods. Trial 11 was redone due to a malfunction in the linear thermal expansion jig.

Basith

– Bulgarelli 24

Data Analysis and Interpretation

Data was collected using formulas calculating specific heat and the alpha coefficient of linear thermal expansion. Variables of the different equations were measured or calculated to ultimately calculate the value for one specific variable.

The data collected was valid and relevant due to the randomized trials and unbiased procedures. The data was analyzed using three different methods. The data was analyzed using percent error, box plots and a two sample t test.

The first method that was used to analyze the data was percent error.

Percent error compares the experimental values to the true values. A sample calculation concerning percent error can be found in Appendix B.

Table 11

Specific Heat of Nickel Percent Error

True Value

(J/g°C)

Percent Error

(%)

1

2

3

4

A

A

B

B

5 A

6 A

7 B

8 B

9 A

10 A

11 B

12 B

13 A

14 A

15 B

AVERAGE

0.442

0.441

0.438

0.435

0.432

0.443

0.444

0.445

0.445

0.438

0.443

0.443

0.444

0.438

0.431

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.440

0.455

0.227

-0.455

-1.136

-1.818

0.682

0.909

1.136

1.136

-0.455

0.682

0.682

0.909

-0.455

-2.045

0.030

Basith

– Bulgarelli 25

Table 11 shows the percent error table for the specific heat experiments on nickel rods. The experimental values were compared to the true specific heat of nickel, which was 0.440 J/g°C. The average specific heat of the nickel rods was 0.440 J/g•°C. The average percent error is 0.030%. The negative percent errors signify the instances when the experimental value was lower than the true value. The positive percent errors signify the instances when the experimental value was higher than the true value. The experimental value should never exceed the true value. The data was consistent. The trial with the unusually high percent error was trial 15, with a percent error of -2.045%.

Table 12

Unknown Metal Specific Heat Percent Error

Trial Rod Experimental

Value (J/g°C)

True Value

(J/g°C)

Percent Error

(%)

1 A

0.359 0.440 -18.409

2 A

0.345 0.440 -21.591

3 B

0.398 0.440 -9.545

4 B

0.388 0.440 -11.818

5 A

0.371 0.440 -15.682

6 A

0.368 0.440 -16.364

7 B

0.342 0.440 -22.273

8 B

0.333 0.440 -24.318

9 A

0.325 0.440 -26.136

10 A

0.340 0.440 -22.727

11 B

0.375 0.440 -14.773

12 B

0.406 0.440 -7.727

13 A

0.367 0.440 -16.591

14 A

0.354 0.440 -19.545

15 B

AVERAGE

0.376

0.363

0.440

0.440

-14.545

-17.470

Table 12 shows the percent error table for specific heat experiment on the unknown metal rods. The experimental values that were collected had to be

Basith

– Bulgarelli 26 compared to the true specific of nickel, which is 0.440 J/g°C. This is because the true specific heat of the unknown metals in not known. The average specific heat of the unknown rods was 0.363 J/g°C. The average percent error was -17.470%.

The negative symbols on the percent errors signify the fact that those trials fell short of 0.440 J/g°C. These trials had relatively low percent errors, but they were still larger than the percent errors for the nickel rods. This hints at the conclusion that the unknown rods are different than nickel, but they have some similar properties. The data was fairly consistent, with a relatively low amount of variability.

Table 13

Linear Thermal Expansion Nickel Percent Error

Trial Rod

Experimental

Value (10 -6 /°C)

True Value

(10 -6 /°C)

Percent

Error (%)

1 B

2 B

3 A

4 B

5 A

6 B

7 A

8 A

9 B

10 A

11 B

12 B

13 A

14 A

15 A

13.246

13.211

13.157

13.246

13.193

13.193

13.175

13.157

13.228

13.211

13.228

13.122

13.246

13.228

13.246

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

-0.406

-0.669

-1.075

-0.406

-0.805

-0.805

-0.940

-0.011

-0.541

-0.669

-0.541

-1.338

-0.406

-0.541

-0.406

AVERAGE 13.206 13.3 -0.637

Table 13 shows the percent errors from the linear thermal expansion experiment conducted on the nickel rods. The average alpha coefficient of linear

Basith

– Bulgarelli 27 thermal expansion was 13.206•10 -6 /°C. The actual unit is 1/°C, which is the unit of the alpha coefficient without any modifications. However, to make the data more manageable, the value was multiplied by 10 6 . This value was then compared to the true alpha coefficient of linear thermal expansion of nickel, whi ch was 13.3•10 -6 /°C. The average percent error was -0.637%. This is a very low percent error. The negative symbols show that the true value of 13.3•10 -6 /°C was never reached. The experimental values always fell short of the true alpha coefficient of linear thermal expansion of nickel, which was 13.3•10 -6 /°C. The data was consistent with a small amount of variability.

