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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


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


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 sw represents specific heat

of water, measured in J/g•°C, mw represents the mass of the water, measured in

mL, ΔTw represents the change in temperature of the water in the calorimeter,

measured in °C, sm represents the specific heat of the metal sample, measured

in J/g•°C, mm represents the mass of the metal sample, measured in g, and ΔTm

represents the change in temperature of the metal sample, measured in °C.

(Chang 240).

swmw∆Tw=smmm ∆T m

Table 1. The Specific Heats of Common Materials

Material Specific Heat(J/g•°C)

Aluminum 0.900Glass 0.880Brass 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


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. Temperature 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 Li represents the initial length of the metal measured in

mm. The variable ∆T represents the change in temperature of the metal in °C.

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

ΔL=∝Li∆T


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

Material Alpha Coefficient (10-6/K)

Aluminum 0.900Glass 0.880Brass 0.380Table 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


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 change 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) Calorimeters500 ml Loaf PanThermometer (0.1 °C)500 mL Graduated CylinderTongs(2) Unknown Metal Rods(2) Nickel RodsVernier LabQuestTemperature Probe (0.01 °C)Scale (0.01g Precision)TI - NspireHot MittHot 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.


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 temperature 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.


Diagram:

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.

Hot Plate

Graduated Cylinder

Nickel RodsUnknown Rods

Temperature Probe

Vernier TM

LabQuestThermometer

Hot Mitt

Loaf Pan

Tongs

Calorimeters

Basith – Bulgarelli 13

Experimental Design for Linear Thermal Expansion

Materials:

(2) Unknown Metal Rods(2) Nickel Rods(2) 500 mL Loaf PanTongsGraduated CylinderLinear Thermal Expansion Apparatus (0.001 in Precision)Thermometer (0.1 °C Precision)Caliper (0.01 mm Precision)TI – NspireHot PlateHot MittBlow-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.


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:

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.

Hot Plate

Blow-Off Compressed Air

Graduated Cylinder

Unknown RodsNickel Rods

Thermometer

LTE Apparatus

Hot Mitt Tongs

Loaf Pan


Data and Observations for Specific Heat

Table 3

Trial Rod Cal

Initial Temp. (°C)

Equil. Temp.(°C)

Change in Temp. (°C)

Mass(g) Correction

Factor (J/g°C)

Specific Heat

(J/g°C)W M W M M W

1 1 A 23.1 100.0 29 5.9 -71.0 36.028 45 0.008 0.4422 2 A 22.6 100.0 28.5 5.9 -71.5 36.028 45 0.010 0.4413 1 B 22.4 100.0 28.3 5.9 -71.7 36.028 45 0.008 0.4384 2 B 22.8 100.0 28.6 5.8 -71.4 36.028 45 0.010 0.4355 1 A 22.7 100.0 28.5 5.8 -71.5 36.028 45 0.008 0.4326 2 A 22.9 100.0 28.8 5.9 -71.2 36.028 45 0.010 0.4437 1 B 23.4 100.0 29.3 5.9 -70.7 36.028 45 0.008 0.4448 2 B 23.2 100.0 29.1 5.9 -70.9 36.028 45 0.010 0.4459 1 A 23.5 100.0 29.4 5.9 -70.6 36.028 45 0.008 0.445

10 2 A 23.4 100.0 29.2 5.8 -70.8 36.028 45 0.010 0.43811 1 B 23.2 100.0 29.1 5.9 -70.9 36.028 45 0.008 0.44312 2 B 22.9 100.0 28.8 5.9 -71.2 36.028 45 0.010 0.44313 1 A 23.4 100.0 29.3 5.9 -70.7 36.028 45 0.008 0.44414 2 A 23.3 100.0 29.1 5.8 -70.9 36.028 45 0.010 0.43815 1 B 22.6 100.0 28.4 5.8 -71.6 36.028 45 0.008 0.431

Average 26.7 100.0 28.9 5.9 -71.1 36.028 45 0.009 0.440Specific Heat of Nickel Data

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

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.


Table 4 Observations for Nickel Specific Heat

Trial Observations

1 LabQuest operated by Researcher 1. Tongs operated by Researcher 2. Trial conducted directly under light.

2 LabQuest reset by Researcher 1. Tongs operated by Researcher 2. Trial conducted directly under light.

3LabQuest reset by Researcher 2. Small spill cleaned up by Researcher 1. Tongs operated by Researcher 1. Trials conducted directly under light.

