BLAH BL

Determining Whether an Unknown Metal is Aluminum Using the Intensive Properties of Specific Heat and Linear Thermal Expansion

Thanvir Ahmed and James VanWagnen

Macomb Mathematics Science Technology Center

Honors Chemistry – 10 A

Mrs. Hilliard / Mr. Supal / Mrs. Dewey

May 20, 2014

BLAH BLAH BLAH

Table of Contents

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

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

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

Experimental Design.................................................................................10

Data and Observations.............................................................................14

Data Analysis and Interpretation...............................................................22

Conclusion................................................................................................30

Application................................................................................................33

Appendix A: Instructions for Calorimeter..................................................35

Appendix B: Sample Calculations.............................................................37

Works Cited..............................................................................................44

Introduction

Aluminum is one of the most abundant elements in the earth’s crust. The

abundance of aluminum makes it very cheap to industrialize and use in everyday

life (Gagnon). Aluminum expands easily and is very malleable. Aluminum does

not easily corrode. Aluminum is the most commonly used element and appears

in almost everything, from kitchen utensils and house siding to electric wires and

soda cans. Aluminum alloys are used in aircraft because they are exceptionally

strong for their light weight (Ophardt).

The purpose of this experiment was to see if a set of unidentified metal

rods were made up of aluminum. A set of aluminum rods were used as a

reference point to compare with the unknown metal rods. In order to identify the

metal rods the intensive properties of specific heat and the linear thermal

expansion needed to be measured for both sets of metal rods. If the specific

heat and the linear thermal expansion coefficients were the same for both sets of

metal rods the unknown metal could identified as aluminum.

To determine the specific heat of the metal, an isolated system called a

calorimeter was constructed. Specific heat is calculated through the First Law of

Thermodynamics. The First Law of Thermodynamics states that energy can

neither be created nor destroyed only transferred. The mass of the metal rods

was recorded and then the rods were heated. The rods were placed in the

calorimeters and the initial and final temperatures were taken using a

temperature probe.

To find the linear thermal expansion of the metal rods, a thermal

expansion jig was used. The initial lengths of the rods were found using a

caliper. The rods were then heated in boiling water. The rods were then quickly

placed inside the thermal expansion jig and cooled. The jig measured the change

in length.

The percent error between known and experimental values was calculated

to validate the data. After this, a two sample t test was used analyze the data.

The two sample t test compared the means of the specific heat and linear

thermal expansion data for both the aluminum and unknown metal rods, and was

used to conclude whether or not the rods were the same metal.

Review of Literature

There many different substances throughout the world and to distinguish

between these substances, intensive physical properties such as specific heat

and linear thermal expansion are used. They can be used because they are

unique to each metal. The amount of heat required to raise one gram of a

substance by one degree Celsius is known as the specific heat of that substance

(Chang 239). Specific heat is measured in J/(g°C) or Joules per gram degrees

Celsius. Linear Thermal Expansion is the extent to which the length of a

substance changes due to a change in temperature (Elert).

A previous experiment was conducted at Clinton Community College to

use specific heat to determine an unknown substance. They indirectly

determined the specific heat by heating up the metal and then submerging it

within water, observing the temperature change to determine the change in

energy (Lawliss). The experimenters then used the equation for specific heat to

calculate the amount of energy released by the metal and absorbed by the water.

The variable s is the specific heat of the metal rod (J/g°C). The specific heat

varies for each metal. The specific heat of aluminum is 0.900 J/g°C, while the

specific heat of gold is 0.129 J/g°C (Stretton). The variable mw is the mass of the

water with the mass of the unknown metal added on (g). The tfw variable is the

final temperature of the water while tiw is the initial temperature of the water

measured in Celsius (Lawliss). The difference between the final temperature and

the initial temperature is the change in temperature (∆T) measured in Celsius.

The specific heat of water is 4.184 J/g°C. The variable mm is the mass of the

metal (g). The tfm variable is the final temperature of the metal while the tim is the

initial temperature of the metal in degrees Celsius. This equation was used to

calculate the specific heat of the metal in J/g°C.

s=4.184 J ×mw×(t fw−t iw)

mm×( t fm−tℑ)

The Clinton research team then searched for a table of specific heat for different

elements and compared the lab value to a known value. The research team built

a Styrofoam calorimeter. A calorimeter is an isolated system which prevents

energy from escaping. %Error=(LabValue )−(KnownValue )

KnownValue∗100

The research team then found that the specific heat of the unknown metal

deviated from that of copper by 1.05% and concluded that the unknown metal

was copper (Lawliss).

There are many different ways of finding the specific heat of a metal. The

most commonly used method for calculating specific heat is known as the

mixture method. This is done by heating a sample of a known mass to a given

temperature, then placing the heated sample into a calorimeter with a known

mass of water at a known temperature inside a calorimeter. The new

temperature of the water is then taken when the metal and the water reach the

level of equilibrium, meaning the temperatures were the same. The water and the

metal will reach the same temperature because the heat from the warmed metal

will release energy in the form of heat into the cooler water until both the water

and the metal have an equal amount of energy (Dziembowski). This is an

exothermic process because energy is lost by the metal and absorbed by the

water. This is based on the First Law of Thermodynamics, which states that

energy cannot be created nor destroyed only transferred (Dziembowski).

Another experiment was done at the University of Cincinnati, which used

the same mixture method. The experimenters at the University of Cincinnati were

trying to determine the specific heat of an unknown metal rod. They placed this

unknown metal rod within a beaker and heated the metal rod. This was placed

within a calorimeter. The experimenters at the University of Cincinnati used the

same calculations as the experimenters at Clinton Community College and

concluded that the unknown metal rod was aluminum with a 1.1% error (Bortner).

Both of these experiments can be applied to this experiment because they

have provided background information, and basic starting ground for the

experiment. They used the mixing method and that can be applied to this

experiment. They also heated up a metal rod and that can be applied to the

procedures in this experiment. The calorimeters within these experiment used

different types of Styrofoam. The Styrofoam calorimeter can be used because

both of these experiments resulted in a 1.05% to 1.1% error range.

At an atomic level, when heat is added to a substance, the vibration of the

molecules increases, causing them to spread out. Specific heat measures the

amount of heat that is required to increase the vibrations of molecules by 1°C.

When the heat is transferred to the water, the water’s molecules begin to vibrate

as well. When the heated substance is placed within water the vibrating

molecules of the substance begin to vibrate the water molecules, thus

transferring energy from the substance to the water. This is an exothermic

process because it transfers heat from the substance to the water (Bortner). This

is supported by the First Law of Thermodynamics.

