An Inside Look at Intensive Properties of Metals

Desiree Boyd – Jacob Reno

Macomb Mathematics Science Technology Center

Chemistry 1, FST, IDS 2 – 10A

Hilliard / Dewey/ Supal

20 May 2013

Boyd – Reno 1

Table of Contents

Introduction……………………………………………………………………………….......... 2

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

Problem Statement and Hypothesis…………………………………………………………. 9

Specific Heat Experimental Design………………………………………………………… 10

Linear Thermal Expansion Experimental Design…………………………………………. 12

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

Data Analysis and Interpretation…………………………………………………….……….25

Conclusion ……………………………………………………………………………………. 35

Appendix A: Calorimeters……………………………………………...……………………. 38

Appendix B: LabQuest………………………………………………………………………...40

Appendix C: Formulas and Equations………………………………………………………42

Works Cited……………………………………………………….……………...…………… 46

Boyd – Reno 2

Introduction

The elements of the periodic table are categorized by their properties and

characteristics. They are distinct substances that behave differently in nature, not one is

the same as the other. Two intensive properties, physical assets that belong to a unique

element, are specific heat and linear thermal expansion. When given the correct

variables, the specific heat and linear thermal expansion equations can be manipulated

to find the specific heat and linear thermal expansion alpha coefficient of a material.

The metal rods were to be identified as either the same or different from

the known metal rods. Both sets of rods were identical. The known metal rods of this

experiment were made of copper (Cu), an effectual heat conductor.

In the following experiment, two metal rods were given. The experimenters

discovered the type of element that the rods were made of. Later on, the experimenters

were given two more metal rods but were unaware of its material. The goal was to

either identify the unknown metal rods as the same type of element or to find that they

were not. To do this, the rods were placed in calorimeters and the temperature change

from 100°C to the temperature of the calorimeter was recorded. The specific heat was

then calculated from the data and compared to the specific heat of the known metal

rods. The rods were then placed in a linear thermal expansion jig. The jig calculated the

change in length of the metal rod in millimeters from 100°C to the temperature of cold

water in a spray bottle used. The linear thermal expansion alpha coefficient was then

calculated and compared to the linear thermal expansion coefficient of the known metal

rods.

Boyd – Reno 3

Review of Literature

The origins of copper (Cu) cannot be exactly pinpointed. However, there are

dates of copper’s earlier sightings in Iraq back in 9,000 BC. Copper’s name originated

from the Latin word “cuprum” which means the island of “Cyprus”. In the earlier years,

around 5,000 years before us, copper (Cu) was popularly used. Its malleable

characteristics allowed early humans to construct tools and weapons by mixing copper

with zinc to form brass and tin to form bronze.

In nature, copper’s raw state is mined. Over 750 thousand tons of copper (Cu)

are mined and collected in the United States per year (Butts 936). Normally copper (Cu)

is found along with sulfur. The copper ores (Cu) usually only contain 1% of copper (Cu)

because it is found with other elements. The ores go through a process called smelting

which puts copper (Cu) into a bar or ingot. Bars and ingots are more easily stored and

transportable. They are put through extreme heat where the copper can be extracted

from the ores ("Mining of Minerals and Methods of Extracting of Metals"). Here is an

example of one of the reactions copper could go through:

CuS (s) + O2 (g) Cu (s) + SO2 (g)

In the reaction shown, copper (Cu) is found with sulfur in its solid state, which is very

common in nature. The copper sulfide ore (CuS) containing the elements is exposed to

oxygen (O2) and extreme heat. Since the compound is absorbing the energy, this is an

endothermic process. The result is copper (Cu) in a solid lustrous form and sulfur

dioxide gas.

A common and generally popular metal, copper (Cu) has many different

purposes that it is used for. In industrial buildings and modern homes, copper is used

Boyd – Reno 4

for pipelines and plating electrical wires. One of copper’s dominant properties is its

conductivity of electricity which is why it is so widely used for wiring. Copper (Cu) has a

strong structure with electrons that move fluidly about the molecules. This is called its

“electron sea” which is rather easily ductile due to the fluidity in which the electrons

move, giving the copper (Cu) molecules a high conductivity level. Electronic products,

transportation equipment, pigments, and medical process execute this important

characteristic. Copper (Cu) can be used for chemical purposes, this includes

compounds containing copper (Cu) in fungicides, wood preservations, anti-fouling

compounds, and chemical reactants.

Copper’s density is about 8.96 g/cm3 with a specific heat of 0.38 J/g*K.

Compared to the specific heat of water, copper’s specific heat is much lower. Specific

heat is the amount of energy it takes to raise one unit of mass of a substance by one

degree Celsius. The specific heat of water is 4.186 J/g*C. It takes much more energy to

raise the temperature of water than it does to raise the temperature of copper. Copper is

an excellent and sufficient heat as well as electricity conductor. Copper’s electrical

conductivity is 58.1 106/m and has a thermal conductivity of 4.01 W/cm*K. These

numbers are relatively high, proving that copper’s conductivity of electricity and heat is

sufficient and much more than water’s (“Specific Heat”).

The atomic mass of the element copper is 63.546 u with 29 protons, 35 neutrons,

and 29 electrons. It is a generally stable element when in its free metallic state. Copper

is relevant because of its abundance in the world and importance ("copper (Cu)").

Copper (Cu) is one the great conductors and metallic elements. The electron

configuration of copper is unique compared to other elements on the periodic table. It is

Boyd – Reno 5

one of the exceptions of electron configurations. The electrons in copper prefer to fill the

3d block, so instead of two electrons in the 4s block, it places one of those two in the 3d

block, filling it with the full 10 electrons. The reasoning behind this is because of the less

energy it takes to fill the 3d block and the more paired electrons it has with that.

Figure 1. Copper Orbital Diagram

In Figure 1, the orbital diagram of copper (Cu) is illustrated. As previously

explained, copper has 29 electrons. Unlike most elements in the periodic table, copper’s

electrons do not arrange themselves in the typical order. Instead, one electron in the 4s

block prefers to place itself in the 3d block, filling the 3d block with its capacity of 10

electrons, pairing all of the electrons in the 3d block. Another element with a different

arrangement of electrons would be chromium (Cr).

The uniqueness of copper lies within its ductility, malleability, and high melting

point which allows great electrical conductivity (Young). These characteristics make

copper (Cu) so incredibly important. Copper (Cu) has been in the lives of humans since

the ancient times. In those times, copper (Cu) was used for tools, plates, bowls,

weapons and even currency. Now, in present time, copper (Cu) is still used for tools and

many other important uses that cannot always be replaced by another element.

