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Identifying a Metal Using Specific Heat and Linear Thermal ExpansionDanny Havern and Andrew Rouditchenko Macomb Mathematics Science Technology CenterChemistry – 10BMrs. Hilliard, Mr. Supal, Mrs. DeweyMay 20, 2013Table of ContentsIntroduction2Background4Review of Literature: Specific Heat6Review of Literature: Linear Thermal Expansion8Problem Statement10Experimental Design11Data and Observations15Data Analysis and Interpretation25Conclusion39Appendix A42Appendix B43Appendix C44Appendix D46Appendix E48Works Cited51IntroductionExperimentation on metals is conducted daily to discover new and unique properties that could be used to benefit society. If two similar metals were mixed up in a laboratory, how would the scientists determine which metal was which? Using the procedures from this experiment, the scientists would be able to confidently identify the metals. The purpose of this experiment was to determine if an unknown metal was the same or different compared to another metal using the intensive properties of specific heat and linear thermal expansion. The known metal was correctly identified from a previous experiment as Vanadium using the property of density. In this experiment, the researchers correctly observed that two unknown metal rods were not the same as the Vanadium rods. The metals were compared by calculating intensive properties, or properties that do not depend on the sample size, of the metals. The specific heats and linear thermal expansion coefficients were found using various tools. The researchers had limited background knowledge on the subject, therefore extensive research was performed. This knowledge was used to design a procedure for an experiment that would provide accurate results. The experiment for specific heat involved observing several changes of temperatures including for the metals and water while the experiment for linear thermal expansion involved observing the change in length of the metal rods and the change in temperature. For specific heat, the researchers used calorimeters that were designed and built specifically for the dimensions of the metal rods. The calorimeters were built from PVC piping, which is an excellent insulator. Computer software was used to analyze the change in temperatures and calculate the specific heat. The most important tools used for the linear thermal expansion procedure were the expansion jigs. These jigs were built in previous years by a skilled professional. These jigs measured the miniscule change of length of the metal rods as they cooled. The researchers used statistical tools to analyze the data, including two sample t-tests and percent error. The experimental averages were compared to the known values for Vanadium. The data proved to be precise as the researchers made an accurate conclusion. Vanadium Metal BackgroundVanadium, V, is a greyish white metal which is very hard, yet ductile. It was first discovered in 1801 by Andrés Manuel del Rio, a Mexican chemist. He later withdrew his claim, but Vanadium was rediscovered in 1830 by the Swedish chemist Gabriel Sefstr?m. The metal was isolated and made pure in 1867 by Sir Henry Enfield Roscoe by reducing Vanadium chloride with hydrogen (Gerhartz).Mining for titanomagnetite ore is the first step in the process of extracting the element to its pure state. Although this mineral only contains about 1.5 percent Vanadium, it is the most common mineral used for the production of Vanadium. Vanadium is in such a small amount in the mineral, it is not included in the formula, as seen below. It yields only 0.24 percent Vanadium from the total extracted materials (Carlson).Mg3[Si2O5](OH)4(s)+(MgAlFe)3[Si2O5](OH)4(s)→ (MgFe)2SiO4(s)+2MgOAl2O3(s)+5SiO2(s)Titanomagnetite ore is reduced in flaming kilns and later melted in a furnace. Olivine, (Mg,Fe)2SiO4, and Cordierite, 2MgOAl2O3+5SiO2 are produced along with a slag of titanium and pig iron, the product of smelting iron ore with high-carbon fuel, with high Vanadium content (Gerhartz). The separated molten pig iron is then blown with oxygen to form a new compound which contains 12-24 percent Vanadium pentoxide (V2O5) (Processing). Pure Vanadium is produced by reducing Vanadium pentoxide with aluminum powder.Vanadium has many practical uses, but 99 percent of all Vanadium produced worldwide was for use as a metal alloy (Vanadium). Pure Vanadium and Vanadium oxides are combined with steel and titanium respectively in order to increase their toughness, ductility, and strength. Vanadium has properties not unlike most other metals. Vanadium’s elemental symbol is V and its atomic number is 23. Vanadium has an atomic weight of 50.942 amu and a density of 6.1g/cm3, which is semi-dense. The specific heat of Vanadium is 0.485J/mol and the thermal expansion coefficient of 8.4 x?10-6?K-1. The boiling point is 3653.15 K (O’Leary). Vanadium is fairly similar to other metals. Compared to water, the values for each property are very different and are not within the same range.19050109664500Vanadium is shaped just like every atom. Protons and neutrons build the nucleus while electrons surround it in the electron cloud. Electrons fill each orbital up to the 3d orbital, as seen below (Greenwood). -88265170180Figure 1. Vanadium Electron Orbital Diagram020000Figure 1. Vanadium Electron Orbital Diagram Electron structure is relevant to this project because electrons deal with energy. The electrons shifting between orbitals require energy, which in turn can affect specific heat, and the size of the orbital directly affects thermal expansion because a larger atomic radius results in expansion of the material (Greenwood).Vanadium’s use as an alloy allows materials to increase strength while decrease weight (Vanadium). Essentially, this characteristic makes Vanadium useful and practical for making products such as buildings and automobiles.Review of Literature: Specific HeatHeat is the transfer of thermal energy (“Experiment 2”). At constant pressure, this transfer is equal to enthalpy, the flow of heat energy (“Enthalpy”). As heat energy flows in or out of a system to its surroundings, the temperature changes, which is a measure of thermal energy. According to the kinetic molecular theory, the kinetic energy of the molecules is directly proportional to absolute temperature, therefore, as heat energy is added to a substance, the kinetic energy of the molecules increases (“Kinetic-Molecular Theory”). If the enthalpy is negative, the reaction is exothermic; heat energy is lost from the system and is transferred to the surroundings (“Enthalpy”). If the enthalpy is positive, the reaction is endothermic; heat energy is lost from the surroundings and is transferred to the system. The relation between the change in heat energy and the change in temperature is the specific heat. Specific heat is an intensive property which means that the property is not dependent on sample size. Specific heat is the heat energy required to raise the temperature of one gram of a substance by one Kelvin; the unit of measure is J/g*K (Schreck). The relationship between heat energy and specific heat is that the heat energy (Q) in J is equal to the mass of the substance (m) in grams times the specific heat (c) in J/g*K times the change in temperature ?T in Kelvin (“Experiment 2”). Q=mc?TSpecific heat is important in the design of materials and products in modern industry. Extensive research is done to determine the right metal to use in products like heat exchangers and piping as they have different