Table 14

Linear Thermal Expansion Unknown Metal Percent Error

Trial Rod

Experimental

Value (10 -6 /°C)

True Value

(10 -6 /°C)

Percent

Error (%)

1 B

2 B

3 A

4 B

5 A

6 B

7 A

8 A

9 B

10 A

11 B

12 B

13 A

14 A

15 A

0.312

0.312

0.311

0.312

0.312

0.312

0.312

0.313

0.311

0.312

0.313

0.311

0.310

0.312

0.313

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

13.3

-97.654

-97.654

-97.662

-97.654

-97.654

-97.654

-97.654

-97.647

-97.662

-97.654

-97.647

-97.662

-97.670

-97.654

-97.647

AVERAGE 0.312 13.3 -97.655

Table 14 shows the percent errors for the linear thermal expansion experiment conducted on the unknown metal rods. The experimental values

Basith

– Bulgarelli 28 were very different from the experimental values collected from the nickel rods.

The average alpha coefficient of linear thermal expansion for the unknown rods was 0.312•10 -6 /°C. The rods were compared to the true alpha coefficient of nickel. This was to determine whether the unknown metal rods were composed of nickel or not. The average percent error was -97.655%. The negative symbols represent the fact that the experimental values of the unknown rods always fell short of the true value of 13.3•10 -6 /°C. The data was very consistent with the percent error never exceeding 98%.

The second method that was used to analyze the data was box plots. The box plots were used to see if the medians of the known and unknown experiments overlapped. The box plots also served as a preliminary test of regularity for the data.

Specific Heat

0.438

0.444

0.431 0.445

0.376

0.442

0.325

0.342

0.406

0.367

Specific Heat of Nickel (J/g°C)

Figure 3. Box Plots of the Specific Heat Experiments

Figure 3 shows the box plots of the specific heat nickel and unknown experiments. The box plot of the unknown metal varies more than the box plot of

Basith

– Bulgarelli 29 the nickel rods. Both box plots were fairly normal, with slight skews to the left.

The box plots do not overlap at all. The medians of the two box plots do not overlap, meaning that there was no crossing of values in the two sets of data.

Values from the experimental data of the nickel rods did not occur as one of the experimental values of the unknown rods.

Linear Thermal Expansion

Min: 0.310

Q1: 0.311

Median: 0.312

Q3: 0.312

Max: 0.313

Min: 13.122

Q1: 13.175

Median: 13.211

Q3: 13.246

Max: 13.246

Alpha Coefficient of Linear Thermal Expansion (10 -6 /°C)

Figure 4. Box Plots of the Linear Thermal Expansion Experiments

Figure 4 shows the box plots of the linear thermal expansion experiments conducted on the nickel and unknown rods. The box plots were very small due to the fact that the scales of the two box plots are very far apart. Due to the size of the box plots, the skewness of the data cannot be seen. Both box plots are skewed to the left. The box plots do not overlap at all. The box plot of the nickel rods varies more than the box plot of the unknown rods. The medians of the box plots do not overlap, meaning that there is a large and distinctive difference between the two sets of data.

Linear Thermal Expansion Nickel

13.175 13.246

Basith

– Bulgarelli 30

13.122

13.211

Alpha Coefficient Linear Thermal Expansion (10 -6 /°C

Figure 5. Linear Thermal Expansion of Nickel Box Plot

Figure 5 shows the box plot of the linear thermal expansion experiment that was concerned with the nickel rods. The data is skewed to the left, as the third quartile and the maximum value are the same. The data is not normal.

Linear Thermal Expansion Unknown Metal

0.311

0.312

0.310 0.313

Alpha Coefficient Linear Thermal Expansion (10 -6 /°C

Figure 6. Linear Thermal Expansion Unknown Metal Box Plot

Basith

– Bulgarelli 31

Figure 6 shows the box plot of the linear thermal expansion experiment that was conducted on the unknown metal rods. The median and the third quartile are the same value. The data is not normal because it is skewed to the left.