There are many industrial uses for specific heat, but it is prominent within

the food market. Food is most likely to spoil because of extreme temperatures

which can occur during shipping. Most food manufacturers would not want their

product to spoil so they look into insulation, constantly having tests that use

specific heat to determine the substances that will absorb heat the slowest

(Violeta). This allows food manufacturers to keep food cold and fresh. These

materials that absorb heat the slowest are often used in the packaging material

and shipping trucks for food to prevent spoiling (Violeta). Food manufacturers

pour thousands of dollars into research for insulation using specific heat to avoid

millions of dollars because of losses due to spoiled food (Violeta).

For linear thermal expansion, an experiment was done by PerkinElmer

Inc. in order to see if an unknown metal could be determined using linear thermal

expansion with a 0.5% error, using a TMA 4000 to measure any length changes

within the metal rod as it underwent temperature changes, by measuring to the

nearest micrometer (Cassel). The team at PerkinElmer heated up the metal rod

from 0°C to 300°C (Cassel). The team at PerkinElmer then used the equation for

linear thermal expansion to determine the coefficient of thermal expansion. The ⍺ represents the coefficient of thermal expansion (1/°C). The coefficient of thermal

expansion varies for different substances. The coefficient of aluminum is 23.1×

10-6 1/°C, while the coefficient of gold is 14.2×10-6 1/°C (Elert). The Lf is the final

length of the metal rod (mm) while the Li is the initial length of the metal rod

(mm). The difference between the final and initial length is the change in length

or ∆L (mm). The Tf variable is the final temperature of the rod while T i is the initial

temperature of the rod measured in degrees Celsius (Cassel). The difference

between the final temperature and the initial temperature is the change in

temperature or ∆T measured in degrees Celsius. This is the equation that the

team used:

⍺=(Lf−Li)Li(T f−T i)

The research team used the percent error equation to determine that the metal

was aluminum with a 0.45% error (Cassel).

Another experiment performed by Tong Wa Chao used a displacement

probe and temperature to find the linear thermal expansion coefficient of a

printed wiring board (Chao). The metal was slowly heated from room

temperature to 230°C over a period of an hour. Labview software was used to

graph the data (Chao). Chao used a caliper and thermometer to measure basic

changes in length.

Both of these experiments can be applied to this experiment. Both

experiments heated the metal rod and measured the change in length using an

apparatus. This protocol can be mimicked in this experiment. The second

experiment used software to graph the data, while the first experiment used the

linear thermal expansion equation and percent error to identify the unknown

metal. This experiment can do both.

On an atomic level, there is a simple explanation for linear thermal

expansion. The length of an object depends on temperature of that object

(Duffy). This is explained with Kinetic Molecular Theory. This theory states that

as the material is heated, the particles move faster. As thermal energy is added

in the form of increased temperatures, the molecules will gain a larger amount of

kinetic energy. The increase in kinetic energy causes the molecules to move

more rapidly. They begin to spread out, increasing the size of the metal rod

(Duffy).

Thermal Expansion is also used in developing computer parts. If computer

parts were allowed to expand when they begin to heat it would cause problems

for the consumer. Computer manufacturers use certain substances that do not

change much when heated to deliver the best performing computers to the user

(Straley).

Problem Statement

Problem Statement:

To determine whether an unknown metal is aluminum using the intensive

properties of specific heat and linear thermal expansion.

Hypothesis:

If the specific heat and linear thermal expansion coefficient of the

unknown metal is found, then the metal will be identified correctly with a 5% error

for specific heat and a 1% error for linear thermal expansion.

Data Measured:

For specific heat, the independent variables measured are the initial

temperatures and masses of the water and the metal. The dependent variables

are the final temperatures of the water and the metal. The temperatures are

measured in degrees Celsius, the mass of the metal is measured in grams, and

the mass of the water is measured in milliliters. The final temperature values

should be the same because the temperatures of the water and the metal will

reach equilibrium. The specific heat of water, 4.184 J/g°C, will also be used.

These variables are used to determine the specific heat of the metal in J/g°C. For

linear thermal expansion, the independent variables are the initial and final

temperatures of the rod and the initial length of the rod. The dependent variable

is the final length of the rod. Length is measured in millimeters (mm) and

temperature is measured in degrees Celsius (°C). These variables will be used to

find the linear thermal expansion coefficient in 1/°C. A two sample t test will be

used to analyze the data.

Specific Heat Experimental Design

Materials:

Hot plate (4) 40 mL Calorimeters

100 mL Graduated Cylinder(2) Metal Aluminum Rods (2) Unknown Metal Rods (4) Temperature Probe 0.1 precision)Flinn Scientific Inc. Thermometer (0.1 precision)

Vernier LabQuest 1200mL Loaf PanTongs250 mL Beaker

OHAUS GA 200 Electronic Scale (0.0001 precision)

Hot Mitt

Procedures:

Be aware of safety precautions. Wear gloves, goggles, and appropriate attire.

1. Randomize all trials in order to ensure accurate results, allocating fifteen trials for both the aluminum and unknown metal, and the calorimeters. Calibrate the calorimeters.

2. Set up the Vernier LabQuest to collect data every 0.5 seconds.

3. Measure the mass of the metal rod using an electronic scale. Record the results.

4. Fill the loaf pan with 250 mL of water and turn the hot plate on the highest setting.

5. Use the thermometer to measure the temperature of the water within the loaf pan. Once the water in the loaf pan is approximately 100°C, submerge the rod in the water for five minutes. Assume the rod will become the same temperature as the water (this is the initial temperature of the metal).

6. Pour 30 mL of tap water within the graduated cylinder.

7. Pour the 30 mL of tap water from the graduated cylinder into the calorimeter.

8. Start data collection on the LabQuest. Record the temperature of the water in Celsius within the calorimeter using the temperature probe (this is

the initial temperature of the water). Allow thirty seconds for the temperature probe to calibrate.

9. Place the rod into the calorimeter, place the cap on the calorimeter, and begin gently stirring the temperature probe.

10. After five minutes, stop stirring and stop the data collection on the LabQuest (This is the final temperature of the water and metal rod).

11. Calculate the specific heat using the specific heat formula and the data collected.

12. Repeat Steps 3-11 fifteen times for the aluminum rod and fifteen times for the unknown metal rod. Refill hot water as it evaporates.