specific heats and some insulate heat better than others (Violeta). Specific heat was a useful property in this experiment because each element has a unique specific heat (“Experiment 2”). The specific heats of both metals were determined using calorimetry, which is the process of measuring the enthalpy change during a reaction (“Calorimetry”). A calorimeter is a tool that involves an isolated system. In an isolated system, no energy or mass can be transferred to the surroundings. One experiment that the researchers found was to find the specific heat of aluminum using a Styrofoam cup to simulate the calorimeter as it insulates heat fairly well (“Experiment 2”). The metal was massed and then heated to 100° C in boiling water. The mass and initial temperature of the water was also recorded. The metal was placed into the cup full of water and the temperature was recorded when equilibrium was reached, when the water and the metal reached the same temperature and the graph of the temperature of the water reached a stable plateau. According to the First Law of Thermodynamics which states that energy is neither created nor destroyed but only converted from one form to another, the heat energy lost by the hotter substance was equal to the energy gained by the colder substance (“Kinetic-Molecular Theory”). The recorded data was then manipulated in the specific heat formula to find the specific heat of the metal. In another experiment, various metals were tested to find their specific heats (Schreck). The calorimeter was an insulated thermos jar which may insulate heat better than a cup. The other elements of the experiment were very similar to others, the specific heat was found by placing the metal into water and observing the masses and temperatures. Both experiments were simple in design and easy to replicate.Review of Literature: Linear Thermal ExpansionThe property of linear thermal expansion refers to the tendency of a metal to expand in length when heated. With small temperature changes, the thermal expansion of regular linear objects is proportional to the change in temperature (Furrer). In a system, the thermal energy increase within the atoms results in an increase in atomic radius of those atoms, and thus a larger distance between the atoms that is signified by an increase in the dimensions of the material (Sutara). Thermal energy increase also results in more kinetic energy of the atoms. Larger volume is a result of the atoms bouncing off of each other at a faster rate. These materials expand volumetrically, but linear thermal expansion refers to the length. This intensive property was used to determine the identity of the unknown metal because each element expands and contracts at different rates, therefore making linear thermal expansion applicable (Nave). Once the experimenters compared their experimental linear thermal expansion coefficient to an accepted value and calculated percent error, they were able to determine the identity of the metal. The linear coefficient of thermal expansion (α) describes a change in length of the metal per degree temperature change (Davis). To calculate the linear thermal expansion coefficient, the experimenters used the equation where the change in length, ΔL, of the metal rod is equal to the expansion coefficient, α, times the initial length, Li, times the change in temperature, ?T (Linear).?L= α?Li??TThe experimenters calculated the coefficient by manipulating the equation above and substituting in their values recorded during the experiment. The linear thermal expansion coefficient is expressed in units defined as the reciprocal temperature, ?C-1. Another common expression of the numerical coefficient is in terms of 10-6/℃ (Gale). There are multiple linear thermal expansion labs, but this experiment was a simple one. Some colleges and universities use steam generators and a thermal expansion apparatus, but they do have similar parts of the procedure. The procedure was applied to more than one rod and it was repeated several times to eliminate outliers (Linear). The length temperature of the metal rod was recorded. It was then placed in boiling water. Once the rod reached equilibrium with the temperature of the water, the temperature was recorded and the rod was taken out and the length was quickly measured (Sutara). Using this data, the experimenters calculated the linear thermal expansion coefficient of their metal.Linear thermal expansion has applications in many fields, but it is most often used in engineering. Designing bridges, buildings, and aircraft or spacecraft requires the science of thermal expansion. The expanding and contracting of the metal that makes these objects may cause some serious problems if not taken into account. Bridges have expansion joints which allow the bridge to expand and contract according to the temperature without collapsing (Gale). Rebar used in buildings sidewalks could expand or contract at a greater rate than the concrete and result in damage (Sutara). A simple application of linear thermal expansion would be in the kitchen with a jar. If the lid was on a jar, running it under hot water allowed the lid to expand, therefore making it easier to remove (Nave). Essentially, linear thermal expansion is used in a multitude of fields.Problem Statement Problem Statement:Can the material properties of specific heat and linear thermal expansion be used to correctly identify an unknown metal as Vanadium? Hypothesis:The experimental data will provide an approximate value of specific heat and the linear thermal expansion coefficient with which the experimenters will be able to correctly identify the unknown metal as Vanadium with one percent error.Data measured:Specific Heat was measured in J/g°C. To calculate specific heat, the mass of the rod and the mass of water was measured in grams, and the initial and final temperatures of both the rod and water were measured in degrees Celsius. Also, the specific heat of water, 4.184 J/g°C, was used in the calculation of the specific heat value.Linear Thermal Expansion Coefficient was measured in 10-6 °C-1. To calculate the linear thermal expansion coefficient, the original and final lengths of the metal rods were measured in millimeters. The initial and final temperatures of the rods were measured in degrees Celsius. Specific Heat Experimental DesignMaterials:Logger ProLogger Pro thermometer probe, 0.1?C Digital thermometer, 0.1?C TI-nspire CX graphing calculatorElectronic timerCalorimeter (2) Unknown metal rods(2) Vanadium, V, rod(2) 20.3 cm x 9.8 cm x 6.3 cm loaf panHotplateTongsScout Pro electronic scale, 0.1g100 ml graduated cylinder300 ml BeakerWork GloveProcedure:1. Use the TI-nspire CX graphing calculator to randomize the order of the trials and the order of the rods. See Appendix A for directions on how to randomize. Make sure to assign fifteen trials to the known metal and fifteen trials to the unknown metal.2. Turn on and set up Logger Pro. Plug in Logger Pro thermometer probe and adjust data gathering information. See Appendix B for further instructions.3. Using the 300 ml beaker, pour 150 ml of water into the loaf pan and set it on the hot plate. Turn the hot plate on and place the second loaf pan on top of the first as a lid.4. Fill the graduated cylinder with 50 ml of water. Record the mass of the water as 50 g.5.Pour this amount of water into each calorimeter. See Appendix C for instructions on building the calorimeter.6. Insert the Logger Pro thermometer probe through the hole in the lid of the calorimeter and place the thermometer in the water of the calorimeter. 