The third and final method of analyzing the data was a two sample t test. A two sample t test is used when there are two different samples that are being compared. A two sample t test is used to determine whether or not there is a significant difference between the two samples. Two of these tests were used, one for specific heat and one for linear thermal expansion. The following equation was used to conduct a two sample t test. In the following equation, x

1 represents the mean of sample one, x

2 represents the mean of sample two, s

1 represents the standard deviation of sample one, s

2

represents the standard deviation of sample two, n

1 represents the number of data points in sample one, n

2

represents the number of data points in sample two, and t represents the number of standard deviations above or below the sample mean the data lies. A sample calculation can be found in Appendix B. 𝑡 = x

1

−

√

(𝑠 𝑛

1

1

) 2

+ x

2

(𝑠

2 𝑛

2

) 2

To conduct a two sample t test, certain assumptions had to be met. First off, the two sets of data being compared had to be simple random samples. A simple random sample is a sample in which all trials were randomized and there was no bias towards one outcome. The data collected was random because the

Basith

– Bulgarelli 32 trials had been randomized prior to experimentation. The second assumption was that the two sets of data must be independent samples. Independent samples are samples that have no effect on the other sample(s). These experiments were conducted using procedures that were unique to the specific experiment. The equipment had also been recalibrated between experiments.

This insured that the results of the previous trial would not affect the next trial.

The third assumption that had to be met was that the population standard deviation is not known. The population standard deviation was not known, as this would require the specific heats and alpha coefficients of every nickel and metal rod in the world. With the resources provided, this was not possible. The fourth assumption that had to be met was that the data sets should consist of normal data. Normality was checked using normal probability plots. If the data turned out to not be normal, than the sample should consist of 30 data points.

Figure 7. Specific Heat Nickel Normal Probability Plot

Figure 7 shows the normal probability plot for the specific heat experiment on the nickel rods. The more closer the data points are to the line in the middle, the more normal the data is. The point in the far upper right hand corner strayed away slightly from the line, as well as three vertical points close to the middle of

Basith

– Bulgarelli 33 the graph. With the exception of these few data points, the data was fairly normal.

Figure 8. Specific Heat Unknown Metal Normal Probability Plot

Figure 8 shows the normal probability plot of the data collected during the specific heat experiment on the unknown rods. The data is fairly close to the line in the middle. Therefore, the data was fairly normal.

Figure 9. Linear Thermal Expansion Nickel Rods Normal Probability Plot

Figure 9 shows the normal probability plots for the linear thermal expansion experiments on the nickel and unknown rods. The probability plot on the left is the plot for the nickel rods, while the plot on the right is the plot for the unknown rods. For the plot of the data for the unknown rods, the points arranged in a vertical manner suggest results that were the same, or in a very small range. The data points were not close to the lines. The data was not normal and extra trials had to be conducted. However, due to time constraints, extra trials could not be

Basith

– Bulgarelli 34 conducted. The two sample t test for linear thermal expansion was conducted using data that was not normal for both samples. This suggests that the results gathered from the two sample t test will not be very reliable.

A two sample t test requires two hypotheses. The first hypothesis, known as the null hypothesis, is assumed to be true. The second hypothesis, known as the alternate hypothesis, is the hypothesis that is being tested. The two hypotheses for the two sample t test for specific heat and linear thermal expansion were:

𝐻 𝑜

: 𝜇

𝑁𝑖𝑐𝑘𝑒𝑙

= 𝜇

𝑈𝑛𝑘𝑛𝑜𝑤𝑛

𝐻 𝑎

: 𝜇

𝑁𝑖𝑐𝑘𝑒𝑙

≠ 𝜇

𝑈𝑛𝑘𝑛𝑜𝑤𝑛

The first hypothesis, the null, denoted by H o

, said that the average specific heat/alpha coefficient for nickel, denoted by

µ

Nickel

, was equal to the average specific heat/alpha coefficient of the unknown metal, denoted by µ

Unknown

. This alternate hypothesis states that the average specific heat/alpha coefficient of the nickel rods was not equal to the average specific heat/alpha coefficient of the unknown rods. Two sample t tests yield a t -value that is converted to a p -value using a p -value table. This p -value is then tested against an alpha level of 0.10. If the p -value is lower than the alpha level, the null hypothesis is rejected. If the p -value is greater than the alpha level of 0.10, the null hypothesis is not rejected.