4 Labquest reset by Researcher 1. Tongs operated by Researcher 2. Trial conducted directly under light.

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

Table 5


Specific Heat of Unknown Metal Data

Trial Rod Cal

Initial Temp. (°C)

Equil. Temp.(°C)

Change in Temp. (°C)

Mass(g)

Correction Factor (J/g°C)

Specific Heat

(J/g°C)W M W M M W

1 1 A 24.9 100.0 29.3 4.4 -70.7 33.375 45 0.008 0.3592 2 B 23.8 100.0 28.2 4.4 -74.2 33.375 45 0.010 0.3453 2 A 23.9 100.0 28.8 4.9 -71.2 33.375 45 0.010 0.3984 1 B 25.5 100.0 30.2 4.7 -69.8 33.375 45 0.008 0.3885 1 A 25.6 100.0 30.1 4.5 -69.9 33.375 45 0.008 0.3716 2 B 24.5 100.0 29.0 4.5 -71.0 33.375 45 0.010 0.3687 2 A 22.6 100.0 26.9 4.3 -73.1 33.375 45 0.010 0.3428 2 B 22.4 100.0 26.6 4.2 -73.4 33.375 45 0.010 0.3339 1 A 23.0 100.0 27.1 4.1 -72.9 33.375 45 0.008 0.32510 1 B 22.6 100.0 26.9 4.3 -73.1 33.375 45 0.008 0.34011 2 A 24.4 100.0 29.0 4.6 -71.0 33.375 45 0.010 0.37512 2 B 23.8 100.0 28.8 5.0 -71.2 33.375 45 0.010 0.40613 1 A 24.8 100.0 29.3 4.5 -70.7 33.375 45 0.008 0.36714 1 B 23.9 100.0 28.3 4.4 -71.7 33.375 45 0.008 0.35415 2 A 24.5 100.0 29.1 4.6 -70.9 33.375 45 0.010 0.376

Average 24.0 100.0 28.5 4.5 -71.7 33.375 45 0.009 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.

Table 6


Observations for Unknown Rods Specific HeatTrial Observations

1LabQuest operated by Researcher 2. Tongs operated by Researcher 1. Trial conducted directly under light. Rod was not massed, so trial was redone.

2 LabQuest reset by Researcher 2. Tongs operated by Researcher 1. Trial conducted directly under light.

3LabQuest reset by Researcher 1. Tongs operated by Researcher 2. Trials conducted directly under light. Redone due to mix-up of the metal rods.


5 LabQuest reset by Researcher 1. Tongs operated by Researcher 2. Trial redone due to rod mix-up. Trial conducted directly under light.











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.

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)

Change in Temp.

(°C)

AlphaCoefficient(10-6/°C)

1 B 0.127 129.388 25.9 100.0 74.1 13.2462 B 0.127 129.388 25.7 100.0 74.3 13.2113 A 0.127 129.388 25.4 100.0 74.6 13.1574 B 0.127 129.388 25.9 100.0 74.1 13.2465 A 0.127 129.388 25.6 100.0 74.4 13.1936 B 0.127 129.388 25.6 100.0 74.4 13.1937 A 0.127 129.388 25.5 100.0 74.5 13.1758 A 0.127 129.388 25.4 100.0 74.6 13.1579 B 0.127 129.388 25.8 100.0 74.2 13.22810 A 0.127 129.388 25.7 100.0 74.3 13.21111 B 0.127 129.388 25.8 100.0 74.2 13.22812 B 0.127 129.388 25.2 100.0 74.8 13.12213 A 0.127 129.388 25.9 100.0 74.1 13.24614 A 0.127 129.388 25.8 100.0 74.2 13.22815 A 0.127 129.388 25.9 100.0 74.1 13.246Average 0.127 129.388 25.7 100.0 74.3 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 8Observations from Linear Thermal Expansion for Nickel Rods

Trial Observations

1 Tongs used by Researcher 1. Jig operated and metal cooled by Researcher 2. Room unusually warm due to other experiments.

2Tongs 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.




Trial Observations6 Tongs used by Researcher 2. Jig operated and metal cooled by


Researcher 1. Room unusually warm due to other experiments.

7Tongs used by Researcher 1. Jig operated and metal cooled by 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 9Linear Thermal Expansion for Unknown Rods Data

Trial Rod ΔL (mm)

Initial Length (mm)

Initial Temp. (°C)

Final Temp (°C)

Change in Temp.