7. Mass the metal rod using the scale and record.8. Lift the lid of the loaf pan with the work glove. Insert the digital thermometer into the beaker. Continue to boil the water until it reaches a temperature of about 100℃. 9. Insert the rod(s) into the boiling water. Make sure the entire rod is submerged in the water and begin the electronic timer.10. After two and a half minutes, stop the timer. Assume that the temperature of the water is the same as the temperature of the metal. Insert the digital thermometer into the boiling water around the rod or between the two rods. Record this temperature as the initial temperature of the metals.11. Begin collecting temperature measurements of the water in the calorimeter using the Logger Pro temperature probe. 12. Use the tongs to carefully remove the metal from the boiling water and place it in the calorimeter. Quickly, attach the top of the calorimeter. The Logger Pro temperature probe should be in the calorimeter through the hole in the lid.11. Data collection should stop after three minutes if the Logger Pro was set up correctly. Save data into the appropriate file and record results into the data table.12. Remove the cap and empty out the calorimeter. Start a new set of data collection on the Logger Pro by selecting the File Cabinet icon in the top right corner of the screen.13. Repeat steps 3 through 12 for each trial.3698875288290004701540151765TI-nspire Calculator4000020000TI-nspire CalculatorDiagram:659130177800000-533401631315Work Gloves020000Work Gloves265239510172700026523951432560004747895909320Digital Thermometer4000020000Digital Thermometer47986951327785Tongs4000020000Tongs2911475225298000369887517780000036988756578600027730453879850048342552252980Metal Rods020000Metal Rods-806452383790Lab Quest020000Lab Quest7772402062480006591301432560007772401017270005759456578600087312524320500-49530497840Beaker4000020000Beaker-49530868045Hot Plate020000Hot Plate4705350238125Thermometer Probes4000020000Thermometer Probes-177800116840Calorimeters4000020000Calorimeters4703445531495Loaf Pans4000020000Loaf Pans47929801778000Scale020000Scale908051279525Timer020000TimerFigure 2. Specific Heat MaterialsFigure 2 above shows most of the materials used in the specific heat experiment. Not pictured is the graduated cylinder. Linear Thermal Expansion Experimental DesignMaterials:Digital thermometer 0.1?C TI-nspire CX graphing calculatorElectronic timerLinear thermal expansion jig(2) Unknown metal rods(2) Vanadium, V, rods (2) 20.3 cm x 9.8 cm x 6.3 cm loaf panHotplateTongs300 ml BeakerThick glovesDigital Calipers 0.1 mm50 ml Spray BottleProcedure:1. Use the TI-nspire CX graphing calculator to randomize the order of the trials and the order of the rods. See Appendix A for directions on how to randomize. Make sure to assign fifteen trials to the known metal and fifteen trials to the unknown metal.2. Using the 300 ml beaker, pour 150 ml of water into the loaf pan and set it on the hot plate. Turn the hot plate on and set the second loaf pan on top of the loaf pan with the boiling water to act as a lid. Boil the water until it reaches a temperature of about 100℃.3. Measure the length of the metal rod using the digital calipers and record as the initial length. 4. Using the tongs, insert the rod into the boiling water. Make sure the entire rod is submerged in the water and begin the electronic timer.5. After two and a half minutes, stop the timer. Assume that the temperature of the metal is equal to the temperature of the water. Lift the top loaf pan using the work glove. Insert the digital thermometer into the loaf pan. Record this as the initial temperature.6. See Appendix D for information on how to use the linear thermal expansion jig. Pull back the pin in the jig to allow the metal to be placed in the jig. 6. Carefully remove the rod from the boiling water using the work glove and tongs and place the rod in the thermal expansion jig. Start the timer and move the needle of the gauge to the starting position. 7. Use the spray bottle filled with water to speed up the cooling process. After three minutes, stop the timer and record the change in length.8. Use the digital thermometer to measure the temperature of each metal and record it as the final temperature.9. Repeat steps two through nine for each trial.Diagram:475678525400Digital Thermometer4000020000Digital Thermometer457835284099000-1104902687955Towel020000Towel265239525723850049885602687955Expansion Jigs020000Expansion Jigs414718520021550052451002002155Scale020000Scale329565013074650052451001206500Tongs4000020000Tongs3948430788670004988560645160Loaf Pans4000020000Loaf Pans237172514986000299085189484000-2692401741805Timer020000Timer557530120650000-2692401061085Hot Plate020000Hot Plate-215900506730Beaker4000020000Beaker4095756667500057594527622500-499110149860Spray Bottle4000020000Spray BottleFigure 3. Linear Thermal Expansion MaterialsFigure 3 above shows the materials used on the linear thermal expansion experiment. Not included in the picture are the metal rods and the work gloves. An image of these materials can be seen in Figure 2. The picture contains a towel that is not included in the materials because it is not mandatory, but it was used to keep the work area dry during the experiment.Data and ObservationsData:Table 1Vanadium Specific Heat Experiment DataTrialRodInitial Temp. (°C )Equilibrium Temp. (°C) Change in Temp. (°C )Mass(g)Specific Heat (J/g°C)Metal WaterMetal WaterMetal Water1B98.026.831.0-67.04.2025.50500.5142B99.022.626.3-72.73.7025.50500.4183A99.324.629.6-69.75.0025.50500.5894A99.528.333.0-66.54.7025.50500.5805B99.530.633.9-65.63.3025.50500.4136A98.530.735.1-63.44.4025.50500.5697A99.327.431.1-68.23.7025.50500.4458B98.528.932.7-65.83.8025.50500.4749A99.327.831.5-67.83.7025.50500.44810B99.323.727.0-72.33.3025.50500.37411B99.331.133.8-65.52.7025.50500.33812A100.026.029.5-70.53.5025.50500.40713A100.027.734.2-65.86.5025.50500.81014B100.020.525.2-74.84.7025.50500.51515B100.026.229.6-70.43.4025.50500.396Avg.?99.326.930.9-68.44.0025.50500.485Table 1 above shows the data recorded during the trials for the specific heat of the Vanadium rods. Most trials were performed two at a time as indicated in the observations tables. The change in temp represents the change of temperature for both the metal and the water. All temperature measuring devices had three significant figures. The mass of the water has only one significant figure because the measuring device was a graduated cylinder. The specific heat was calculated with three significant figures after the decimal. For a sample calculation, see Appendix D. Table 2Unknown Metal Specific Heat Experiment DataTrialRodInitial Temp. (°C )Equilibrium Temp. (°C) Change in Temp. (°C )Mass(g)Specific Heat (J/g°C)Metal WaterMetal WaterMetal Water1B99.022.626.3-72.73.7024.50500.4352A99.022.626.1-72.93.5024.50500.4103B99.020.423.9-75.13.5024.50500.3984B99.317.020.8-78.53.8024.50500.4135A99.019.923.8-75.23.9024.50500.4436A99.527.730.7-68.83.0024.50500.3727B99.521.525.5-74.04.0024.60500.4608B99.426.030.4-69.04.4024.60500.5429B99.421.224.8-74.63.6024.60500.41010A99.519.223.3-76.24.1024.50500.45911A99.426.629.9-69.53.3024.50500.40512B99.923.727.1-72.83.4024.60500.39713A99.419.623.3-76.13.7024.50500.41514A99.918.022.0-77.94.0024.50500.43815A99.316.821.0-78.34.2024.50500.458Avg.?99.421.525.3-74.13.7024.53500.430Table 2 above shows the data recorded during the trials for the specific heat of the unknown metal rods. Most trials were performed two at a time as indicated in the observations