Specific heat is the amount of heat required to raise the temperature of one unit

mass of a substance by one degree Celsius ("Measurement of Specific Heat"). Below is

the equation to find the change in enthalpy (q), heat content present in a system at

constant pressure. To find q, the specific heat of the substance being measured, the

Boyd – Reno 6

mass of said substance, and ΔT are required. Any of the four variables could be

calculated if the remaining three are known. The relationship is show below:

q=smΔT

The transfer of heat is accompanied by a change in temperature of the object.

The change in temperature of the object is defined as the final temperature of the object

minus the initial temperature of the object. The relations are shown below:

ΔT = Tf – Ti

The sign of the energy associated with q will depend on ΔT. If the final

temperature is warmer than the initial temperature, the value of ΔT will have a positive

sign. Thus, the sign of the value of q will be positive (+q). This is the same as with the

sign for an endothermic process in which heat flows into a substance from the system. If

the final temperature is cooler than the initial temperature, the value of ΔT will have a

negative sign. In this case, the sign of the value of q will be negative (-q). This is the

same as the sign convention for an exothermic process in which heat flows out of a

substance into a system ("Measurement of Specific Heat").

This process is consistent with the First Law of Thermodynamics, which states

that energy is neither created nor destroyed, only changed from one form to another.

A calorimeter is a device designed to measure heat changes during physical or

chemical transformations ("Measurement of Specific Heat"). They are well insulated so

that no heat is gained from the surroundings or lost to the surroundings ("Calorimetry:

Specific Heat Capacity of Copper"). A simple calorimeter can be made from a

Styrofoam coffee cup and water ("Measurement of Specific Heat").

Boyd – Reno 7

Using the calorimeter, the specific heat of the unknown metal rod can be

calculated by setting the enthalpy of the unknown metal rod equal to the enthalpy of the

water. Specific heat can be found by manipulating the equations of the enthalpy of the

unknown metal rod and the enthalpy of the water, shown below ("Measurement of

Specific Heat"). The specific heat of water is shown below:

4.184 Jg∗°C

The specific heat of copper, which the researchers have identified the first pair of

rods to be, is also shown below ("Measurement of Specific Heat"):

0.385 Jg∗° C

Cause and effect relationships are proposed to explain the differences in heat

capacity between copper and other metals such as aluminum.

Fourteen other metallic elements could be tested in a second experiment from

the first where copper was identified. Multiple trials must be conducted in order to

obtain sufficient data. The metallic elements arrange on the periodic table have similar

properties. However, during experiments, the specific heats of elements are never the

same (Bindel, and Fochi 955-57).

Thermal expansion is the tendency of matter changing its volume in depending

on the change of temperature (Tipler 666-670). Thermal expansion involves an

important variable: heat. Heat causes molecules to move at a more rapid pace and

spread farther apart from each other. Gases simply move out a farther distance; liquids

warm up and eventually evaporate; solids expand. Linear thermal expansion is the

continuous expanding of molecules with applied heat and constant pressure. This

Boyd – Reno 8

process is consistent with the kinetic molecular theory, which concerns the behavior of

matter and relationship between volume, temperature, and pressure.

Linear thermal expansion is an intensive property specific to each individual

element. Each is different and solely for each element. The thermal expansion of copper

(Cu) is 16.5 (Linear Coefficient, α, at 20°C) (“Engineering Tool Box”). No other element

has the same thermal expansion as copper (Cu). This means that with the right given

variables, the thermal expansion of any element, such as copper (Cu) can be calculated

but will not expand with applied heat the same amount as another. The equation for

finding the linear thermal expansion of any element is shown below where ∆L is the

change in length (Lf – Li), α is the linear expansion coefficient, L is the original length,

and ∆T is the change in temperature (Tf – Ti):

∆ L=αL∆T

During the experiment, the change in length (millimeters) of the unknown metal

rod, the change in temperature of the metal rod (Celsius), and the original length of the

unknown metal rod (millimeters) of the rod are being measured.

In previous experiments, copper (Cu) is sometimes measured or measured with

other similar metals such as aluminum oxide (Al2O3). In one such experiment

(“Measuring the coefficient of Linear Expansion of Copper, Steel, and Aluminum”),

copper, steel, and aluminum tested for their linear thermal expansion coefficient.

Both thermal expansion and specific heat are unique qualities of each element.

The behavior of the molecules is based off of the temperature in which the molecules

are surrounded by and the pressure which will be kept constant during the future

experiment and trials.

Problem Statement

Boyd – Reno 9

Problem Statement:

If intensive properties, such as specific heat and linear thermal expansion are

calculated the unknown metal rod will be identified.

Hypothesis:

After calculating the specific heat and linear thermal expansion, the unknown

metal rod will be identified as copper (Cu) within a 1.81% margin of error.

Data:

In each of the experimental trials, the variables of specific heat and linear thermal

expansion are measured and collected. For specific heat, the variable q represents the

enthalpy; s represents the specific heat of the unknown metal; For linear thermal

expansion, the variable α stands for the linear thermal expansion coefficient which for

copper (Cu) is 16.5 α (10−6 C−1); ΔT, similar to that in the equation of specific heat, is the

change in temperature of the water in the calorimeter in degrees Celsius; L is equal to

that of the original length of the unknown metal rod in millimeters; ΔL, also similar to that

of ΔT, is the change in length of the unknown metal rod in millimeters.

Specific Heat Experimental Design

Boyd – Reno 10

Materials:

(2) unknown metal rods TI-Nspire CX graphing calculator(2) copper rods hot platecalorimeter LabQuestScout Pro scale (0.1g) temperature probe (0.01ºC)X mL graduated cylinder 250mL beaker22 x 12 x 6 cm loaf pan tongstimer

Procedures:

*Note: Goggles and apron are to be worn at all times for safety of boiling water and

steam.

1. Using the Random Integer function on the TI-Nspire CX calculator, randomize all

trials.

2. Fill the loaf pan with water and place on hot plate and bring to 100 ºC.

3. Insert the selected metal rod into the loaf pan for approximately three minutes

after it reaches 100 ºC. Record its temperature in Celsius.

4. Measure 50 - 60mL water with the graduated cylinder and transfer it to the

constructed calorimeter (see Appendix 1). Record amount of water in grams.

5. Set up the LabQuest data auto collector graph (see Appendix 2).

6. Insert the metal rod into the 100 ºC water of the calorimeter.

7. Watch thermometer for the equilibrium point, the temperature will cease

changing. Record the first equilibrium temperature in Celsius after the highest

point on the LabQuest graph.

8. Record the final temperature of the water in Celsius.

9. Repeat steps 2 - 8 for thirty trials.

Diagram: X mL graduated cylinder Jigs Caliper (0.01 mm)

Boyd – Reno 11

Figure 2. Experimental Setup for Trials

In Figure 2, the setup is ready for trials to be executed. The hot plate is turned

on, boiling the water that will heat the metal rods to approximately 100°C. The Scout

Pro scale (0.1 g) has measured the weight of the rods and the caliper (0.01 mm) has

measured the lengths. The tongs will place the metal rods in the calorimeters or jigs.