Diagram:

Figure 1. Specific Heat Materials

Figure 1 shows all of the materials that were used in the specific heat

experiment. The OHAUS GA 200 Electronic Scale is not included. The unique

materials for this experiment include the calorimeters and the metal rods.

Linear Thermal Expansion Experimental Design

Calorimeters

Hot Plate Beaker

Loaf Pan

Hot Mitt

Tongs

Graduated Cylinder

Thermometer

Temperature Probe

Vernier LabQuest

Aluminum Rods

Unknown Metal Rods

Materials:

TESR Caliper 00530085 (0.01 mm precision)Linear Thermal Expansion Jig (0.001 mm precision)(2) Aluminum Rod (2) Unknown Metal Rod

100 mL Graduated Cylinder TongsFlinn Scientific Inc. Thermometer (0.1°C precision)Hot PlateLoaf pan (1200 mL)

Procedures:

Be aware of safety precautions. Wear gloves, goggles, and appropriate attire.

1. Randomize all trials in order to ensure accurate results, allocating fifteentrials for both the aluminum and unknown metal.

2. Use the caliper to measure the original length of the metal rod (this is the initial length). Record the results.

3. Turn the hot plate on the highest setting and pour 250 mL of hot water into the other loaf pan.

4. Place the loaf pan onto the hot plate.

5. Once the temperature reaches close to 100°C, submerge the metal rod into the boiling water for three minutes. Assume the metal rod reaches equilibrium with the water. Use the thermometer to find the temperature. Record the results (this is the initial temperature).

6. Use the tongs to remove the metal rod from the water.

7. Place the rod within the linear thermal expansion jig and zero the jig out.

8 Use the linear thermal expansion jig to measure the change in length of the metal rod by leaving the rod within the jig and allow it to cool down for 3 minutes. Record the results (this is the change in length).

9 Place temperature probe in the air and measure the temperature of the room (this is the final temperature of the metal rod).

10. Use the linear thermal expansion formula to find the linear thermal expansion coefficient of aluminum and the unknown metal.

11. Repeat steps 2-10 until fifteen trials have been conducted for aluminum and fifteen trials have been conducted for the unknown metal. Refill hot water as it evaporates. Replace room temperature water as it heats up.

Diagram:

Figure 2. Materials Used

Figure 2 shows all the materials used in the linear thermal expansion

experiment. The unique materials for this experiment include the two sets of

metal rods and the linear thermal expansion jig.

Data and Observations

Hot Mitt

Linear Thermal Expansion Jig

Loaf Pan Hot Plate

Aluminum Rods

Unknown Metal Rods

Hot Mitt

Beaker

Caliper Tongs

Table 1Specific Heat Data for Aluminum

Trial Cal, Rod

Initial Temp. (°C) Final

Temp.(°C)

Change in Temp. (°C)

Mass Correction Factor(J/g°C)

Specific Heat

(J/g°C)W M W M M(g)

W (mL)

1 B,1 22.8 96.9 26.1 3.3 -70.8 9.881 30 0.307 0.899

2 W,2 23.1 96.9 26.4 3.3 -70.5 9.947 30 0.304 0.895

3 G,1 23.5 95.0 26.7 3.2 -68.3 9.881 30 0.293 0.888

4 Y,2 23.2 95.0 26.5 3.3 -68.5 9.947 30 0.267 0.875

5 B,2 22.9 97.3 26.4 3.5 -70.9 9.948 30 0.307 0.930

6 W,1 22.9 97.3 26.2 3.3 -71.1 9.880 30 0.304 0.894

7 G,1 23.1 98.2 26.6 3.5 -71.6 9.885 30 0.293 0.914

8 Y,2 22.8 97.4 26.5 3.7 -70.9 9.947 30 0.267 0.925

9 B,1 22.9 97.5 26.1 3.2 -71.4 9.879 30 0.307 0.876

10 W,2 23.1 98.5 26.7 3.6 -71.8 9.974 30 0.304 0.935

11 G,1 23.2 96.7 26.5 3.3 -70.2 9.884 30 0.293 0.890

12 Y,2 23.4 96.7 26.9 3.5 -69.8 9.947 30 0.267 0.900

13 B,2 23.8 98.0 27.1 3.3 -70.9 9.948 30 0.307 0.894

14 W,1 23.8 98.0 27.0 3.2 -71.0 9.883 30 0.304 0.877

15 G,2 23.9 95.0 27.2 3.3 -67.8 9.946 30 0.293 0.907

Average 23.2 97.0 26.6 3.4 -70.4 9.918 30 0.294 0.900

Table 1 shows the data used to calculate the specific heat for the known

aluminum sample. Correction factors were calculated for all of the calorimeters

(see Appendix 2 for sample calculation.) The average specific heat of the

aluminum sample was calculated to be 0.900 J/g x °C. The actual specific heat of

aluminum is 0.9 J/g x °C. The mass of the water slightly deviated from 30 mL for

every single trial. The rod and calorimeter column shows what rod and

calorimeter were used for each trial. For example, B1 stands for black

calorimeter and rod 1. B is black, W is white, G is green, and Y is yellow.

Table 2Specific Heat Data for Unknown

Trial Cal, Rod

Initial Temp. (°C) Final

Temp.(°C)

Change in Temp. (°C)

Mass Correction Factor(J/g°C)

Specific Heat

(J/g°C)W M W M M(g)

W (mL)

1 B,2 22.5 96.6 32.0 9.5 -64.6 27.086 25 0.307 0.875

2 G,1 22.6 95.4 32.1 9.5 -63.3 26.992 25 0.293 0.875

3 Y,1 23.1 94.5 32.8 9.7 -61.7 26.992 25 0.267 0.876

4 W,2 23.1 96.5 33.2 10.1 -63.3 27.085 25 0.304 0.920

5 B,1 24.0 96.6 33.3 9.3 -63.3 27.004 25 0.307 0.876

6 W,2 23.1 96.6 32.6 9.5 -64.0 27.101 25 0.304 0.877

7 Y,1 23.1 95.4 32.9 9.8 -62.5 26.995 25 0.267 0.875

8 G,2 22.6 95.6 32.2 9.6 -63.4 27.086 25 0.293 0.878

9 B,2 23.2 96.3 32.6 9.4 -63.7 26.992 25 0.307 0.879

10 W,1 22.8 96.3 32.3 9.5 -64.0 27.083 25 0.304 0.877

11 G,2 23.4 97.1 33.5 10.1 -63.6 27.085 25 0.293 0.907

12 B,1 24.9 98.1 34.9 10.0 -63.2 26.993 25 0.307 0.920

13 W,1 24.0 94.8 33.1 9.1 -61.7 26.999 25 0.304 0.875

14 G,2 24.7 94.8 33.8 9.1 -61.0 27.085 25 0.293 0.869

15 Y,1 24.5 94.8 34.8 10.3 -60.0 26.992 25 0.267 0.932

Average 23.4 96.0 33.1 9.6 -62.9 27.038 25 0.294 0.887

Table 2 shows the data used to calculate specific heat for the unknown

metal sample. The corrections factors that were used in Table 1 were also used

here. The average specific heat of the unknown sample was calculated to be

0.887 J/g x °C. The specific heat of aluminum is 0.9 J/g x °C. The volume of the

water deviated slightly from 25 mL for each trial.