tables. The change in temp represents the change of temperature for both the metal and the water. All temperature measuring devices had three significant figures. The mass of the water has only one significant figure because the measuring device was a graduated cylinder. The specific heat was calculated with three significant figures after the decimal. For a sample calculation, see Appendix D.Table 3Vanadium Linear Thermal Expansion DataTrialRodOriginal Length (mm)Change in Length(mm)Initial Temp. (°C )Final Temp. (°C )Change in Temp. (°C)Alpha Coefficient (10-6°C-1)1B127.310.0598.730.168.65.7252A127.390.0699.329.969.46.2213B127.340.0599.330.568.85.7074A127.410.0498.327.770.64.4475B127.290.0498.928.470.54.4576B127.370.0598.929.269.75.6327A127.360.0498.928.470.54.4558B127.280.0598.630.468.25.7609B127.370.0598.631.567.15.85010A127.440.0598.929.269.75.62911A127.330.0598.629.569.15.68312A127.40.0598.630.668.05.77213B127.370.0699.029.769.36.23114A127.440.0699.030.069.06.25515A127.370.0698.730.168.66.295Avg.?127.360.0598.829.769.15.608Table 3 above shows the data recorded during the trials for the linear thermal expansion coefficient of the Vanadium rods. Most trials were performed two at a time as indicated in the observations tables. The change in length represents the change of length of the metal as it cooled down in the jig after being heated. The dial on the jig was accurate to two significant figures. The linear thermal expansion coefficient was calculated with three significant figures after the decimal. For a sample calculation, see Appendix D.Table 4Vanadium Linear Thermal Expansion DataTrialRodOriginal Length (mm)Change in Length(mm)Initial Temp. (°C )Final Temp. (°C )Change in Temp. (°C)Alpha Coefficient (10-6°C-1)1A128.280.07100.230.270.07.7952B128.810.07100.228.571.77.5793B128.880.07101.026.574.57.2904A128.260.08100.426.274.28.4065B128.770.08100.429.770.78.7876A128.400.08100.429.171.38.7387A128.300.08100.828.172.78.5778B128.870.07100.828.572.37.5139B128.790.07100.428.272.27.52810A128.280.08100.426.973.58.48511B128.810.08100.526.374.28.37012A128.310.08100.526.074.58.36913B128.760.07100.225.175.17.23914A128.290.08100.224.575.78.23815A128.350.08101.026.174.98.322Avg.?128.540.08100.527.373.28.082Table 4 above shows the data recorded during the trials for the linear thermal expansion coefficient of the unknown metal rods. Most trials were performed two at a time as indicated in the observations tables. The change in length represents the change of length of the metal as it cooled down in the jig after being heated. The dial on the jig was accurate to two significant figures. The linear thermal expansion coefficient was calculated with three significant figures after the decimal. For a sample calculation, see Appendix D.ObservationsTable 5Vanadium Specific Heat ObservationsTrialRodDateCal.Observations1B15-Apr1First day of experimenting. Pre trials were run and the procedures were adjusted. 2B17-Apr1Both researchers were present. This trial was run by itself. The window was open, and there was cold air blowing in making the room very chilly. The researchers forgot to mass the metal rod and it was massed after being removed from the calorimeter.3A17-Apr2The researchers forgot to mass the rod again. It was once again massed after being removed from the calorimeter. The water from the loaf pan was emptied.4A17-Apr1The researches decided to start putting two metals in the loaf pan during the heating process due to time constraints. This trial was done with trial five.5B17-Apr2The Vanadium rod was dropped on the floor. The water in the loaf pan was emptied and refilled. 6A17-Apr1This trial was performed along with trial eight. The temperature of the water deviated slightly on the thermometer about 0.1 – 0.3°C. 7A17-Apr1This trial was performed along with trial 10. The Vanadium rod dropped in and out of the water two times in the loaf pan while attempting to pick it up using the tongs. The temperatures of the water in the calorimeters varied. 8B17-Apr2The water in the loaf pan was emptied and refilled after removing the metal rods from it. The amount was slightly more than 150 ml.9A17-Apr1This trial was performed with trial 11. The researchers had difficulty placing the metal rod into the calorimeter. 10B17-Apr2The water in the loaf pan was emptied and refilled after removing the metal rods from it. 11B17-Apr2The researchers began to transfer the metal rods slightly late and this rod spent more time than usual in the loaf pan.12A17-Apr1This trial was performed with trial 14. The researchers began to transfer the metal right on time. 13A17-Apr1The metal rod slipped from the tongs while being transferred to the calorimeter and touched one of the researcher's hands. The calorimeter was spilled and this trial had to be redone. The redone trial was performed as usual.14B17-Apr2The transfer of the metal to the calorimeter from the loaf pan was very smooth. The water in the loaf pan was emptied and refilled after removing the metal rods from it. 15B17-Apr2This was the final trial of the day. The water poured into the calorimeter was slightly more than 50 ml. After this trial the researchers began to clean up.Table 5 shows the observations recorded during the trials for the specific heat of the Vanadium rods. The table indicates observations for each trial, the date, which rod and calorimeter were being used, and which trials were done together. All trials for the A rod were done with calorimeter 1and all trials for the B rod were done with calorimeter 2.Table 6Unknown Metal Specific Heat ObservationsTrialRodDateCal.Observations1B18-Apr2Both researchers were present during the second day of trials. This trial was performed along with trial two. The water in the loaf pan spent a longer duration of time than usual on the hot plate. The wrong data collection method was set up on logger pro, but the data was still valid. The researches forgot to measure the initial temperature of the metal; however they measured the temperature of the water in the loaf pan after removing the metal rods from it.2A18-Apr1The metal rod was dropped before being placed in calorimeter. The water in the loaf pan was emptied and refilled after removing the metal rods from it. 3B18-Apr2This trial was performed with trial five. This trial was redone as the original value gave a large percent error. The metal rods were touching in the loaf pan while being heated. 4B18-Apr2This trial was performed with six. The metal rod touched the side of the calorimeter before being dropped in it.5A18-Apr1The metal wiggled around in the tongs while being transferred from the loaf pan to the calorimeter and the process took more time than usual. 6A18-Apr1The metal spent a longer time in the loaf pan because the transfer of rod A took longer than usual.7B18-Apr2This trial was performed with trial 10. The researcher had difficulty placing the top of the loaf pan on. The metal wobbled in the tongs as it was being transferred.8B18-Apr2This trial was performed with trial 11. The metal dropped on the floor before boiling. A few seconds went by as the rods were out of the boiling water and placed into the calorimeters.9B18-Apr2This trial was performed with trial 13. The metal was dropped while attempting to put in the calorimeter.10A18-Apr1The metal touched the outside edge of the calorimeter while being transferred.11A18-Apr1The researcher had difficulty placing the metal in the loaf pan. The position of the rod had to be adjusted. 