Basith

– Bulgarelli 35

Figure 10. Specific Heat Two Sample t Test Results

Figure 10 shows the results for the two sample t test that was conducted on the data collected from the specific heat experiments. The t -value was 12.362.

This shows how far the data was from the sample mean, measured in standard deviations. The p value was 2.760 • 10 -9 .

PVal = 2.760•10 -9

Figure 11. Specific Heat P-Value Plot

Figure 11 shows a bell curve with the p -value plotted on it. The p -value is

2.760 • 10 -9 . The p -value is very close to zero.

The p value of 2.760 • 10 -9 is less than the alpha level of 0.10. This means that the null hypothesis was rejected. There is significant evidence to suggest that the specific heat of the unknown rods is not equal to the specific heat of

Basith

– Bulgarelli 36 nickel. If the null hypothesis was true, there is a 0.000002671% chance of getting results this extreme by chance alone.

Figure 12. Linear Thermal Expansion Two Sample t Test Results.

Figure 12 shows the results of the two sample t test that was performed on the data from the linear thermal expansion experiments. The t -value was

1287.31. This means that the data collected was 1287.31 standard deviations away from the sample mean. The p value was 6.00 • 10 -37 . However, the results from this two sample t test were not very reliable because the data collected from the linear thermal expansion experiments was not normal.

PVal = 6.00 • 10 -37

Figure 13. Linear Thermal Expansion P-Value Plot

Figure 13 shows a bell curve with the p -value. The p value is 6.00 • 10 -36 , which is why the p -value cannot be seen.

Basith

– Bulgarelli 37

The p value of 6.00 • 10 -36 is less than the alpha level of 0.10. Thus, the null hypothesis was rejected. There is significant evidence to suggest that the alpha coefficient of linear thermal expansion for the unknown rods is not equal to the alpha coefficient of linear thermal expansion of the nickel rods. If the null hypothesis was true, there is essentially a zero percent chance of getting results this extreme by chance alone.

Basith

– Bulgarelli 38

Conclusion

The purpose of the experiment was to use the intensive properties of specific heat and linear thermal expansion to determine whether or not a pair of unknown metal rods were composed of nickel. The hypothesis that stated that the metals would be identified as nickel if the percent error for both specific heat and linear thermal expansion was less than 1% was accepted. The percent errors for specific heat and linear thermal expansion experiments had percent errors larger than 10%.

Various forms of evidence were collected to verify this claim. The physical properties of the rods, the percent error calculations, and the results of the two sample t tests were taken into account. The percent errors for the unknown rods were substantially larger than the percent errors collected from the trials performed on the nickel rods, which were below 1% for both specific heat and linear thermal expansion. The unknown rods had a brighter luster than the nickel rods and had a different resonance as well. The unknown rods also felt lighter than the nickel rods. Two sample t tests were conducted to determine whether or not there was significant mathematical difference between the two pairs of rods.

The t test conducted on the specific heat data yielded results that suggested the unknown metal rods were not composed of nickel. These results were valid due to the fact that the data collected from the specific heat experiments were normal. The t test conducted on the linear thermal expansion data yielded results that also suggested that the unknown metal rods were not composed of nickel. However, the results for the linear thermal expansion t test were not valid. This was due to the fact that the data collected from the linear

Basith

– Bulgarelli 39 thermal expansion experiments was skewed. The abnormality of the data could have been eliminated if additional trials had been conducted. Due to time constraints, however, this could not be done.

A major error during this experiment was that the initial temperature of the metal was assumed to be the temperature of the boiling water. Also, instead of using a thermometer, the boiling water was assumed to be at 100 °C. This would change the values placed into the equation used to calculate specific heat. The water for the calorimeters was measured incorrectly. Instead of rounding to the nearest tenths place on the graduated cylinder, the water was measured to the nearest whole. Since the calorimeters contained a small volume of water, and small changes in temperature were recorded, any change in volume, no matter how small, would yield significantly different results. Other potential sources of error include the construction of the calorimeters and expansion jigs. The calorimeters were constructed using household items and were not truly isolated systems. The First Law of Thermodynamics states that energy is not created or destroyed, only transferred. Due to this law, the heat lost from the heated metal in the calorimeter is gained by the water. If the calorimeter is not truly isolated, heat will escape and all the heat lost from the metal will not be gained by the water. The expansion jigs were constructed using household items as well and there was also no method of preventing the metal rods from cooling down as they were transferred from the boiling water to the jig. During this time, the rods cooled down slightly, causing the change in length to be inaccurate. The inaccurate change in length would contribute to inaccurate results.