The TI-Nspire CX will randomize trials.

Linear Thermal Expansion Experimental Design

22 x 12 x 6 cm loaf pan

calorimetersSpray bottle

Unknown metal rods

LabQuest

TI-Nspire CX graphing calculator

Hot plate

Scout Pro scale (0.1g)

Copper rods

temperature probe (0.01ºC)

tongstimer

Boyd – Reno 12

Materials:

(2) unknown metal rods 22 x 12 x 6 cm loaf pan(2) copper rods tongsTI-Nspire graphing calculator X mL cylinderhot plate linear thermal expansion jigCaliper (0.01 mm) spray bottle

Procedure:

*Note: Goggles and apron are to be worn at all times for safety of the boiling water and

steam.

1. Randomize the trials using the TI-Nspire graphing calculator.

2. Ready the linear thermal expansion jig for use.

3. Fill the loaf pan with water. Set it on the hot plate and bring to 100°C.

4. Measure and record the initial length of the selected metal rod (mm).

5. Insert the metal rod into the loaf pan. Measure and record the temperature in °C.

6. After approximately three minutes, move the metal rod with the tongs to the jig.

7. Mark where the beginning dial is at on the jig. Record the mark.

8. Spray the rod from bottom to top in order to cool the rod faster. Record the

temperature of the water in the spray bottle in °C.

9. After 3-5 minutes, mark final dial reading.

10.Record the change from the first dial to the second (ΔL mm).

11.Remove the metal rod from the jig.

12.Repeat steps 2 – 10 for the remaining trials.

Diagram:

jigs

Boyd – Reno 13

Figure 3. Linear Thermal Expansion Jig Containing Metal Rod

Figure 3 shows the linear thermal expansion jigs. The linear thermal expansion

jig measures the change in length of the metal rod from when it is at its longest to when

it is at its shortest. The measurements are measured in millimeters to a 0.01 mm

precision. The appearance does not seem to change but is required to be measured in

order to calculate the alpha coefficient. Each alpha coefficient is specific to its element.

A type of element will only expand according to its alpha coefficient, no more and no

less. Along with the jigs, the TI-Nspire calculator randomizes trials, the timer times five

minutes per measurement and the temperature probe measures the temperature in °C

of the spray bottle which cools the rod faster to its final temperature.

Data and Observations

Table 1

Spray bottle

Temperature probe

timer

TI-Nspire graphing calculator

Boyd – Reno 14

Copper Specific Heat Data

Trial Rod

Initial Temp. (⁰C) Equilibriu

m Temp. (⁰C)

Change in Temp. (⁰C) Mass (g) Specifi

c Heat(J/ g⁰C)Water Metal Water Metal

Metal

Water

1 A 23.60 95.50 26.60 3.00 -68.90 28.5 60.0 0.3842 B 24.30 95.50 27.30 3.00 -68.20 28.4 60.0 0.3893 A 23.40 95.50 26.40 3.00 -69.10 28.5 60.0 0.3824 B 24.10 96.60 27.10 3.00 -69.50 28.4 60.0 0.3825 A 23.20 98.20 26.45 3.25 -71.75 28.5 60.0 0.3996 A 23.90 97.50 27.00 3.10 -70.50 28.5 60.0 0.3877 B 24.10 98.20 27.20 3.10 -71.00 28.4 60.0 0.3868 A 23.70 97.70 26.70 3.00 -71.00 28.4 60.0 0.3739 B 24.20 96.60 27.40 3.20 -69.20 28.4 60.0 0.409

10 B 24.60 97.40 27.70 3.10 -69.70 28.4 60.0 0.39311 B 24.10 97.60 27.20 3.10 -70.40 28.4 60.0 0.38912 A 24.00 97.80 27.10 3.10 -70.70 28.5 60.0 0.38613 B 23.90 98.10 27.10 -3.20 -71.00 28.4 60.0 0.39814 B 24.10 98.10 27.20 3.10 -70.90 28.4 60.0 0.38615 A 24.50 97.90 27.60 3.10 -70.30 28.5 60.0 0.388

Average

____ 23.98 97.21 27.07 2.66 -70.14 28.4 60.0 0.389

In Table 1, the raw data trials collected from the known copper metal rods trials

are shown. It can be observed that the temperatures varied by no more than three

degrees at most in each column. The mass of the water used in the calorimeters

(Appendix A) was consistent, a total of 60 mL of water in each calorimeter and trial. The

specific heat was calculated via Microsoft Excel equations. The real specific heat of

copper is 0.386 J/g°C. The calculated specific heats from the experiment varied within a

low margin.

Table 2Copper Specific Heat Observations

Boyd – Reno 15

Trial Rod Date Observations

1 A 19-AprRedone; used calorimeter #1; metal rod is smooth. Lustrous color and cool to the touch before submersing in boiling water. Smells like pennies. Lab window let in a breeze.

2 B 19-Apr Redone; used calorimeter #2; metal rod was left in boiling water for an extra 30 seconds longer. Lab window let in a breeze.

3 A 17-AprUsed calorimeter #1; Thermometer tapped the bottom of the metal loaf pan. Temperature read a few degrees higher. Room temperature rises.

4 B 19-AprRedone; used calorimeter #2; metal rod was difficult to place in calorimeter for a brief few moments. Temperature could have been lost by a few degrees. Lab window let in a breeze

5 A 17-Apr Used calorimeter #1; Water in loaf pan appeared to be losing a lot of water through steam. Rods may not have heated enough.

6 A 17-Apr Used calorimeter #3; metal rod was difficult to grab with tongs due to a slip of the hand.

7 B 17-Apr Used calorimeter #2; more water was added to the loaf pan, took longer to heat up.

8 A 17-Apr Used calorimeter #3; metal rod stayed in calorimeter for an extra 20 seconds due to data table entries

9 B 19-Apr Redone; used calorimeter #2; trial was conducted with no issues; lab window let in a draft.

10 B 19-Apr Redone; used calorimeter #3; trial was conducted with no issues. Lab window let in a draft.

11 A 19-Apr Redone; used calorimeter #1; trial took a few moments longer than timed. Lab window was opened, let in a draft.

12 A 17-AprUsed calorimeter #1; the room temperature rose with steam collecting from other lab stations. More water was added to loaf pan.

13 A 17-Apr Used calorimeter #1; the room temperature rose with steam collecting from other lab stations.

14 B 17-Apr Used calorimeter #3; metal rod was taken from boiling water 20 seconds longer than three minutes.

15 A 17-Apr Used calorimeter #1; trial was conducted with no issues. Experimenters fatigued.

Table 2 contains the recorded observations made during the trials of the copper

rod specific heat experiment. Some of the trials had to be redone due to the loss of heat

during the experiment. The opened lab window caused some difficulties maintaining

Boyd – Reno 16

heat in the rods. Sometimes the rod was left in the water for a little extra time. This

allowed the metal to keep or regain the heat lost from the window.