12B18-Apr2This trial was performed with trial 14. The exteriors of the calorimeters were slightly warm; this may have been from the heat of the hot plate.13A18-Apr1The window in the experimenting area was opened.14A18-Apr1The metal was removed from the loaf pan right on time.15A18-Apr1This trial was done along with a redo of trial 3 as well; the original value gave a large percent error. This was the last trial of the day. Table 6 shows the observations recorded during the trials for the specific heat of the unknown metal rods. The table indicates observations for each trial, the date, which rod and calorimeter were being used, and which trials were done together. All trials for the A rod were done with calorimeter 1and all trials for the B rod were done with calorimeter 2.Table 7Vanadium Linear Thermal Expansion ObservationsTrialRodDateJigObservations1B19-AprS3Both researchers were present during the third day of trials. The linear thermal expansion jigs S1 and S3 were used and they provided a metric measurement. The heat on the hot plate was turned to 10.2A19-AprS1The window was closed and it was slightly humid in the room.3B19-AprS3This trial was performed with trial four. The length of the rod was measured on the rag as opposed to the face of the table. The metal was removed from the loaf pan slightly later than usual, at about three minutes. 4A19-AprS1The cooling time of the metal was cut short to about two minutes. The water in the loaf pan was refilled.5B19-AprS3This trial was performed with trial seven. The length of the metal was taken twice as the first measurement was an odd value. The calipers were used on the table. 6B19-AprS3This trial was performed with trial 10. The length was measured on the table. The orientation of the metals was switched when they were placed in loaf pan. 7A19-AprS1The transfer of the metals began slightly late at about three minutes. The temperature of the spraying water was 29 ?C; cold water was added to drop it to 25 ?C. The water in the loaf pan was emptied and refilled.8B19-AprS3This trial was performed with trial 11. The metal was measured on the table. The metals were placed correctly in the center of the loaf pan. The water in the loaf pan seemed hotter to the researcher that placed metal in the water. The metal spent about three minutes in the loaf pan.9B19-AprS3This trial was performed with trial 12. It took several attempts to get an accurate measure. The researchers began feel very hot and took a brief break.10A19-AprS1The temperature of the water in the loaf pan was measured slightly earlier than usual. More than three minutes passed as the researchers waited for the metal to change length.11A19-AprS1The temperature reading of the water in the loaf pan started late the metal was placed in the jig after about three minutes. 12A19-AprS1The length of the metal was measured on the table. The water in the spraying bottle was at 28.7? C.13B19-AprS3This trial was performed with trial 13. The rods were touching in the loaf pan. 14A19-AprS1The metal was sprayed with a larger quantity of water than usual to cool it down. The change in length of the metal was greater than usual.15A19-AprS1This trial was performed with trial one. The tips of the rods were touching again in the loaf pan. The rods were dropped crooked in the jigs and then straightened. The transfer of the metals began right on time. This was the last trial of the day.Table 7 shows the observations recorded during the trials for the linear thermal expansion of the Vanadium rods. The table indicates observations for each trial, the date, which rod and jig were being used, and which trials were done together. During these trials, all trials for the A rod were done with jig S1 and all trials for the B rod were done with jig S3.Table 8Vanadium Linear Thermal Expansion ObservationsTrialRodDateJigObservations1A22-AprS1Both researchers were present during the third day of trials. The trial was performed along with trial two. It was noticed that the both metal rods were magnetic to the hot plate. 2B22-AprS3The metals were very hard to remove from the loaf pan using the tongs as they were attracted to the surface of the plate. The metal rod was put in the jig late at about three minutes. The rod slipped from the slot while trying to place it in the jig.3B22-AprB1The jigs were switched with another group, but one had a faulty dial. The dial was replaced and both jigs provided a metric measurement. The transfer of the metal from the loaf pan to the calorimeter began late.4A22-AprA1This trial was performed along with trial three. The water in the loaf pan was refilled before the metals were heated.5B22-AprB1This trial was performed along with trial six. The loaf pan was adjusted on the hot plate, while the rods stuck to the bottom. The researchers began to indicate the original length of the metals with a marker as opposed to the previous method because the dial moved if touched.6A22-AprA1The water in the spray bottle was refilled. The water seemed quite cold. The amount of time that the metal spent in the jig was longer than usual.7A22-AprA1This trial was performed along with trial eight. The water in loaf pans was refilled. The metals were touching while being heated in the loaf pan.8B22-AprB1The metal rod was transferred to the jig late and the cooling time was less than usual. The Metals had to be adjusted when placed into the jigs. 9B22-AprB1This trial was performed along with trial 10. The cooling time was about three and a half minutes as the researchers forgot to pay attention to the stopwatch. The jigs were very wet and the wood was soaked with water.10A22-AprA1The water used in the spray bottle was refilled again. 11B22-AprB1This trial was performed along with trial 12. The water in the loaf pan was refilled and slightly exceeded 150 ml.12A22-AprA1The metal was dropped back into loaf pan while being transferred into the jig. The metals received a heavy dousing of water from the bottle during the cooling process.13B22-AprB1Trial performed with trial 14. The water in the loaf pan was refilled. The researcher had some trouble with the timer and the transfer of the metals started late. 14A22-AprA1The water in the spray bottle was refilled with water at approximately room temperature. The metal had to be adjusted when placed into the jig.15A22-AprA1This trial was performed along with a redo of trial three. The top of the loaf pan put was put on after about 10 seconds from when the metals were placed in the water. This was the final trial of the day.Table 8 shows the observations recorded during the trials for the linear thermal expansion of the unknown metal rods. The table indicates observations for each trial, the date, which rod and jig were being used, and which trials were done together. Data Analysis and InterpretationSeveral statistical graphs and tests were used to analyze the data from the experiment. The data was both quantitative and continuous as the data was numerical and could take any value in a range. The data measured included values for specific heat and linear thermal expansion coefficients of the metals. These values were calculated using data collected from the experiment including temperatures, masses, and lengths of the metal rods. The two sample t- test was used for the analysis. This test was appropriate as two different samples were being compared. The first sample was the data for the Vanadium rods and the second sample was the data for the unknown metal rods. Two t- tests were conducted; one for specific heat and the other for linear thermal expansion.Before the tests were conducted, several assumptions had to be checked. If these assumptions are not met for a two sample t- test, the validity of the results of the test is questionable. The first assumption was that both samples were simple random samples taken from two distinct populations. This condition was met because the trials were randomized. The samples were also independent because the metal rods were placed in separate calorimeters and jigs. Another assumption was that both the means and the standard deviations of the populations were not known. In this experiment, the researchers did not know the true values of these parameters. The final assumption was that both samples were normally distributed. According to the Central Limit Theory, if the simple size is 30 or greater the samples are normal. Unfortunately, each sample size was only 15 trials which meant that the distributions of the samples had to be checked. The purpose of conducting the two sample t- test was to compare two different sample means and check to see whether the null hypothesis was true or not. The alternate hypothesis was checked against a null hypothesis. In this scenario, the null hypothesis was that the known and the unknown metal were the same while the alternate hypothesis was that the metals were different. For the specific heat experiment, the null hypothesis was that the mean specific heat for the known metal, SX1 was the same as the mean specific heat for the unknown metal, SX2. The alternative hypothesis was that the mean specific heats were different. Ho: SX1 = SX2Ha: SX1 ≠ SX2For the linear thermal expansion experiment, the null hypothesis was that the mean linear thermal expansion coefficient for the known metal was the same as the mean linear thermal expansion coefficient for the unknown metal. The alternative hypothesis was that the mean coefficients were different. Ho: LX1 = LX2Ha: LX1 ≠ LX2Before the t-Test was completed, percent error was calculated for each trial in order to check the validity of the results. Percent error was calculated while the experiment was being run in order for the researchers to make sure the results were consistent.Table 9. Vanadium Specific Heat Percent ErrorTrialExperimental ValueTrue ValuePercent Error10.5140.4856.03620.4180.485-13.91130.5890.48521.34440.5800.48519.55250.4130.485-14.90860.5690.48517.39370.4450.485-8.23180.4740.485-2.31390.4480.485-7.689100.3740.485-22.793110.3380.485-30.273120.4070.485-16.023130.8100.48567.096140.5150.4856.286150.3960.485-18.307Avg.0.4860.4850.217Table 9 above shows the percent error data for each trial of the specific heat experiment for Vanadium. The averages are indicated on the bottom of the table. The specific heat of Vanadium was used as the true value. The percent error was then calculated for each trial and shows the percentage of difference between the experimental and true value. See Appendix E for a sample calculation. Negative percent error means that the value for specific heat was lower than the true value. The percent error was constantly being checked during the experiment to make sure that the results were consistent. Any trials that were not consistent with the other trials were redone as indicated in the observations tables. The range of the data was from about 67% to -30%. This large range and variability indicates a flaw or an error in the experimental design. Using the absolute values of each trial’s percent error, the average percent error of this data would have been about 18%. This method was not used because percent error is actually calculated using the true values. The actual average percent error was calculated to be 0.217%. Because this value is less than one percent, it is excellent evidence that the metal is actually Vanadium.Table 10. Unknown metal Specific Heat Percent ErrorTrialExperimental ValueTrue ValuePercent Error10.4350.485-10.39720.4100.485-15.47330.3980.485-17.94940.4130.485-14.77550.4430.485-8.69460.3720.485-23.23170.4600.485-5.22180.5420.48511.81290.4100.485-15.385100.4590.485-5.271110.4050.485-16.404120.3970.485-18.110130.4150.485-14.401140.4380.485-9.598150.4580.485-5.563Avg.0.4300.485-11.244Table 10 above shows the percent error data for each trial of the specific heat experiment for the unknown metal. At the bottom of the table, the averages of the data can be found. The researchers used the specific heat of Vanadium as the true value while calculating percent error. The percent error was calculated for each trial. Negative percent error shows that the value for specific heat was lower than the true value. The range of the data was only from about -23% to about 12%. This range is significantly smaller than the range of the data for Vanadium. The average percent error was calculated to be approximately -11.244%. This value is somewhat far away from the average percent error for Vanadium, suggesting that the metals are different.Table 11. Vanadium Linear Thermal Expansion Coefficient Percent ErrorTrialExperimental ValueTrue ValuePercent Error15.7258.4-31.844%26.2218.4-25.939%35.7078.4-32.058%44.4478.4-47.061%54.4578.4-46.936%65.6328.4-32.951%74.4558.4-46.965%85.7608.4-31.428%95.8508.4-30.353%105.6298.4-32.988%115.6838.4-32.348%125.7728.4-31.291%136.2318.4-25.821%146.2558.4-25.539%156.2958.4-25.064%Avg.5.6088.4-33.239%Table 11 above shows the percent error data for each trial of the linear thermal expansion coefficient for Vanadium. The average results are found at the bottom of the table. The true value is the linear thermal expansion coefficient of Vanadium. The percent error column shows the percent difference between each experimental value compared to the true value. All of these values are negative which means each experimental value was lower than the true value. The range of the data is from about -46% to about -25%. This small range suggests that there is small variability in the data. The average percent error was calculated to be approximately -33.239%. This indicates a flaw in the experimental design. The percent error should have been close to zero, or very little difference like the average of the specific heat trials. Table 12. Unknown Metal Linear Thermal Expansion Coefficient Percent ErrorTrialExperimental ValueTrue ValuePercent Error17.7958.4-7.19727.5798.4-9.77037.2908.4-13.20948.4068.40.07358.7878.44.61168.7388.44.02978.5778.42.10687.5138.4-10.56197.5288.4-10.381108.4858.41.010118.3708.4-0.355128.3698.4-0.369137.2398.4-13.822148.2388.4-1.933158.3228.4-0.932Avg.8.0828.4-3.780 Table 12 above shows the percent error data for each trial of the linear thermal expansion coefficient for the unknown metals. The averages are shown at the bottom of the table. The true value is the linear thermal expansion coefficient of Vanadium. The percent error column shows the percent difference between each experimental value compared to the true value. The range of the data is from about -14% to about 4% which suggest small variability ion the data. The average percent error was calculated to be approximately -3.780%. This average is conflicting because it is closer to the true value than the Vanadium used in the experiment. This is more evidence of an experimental design flaw. The difference between the average coefficients of the two metals is 29.459, which is a significant amount.Figure 4.Normal Probability Plot of Vanadium Specific HeatFigure 4 above is the normal probability plot of the Vanadium specific heat data. The graph suggests that the data is normally distributed because the data points are relatively close to the expected z-value line. The first and last data points stray far from the