Basith

– Bulgarelli 40

To eliminate these errors, more trials could be conducted. This would increase the normality of the data, eliminating any skewness. Also, calorimeters and expansion jigs with improved designs could be used. For specific heat, this would minimize the loss of heat of the calorimeter. An improved jig would produce a more accurate change in length measurement for linear thermal expansion. The temperature of the boiling water could be taken using a thermometer instead of assuming that the water is at 100 °C. This would insure that the change in temperature in the calorimeter was more accurate, which in turn would yield more accurate results. The volume of the water could be rounded to the nearest tenth instead of to the nearest whole number.

The experiment could be modified or expanded to include other intensive properties such as density, melting point, and tensile strength. These properties were not tested as many of them, such as melting point and tensile strength, would require the rods to be physically destroyed. The relevancy of this experiment can be seen in the industrial world, as intensive properties such as specific heat and linear thermal expansion are taken into consideration before choosing the material for a product. An example would be when constructing girders for a building, as expansion of the material wants to be minimized. If the girders or supports expand in increased temperatures, there is an increased chance of collapse. If the material used to construct the girders has a high specific heat, the girders will absorb more heat, and then release the heat into the building. This will increase the temperature inside the building and make it extremely hot.

Application

Basith

– Bulgarelli 41

Figure 14. Nickel Key

Figure 14 above shows a key made of nickel. Nickel is useful when making keys because the nickel is very durable. Keys made of nickel are more durable than keys made of other metals. Nickel is the metal of choice for locksmiths as the key will not get deformed or bent easily. If the key were made of nickel, the mass would be 0.02 pounds and the cost would be $0.17 due to nickel costing approximately $8.37 per pound. Nickel is an excellent material for keys due to being extremely strong, as well as haven’t little variability in size due to having nearly no significant change in size due to linear thermal expansion.

Figure 15. Mechanical Drawing of Nickel Key.

Figure 15 above shows a mechanical drawing of the nickel key. This drawing is necessary for any locksmith to produce a key of this model. Due to all keys being custom, measurements for the “teeth” of the key have not been

Basith

– Bulgarelli 42 recorded. Keys would be produced as blanks (without teeth), and would be created custom for the customer’s lock.

Basith

– Bulgarelli 43

Appendix A: Calorimeter Instructions

Materials:

(2) ½” diameter x 6” PVC

(2) ¾” diameter x 6” PVC

(2) ¾” diameter x 6” Polymer pipe insulation

(4) ¾” PVC pipe caps

Purple PVC primer

Orange PVC cement

X-Acto Knife

Belt Sander

Chop Saw

Drill Press

Instructions to construct 1 Calorimeter:

1.

Apply a thin layer of PVC primer on one end of the ¾” PVC pipe. Do the same for the inside for one of the caps.

2. Apply a thin l ayer of PVC cement to both the ¾” PVC pipe and the inside of the cap. Be sure to completely cover the primer with the cement

3. Firmly press the cap on the pipe whilst slowly twisting until a solid bond can be felt between the cap and the pipe.

4. Allow to the cement to set for about 5 minutes

5. Use the X-Acto knife to cut the inside of the insulation tube around the perimeter of the cap, increasing the internal diameter until it is large enough to fit a PVC cap.

6. Slide the PVC pipe into the insulation sleeve (Be sure to insert the open end of the tube into the side of the insulation that was opened in the previous step)

7. Cut off ½” to 1” of the insulation on the open end of the pipe to create room for the 2 nd cap

8. Force the ½” diameter PVC pipe over the ¾” insulated pipe ( note: the ½” pipe will not fit over the ¾” pipe easily; it will have to be forced on . Any sort of blunt instrument will be sufficient.

)

Basith

– Bulgarelli 44

9. Drill a hole into the second cap that is the size of the temperature probe that will be used.

10. Place second cap on the finished calorimeter. This cap will not be cemented on in order to provide access to the inside of the calorimeter

11. Repeat steps 1-10 for another calorimeter.

Figure 1. Calorimeter Materials

Figure 1 shows the materials that were used to construct one calorimeter.