Table 3Unknown Specific Heat Data

Trial RodInitial Temp.

(⁰C)Equilibrium Temp.

(⁰C)

Change in Temp. (⁰C) Mass (g)

Specific Heat

(J/ g⁰C)Water Metal Water Metal Metal Water

1 C 23.60 98.00 38.30 14.70 -59.70 106.90 50.00 0.4822 D 26.20 96.40 42.10 15.90 -54.30 106.90 50.00 0.5733 D 25.70 96.40 40.30 14.60 -56.10 106.90 50.00 0.5094 D 24.80 98.20 42.10 17.30 -56.10 106.90 50.00 0.6035 C 24.20 97.40 39.93 15.73 -57.47 106.90 50.00 0.5366 C 26.00 96.30 41.80 15.80 -54.50 106.90 50.00 0.5677 C 23.50 96.60 38.20 14.70 -58.40 106.90 50.00 0.4938 C 20.50 96.40 35.40 14.90 -61.00 106.90 50.00 0.4789 D 25.00 96.30 39.30 14.30 -57.00 106.90 50.00 0.491

10 D 23.90 97.20 39.00 15.10 -58.20 106.90 50.00 0.50811 C 22.60 96.60 39.60 17.00 -57.00 106.90 50.00 0.58412 C 23.30 98.00 39.60 16.30 -58.40 106.90 50.00 0.54613 D 30.10 99.20 44.80 14.70 -54.40 106.90 50.00 0.52914 C 23.70 97.10 39.80 16.10 -57.30 106.90 50.00 0.55015 D 24.60 97.20 40.30 15.70 -56.90 106.90 50.00 0.540

Average _ -_ 24.51 97.15 40.04 15.52 -57.12 106.90 50.00 0.533

Table 3 shows the raw data of the unknown metal rods. The temperatures varied

more than the copper metal rods. Less water was used for the trials than the copper

rods due to the longer diameter and more mass. The specific heat of the unknown metal

rods differed from the copper rods. The calculated specific heats vary but are averaged

at 0.533 J/g°C. The specific heat of the unknown metal rods is not the same as the

copper metal rods.

Table 4Unknown Specific Heat Observations

Boyd – Reno 17

Trial Rod Date Observations

1 C 17-Apr

Used calorimeter #1; metal rod appears lustrous and smooth to the touch, resembling copper. More mass than known copper rods and smells like pennies. Slightly bounces when placed in boiling water.

2 D 19-Apr Redone; used calorimeter #2; lab window was opened during trial. Metal rod caused calorimeter to overflow with water.

3 D 19-Apr Redone; used calorimeter #2; metal rod was dropped and replaced in boiling water. Lab window let in a draft.

4 D 19-Apr Redone; used calorimeter #3; metal rod bounced when dropped into calorimeter. Lab window let in a draft.

5 C 17-Apr Used calorimeter #1; room temperature increasing from steam. Metal rod was left in water for 30 extra seconds.

6 C 19-Apr Redone; used calorimeter #3; lab window let in a draft; metal rod was dropped by slippery tongs.

7 C 17-Apr Used calorimeter #1; trial was conducted without issues.

8 C 17-Apr Used calorimeter #1; trial was conducted without issues.

9 D 17-Apr Used calorimeter #2; trial was conducted without issues.

10 D 19-Apr Redone; lab window let in a breeze. Metal rod taken from boiling water ten seconds early.

Trial Rod Date Observations

11 C 17-Apr Used calorimeter #1; trial was conducted without issues.

Boyd – Reno 18

12 C 17-Apr Used calorimeter #3; metal rod was dropped and replaced in boiling water for 30 seconds.

13 D 17-Apr Used calorimeter #3; experimenters fatigued

14 C 17-Apr Used calorimeter #1; metal rod bounced when dropped into calorimeter.

15 D 19-Apr Redone; used calorimeter #2; trial was conducted without issues.

In Table 4, the observations made during the specific heat trials for the unknown

metal are shown. During these trials, there were fewer errors made than when

conducting the trials for the copper rods. The experimenters conducted the experiment

more efficiently as time progressed. Less water was used in the calorimeters because

of the more mass of the unknown metal rods than the known copper rods.

Table 5Copper Linear Thermal Expansion Data

Trial Rod Initial Temp. (°C)

Initial Length (mm)

Final Temp. (°C)

∆L (mm)

Alpha Coefficient

(mm x10^-6)1 B 96.60 124.24 22.20 0.05 5.409

Boyd – Reno 19

2 B 97.40 124.24 21.40 0.06 6.3543 A 97.40 124.63 21.40 0.05 5.2794 B 95.10 124.24 23.20 0.05 5.5975 A 95.10 124.63 23.50 0.09 1.0096 B 96.60 124.24 23.00 0.06 6.5627 A 96.60 124.63 23.00 0.09 9.8128 A 96.20 124.63 23.60 0.09 9.9479 B 96.20 124.24 23.60 0.07 7.76110 A 96.20 124.63 23.50 0.05 5.51811 B 96.20 124.24 23.50 0.08 8.85712 B 97.50 124.24 23.30 0.07 7.59313 A 97.50 124.63 23.30 0.06 6.48814 B 97.50 124.24 23.30 0.06 6.50915 A 97.50 124.63 23.30 0.08 8.651

Average __ 96.64 124.42 23.01 0.07 7.362

In Table 5, the raw data used to calculate the linear thermal expansion of copper

is presented. The temperatures in the table are within 3 degrees of each other. The

changes in length of the copper rods were measured on a small scale. The length

changed within 0.04 mm, unnoticeable to the eye.

Table 6Copper Linear Thermal Expansion Observations

Trial Rod Date Observations

Boyd – Reno 20

1 B 22-Apr Redone; used jig #9; the spring on the jig slipped from the experimenter's hand when pulled back. Lab window was opened.

2 B 18-Apr Redone; used jig #9; metal rod was dropped and replaced in boiling water. Lab window was open and let in a cool breeze.