expected z-value, so they might be considered as outliers. Most of the data is near or on the line, so they data as a whole can be confirmed as normally distributed. Figure 5. Normal Probability Plot of the Unknown Metal Specific HeatFigure 5 above is the normal probability plot of the unknown metal specific heat data. The last data point varies far from the expected z-value line, so it may be considered as an outlier. Overall, the data is fairly close to the expected z-value which suggests that the data of the unknown metal for specific heat is normally distributed.Figure 6. Specific Heat Data Box PlotFigure 6 above is the box plot of the specific heat data. The overlapping of the box advocates evidence that the specific heats of the two metals are the same or very similar. The unknown metal data appears to be precise because it is contained in such a small area or box. The Vanadium data has high variability because it occupies a larger area. The drastic differences between the two boxes, very low and very high variability, suggest that the data may be unreliable. The outlier of the unknown metal data also suggests the data may be skewed. In general, the validity of the data is questionable based on high variability and outliers of the box plot. As stated earlier, the statistical test used to analyze the data of the experiment was a two sample t-Test. To determine whether there was a significant difference between the metals or not, the p-value was calculated. In a two Sample t-Test, the t-value must be calculated in order to find the p-value. The t-value was calculated using the formula where this value is equal to the sample mean of the Vanadium data, EQ \O(x1,ˉ) , minus the sample mean of the unknown metal data, EQ \O(x2,ˉ), divided by the square root of the Vanadium sample standard deviation, s1, squared which is divided by the sample size of the Vanadium data, n1, and this quotient is added to a second quotient of the unknown metal sample standard deviation, s2, squared and divided by the unknown metal sample size, n2. t=( EQ \O(x,ˉ) 1 - EQ \O(x,ˉ) 2 )s12n1+s22n2A sample calculation of this test statistic can be found in Appendix E. Once the t-value is calculated, the p-value is found using either the calculator software or a statistics table, for this experiment a calculator was used. The p-value was then compared to the alpha value to determine significance.Figure 7. Calculator Specific Heat Statistical Test ResultsFigure 7 above is the result of the calculator statistical test of the specific heat data. Each value used in the test statistic has been calculated and defined. The p-value produced from the data is a value of 0.101697, which is a slightly large number. A larger number would support the hypothesis that there is no significant difference between the two metals.Figure 8. P-Value Plot of Specific Heat DataFigure 8 above is a p-value plot of the specific heat data gathered in this experiment. The shaded area of the graph shows the specific heats greater than the t-value of 1.72897 and less than -1.7289. The negative t-value is shown because the alternative hypothesis is not equal to. This area accounts for 10.1697% of the bell curve.With the p-value calculated, the experimenters could review the results of the statistical test. The results led them to fail to reject the null hypothesis, Ho, because the p-value of 0.101697 is greater than the alpha level of 0.10. This means that there is no significant evidence that the specific heat of Vanadium is different from the specific heat of the unknown metal. There is only a 10.1697% chance of getting results as extreme as these by chance alone if the null hypothesis, that the metals had the same average specific heat, were true. Figure 9. Normal Probability Plot of Vanadium Linear Thermal ExpansionFigure 9 above is the normal probability plot of the Vanadium data for the linear thermal expansion portion of the experiment. A large number of the data points vary far from the expected z-value line. This suggests that the data is not very reliable as it does not seem to be normal. With a small number of outliers, they could be taken out of the data set and the tests could be run again, but there are too many here to take out and yield accurately represented data. Overall, the data cannot be very reliable based on its heavy variability. The results of thet-Test might not be conclusive due to the fact that these data points are not normally distributed.Figure 10. Normal Probability Plot of Unknown Metal Linear Thermal ExpansionFigure 10 above is the normal probability plot of the unknown metal linear thermal expansion data. For the most part, the data points lie close to the expected z-value line, which suggest that the data is valid and reliable.Figure 11. Linear Thermal Expansion Data Box PlotFigure 11 above shows a box plot of the linear thermal expansion data, Vanadium is the top box and the unknown metal is the bottom box. The two boxes do not overlap at any measurement which suggests that the linear thermal expansion coefficient is different between the two metals. The unknown metal box is wider which indicates larger variability in the data. A thinner box, such as the Vanadium box, indicates low variability in the data. There is a slight skew of the data in the box plot. The Vanadium data box plot is right skewed which suggests that the sample mean is greater than the median. The unknown metal data box plot is left skewed which suggests that the sample mean will be less than the median. Essentially, this graph shows that the data is reliable because of few outliers and small variability in the data itself.The same test was used to analyze the data of the linear thermal expansion portion of this experiment as was the specific heat portion, a two sample t-Test. All variables in the formula refer to the same values of the data, just using a different set of data. The formula can be found above Figure 7 and a sample calculation of the test statistic can be found in Appendix E.Figure 12. Calculator Linear Thermal Expansion Statistical Test ResultsFigure 12 above shows the calculator statistical test results of the linear thermal expansion data. Each value used to calculate the test statistic, or t-value, was recorded by the calculator. The p-value is so miniscule that it was recorded in scientific notation. The p-value of 6.76291E-12 represents a value of 0.00000000000676291. This very small value will provide significance against the null hypothesis.Figure 13. P-Value Plot of Linear Thermal Expansion DataFigure 13 above shows the p-value plot of the test statistic run for the linear thermal expansion data. The p-value is so small it basically cannot be seen on the graph. The area that is shaded under this region is 0.000000000676291% of all the data.With the p-value calculated, the experimenters could evaluate the results of the statistical test. The results led them to reject the null hypothesis, Ho, because the p-value of 6.76291E-12 is less than the alpha level of 0.10. This means that there is significant evidence that the linear thermal expansion coefficient of Vanadium is different from the linear thermal expansion coefficient of the unknown metal. There is only a 6.76291E-12 % chance of getting results as extreme as these by chance alone if the null hypothesis, that the metals had the same linear thermal expansion coefficient, were true. This result is conflicting because the t-Test for specific heat provided evidence that the metals were the same. ConclusionThe initial objective of this