Table 3Specific Heat Observations for Aluminum Trial Observations

1Heated at the top of the loaf pan. Some water spilled on table. Removed after 3 minutes. Stirred by researcher 1.

2Heated at the bottom of the loaf pan. Pan slightly moved. Removed after 3.5 minutes. Stirred by researcher 1.

3 Heated at the top of the loaf pan. Removed after 3 minutes. Stirred by researcher 1.

4 Heated at the middle of the loaf pan. Water began to evaporate. Stirred by researcher 1.

5Water evaporated. 250 mL added to the loaf ban and allowed to boil. Rod placed at bottom of loaf pan and removed after 3 minutes. Stirred by researcher 1.

6 Placed at the middle of the loaf pan. Removed after 2.75 minutes. Stirred by researcher 1.

7Placed at the bottom of the loaf pan. Timer was started 10 seconds late. Removed after 3 minutes. Stirred by researcher 1.

8Placed at the bottom of the loaf pan. Calorimeter's cap was stuck on, but removed. Taken out after 3.25 minutes. Stirred by researcher 1.

9Heated at the top of the loaf pan. Some water spilled so it had to be refilled within the calorimeter. Removed after 3 minutes. Stirred by researcher 1.

10This trial was redone. Heated in the middle of the loaf pan. Loaf pan moved a little bit. Removed after 2.75 minutes. Stirred by researcher 1.

11Heated at the bottom of the loaf pan. Loaf pan moved a little bit. Removed after 3 minutes. Stirred by researcher 1.

12Heated at the top of the loaf pan. Loaf pan moved a little bit. Removed after 3 minutes. Stirred by researcher 1.

13Metal heated up longer than other trials. Heated in the middle. Removed after 3.75 minutes. Stirred by researcher 1.

14Metal heated up longer than other trials. Heated at the top of the loaf pan. Removed after 3.75 minutes. Stirred by researcher 1.

15 Heated in middle of loaf pan. Removed after 3 minutes. Stirred by researcher 1.

Table 3 shows the observations for specific heat of aluminum. In most of

these trials the rod was heated for 3 minutes and then stirred by researcher 1.

Trial 10 was redone because the original result did not agree with the rest of the

data. The original results were substituted with the results of the new trial.

Table 4Specific Heat Observations for UnknownTrial Observations

1 Heated at top of loaf pan. Removed after 3 minutes. Stirred by researcher 1.

2 Heated at bottom of loaf pan. Removed after 3.25 minutes. Stirred by researcher 2.

3Heated at the top of the loaf pan. Removed after 3 minutes. Stirred by researcher 2. Water began to evaporate.

4 Heated at the bottom of the loaf pan. Removed after 3 minutes. Stirred by researcher 2.

5Water refilled with 250 mL. Heated in middle of the loaf pan and removed after 3 minutes. Researcher 1 stirred.

6 Heated at the top of the loaf pan. Removed after 3.25 minutes. Stirred by researcher 1.

7 Heated at the bottom of the loaf pan. Removed after 3 minutes. Stirred by researcher 2.

8This trial was redone. Heated at the top of the loaf pan. Removed after 3 minutes. Stirred by researcher 2.

9 Heated at the top of the loaf pan. Removed after 3.5 minutes. Stirred by researcher 2.

10 Heated at the bottom of the loaf. Removed after 3.5 minutes. Stirred by researcher 2.

11Heated at the top of the loaf pan. Removed after 3 minutes. Stirred by researcher 2. Water began to evaporate.

12 Heated at the top of the loaf pan. Removed after 3 minutes. Stirred by researcher 2.

13Water refilled with 250 mL. Heated at the top of the loaf pan. Removed after 3.5 minutes. Stirred by researcher 2.

14 Heated at the top of the loaf pan. Removed after 3 minutes. Stirred by researcher 2.

15Heated in the middle of the loaf pan. Removed after 3 minutes. Stirred by researcher 2. This trial was videotaped.

Table 4 shows the observations for specific heat of the unknown metal. In

most of these trials the rod was heated for 3 minutes and then stirred by

researcher 2. Trial 8 was redone because the original result did not agree with

the rest of the data. The original results were substituted with the results of the

new trial.

Table 5Linear Thermal Expansion Data for Aluminum

Trial Rod ΔL (mm)

Initial Length(mm)

Initial Temp.

(ºC)

Final Temp.

(ºC)

Change in

Temp.(ºC)

Alpha Coefficient

(°C-1)

1 1 0.229 131.286 96.7 22.3 74.4 2.344 X10-5

2 2 0.229 132.131 96.4 22.3 74.1 2.335 X10-5

3 2 0.229 132.144 97.6 22.6 75.0 2.307 X10-5

4 1 0.229 130.121 97.2 22.7 74.5 2.363 X10-5

5 2 0.230 132.156 97.9 23.1 74.8 2.326 X10-5

6 1 0.229 132.261 97.1 22.6 74.5 2.320 X10-5

7 2 0.229 132.258 96.6 22.4 74.2 2.329 X10-5

8 1 0.229 130.223 96.6 22.5 74.1 2.368 X10-5

9 2 0.229 132.232 98.0 22.1 75.9 2.278 X10-5

10 1 0.229 129.299 98.7 22.4 76.3 2.317 X10-5

11 1 0.229 131.324 95.8 22.1 73.7 2.365 X10-5

12 2 0.229 132.105 96.3 22.2 74.1 2.335 X10-5

13 1 0.229 129.299 97.1 22.3 74.8 2.364 X10-5

14 2 0.229 132.220 96.3 22.3 74.0 2.336 X10-5

15 1 0.229 129.286 96.8 22.3 74.5 2.373 X10-5

Average 0.229 131.223 97.0 22.4 74.6 2.337 X10-5

Table 5 shows the data used to calculate the alpha coefficient of the

known aluminum sample. The average alpha coefficient was calculated to be

2.337X10-5°C-1. The actual alpha coefficient of aluminum is 2.31X10-5°C-1.