3 A 18-Apr Used jig #8; trial was slow due to difficulty of slippery spring.

4 B 18-Apr Used jig #9; water coated the table, made things slippery and cooler

5 A 18-Apr Used jig #8; trial was conducted without issues.

6 B 18-Apr Used jig #9; jig was sopping wet, cooled rod fast.

7 A 18-Apr Used jig #8; jig was sopping wet, cooled rod fast.

8 A 18-Apr Used jig #8; jig was sopping wet, cooled rod fast.

9 B 18-Apr Used jig #9; jig was sopping wet, cooled rod fast.

10 A 18-Apr Used jig #8; metal rod nearly slipped from tongs, lost heat in process

11 B 18-Apr Used jig #9; trial conducted without issues.

12 B 22-Apr Redone; used jig #9; experimenters fatigued.

Trial Rod Date Observations

13 A 18-Apr Used jig #8; trial was conducted without issues.

14 B 18-Apr Used jig #9; jig and table were sopping wet from spray bottle

Boyd – Reno 21

15 A 18-Apr Used jig #8; experimenters fatigued, metal rod was dropped and replaced in boiling water for less than a minute.

In Table 6, the linear thermal expansion observations are presented. The linear

thermal expansion jig was often soaked with water and cooled the rod quickly. This

helped with time management and the data was not negatively altered. The temperature

of the metal rod was assumed to be the same temperature as the water in the spray

bottle.

Table 7Unknown Linear Thermal Expansion Data

Trial Rod Initial Temp. (°C) Initial Length (mm) Final Temp. (°C) ∆L

(mm)Alpha

Coefficient(mm x 10^-6)

1 D 97.20 118.50 21.40 0.03 3.1792 D 94.60 118.50 21.70 0.03 3.2763 C 94.60 118.51 21.70 0.02 2.1844 D 96.40 118.50 22.90 0.04 4.340

Boyd – Reno 22

5 D 97.20 118.50 21.40 0.03 3.1796 D 94.90 118.50 25.10 0.03 3.3857 C 94.90 118.51 25.10 0.02 2.2568 C 96.20 118.51 21.50 0.04 4.2869 C 96.20 118.51 21.50 0.03 3.215

10 C 95.60 118.51 22.10 0.03 3.25511 D 95.60 118.50 22.10 0.03 3.25512 C 96.20 118.51 21.80 0.03 3.22513 C 96.40 118.51 21.60 0.03 3.21114 C 96.50 118.51 21.60 0.03 3.20815 D 96.20 118.50 21.80 0.03 3.225

Average ____ 95.91 118.51 22.22 0.03 3.245

Table 7 shows the unknown metal rods’ linear thermal expansion data, the

calculated and the collected to do so. The change in length of the unknown metal was

less than that of the copper rods. The change in temperature varied as well ranging

from 25.1°C to 21.4°C. Although a seemingly small difference, this changed the length

alpha coefficient. The alpha coefficient of the copper rods is larger than that of the

unknown metal rods.

Table 8Unknown Linear Thermal Expansion Observations

Trial Rod Date Observations

1 D 22-Apr Redone; used jig #9; lab window was opened and let in a breeze.

Boyd – Reno 23

2 C 18-Apr Used jig #8; metal rod dropped due to slippery tongs and more mass of rod.

3 C 18-Apr Used jig #8; metal rod stayed in boiling water for 20 seconds longer.

4 D 18-Apr Used jig #9; water is evaporating quickly; tops of metal rods are protruding from water.

5 D 18-Apr Used jig #9; more water was added to loaf pan, took longer to boil once again.

6 D 22-Apr Redone; lab window was opened and let in a breeze. Metal rods stayed in loaf pan an extra minute to reheat.

7 C 22-Apr Redone; used jig #8; lab window was opened and let in a breeze. Trial conducted without issues.

8 C 22-Apr Redone; used jig #8; lab window was opened and let in a breeze. Trial conducted without issues.

9 C 22-Apr Redone; used jig #8; lab window was opened and let in a breeze. Trial conducted without issues.

10 C 22-AprUsed jig #8; lab window was opened. Metal rod slipped from tongs again. Replaced in boiling water for 20 seconds.

11 D 22-Apr Used jig #9; lab window was opened. Trial was conducted without issues.

12 C 22-Apr Used jig #8; spring on jig slipped from experimenter's hand. Few moments before placing rod into jig.

Trial Rod Date Observations

13 C 22-Apr Used jig #8; metal rod stayed in boiling water for 20 seconds longer.

14 C 22-Apr Used jig #8; window was closed. Temperature rose.

15 D 22-Apr Used jig #9; left in jig for 45 extra seconds due to data entry.

Boyd – Reno 24

Table 8 exhibits the unknown metal rod linear thermal expansion observations.

The experimenters took extra care to mop up the water that soaked the jig to keep from

losing heat too quickly to analyze. The loaf plan was refilled multiple times and took

longer to reheat. Some of the trials had to be redone in order to correctly heat the metal

rods.

Data Analysis and Interpretation

Specific Heat:

The data was collected during experimentation to calculate the specific heat of

the copper (Cu) rods and the unknown metal rods. The data was manipulated into the

specific heat equation (see Appendix C) in order to find the specific heat values used in

Boyd – Reno 25

this analysis. It is known to be valid based on the percent error tables below. The data

is quantitative.

Table. 9Percent Error Copper.

Trial Percent Error

1 -0.6402 0.7343 -0.9274 -1.1515 3.3656 0.3427 -0.0148 -3.2399 5.89610 1.85111 0.83812 0.05813 3.21214 0.12715 0.628

Average 0.739

The table above shows the percent error, calculated using the percent error

equation (see Appendix C) for the trials conducted to find the specific heat of the known

metal rods. The acceptable margin of error for the specific heat of the element copper

(Cu) is ±1.8% which has been used in previous experiments. The average of the

percent errors displayed in the table above is 0.739%, well within this range; therefore it

can be assured that the known element is copper and the experiments were conducted

without flaw.

Table 10Percent Error Unknown

TrialPerce

nt Error

1 24.836

Boyd – Reno 26

2 48.4553 31.9434 56.3445 38.7666 46.9797 27.6158 23.8389 27.191

10 31.53811 51.20712 41.50513 36.99814 42.45215 39.889

Average 37.97

The table above shows the percent error for the trials conducted to find the

specific heat of the unknown metal rods. The acceptable margin of error for the specific

heat of the element copper is ±1.8%. The average of the percent errors displayed in the

table above is 37.97%, well outside this range; therefore it can be assured that the

unknown element is not copper. The range varies between 23.838% and 56.344%, a

large difference. This suggests not only a difference between the two metals but also an

error in the experiment. Even though the metals can be proven to be different, the range

plays a huge role with many outliers in the wide range.

Boyd – Reno 27

Figure 4. Specific Heat Probability Plots

Figure 4 shows probability plots for the copper rods and the unknown metal rods.

The calculated specific heats of the copper rods are relatively close to the line of

regression. This is because of the consistency of the specific heats and the closeness

of the targeted specific heat. The calculated specific heats from the trials are near that

of coppers, meaning the trials and the equations used are valid. The unknown specific

heats are also relatively close to the line of regression, although the line has a different

equation. Since the line has a different equation and the data falls close to the line of

regression, the specific heat of the unknown metal rods is different than that of copper.