The belt sander, the chop saw and the drill press are not shown.

Basith

– Bulgarelli 45

Appendix B: Formulas and Sample Calculations

Specific Heat

In the following equation, s water

represents the specific heat of water, m water represents the mass of the water, Δ t water

represents the change in temperature of the water, s metal

represents the specific heat of the metal, m metal represents the mass of the metal, Δ t metal represents the change in temperature of the metal and

CF represents the correction factor of the calorimeter for that particular trial. The specific heat of the metal, s metal

, is what the equation was used to solve for. The absolute value will be used for the change in tem perature of the metal, Δ t metal

.

This is because the negative symbol does not change the value. The negative symbol is only to show the direction of the flow of heat. The product of the specific heat of water, the mass of the water, and the change in temperature of the water is divided by the product of the mass of the metal and the change in temperature of the metal. The correction factor of the calorimeter that was used is then added on to the answer.

( 𝑠 𝑤𝑎𝑡𝑒𝑟 𝑚 𝑤𝑎𝑡𝑒𝑟 𝑚 𝑚𝑒𝑡𝑎𝑙

∆𝑡 𝑤𝑎𝑡𝑒𝑟

∆𝑡 𝑚𝑒𝑡𝑎𝑙

) + 𝐶𝐹 = 𝑠 𝑚𝑒𝑡𝑎𝑙

This equation was used to calculate the specific heats of all the trials for the specific heat experiments. A sample calculation is shown below.

( 𝑠 𝑤𝑎𝑡𝑒𝑟 𝑚 𝑚 𝑤𝑎𝑡𝑒𝑟

∆𝑡 𝑚𝑒𝑡𝑎𝑙

∆𝑡 𝑤𝑎𝑡𝑒𝑟 𝑚𝑒𝑡𝑎𝑙

) + 𝐶𝐹 = 𝑠 𝑚𝑒𝑡𝑎𝑙

(

4.184 J/g°C • 45 g • 5.9 °C

36.028 g • 71 °C

) + 0.008 J/g°C = 𝑠 𝑚𝑒𝑡𝑎𝑙

0.442 J/g°C = 𝑠 𝑚𝑒𝑡𝑎𝑙

Figure 1. Specific Heat Sample Calculation

Basith

– Bulgarelli 46

Figure 1 shows a sample calculation using the specific heat equation. The values used in this sample calculation are from the first trial of the specific heat experiment that dealt with the nickel metal rods.

Correction Factor

The calorimeters that were constructed for the specific heat experiment were not perfect. Heat escaped from the calorimeters. To get accurate results, a correction factor for each calorimeter had to be calculated and then added onto the original result. Afterwards, the individual correction factors were added together and then divided by the number of trials to get the average correction factor for the calorimeter. In the following equation, V

Tn represents the true specific heat of the material in J/g°C,

V

E

represents the experimental value c ollected during experimentation in J/g°C, n represents the number of trials and

CF represents the correction factor in J/g°C. For every trial, the true value of nickel is subtracted from the experimental value. The capital sigma denotes the sum of the individual correction factors, which is then divided by the number of trials to find the average.

∑ 𝑛 𝑖=1

(𝑉

𝑇𝑛 𝑛

− 𝑉

𝐸𝑛

)

= 𝐶𝐹

To find the average correction factor of each calorimeter, the averages of all the trials that used that calorimeter are taken. A sample calculation is shown below.

∑ 𝑛 𝑖=1

(𝑉

𝑇𝑛

− 𝑉 𝑛

0.064 J/g°C

𝐸𝑛

)

= 𝐶𝐹

= 𝐶𝐹

8 trials

Basith

– Bulgarelli 47

0.008 J/g°C = 𝐶𝐹

Figure 2. Correction Factor Sample Calculation.

Figure 2 shows a sample calculation for the average correction factor of a calorimeter. The values used in the sample calculation are from trials of the specific heat experiment of the nickel rods that used calorimeter 1. The final answer shown is the average correction factor of calorimeter 1.