(°C)

AlphaCoefficient(10-6/°C)

1 A 0.003 129.28 25.7 100.0 74.3 0.3122 B 0.003 129.28 25.6 10.00 74.4 0.3123 A 0.003 129.28 25.5 100.0 74.5 0.3114 B 0.003 129.28 25.7 100.0 74.3 0.3125 B 0.003 129.28 25.6 100.0 74.4 0.3126 B 0.003 129.28 25.7 100.0 74.3 0.3127 A 0.003 129.28 25.6 100.0 74.4 0.3128 A 0.003 129.28 25.9 100.0 74.1 0.3139 A 0.003 129.28 25.5 100.0 74.5 0.31110 B 0.003 129.28 25.7 100.0 74.3 0.312

Trial Rod ΔL (mm)

Initial Length

Initial Temp.

Final Temp

Change in Temp.

AlphaCoefficient


(mm) (°C) (°C) (°C) (10-6/°C)11 B 0.003 129.28 25.9 100.0 74.1 0.31312 A 0.003 129.28 25.4 100.0 74.6 0.31113 B 0.003 129.28 25.1 100.0 74.9 0.31014 A 0.003 129.28 25.7 100.0 74.3 0.31215 B 0.003 129.28 25.9 100.0 74.1 0.313Average 0.003 129.28 25.6 100.0 74.4 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 10Observations from Linear Thermal Expansion for Unknown Metal Rods.

Trial Observations


2 Tongs used by Researcher 2. Jig and metal cooled by Researcher 1. Room unusually warm due to other experiments.





7Tongs used by Researcher 1. Jig operated and metal cooled by Researcher 2. Room unusually warm due to other experiments. Trial redone due to rod mix-up.




11

Tongs used by Researcher 1. Jig operated and metal cooled by Researcher 2. Room unusually warm due to other experiments. Trial redone due to jig malfunction.

Trial Observations12 Tongs used by Researcher 1. Jig operated and metal cooled by


Researcher 2. Room unusually warm due to other experiments.




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.

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 11Specific Heat of Nickel Percent Error

Trial Rod Experimental Value (J/g°C)

True Value (J/g°C)

Percent Error (%)

1 A 0.442 0.440 0.4552 A 0.441 0.440 0.2273 B 0.438 0.440 -0.4554 B 0.435 0.440 -1.1365 A 0.432 0.440 -1.8186 A 0.443 0.440 0.6827 B 0.444 0.440 0.9098 B 0.445 0.440 1.1369 A 0.445 0.440 1.13610 A 0.438 0.440 -0.45511 B 0.443 0.440 0.68212 B 0.443 0.440 0.68213 A 0.444 0.440 0.90914 A 0.438 0.440 -0.45515 B 0.431 0.440 -2.045

AVERAGE 0.440 0.440 0.030

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 12Unknown 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 -18.4092 A 0.345 0.440 -21.591

3 B 0.398 0.440 -9.5454 B 0.388 0.440 -11.8185 A 0.371 0.440 -15.6826 A 0.368 0.440 -16.3647 B 0.342 0.440 -22.2738 B 0.333 0.440 -24.3189 A 0.325 0.440 -26.13610 A 0.340 0.440 -22.72711 B 0.375 0.440 -14.77312 B 0.406 0.440 -7.72713 A 0.367 0.440 -16.59114 A 0.354 0.440 -19.54515 B 0.376 0.440 -14.545AVERAG

E 0.363 0.440 -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

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 13Linear Thermal Expansion Nickel Percent Error

Trial Rod Experimental Value (10-6/°C)

True Value (10-6/°C)

Percent Error (%)

1 B 13.246 13.3 -0.4062 B 13.211 13.3 -0.6693 A 13.157 13.3 -1.0754 B 13.246 13.3 -0.4065 A 13.193 13.3 -0.8056 B 13.193 13.3 -0.8057 A 13.175 13.3 -0.9408 A 13.157 13.3 -0.0119 B 13.228 13.3 -0.541

10 A 13.211 13.3 -0.66911 B 13.228 13.3 -0.54112 B 13.122 13.3 -1.33813 A 13.246 13.3 -0.40614 A 13.228 13.3 -0.54115 A 13.246 13.3 -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

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 106. This value was then

compared to the true alpha coefficient of linear thermal expansion of nickel,

which 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 14Linear Thermal Expansion Unknown Metal Percent Error

Trial Rod Experimental Value (10-6/°C)

True Value (10-6/°C)

Percent Error (%)

1 B 0.312 13.3 -97.6542 B 0.312 13.3 -97.6543 A 0.311 13.3 -97.6624 B 0.312 13.3 -97.6545 A 0.312 13.3 -97.6546 B 0.312 13.3 -97.6547 A 0.312 13.3 -97.6548 A 0.313 13.3 -97.6479 B 0.311 13.3 -97.662

10 A 0.312 13.3 -97.65411 B 0.313 13.3 -97.64712 B 0.311 13.3 -97.66213 A 0.310 13.3 -97.67014 A 0.312 13.3 -97.65415 A 0.313 13.3 -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

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.

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

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

Nickel Rods

Unknown Rods

Specific Heat of Nickel (J/g°C)

Specific Heat

0.431

0.442

0.444

0.445

0.438

0.376

0.406

0.367

0.342

0.325


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.

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.

Nickel Rods

Unknown Rods Min: 0.310Q1: 0.311Median: 0.312Q3: 0.312Max: 0.313

Min: 13.122Q1: 13.175Median: 13.211Q3: 13.246Max: 13.246


Alpha Coefficient of 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.

Figure 6. Linear Thermal Expansion Unknown Metal Box Plot

Linear Thermal Expansion Nickel

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

13.211

13.175 13.246

13.122

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

0.310

0.311

0.313

0.312

Linear Thermal Expansion Unknown Metal


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, x̄1

represents the mean of sample one, x̄2 represents the mean of sample two, s1

represents the standard deviation of sample one, s2 represents the standard

deviation of sample two, n1 represents the number of data points in sample one,

n2 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= −¿

√ ( s1 )2

n1+

( s2)2

n2

¿

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


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


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


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 : μNickel=μUnknown

H a : μNickel ≠μUnknown

The first hypothesis, the null, denoted by Ho, said that the average specific

heat/alpha coefficient for nickel, denoted by µNickel, was equal to the average

specific heat/alpha coefficient of the unknown metal, denoted by µUnknown. 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.


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.

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

PVal = 2.760•10-9


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.

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.

PVal = 6.00 • 10-37


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.


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


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.


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

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


recorded. Keys would be produced as blanks (without teeth), and would be

created custom for the customer’s lock.


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 capsPurple PVC primerOrange PVC cementX-Acto KnifeBelt SanderChop SawDrill 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 layer 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 2nd 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.)


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.


Appendix B: Formulas and Sample Calculations

Specific Heat

In the following equation, swater represents the specific heat of water, mwater

represents the mass of the water, Δtwater represents the change in temperature of

the water, smetal represents the specific heat of the metal, mmetal represents the

mass of the metal, Δtmetal 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, smetal, is what the equation was used to solve for. The

absolute value will be used for the change in temperature of the metal, Δtmetal.

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.

( swatermwater ∆twatermmetal∆ tmetal

)+CF=smetal

This equation was used to calculate the specific heats of all the trials for

the specific heat experiments. A sample calculation is shown below.

( swatermwater ∆twatermmetal∆ tmetal )+CF=smetal

( 4.184 J /g° C •45g•5.9 ° C36.028 g•71° C )+0.008 J /g°C=smetal

0.442J /g°C=smetal


Figure 1. Specific Heat Sample Calculation

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, VTn represents the true

specific heat of the material in J/g°C, VE represents the experimental value

collected 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.

∑i=1

n

(V Tn−V En)

n=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.


∑i=1

n

(V Tn−V En)

n=CF

0.064 J / g°C8 trials

=CF

0.008J / 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, Li represents the initial length of the metal rods

in mm, ΔL represents the change in length of the metal rods in mm, ΔT

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 106 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 106 to get

an answer that is manageable.

( ∆ LLi∆T )•106=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.

( ∆ LLi∆T )•106=α

( 0.127mm129.388mm• 74.1° C )•106=α

13.246 106

°C=α

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, VE represents the experimental value, VT

represents the true value, and PError 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.

PError=V E−V T

V T∙100

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.


PError=V E−V T

V T∙100

PError=0.442 J /g °C−0.440 J /g°C

0.440 J /g°C∙100

PError=0.002 J /g °C0.440 J /g °C

∙100

PError=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, µ1 represents

the mean of sample one, µ2 represents the mean of sample two, s1 represents

the standard deviation of sample one, s2 represents the standard deviation of

sample two, n1 represents the number of data points in sample one, n2 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= −¿

√ ( s1 )2

n1+

( s2)2

n2

¿

This equation was used to compare the nickel specific heat to the

unknown specific heat and the nickel linear thermal expansion to the unknown

linear thermal expansion. The units that were used in data collection become

meaningless in this equation. A sample calculation is shown below.

t= −¿

√ s1n1

+s2n2

¿

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

√ (0.0045960645691515)2

15−

(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.


Works Cited

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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.

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