A two sample T-test was chosen to analyze this data because there are two

samples from two separate populations, testing the specific heat of two copper rods

compared to the specific heat of two unknown metal rods. Both samples are Simple

Random Samples randomized by the Random Integer Function on the TI-Nspire

graphing calculator and have been chosen from two different populations. Both are

normally distributed, shown in the box plots below. The population means and standard

deviations are not known.

Figure 5. Box Plots

Boyd – Reno 28

Figure 5 shows the box plots of both samples. The box plots are both slightly

skewed. The unknown specific heat is slightly skewed to the left whereas the copper

specific heat is slightly skewed to the right. The known metal has a lower median and a

much lower standard deviation, the distance data lies above and below the mean. The

unknown has a larger standard deviation and with larger quantitative data. There is an

outlier for the known specific heat, which means this information tells us to still run the

test but take caution when looking at the results. There is one outlier in the copper

specific heat box plot. This does not alter the data. When the outlier is removed, the

skew is not significantly changed.

H 0 :µ1=µ2

H a :µ1≠µ2

The null hypothesis states that there is no difference in the mean specific heat of

the known and the unknown metal. The alternative hypothesis states that there is a

difference in the mean specific heat of the known and unknown metal. The copper

specific heat is represented by µ1 and the unknown metal is represented by µ2.

t=x̄1− x̄2

√ s12

n1+s2

2

n2

Above is shown the equation used to calculate the t and p values. In the

equation x̅₁ represents the sample mean of the known metal, and x ̅₂ represents the

sample mean of the unknown metal. The sample standard deviations from the two

groups are respectively s₁ and s₂, along with the respective sample sizes n₁ and n₂. the

t-value calculated represents the number of standard deviations above or below zero.

In this situation, the t-value was found to be -17.2339.

Boyd – Reno 29

Figure 6. Results of Two Sample T-Test

The figure above shows the p-value on both sides of the distribution, providing

good evidence against the null hypothesis. The p-value on the distribution is difficult to

see, showing how small it is and its significance.

We reject the null hypothesis because the p-value of 2.70761E-11is less than the

alpha level of 0.10. There is evidence to suggest that the known metal is not the same

as the unknown metal. The probability of getting a difference in the mean values the

same as these by chance alone is 0.0000000027% if the null hypothesis true.

Linear Thermal Expansion:

Data was collected during experimentation to calculate the linear thermal

expansion coefficient of the copper (Cu) rods and the unknown metal rods. The

calculations were manipulated in the linear thermal expansion equation (see Appendix

3) to find the linear thermal expansion coefficient values used in this analysis.

Table 11

Boyd – Reno 30

Percent Error of Copper Rods

Trial Percent Error

1 -67.217%2 -61.488%3 -68.007%4 -66.077%5 -38.874%6 -60.233%7 -40.535%8 -39.716%9 -52.966%10 -66.555%11 -46.320%12 -53.980%13 -60.678%14 -60.554%15 -47.570%

Average -55.385%

Table 11 shows the percent error for the trials conducted to find the linear

expansion alpha coefficient of the copper (Cu) metal rods. The acceptable margin of

error for the linear expansion coefficient of the element copper is ± 18.75%. The

average of the percent errors of the alpha coefficient is -55.385%. The known metal rod

was indeed copper (Cu) based off of the specific heat data and the analysis of that data.

The percent error was fairly consistent; meaning the conduction of the trials was flawed,

but consistent.

Table 12Percent Error of Unknown Rods

Trial Percent Error

1 -80.736%

Boyd – Reno 31

2 -80.145%3 -86.765%4 -73.695%5 -80.736%6 -79.487%7 -86.326%8 -74.023%9 -80.517%10 -80.273%11 -80.271%12 -80.457%13 -80.538%14 -80.558%15 -80.455%

Average -80.332%

In Table 12 are the calculated percent error values for the trials of the unknown

metal rods. For copper (Cu), the margin of error for the linear thermal expansion

coefficient is ± 18.75%, as previously explained. The average percent error is -80.332%

and varied little throughout trials. They were consistent. These percentages were much

larger than that of the known metal rods. This suggests that the linear thermal

expansion coefficient of the unknown metal rods is different from that of copper’s (Cu)

coefficient.

Boyd – Reno 32

Figure 7. Linear Thermal Expansion Probability Plots

Figure 7 shows the probability plots of the calculated linear thermal expansion

coefficients. The coefficients of the copper rods lie close to the line of regression. There

is an exception for an outlier that lies closer to the bottom of the regression line shown.

This outlier may be farther from the others but it still is near the regression line, meaning

it is still relevant. The coefficients calculated from the unknown metal rods have a

different pattern. These coefficients lie vertically. This is because of how close the

coefficients are to the same number. This shows a difference from the copper rods.

However, it can also be noted that the lines of regression are a different equation. This

means that the linear

A two-sample t test was chosen to analyze this data because there are two

samples from two separate populations, testing the linear thermal expansion coefficient

of the two copper (Cu) rods compared to the linear thermal expansion coefficient of two

unknown metal rods. Both samples of rods are Simple Random Samples randomized

by the Random Integer Function on the TI-Nspire graphing calculator and have been

chosen from two different and independent populations. Both are normally distributed,

shown in the box plots below. The population means and standard deviations are also

unknown.

Boyd – Reno 33

Figure 8. Box Plots

Figure 8 shows the box plots of both samples and their distributions. The

unknown metal has a lower median and a much lower standard deviation. There are a

few outliers for the unknown data, which means this information tells us to still run the

test but take caution when looking at the results and watch more carefully for trends that

may appear or flaws in the experiment. These outliers are important to the data

because they suggest flaws in the experiment and a difference between the coefficients.

The coefficient of the copper rods has a larger standard deviation with a skew to the

right. This also suggests flaws in the data.

H 0 :µ1=µ2

H a :µ1≠µ2

The null hypothesis states that there is no difference in the mean of the linear

thermal expansion coefficient of the known and the unknown metal rods. The alternative

hypothesis states that there is a difference in the mean of the linear thermal expansion

coefficient of the known and unknown metal rods. The linear expansion coefficient of

copper is represented by µ1 and the unknown metal is represented by µ2.

Boyd – Reno 34

t=x̄1− x̄2

√ s12

n1+s2

2

n2

Above is shown the equation used to calculate the t and p values. In the

equation x̅₁ represents the sample mean of the known metal, and x ̅₂ represents the

sample mean of the unknown metal. The sample standard deviations from the two

groups are respectively s₁ and s₂, along with the respective sample sizes n₁ and n₂. The

t-value calculated represents the number of standard deviations above or below zero.

In this situation, the t-value was found to be 4.6332.