In the following equation, L i represents the initial length of the metal rods in mm, Δ L represents the change in length of the metal rods in mm, Δ T represents the change in temperature of the metal rods in °C, and α represents the coefficient of linear thermal expansion in 1/°C. After the value of the alpha coefficient is found, the value is multiplied by 10 6 to make the value more manageable. This changes the units of the alpha coefficient into 10 -6 /°C. The coefficient of linear thermal expansion, also known as the alpha coefficient of linear expansion, denoted by

α

is what this equation was used to solve for. The initial length is multiplied by the change in temperature. The change in length is then divided by that number. Finally, the quotient is then multiplied by 10 6 to get an answer that is manageable.

(

𝐿 𝑖

∆𝐿

∆𝑇

) • 10 6 = 𝛼

The equation above was used to solve for alpha coefficient of linear expansion for all of the trials in the linear thermal expansion experiments. A sample calculation is shown below.

Basith

– Bulgarelli 48

(

𝐿 𝑖

∆𝐿

∆𝑇

) • 10 6 = 𝛼

0.127 mm

(

129.388 mm • 74.1 °C

) • 10 6 = 𝛼

10 6

13.246 = 𝛼

°C

Figure 3. Linear Thermal Expansion Sample Calculation

Figure 3 shows a sample calculation using the linear thermal expansion equation. The values used in this sample calculation are from the first trial of the linear thermal expansion lab that was conducted on the nickel rods.

Percent Error

In the following equation, V

E represents the experimental value, V

T represents the true value, and P

Error

represents percent error. The experimental value is subtracted from the true value. The difference is then divided by the true value. To obtain a percentage, the quotient is then multiplied by 100.

𝑃

𝐸𝑟𝑟𝑜𝑟

=

𝑉

𝐸

− 𝑉

𝑇

∙ 100

𝑉

𝑇

This equation was used in all four of the experiments: specific heat for the nickel rods, specific heat for the unknown rods, linear thermal expansion of the nickel rods, and linear thermal expansion of the unknown rods. It was used as one of the three methods of statistical analysis. A sample calculation is shown below.

𝑃


=

𝑉

𝐸

− 𝑉

𝑇

∙ 100

𝑉

𝑇

𝑃


=

0.442 J/g°C − 0.440 J/g°C

∙ 100

0.440 J/g°C

𝑃


=

0.002 J/g°C

0.440 J/g°C

∙ 100

Basith

– Bulgarelli 49

𝑃


= 0.455

%

Figure 4. Percent Error Sample Calculation.

Figure 4 shows a sample calculation using the percent error equation. The values used in the sample calculation are from the first trial of the specific heat experiment of the nickel rods.

Two Sample t Test

A two sample t test is a method of statistical analysis when there are two sets of data that need to be compared. In the following equation, µ

1

represents the mean of sample one, µ

2 represents the mean of sample two, s

1 represents the standard deviation of sample one, s

2

represents the standard deviation of sample two, n

1 represents the number of data points in sample one, n

2 represents the number of data points in sample two, and t represents the number of standard deviations above or below the sample mean the data lies. The sample means are subtracted from each other. The standard deviations are squared and then divided by the number of trials for each respective experiment.

The answers are added and the square root of the sum is taken. The difference of the means is then divided by the root answer. 𝑡 = x

1

−

√

(𝑠 𝑛

1

1

) 2

+ x

2

(𝑠

2 𝑛

2

) 2

This equation was used to compare the nickel specific heat to the unknown specific heat and the nickel linear thermal expansion to the unknown

Basith

– Bulgarelli 50 linear thermal expansion. The units that were used in data collection become meaningless in this equation. A sample calculation is shown below. 𝑡 =

√ x

1

− 𝑠 𝑛

1

1

+ x

2 𝑠

2 𝑛

2 𝑡 =

0.44013333333333 J/g°C − 0.36313333333333 J/°C

√(0.0045960645691515)

15

2

−

(0.02368202050582) 2

15 𝑡 = 12.362

Figure 5. Two Sample t Test Sample Calculation

Figure 5 shows a sample calculation using the equation for a two sample t test. The values used were from the specific heat experiment that was conducted on the nickel and unknown rods. The reason there are extra decimal places shown is because they are required to get accurate results. The t value is then converted to a p value by using a table.

Basith

– Bulgarelli 51

Works Cited

De Leon, Prof. "Specific Heat." Specific Heat and Heat Capacity . Indiana

University, 30 Jan. 2001. Web. 24 Mar. 2014.

<http://www.iun.edu/~cpanhd/C101webnotes/matter-and

-energy/specificheat.html>.