experiment was to determine if the material properties of specific heat and linear thermal expansion could be used to correctly identify an unknown metal as Vanadium. After all of the data collection and analysis, the researchers were finally able to propose a conclusion. The hypothesis that, “The experimental data will provide an approximate value of specific heat and the linear thermal expansion coefficient with which the experimenters will be able to correctly identify the unknown metal as Vanadium with one percent error”, was rejected. The data that supports the rejection of the hypothesis is the data representing percent error. Each of the average percent errors of the data of the unknown metals were above or below the hypothesized value of one percent error. The average percent errors of the known metals were 0.217% for the specific heat and -33.239% for linear thermal expansion. The average percent errors of the unknown metals were -11.244% for specific heat and -3.780% for the linear thermal expansion. Even though the hypothesis was rejected, the experimenters were still able to correctly identify the unknown metal. The researchers observed that the unknown metal was not Vanadium. The researchers concluded that the metals were different based on several observations. The p-value for linear thermal expansion coefficient was extremely small, much less than the alpha value of 0.10. This suggests that the metals were different. The p value for specific heat was only above the alpha level by 0.001697. This p value is extremely close to the alpha level but because it is above the alpha level of 0.10, it suggests that the metals are the same. Although the results were slightly conflicting, the researchers made an important observation during the experimental procedure. While placing the unknown metals in the loaf pan on the hot plate to heat up, the metals were magnetically attracted to the hot plate. This meant that the unknown metal rods were magnetic. The known metal, Vanadium, is not magnetic. This means that the unknown metal is either a magnetic element or an alloy with magnetic properties. Based on all of these observations, the experimenters were certain that the unknown metal was not Vanadium.There were some problems that the experimenters encountered during their research. The experimenters were not familiar with the tools or the procedures used in the experiment before beginning the trials. Also, the tools limited the quality of the researcher’s results. The calorimeters and expansion jigs were built in house and were not exactly laboratory grade. The dial on the expansion jig was not very accurate and was difficult to get a correct reading. Many of the values for temperature were not accurate due to the fact that the procedure was not reproduced exactly the same for each trial. The transfer time for the metals from the loaf pan to the calorimeter or jig varied greatly each time. The amount of time the metals spent in the loaf pan was not the same; sometimes the metals spent almost an extra minute within the pan. The rods were dropped in and out of the pan and the calorimeters. These chronological errors were due mostly to human errors. Because these factors changed often, the data could have been easily skewed. The percent error for the linear thermal expansion for Vanadium was -33.239%. This value should have been close to one which shows that there were flaws in the experimental design. To minimize error, there could have been several improvements to the procedure. The experiment could be done in an isolated environment such as a temperature controlled room without doors and windows being opened. This could minimize confounding variables such as other experiments being conducted and large changes in ambient room temperature. The researchers would have benefitted from a more spacious work space as the small table was very cramped with materials. A larger work space would allow the researchers to be more organized and not have to sort through materials. The probability of spilling the calorimeters would also be reduced. Future researchers could run each trial completely independently. The researchers ran almost all the trials in pairs due to time constraints. The greatest investment for the experiment would be better technology. A bomb calorimeter and a steam expansion jig would produce more accurate results. Several industries would benefit from using these procedures to identify metals. Companies that use pure elements in their productions would be able to check to see if their suppliers are selling them what they are paying for. Random quality control checks could be done to see if the metals are pure elements and not alloys or elements with similar properties. Businesses such as construction and plumbing that use metals often could conduct research to find if there are cheaper options. They could run similar experiments to observe if there are metals with values of specific heat and linear thermal expansion coefficient that are alike to the current metals being used. Essentially, this experiment this could be adapted for many uses. Works Cited"Calorimetry - Measuring Heats of Reactions."?. University of Waterloo, n.d. Web. 25 Mar. 2013. <, O. N., and C. V. Owen. "Preparation of High‐Purity Vanadium Metal by the Iodide Refining Process."?Journal of the Electrochemical Society?108.1 (1960): 88-93. Print. 24 Mar. 2013. <;. Davis, Joseph R. "Linear Coefficient of Thermal Expansion."?Thermal Expansion. National Science Foundation, n.d. Web. 07 Apr. 2013. <;."Enthalpy."?. Texas A&M University, n.d. Web. 25 Mar. 2013. <;."Experiment 2” . University of Massachusetts Boston, n.d. Web. 25 Mar. 2013. <;. Furrer, David, and S. L. Semiatin, eds. "Effect of Thermal Stresses on the Thermal Expansion and Damping Behavior of ZA-27/Aluminite Metal Matrix Composites."?Journal of Materials Engineering and Performance?10.2 (2001): 220-24.?ASM. Print. 7 Apr. 2013. <, Thomas. "Thermal Expansion Forum."?Science Clarified. JRank, 2006. Web. 07 Apr. 2013. <, Wolfgang, ed. "Encyclopedia Britannica Online."?Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Print. 24 Mar. 2013. <, and Ershaw. "Chemistry of Vanadium."?Vanadium. Prof. Robert J. Lancashire, 1 Apr. 2011. Web. 24 Mar. 2013. <;."Kinetic-Molecular Theory."?. University of Florida, n.d. Web. 25 Mar. 2013. <;."Linear Thermal Expansion Lab."?Georgia Primeter College. Physical Sciences Lab at GPC Decatur, 2011. Web. 07 Apr. 2013. <, et al. "Thermal Expansion."?HyperPhysics Concepts. National Science Teachers Association, 2001. Web. 07 Apr. 2013. <'Leary, Donal. "Vanadium."?Vanadium. Hypertext, 2000. Web. 24 Mar. 2013. <, Charles L. "The Extraction of Radium Etc. By the US Bureau of Mines." The Journal of Industrial and Engineering Chemistry 8.5 (1916): 469-73. Google Books. Print. 26 Mar. 2013. <;."Processing of Vanadium: A Review."?Minerals Engineering?16.9 (2003): 793-805.. Web. 24 Mar. 2013. <, James O. "Guided Discovery: Law of Specific Heats."?Specifc Heat?74.8 (1997): n. pag.?Journal of Chemical Education. Print. 25 Mar. 2013.Sutara, Jessica. "The Thermodynamics of Everyday Materials."?Carnegie Mellon. NSF Grant DMR 0520425: High School Teachers Summer Internship, 2011. Web. 7 Apr. 2013.<;."Vanadium."?Vanitec. FSC Marketing Communications, 2011. Web. 24 Mar.2013. <, Iancu Carmen, Bara Vasile, Sarca Gheorghe, and Timar Adrian. "Solutions of Heat Exchangers Used In Food Industry."?University Of Oradea. , n.d. Web. 25 Mar. 2013. <;. ................
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