Figure 9. Results of Two Sample T-Test

The figure above shows the p-value on both sides of the distribution, providing

good evidence against the null hypothesis. The p-value on the distribution is difficult to

see, showing how small it is and its significance.

We reject the null hypothesis because the p-value of 0.0001579 is less than the

alpha level of 0.10. There is evidence to suggest that the known metal is not the same

as the unknown metal. The probability of getting a difference in the mean values the

same as these by chance alone is 0.01579% if the null hypothesis is true.

Boyd – Reno 35

Conclusion

Using the intensive properties, specific heat and linear thermal expansion, of the

known metal copper (Cu), it was determined that the unknown metal rods were not

copper, proving the hypothesis be rejected. The hypothesis was rejected based on the

p-values found in the two sample t-test being less than the alpha level. Also, the specific

heat of copper (Cu) is 0.386 J/g°C. The averaged calculated specific heat of the

unknown metal rods was 0.533 J/g°C. The difference between these two calculated

specific heats is outside of the acceptable margin of error of 1.81%. The linear thermal

expansion coefficient of copper is 16.5 × 10-6 °C-1. However, the calculated and

averaged linear expansion coefficient of the copper rods was 7.362 × 10-6 °C-1, but it

remained fairly consistent, allowing the data to be considered valid. The linear thermal

expansion coefficient of the unknown metal rods was averaged to be 3.245 × 10-6 °C-1

with an average percent error of -80.332%, well outside of the margin of error, helping

to prove that the two unknown metal rods were not copper (Cu).

These results can be supported by several scientific concepts. It is known that

the transfer of heat is accompanied by a change in the temperature of the object. As the

metal rods were heated and then allowed to cool within the calorimeter, the heat energy

present in the rods were released into the surrounding water. This process is consistent

with The First Law of Thermodynamics which states that energy is neither created nor

destroyed, only changed. In this case the heat energy was transferred from the rods to

the water. Being an intensive property, specific heat is specific to each individual

Boyd – Reno 36

element, so it can be assured that if the known and unknown metal rods do not have the

same specific heat as copper, then they are certainly not the same element.

Thermal expansion is the tendency of matter to change its volume according to a

change in temperature. Similar to specific heat, linear thermal expansion is an intensive

property, specific to each element. Linear thermal expansion is defined as the

continuous expanding of molecules with applied heat and constant pressure. Also

similar to specific heat, the process releases heat energy from the metal rod into the

surroundings. As the metal rod loses heat, it contracts. This is consistent to the Kinetic

Molecular Theory, which concerns the behavior of matter and the relationship between

volume, temperature, and pressure. Heat causes molecules to move at a rapid pace

and spread farther apart from each other. The process was observed during

experimentation, and the vastly different linear thermal expansion coefficients ensure

that the known and unknown metal rods are not the same element.

Despite the high percent errors, discussed earlier in the paper, the data was

consistent during experimentation, meaning that the trials were still viable to be

analyzed. Some problems that had occurred during trials include the rods being

occasionally dropped onto the table; the water in the loaf pan evaporating too much; the

opened lab window cooled the room; in addition to expected human error. However, the

most significant of errors is the assumption that the metal rods were actually heated to

100 degrees Celsius in the boiling water, and that they completely cooled to room

temperature in the time between temperature measurements.

Further experiments that could be conducted to identify the unknown element

include testing for several physical properties such as density, ductility, malleability, and

Boyd – Reno 37

conductivity. Chemical properties such as heat of combustion, enthalpy, toxicity,

flammability, and chemical stability in a given environment could also be tested.

The information presented could be considered helpful to a large quantity of

industries. Copper (Cu) is a common and generally popular metal with many different

purposes. Used in industrial buildings as well as homes, copper is present in the form

of plumbing pipes and electrical wires. This is based on coppers (Cu) high propensity to

conduct electricity, and heat. Another reason copper (Cu) is used as electrical wire due

to it being easily ductile because of the fluidity of the electrons on a molecular level. The

element has the electron configuration “electron sea”, which is what allows the

molecules to freely conduct electrical charge. This property is utilized in electronic

products, transportation equipment, pigments, and medical processes. Copper (Cu) is

also used for chemical purposes including fungicides, wood preservations, anti-fouling

compounds, and chemical reactants. The uniqueness of copper lies within its ductility,

malleability, and high melting point which allows great electrical conductivity.

Boyd – Reno 38

Appendix A: Construction of Calorimeter

Materials:

7 inch piece of 0.75” inner diameter PVC pipe(1) 0.75” PVC pipe cap(1) 0.75” PVC pipe threaded cap female adapter(1) 0.75” PVC pipe threaded cap male adapterPVC pipe gluePVC pipe primer5.75 inch piece of 0.75” foam pipe insulationPlastic water bottleDrill0.25” drill bit

Procedure:

1. Prime both ends of PVC pipe, inside of pipe cap, and inside of threaded cap

female adapter.

2. Glue both ends of PVC pipe, inside of pipe cap, and inside of threaded cap

female adapter.

3. Secure caps to opposite ends of PCV pipe and hold for 30 seconds.

4. Wrap foam pipe insulation around PVC pipe and secure by pealing plastic and

sticking ends together.

5. Using 0.25” drill bit, drill hole in top of threaded cap male adapter.

6. Screw threaded cap male adapter into threaded cap female adapter.

7. Cut top 1/3 off plastic water bottle.

8. Place calorimeter into plastic water bottle as a stand and duct tape to hold.

Boyd – Reno 39

Diagram:

Figure 10.Calorimeter One

Figure 10 shows the first calorimeter constructed. It is 23 cm tall and 5.5 cm

wide, large enough to contain either of the copper rods and unknown metal rods. The

cap screws onto the calorimeter tightly to conceal all water and heat that may rise

during trials. The pipe insulation hugs the body of the PCV pipe to better contain the

heat of the water inside. The calorimeter stands by a plastic water bottle and held to it

with duct tape. A hole in the cap was just large enough for the temperature probe to be

inserted. Two other identical calorimeters were used during the execution of the trials.

All sufficed for the experiment.

Boyd – Reno 40

Appendix B: LabQuest

Materials:

LabQuestLabQuest AppTemperature Probe

Procedure:

1. Connect the Temperature Probe to LabQuest.

2. Place the Temperature Probe into the calorimeter.

3. Press the power button on LabQuest to turn it on. Choose New from the File

menu. If you have an older sensor that does not auto-ID, manually set up the

sensor by choosing Sensor Setup from the Sensors menu.

4. On the Meter screen, select Rate. Change the data-collection rate to 0.5

samples/second (interval of 2 seconds/sample) and the data-collection length to

180 seconds (3 minutes). Select OK.

5. Data collection is ready to begin.

a. Start data collection.

b. After about 20 seconds have elapsed, gently stir the water in the calorimeter.

c. A real-time graph of temperature vs. time will be displayed on the screen during

data collection.

d. Temperature readings (in °C) can also be monitored in a display box to the right

of the graph.

6. Dispose of the water down the sink.

Boyd – Reno 41

7. When data collection is complete, a graph of temperature vs. time will be

displayed. To examine the data pairs on the displayed graph, select any data

point. As you select each data point, its temperature and time values are

displayed in the display box to the right of the graph.

8. The data may be stored when selecting the File Cabinet icon. Save the file using

the following format (last name_last name_Exp1) and make note of which

LabQuest was used for future experiments.

Boyd – Reno 42

Appendix C: Formulas & Equations

Linear Thermal Expansion Formula:

∆ L=αL∆T

ΔL is the change in length of the metal rod (L f −Li). The alpha coefficient α is

the variable that is being solved for. Each alpha coefficient is specific to each element.

L, the initial length of metal rod in mm, is not to be confused with ΔL. It is the length of

the rod before participating in its trial. ΔT, similar to that of ΔL, is the change in

temperature of the metal rod (T f – T i ) in Celsius.

∆ L=αL∆T

0.05=α∗124.24mm∗74.4mm

α = 5.40923E-06 °C-1

Figure 11. Sample Thermal Expansion Calculation

In Figure 11, the substitutions for the linear thermal expansion formula are shown

in order to find the alpha coefficient. The linear thermal expansion formula can also be

manipulated. An example of this is shown below:

α= ∆ LL∗∆T

The substitutions are plugged into the equation the same way in either equation used.

Boyd – Reno 43

Two - Sample T Test Equation:

To conduct the two-sample t test, the assumptions must be checked. The sample

populations were Simple Random Samples, randomized by the TI-Nspire calculator.

Since the number of trials was less than 30, the distribution was assumed to be normal

and independent of each other. All of these assumptions were met.

Below is the equation used to calculate the t-value of the data.

t=x̄1− x̄2

√ s12

n1+s2

2

n2

In the equation, x ̄ represents the sample mean. The one and two correspond to the

known metal and unknown metal rods, 1 being known and 2 being unknown. S is the

calculated standard deviation, with 1 and 2 corresponding to the known and unknown.

The same applies to the number of trials, n.

t=

x̄1− x̄2

√ s12

n1+s2

2

n2

t= 0.389−0.533

√ 0.0062

15+ 0.0322

15

t=−17.1299

Figure 12. Sample T-Value Calculation

In Figure 12, the sample means, the sample standard deviations, and the

number of trials are placed into the two-sample t test equation. These numbers are

Boyd – Reno 44

values from the specific heat trials. With this, the t-value can be interpreted, and the p-

value can be found. These can then be interpreted as either significant or not

significant.

Specific Heat Formula:

Specific heat is found by using the equation below:

q=sm∆T

ΔT is the change in temperature (T f−T i) in Celsius, identically to that of the linear

thermal expansion coefficient equation. The mass water being used in the experiment in

grams is needed as well as the enthalpy, heat in the current state of the material being

experimented on, in KJ/mol.

smΔT=smΔT

4.184 gJ∗°C

∗60 g∗3mm=s∗28.25∗−68.9mm

0.384 gJ∗° C

=s

Figure 13. Sample Specific Heat Calculation

To find the specific heat of the unknown metal rod, the specific heat equation of

water must be set equal to that of the unknown metal rods or the copper rods (Cu) as

shown in Figure 13. The equation shows an example from the trials. The equation is

simplified and s is found.

Boyd – Reno 45

Percent Error Formula:

To check that the calculated values are correct, the percent error formula is used.

Using the correct value of the targeted value and the value calculated, the percent error

will give a percentage that describes how much error there is between the theoretical

value and the experimental value that was calculated.

percent error= theoretical value−experimental valuetheoretical value

∗100

percent error=( 0.386−0.3840.386

)∗100

percent error=0.5181%

Figure 14. Sample Percent Error Calculation

The percent error helps to make sure that the experiment is running smoothly. It

tells how correctly the experimental value is compared to the value that is being aimed

towards. Figure 14 is sample from one of the trials of the copper (Cu) rods. The specific

heat of the trials is subtracted from the goal value and then divided by the goal value.

When multiplied by 100, the percentage of inaccuracy is produced.

Boyd – Reno 46

Works Cited

Nave, R. "Specific Heat." Thermodynamics. HyperPhysics. Web. 24 Mar 2013.

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

Winter, Mark. "Copper: historical information." WebElements. The University of

Sheffield, n.d. Web. 24 Mar 2013. <http://www.webelements.com/copper/history.html

Brown, Doc. "Mining of Minerals and Methods of Extracting of Metals ." Doc Brown's

GCSE/IGCSE O Level KS4 science–CHEMISTRY Revision Notes. Dr W P Brown, n.d.

Web. 24 Mar 2013. <http://www.docbrown.info/page04/Mextractc.htm>.

"copper (Cu)." Encyclopedia Britannica: Facts Matter. 2013: n. page. Print.

<http://www.britannica.com/EBchecked/topic/136683/copper-Cu>.

Butts, Allison. "0 COPPER." New York, 1954. xii 936 pp. Illustrated. 16 x 23.5 cm..

American Chemical Society Monograph.122 n. page. Print.

Young, Jay A. "CLIP, Chemical Laboratory Information Profile: Copper." Journal of

Chemical Education. 83.10 (2006): n. page. Print.

Bindel, Thomas H., and John C. Fochi. "Guided Discovery: Law of Specific Heats."

Journal of Chemical Education. 74.8 (1997): 955-57. Web. 26 Mar. 2013.

McCullough, Thomas. "A Specific Heat Analogy." Journal of Chemical Education. 896.

Web. 26 Mar. 2013.

"Calorimetry: Specific Heat Capacity of Copper." physics. N.p.. Web. 24 Mar 2013.

<http://www.physics.unr.edu/181labs/Measuring the Coefficient of Linear Expansion for

Copper, St.pdf>.

Boyd – Reno 47

"Measurement of Specific Heat." acs.org. N.p.. Web. 24 Mar 2013.

<http://pubs.acs.org/doi/pdf/10.1021/ed074p955>.

Paul A., Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1

(6th ed.). New York, NY: Worth Publishers. pp. 666–670. Print.

"Measuring the coefficient of Linear Expansion of Copper, Steel, and Aluminum."

physics.edu. N.p.. Web. 28 Mar 2013.

"Thermal Expansion - Linear." The Engineering Tool Box. N.p.. Web. 28 Mar 2013.

<http://www.engineeringtoolbox.com/linear-thermal-expansion-d_1379.html>.

Gibbs, Keith. "Thermal expansion - demonstration experiments." schoolphysics.

schoolphysics, n.d. Web. 6 Apr 2013.