Table 6Linear Thermal Expansion Data for Unknown

Trial Rod ΔL (mm)

Initial Length(mm)

Initial Temp.

(ºC)

Final Temp.

(ºC)

Change in

Temp.(ºC)

Alpha Coefficient

(°C-1)

1 1 0.230 139.332 96.4 23.9 72.5 2.276X10-5

2 2 0.230 139.598 96.4 23.8 72.6 2.268X10-5

3 2 0.229 139.624 94.6 23.6 71.0 2.306X10-5

4 1 0.229 139.154 94.6 23.6 71.0 2.314 X10-5

5 2 0.229 139.637 96.2 24.1 72.1 2.271 X10-5

6 1 0.229 139.319 95.4 24.1 71.3 2.301 X10-5

7 2 0.229 139.713 95.3 24.0 71.3 2.295 X10-5

8 1 0.229 139.471 94.5 24.1 70.4 2.328 X10-5

9 1 0.230 139.154 96.5 23.8 72.7 2.272 X10-5

10 2 0.231 139.535 96.4 23.5 72.9 2.271 X10-5

11 1 0.230 139.370 96.3 23.6 72.7 2.269 X10-5

12 2 0.230 139.675 96.1 23.8 72.3 2.274 X10-5

13 2 0.230 139.675 96.7 24.1 72.6 2.265 X10-5

14 1 0.231 139.332 96.9 24.2 72.7 2.280 X10-5

15 2 0.230 139.675 96.2 23.8 72.4 2.272 X10-5

Average 0.229 139.484 95.9 23.9 72.0 2.284 X10-5

Table 6 shows the data used to calculate the alpha coefficient of the

unknown metal sample. The average alpha coefficient was calculated to be

2.284X10-5°C-1. The alpha coefficient of aluminum is 2.31X10-5°C-1.

Table 7Linear Thermal Expansion Observations for AluminumTrial Observations

1Heated at the center of the loaf pan. Removed after 3 minutes, when removing loaf pan slightly moved. Allowed to cool down for 3.5 minutes.

2Heated at the bottom of the loaf pan. Removed after 3.25 minutes. Room temperature rose. Allowed to cool down for 3 minutes.

3Heated at the top of the loaf pan. Removed after 3 minutes, when removing loaf pan slightly moved. Allowed to cool down for 2.75 minutes.

4Heated at the bottom of the loaf pan. Removed after 2.75 minutes. Allowed to cool down for 3 minutes. Some water spilled.

5Heated at the top of the loaf pan. Removed after 3 minutes. A window in the room was opened. Allowed to cool down for 3 minutes.

6This trial was redone. Heated at the center of the loaf pan. Removed after 3 minutes. Allowed to cool down for 3 minutes. Room temperature rose.

7 Heated at the top of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

8 Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

9 Heated at the center of the loaf pan. Removed after 2.75 minutes. Cooled for 3 minutes.

10 Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

11Heated at the top of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes. Some water spilled when removing.

12Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes. Room temperature rose.

13Heated at the top of the loaf pan. Removed after 2.75 minutes. Cooled for 3 minutes.

14Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3.25 minutes.

15 Heated at the bottom of the loaf pan. Removed after 2.75 minutes. Cooled for 3 minutes.

Table 7 shows the observations for the linear thermal expansion of

aluminum. In most of the trials the rod was heated for 3 minutes and then cooled

for 3 minutes. Only trial 6 was redone because the original result did not agree

with the rest of the data. The original results were substituted with the results of

the new trial.

Table 8Linear Thermal Expansion Observation for UnknownTrial Observations

1 Heated at the top of the loaf pan. Removed after 3 minutes. Cooled down for 3 minutes.

2Heated at the center of the loaf pan. Removed after 3 minutes. Some water spilled when removing. Allowed to cool down for 3 minutes.

3Heated at the top of the loaf pan. Removed after 3 minutes, when removing loaf pan slightly moved. Allowed to cool down for 3 minutes.

4Heated at the bottom of the loaf pan. Removed after 3 minutes. Allowed to cool down for 3 minutes. Window in room was opened.

5 Heated at the top of the loaf pan. Removed after 3 minutes. Cooled down for 3 minutes.

6Heated at the center of the loaf pan. Removed after 3 minutes. Allowed to cool down for 3 minutes. Room temperature rose.

7 Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3.25 minutes.

8Heated at the top of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes. Room temperature rose.

9This trial was redone. Heated at the center of the loaf pan. Removed after 2.75 minutes. Cooled for 3 minutes.

10 Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

11 Heated at the top of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

12Heated at the center of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes. Room temperature rose.

13 Heated at the bottom of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

14 Heated at the top of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes.

15Heated at the center of the loaf pan. Removed after 3 minutes. Cooled for 3 minutes. This trial was videotaped.

Table 8 shows the observations for the linear thermal expansion of the

unknown metal. In most of the trials the rod was heated for 3 minutes and then

cooled for 3 minutes. Only trial 9 was redone because the original result did not

agree with the rest of the data. The original results were substituted with the

results of the new trial.

Data Analysis and Interpretation

The trials for specific heat and linear thermal expansion were randomized

using a TI-Nspire calculator. This makes the results valid because the

randomization eliminated bias. Each trial was conducted independently, meaning

the results of the trials did not affect each other. The specific heat trials were

conducted using four different calorimeters. These calorimeters were randomized

for each trial. Correction factors for each calorimeter were calculated using the

data from aluminum. The rod used for each trial was randomized for linear

thermal expansion. The data was checked for normality using normal probability

plots for both experiments and was analyzed using percent errors, box plots, and

two sample t tests. Refer to Appendix B for sample calculations and correction

factors.

Table 9Percent Error for Specific Heat

Trial

Percent Error for

Aluminum Specific

Heat(%)

Percent Error for Unknown Specific

Heat(%)

Trial

Percent Error for

Aluminum Specific

Heat(%)

Percent Error for Unknown Specific

Heat(%)

1 -0.094 -2.779 9 -2.611 -2.341

2 -0.587 -2.797 10 3.894 -2.515

3 -1.292 -2.647 11 -1.089 0.723

4 -2.797 2.252 12 -0.032 2.247

5 3.330 -2.648 13 -0.625 -2.726

6 -0.697 -2.557 14 -2.610 -3.406

7 1.552 -2.832 15 0.829 3.576

8 2.829 -2.448 Average 0.000 -1.393

Table 9 shows the percent errors for each specific heat trial. The mean

percent errors of aluminum and the unknown metal are 1.393% away from each

other. Both were close to the actual specific heat of aluminum, with the largest

percent error being 3.576%. Most of the trials were consistent. A few trials had

positive percent errors, while most of the trials had negative percent errors.

Figure 3. Box Plots of Specific Heat

Figure 3 shows the box plots for both sets of specific heat data. The box

plots for the aluminum (top) and the unknown metal (bottom) overlap. Neither

have any outliers. Both box plots center around the true specific heat of

aluminum. The box plot for aluminum appears slightly skewed to the right and the

box plot for the unknown metal appears skewed to the right. If a statistical test

were to be performed on this data, it must be done with caution because the data

is not normal.

True Specific Heat of Aluminum

0.932

0.9070.877

0.875

0.869

0.935

0.9140.8950.888

0.875

Figure 4. Normal Probability Plots for Specific Heat

Figure 4 shows the normal probability plots for both sets of specific heat

data. The normal probability plot for aluminum is on the left. The normal

probability plot for the unknown metal is on the right. The data for aluminum is

mostly linear, so it is close to being normal. The data for the unknown metal is

not linear. It is not normal. This reinforces what the box plot showed.

H 0 : μAluminum=μUnknown

H a : μAluminum≠μUnknown

A two sample t test was used to analyze the data because there were two

sets of data to compare, the aluminum and the unknown metal. Before

conducting the t test, certain assumptions must be met: the trials must be

randomized and independent, the population standard deviation is not known,

and the data is normal. As stated previously, all of the assumptions were met

except the final one. This could possibly make the t test unreliable. The null

hypothesis, H0, stated that the mean specific heat of aluminum was equal to the

mean specific heat of the unknown metal. The alternate hypothesis, Ha, stated

that the mean specific heat of aluminum was not equal to the mean specific heat

of the unknown metal.

Figure 5. Specific Heat Two Sample t Test

Figure 5 shows the results of the specific heat two sample t test. Based on

the results of the t test, fail to reject the null hypothesis because the P-value of

0.096618 is very close to the α level of 0.1. There was no evidence that the

specific heat of aluminum is different from the specific heat of the unknown

metal. If H0 was true, results this extreme would happen 9.6618% of the time by

chance alone. The standard deviation of aluminum is 0.019 and the standard

deviation of the unknown metal is 0.021. This means the data varies by these

amounts, on average. The standard deviation shows that there was not much

variation in the data. The graph on the right shows the bell curve for the specific

heat two sample t test with the P-value shaded in.

Table 10Percent Error Table for Linear Thermal Expansion

Trial

Percent Error for

Aluminum Linear

Thermal Expansion

(%)

Percent Error for Unknown

Linear Thermal

Expansion(%)

Trial

Percent Error for

Aluminum Linear

Thermal Expansion

(%)

Percent Error for Unknown

Linear Thermal

Expansion(%)

1 2.882 -2.971 9 -1.398 -3.767

2 1.075 -3.156 10 0.310 -3.769

3 -0.148 -0.173 11 3.829 -3.655

4 2.875 0.164 12 1.094 -3.341

5 0.109 -1.705 13 2.322 -4.898

6 2.764 -0.376 14 1.143 -4.664

7 0.841 -0.657 15 2.744 -3.473

8 3.349 0.644 Average 1.565 -2.386

Table 10 shows the percent error table for linear thermal expansion. Each

percent error stayed close to the expected value for linear thermal expansion,

with the highest percent error being -4.898%. The mean percent error of the

aluminum trials and the mean percent error of the linear thermal expansion trials

are 3.951% away from each other. The trials were not consistent because of the

large difference between the highest percent error and the lowest percent error

for both the aluminum rods and the unknown rods.

Figure 6. Box Plots of Linear Thermal Expansion

Figure 6 shows the box plots of the linear thermal expansion of aluminum

(top) and the unknown metal (bottom). These box plots overlap slightly. The box

plot for aluminum appears slightly skewed to the left while the box plot for the

unknown metal is skewed to the right. If a statistical test were to be done on this

data, it must be done with caution.

Figure 7. Normal Probability Plots for Linear Thermal Expansion

Figure 7 shows the normal probability plots for both sets of linear thermal

expansion data. The normal probability plot on the left shows the data for

True Alpha Coefficient of Aluminum2.301*10-5

2.271*10-5

2.265*10-5 2.328*10-5

2.274*10-5

2.373*10-5

2.364*10-52.335*10-5

2.320*10-5

2.278*10-5

aluminum. The points are all near the line, showing that this data is normal. The

normal probability plot on the right shows the data for the unknown metal. Not all

the points are on the line, suggesting that the data for the unknown metal may

not be normal.

Like specific heat, a two sample t test was used to analyze the two sets of

data for linear thermal expansion. Like specific heat, all the assumptions are met

except the assumption that the data is normal. This t test may be unreliable.

H 0 : μAluminum=μUnknownH a : μAluminum≠μUnknown

H0 stated that the linear thermal expansion coefficient of aluminum is

equal to the linear thermal expansion coefficient of the unknown metal. Ha stated

that the linear thermal expansion coefficient of aluminum is not equal to the linear

thermal expansion coefficient.

Figure 8. Linear Thermal Expansion Two Sample t Test

Figure 8 shows the results of the linear thermal expansion two sample t

test. Because the P-value of 0.000001 is lower than the α level of 0.1, H0 was

rejected. There was evidence that the linear thermal expansion coefficient of

aluminum is not equal to the linear thermal expansion of the unknown metal. A

result this extreme would happen only 0.0001% of the time by chance alone if H0

was true. The standard deviation of aluminum was 2.651×10-7 and the standard

deviation of the unknown metal was 1.962×10-7. This means the data varied by

these amounts, on average. The standard deviation showed that the data did not

vary by much. The graph on the right shows the bell curve of the two sample

t test for linear thermal expansion with the P-value shaded in. The shaded value

cannot be seen, showing that the means are different.

Conclusion

The purpose of this experiment was to determine whether an unknown

metal is aluminum using the intensive properties of specific heat and linear

thermal expansion. These properties were used because they are unique to each

metal. The hypothesis stating that the metal would be identified correctly with a

5% error for specific heat and a 1% error for linear thermal expansion was

accepted because the unknown metal was identified correctly as aluminum.

Evidence that supports this decision includes the t test for specific heat, percent

errors for specific heat and linear thermal expansion, and the physical properties

of both sets of rods. Specific heat is the amount of energy required to raise one

gram of a material 1°C (Bortner). The t test for specific heat showed that based

on specific heat, there is no proof that the metals are different. The percent errors

for the aluminum rods and the unknown metal rods are within 3% of the true

values of aluminum for specific heat and linear thermal expansion. Both the

aluminum rods and the unknown metal rods had a lustrous shine to them. Both of

the metal rods were also very lightweight and had a similar resonance.

Most of the data and evidence support the claim that the unknown metal is

aluminum, but all the data must be accounted for. The only piece of evidence

that does not support this decision is the t test for linear thermal expansion.

Linear thermal expansion measures the change in length of a metal to identify it.

The results of the t test implied that based on linear thermal expansion, the

metals are different. These results are not as valid because the data used in the

t test was not normal. This abnormal, or skewed data, caused for inaccurate

results. This could be alleviated by conducting more trials. Increasing the number

of trials that were conducted would ensure for normal data, which would produce

more accurate data. The linear thermal expansion jigs used to measure the

change in the length of the metals were not always accurate as they were

handmade out of wood. Another potential source for error was the time in

between taking the metal out of the boiling water and placing it in the jig. The

metal could have cooled during this time. According to Kinetic Molecular Theory,

as the temperature of an object increases the particle begin to move more rapidly

and expand (Chang 452). This theory also works in reverse, and because the rod

was able to cool within this time period the particles moved slower and the

overall size of the metal decreased. A similar error occurred transferring the

metal from the water to the calorimeter in specific heat.

Errors within any experiment can alter and confound the data. A major

error in this experiment was assuming the metal reached the same temperature

as the water while it was in the boiling water. There is no way to confirm that the

metal actually reached the same temperature as the water. This can alter the

results of both the specific heat and linear thermal expansion. Another error was

that the data was not completely normal. This skewed data altered the results of

the two sample t test. The calorimeters were not completely isolated systems and

this resulted in heat being able to escape. Even though a correction factor was

used to conduct more accurate trials some of the escaped heat was still

unaccounted for. The water measurement deviated slightly from 30 mL for the

known, and 25 mL for the unknown. This slight deviation could have altered, and

even have skewed the data.

There are many different ways to further this research. This research

could be expanded by testing different metals. Other intensive properties that

could be used to identify these metals include density and tensile strength.

Tensile strength was not used in this experiment because it would destroy the

material. Tensile strength measures the maximum amount of stress a material

can withstand before breaking (Lawliss). This experiment could have been

improved by conducting more trials. Another way to improve this experiment

would be by using better materials like professional calorimeters and linear

thermal expansion jigs. Professional calorimeters would be better insulated

systems and less heat would escape, which would result in more accurate data.

Research and experiments are always conducted for real life applications.

This experiment is relevant because properties like specific heat and linear

thermal expansion are taken into consideration when choosing the material for a

product. For example a manufacturer would want to use aluminum to construct a

ladder because it is cheap and strong for its lightweight. A material in a product

can also be identified if its intensive properties are known. It is important that

specific materials are used for specific products because of safety concerns and

expenses.

Application

From an industrial standpoint aluminum is cheap and malleable. Based on

the results from this experiment it was observed that aluminum expands easily

when heated at high temperatures so it would not be used to plate stoves or

nuclear reactors. Aluminum would more likely be used in common household

items that are not exposed to extreme temperatures. One of the most common

uses for aluminum is shown below.

Figure 9. Aluminum Ladder

Figure 9 shows an aluminum ladder. Aluminum is a good material for a

ladder because it is lightweight and cheap. It is also quite strong for its weight.

Most aluminum ladders can withstand 250-300 pounds of weight. To make this

ladder out of pure aluminum, 18.75 pounds of aluminum need to be used.

Aluminum is $0.79 per pound, so the cost to make this ladder out of pure

aluminum is $14.81.

Figure 10. Drawing of Aluminum Ladder

Figure 10 is a drawing of the top, front, and side of the aluminum ladder

with dimensions. This model is scaled by 1/5.

Appendix A: Instructions for Calorimeter

Materials:

(4) 3/4” diameter x 6” CPVC Pipe(4) 3/4” PVC pipe capsPurple PVC primerOrange PVC cement

Drill PressPink Fiber Glass Insulation Scotch Duct Tape Scissors

Procedures:

1. Apply a thin layer of PVC primer on one end of the 3/4” CPVC pipe. Do the same for the inside for one of the caps.

2. Apply a thin layer of CPVC cement to both the 3/4” CPVC 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 while slowly twisting until a solid bond can be felt between the cap and the pipe.

4. Allow to the cement to set for about 7 minutes

5. Use the scissors to cut out a strip of insulation that can comfortably warp around the CPVC pipe once.

6. Wrap the insulation strip around the CPVC pipe and use the duct tape to completely surround the insulation.

7. Wrap the calorimeter with construction paper or any other material that can help distinguish between calorimeters.

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

9. 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

10.Repeat steps 1-9 for desired number of calorimeters.

Diagrams:

Figure 1. Calorimeter Materials

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

The drill press is not shown here.

Figure 2. Finished Calorimeter

Figure 2 shows a finished calorimeter. This calorimeter was used in this

experiment.

Appendix B: Sample Calculations

Correction Factor:

A correction factor was calculated to ensure better results. The correction

factor was calculated using the known specific heat of the known aluminum metal

rods. The experimental specific heat values were subtracted from the actual

specific heat of aluminum, 0.9 J/g°C, for each trial. This difference was then

averaged for each calorimeter to calculate the correction factor for each

calorimeter. The table below shows the differences.

Table 1Specific Heat Raw Data

Table 1 shows the raw data for the specific heat of aluminum. The

difference between this raw data and the actual specific heat of aluminum, 0.9

J/g°C, was calculated. These differences were averaged for each calorimeter to

Trial Rod,Cal

Raw Specific

Heat(J/g°C)

Difference(J/g°C)

1 B1 0.592 0.308

2 W2 0.591 0.309

3 G1 0.595 0.305

4 Y2 0.608 0.292

5 B2 0.623 0.277

6 W1 0.590 0.310

7 G1 0.621 0.279

8 Y2 0.659 0.241

Trial Rod,Cal

Raw Specific

Heat(J/g°C)

Difference(J/g°C)

9 B1 0.569 0.331

10 W2 0.631 0.269

11 G1 0.597 0.303

12 Y2 0.633 0.267

13 B2 0.587 0.313

14 W1 0.572 0.328

15 G2 0.614 0.286

calculate the correction factors. For the rod and calorimeter column the letter

indicates the calorimeter’s color: B is black, W is white, G is green, and Y is

yellow. The number that follows the letter is the rod used.

dif=0.9−s

dif=0.9−0.592

dif=0.308 J / g°C

Figure 1. Difference Sample Calculation

Figure 1 shows a sample calculation for the differences using Trial 1 for

specific heat.

CF=∑ difndif

CF=0.308+0.277+0.331+0.3134

CF=0.307J / g°C

Figure 2. Correction Factor Sample Calculation

Figure 2 shows a sample calculation for the correction factor using all the

specific heat trials for the black calorimeter.

Table 2Correction Factors for Calorimeters

Table 2 shows the correction factors for each calorimeter. These

correction factors were added on to the raw experimental specific heat to

calculate the final specific heat.

Specific Heat:

To calculate the specific heat of the data the following equation was used.

It divides the mass and the change in temperature of the metal from the specific

heat, mass, and change in temperature of the water. The absolute value of this is

taken and the correction factor is added. The variable s is the specific heat of the

metal rod (J/g°C). The variable mw is the mass of the water with the mass of the

metal added on (g). The tfw variable is the final temperature of the water while t iw

is the initial temperature of the water measured in degrees Celsius. The

difference between the final temperature and the initial temperature is the change

in temperature (∆T) measured in degrees Celsius. The specific heat of water is

4.184 J/g°C. The variable mm is the mass of the metal (g). The tfm variable is the

final temperature of the metal while the tim is the initial temperature of the metal in

degrees Celsius. The CF variable is the correction factor for the calorimeter. This

equation was used to calculate the specific heat of the metal in J/g°C. The

specific heat is an absolute value because it can never be negative.

s=|4.184 J ×mw×(t fw−t iw)mm×(t fm−tℑ) |+CF

A sample calculation is shown in the figure below using this equation.

s=|4.184 J ×mw×(t fw−t iw)mm×(t fm−tℑ) |+CF

s=|4.184 J ×30 g×(26.1 ° C−22.8° C)9.881 g×(26.1°C−96.9 °C) |+0.307 J / g°C

s=| 414.216−699.59604|+0.307 J / g°C

s=0.899J /g° C

Figure 3. Specific Heat Sample Calculation

Figure 3 shows a sample calculation for specific heat. The data used for

this calculation is from Trial 1 for the specific heat of aluminum, which is shown in

Table 1. The correction factor was 0.307 because the black calorimeter was

used. The specific heat in this trial was calculated to be 0.899 J/g °C. The actual

specific heat of aluminum is 0.9 J/g x °C.

Linear Thermal Expansion:

To calculate the coefficient of thermal expansion the linear thermal

expansion equation had to be used. In this equation, the change in length of the

metal is divided by the initial length and the change in temperature of the metal.

The ⍺ represents the coefficient of thermal expansion (1/°C). The Lf is the final

length of the metal rod (mm) while the Li is the initial length of the metal rod

(mm). The difference between the final and initial length is the change in length

or ∆L (mm). The Tf variable is the final temperature of the rod while T i is the initial

temperature of the rod measured in degrees Celsius. The difference between the

final temperature and the initial temperature is the change in temperature or ∆T

measured in degrees Celsius.

⍺=(Lf−Li)Li(T f−T i)

A sample calculation is shown in the figure below using this equation.

⍺=(∆ L)Li(∆T )

⍺=(0.229mm)

131.286mm(74.4 ° C)

⍺=2.344×10−5°C−1

Figure 4. Linear Thermal Expansion Sample Calculation

Figure 4 shows a sample calculation for linear thermal expansion.

The data used for this calculation is from Trial 1 for the linear thermal expansion

of aluminum, which is shown in Table 5. The coefficient of thermal expansion in

this trial was calculated to be 2.344×10−5° C−1.The actual alpha coefficient of

aluminum is2.31×10−5 °C−1.

Percent Error:

To calculate the percent error, or how much the lab value deviates from

the known, the following equation was used.

%Error= (LabValue )−(KnownValue )KnownValue

∗100

The known value is subtracted from the lab value. This is then divided by the

known value in order to calculate the percent error. A sample calculation of this

equation is shown in the figure below.

%Error= (LabValue )−(KnownValue )KnownV alue

∗100 %

%Error= (0.899 )− (0.9 )0.9

∗100 %

%Error=−0.111%

Figure 5. Percent Error Sample Calculation

Figure 5 shows a sample calculation for percent error. The data used for

this calculation is from Trial 1 for the specific heat of aluminum, which is shown in

Table 1. The percent error in this trial was calculated to be -0.111%. This means

that the specific heat in this trial deviates from that of aluminum by 0.111%. The

actual percent error for this trial was calculated to be -0.094%, which is shown in

Table 9. The percent error in this sample calculation deviates from the one in

Table 9 because the percent errors in Table 9 are calculated using exact

numbers instead of rounded ones.

Two Sample t Test :

To analyze the number of standard deviations away the data lies from the

sample mean, t, a two sample t test was conducted. In this equation, the

difference of the sample means is taken. This is divided by the square root of the

sum of both standard deviations divided by the number of trials. The variable x1 is

the mean of the first sample J/g °C. The variable x2 is the mean of the second

sample J/g °C. The variable s1 is the standard deviation of the first sample. The

variablen1 is the number of trials that were conducted for the first sample. The

variable s2 is the standard deviation of the second sample. The variablen2 is the

number of trials that were conducted for the second sample. The p - value is

calculated by comparing the t - value with the degrees of freedom on a t Table.

An alpha level of 0.1 was used. The following equation was used to calculate the

t value. t=

x1−x2

√ s12

n1+s2

2

n2

A sample calculation of this 2–sample t test is shown in the figure below.

t=x1−x2

√ s12

n1+s2

2

n2

t= 0.89993333333333 J /g°C−0.8874 J / g°C

√ 0.0190242952438142

15+ 0.0208525435241982

15

t=1.720

Figure 6. Two Sample t Test Sample Calculation

Figure 6 shows how the two sample t test was conducted. The t - value

of 1.720 was compared to a t table in order to calculate the p – value. The data

that was used in this sample calculation is from the two specific heat

experiments. This data is shown in Table 1 and Table 2.

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