Lenntech "Nickel-Ni." Lenntech . Lenntech Water Treatment Solutions, 2001.

Web. 13 May 2014. <http://www.lenntech.com/periodic/elements/ni.htm>.

Selby, Mark. "Royal Nickel Corp: Indonesian Nickel Ore Ban." Mining.com

. The

Royal Nickel Corp, 14 Feb. 2014. Web. 13 May 2014.

<http://www.mining.com/web/royal-nickel-corp-indonesian-ore-export-banopens-door-to-the-next-generation-of-nickel-mines/>.

Winter, Mark. "Linear Expansion Coefficient: Periodicity." WebElements Periodic

Table of the Elements . The University of Sheffield, 12 Mar. 2012. Web. 26

Mar. 2014.

<https://www.webelements.com/periodicity /coeff_thermal_expansion/>.

Chang, Raymond. "Chapter 6. Thermochemistry." Chemistry . 9th Ed. New York

City: McGraw Hill Wright Group, 2007. 239-40. Print.

Duffy, Andrew. "Temperature and Thermal Expansion." Boston University

Physics . Boston University, 26 July 2004. Web. 26 Mar. 2014.

<http://physics.bu.edu/~duffy/py105/Temperature.html>.

Basith

– Bulgarelli 52

Helmenstine, Anne M., Ph.D. "Intensive Property Definition." About.com

Chemistry . About.com, 15 Apr. 2012. Web. 29 Mar. 2014.

<http://chemistry.about.com/od/chemistryglossary/g/Intensive-Property-

Definition.htm>

The Physics Classroom. "Calorimeters and Calorimetry." The Physics Classroom

- Thermal Chemistry . The Physics Classroom, 26 Mar. 2014. Web. 26

Mar. 2014.

<http://www.physicsclassroom.com/class/thermalP/Lesson2/Calorimetersand-Calorimetry>.

Kent, Mr. "Specific Heat Capacity." Mr. Kent's Chemistry Page . N.p., 4 Sept.

2012. Web. 24 Mar. 2014

<http://www.kentchemistry.com/links/Energy/SpecificHeat.htm>.

Woodward, Patrick. "1st Law of Thermodynamics." Ohio State Department of

Chemistry . Ohio State University, 27 Apr. 2009. Web. 28 Mar. 2014.

<http://chemistry.osu.edu/~woodward/ch121/ch5_law.htm>002E

Nave, R. "Specific Heat." Hyperphysics . N.p., 7 June 2004. Web. 25 Mar. 2014.

<http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html>.

Bauck, Mrs. Chemistry Lab: Specific Heat of a Metal.

2006. MS Palm Harbor

University High School.

Montbriand, Mr. Experiment 5: Calorimetry and Specific Heat.

2008. MS San

Juan High School.

Basith

– Bulgarelli 53

Cassel, B. & Menard, K. “Coefficient of Thermal Expansion Measurement using the TMA 4000.” Thermal Analysis . Waltham: PerkinElmer Inc, 2013. 1.

Print.

Raymond, Jimmy. "Thermal Expansion Equations and Formulas

Calculator." Thermal Expansion Equations Formulas and Calculations . AJ

Design, 2012. Web. 26 Mar. 2014.

< http://www.ajdesigner.com/phpthermalexpansion/thermal_expansion_equ ation_linear_coefficient.php#ajscroll >.

Ellert, Glenn. "Thermal Expansion." The Physics Hypertextbook . N.p., 1994.

Web. 24 Mar. 2014.

<http://physics.info/expansion/>.

Nave, R. "Thermal Expansion." Hyperphysics . N.p., 15 Oct. 2001. Web. 25 Mar.

2014.

<http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thexp.html>.

Gibbs, Keith. "Thermal Expansion - Demonstration Experiments." Schoolphysics .

N.p., 8 Jan. 2013. Web. 25 Mar. 2014.

<http://www.schoolphysics.co.uk/age14-

16/Heat%2520energy/Expansion/text/Thermal_expansion/index.html>.

Buckhardt, C. Coefficient of Linear Thermal Expansion.

2013. MS St. Louis

Community College

Department of Natural Sciences . Lab 12 Thermal Expansion 2012. MS Lawrence

Technological University

Linear Thermal Expansion and Specific Heat of Nickel

Related documents

Products

Support

Linear Thermal Expansion and Specific Heat of Nickel

Related documents

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib