Identifying an Unknown Set of Metal Rods as Consisting of Niobium or
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- Application
- Acknowledgements
- Works Cited
Data Analysis and Interpretation
In two separate experiments, data was collected in order to determine the specific heat and the coefficients of linear thermal expansion of the known metal rods, which consisted of Niobium, as well as the two unknown metal rods. Based on the physical appearances of the Niobium and the unknown rods, it has been proposed that the metals are not the same, as the unknown rods seem to have a slightly different shade of gray than the known Niobium rods. In order to determine the specific heat of the metal rods, their masses and initial and final temperatures were measured, as well as the masses and initial and final temperatures of the water. This was done using a plethora of equipment, which included a digital thermometer that recorded to the tenths place to record the initial temperature of the rods, two calorimeters along with a Lab Quest temperature probe to measure the temperature of the water as time progressed, and a scale accurate to four decimal places to measure the masses of the rods. In the linear thermal expansion experiment, two jigs which measured the change in length of the rods, a caliper to measure the initial length of the rods, and a digital thermometer accurate to one decimal place to measure the temperature of the metal were used to collect data. Given the high accuracy of the instruments used, the caution taken to complete the experiments, and the randomization and independence of the trials, the data is considered to be valid for use to determine whether or not the unknown metals consist of the same elements. 
Figure 3. Labeled box plot of calculated coefficients of Linear Thermal Expansion Figure 3 above is of the calculated coefficients of linear thermal expansion that were calculated based off of the data collected in the linear thermal expansion experiment. The top box plot is of the known Niobium rods, and the bottom of the unknown rods. The minimum, quartile one, median, quartile three, and the maximum of each plot is labeled, along with the known linear thermal expansion coefficient of Niobium, which is 7.3 x 10-6 °C-1. It should be noted that the unknown rods have strongly right skewed data, which shows that the distribution of the data is not normally distributed. It should also be noted that none of the trials, known or unknown, ever touched the true value of the coefficient of linear thermal expansion of Niobium, which suggests error in the trials; however, it can be seen that the unknown rods comes significantly closer to the known value than the known rods did. Figure 4. Box Plot of Specific Heat Values Figure 4 shows the specific heat box plots with the five number summaries for both the niobium rods and the unknown metal rods, along with the published specific heat value for niobium. It can be seen that the values are fairly normally distributed, and the known niobium rods cluster around the published specific heat value of niobium, while the unknown rods do not. Also, the two plots have no overlap and do not come relatively close to each other. Finally, the plots have a relatively normal distribution. All of these pieces of evidence suggest that the metals may have different identities. Table 9
Percent Error Table of Linear Thermal Expansion Experiment
Table 9 above contains the calculated percent errors for each trial that was completed during the linear thermal expansion experiment for the Niobium rods and the unknown rods. The final row contains the average percent error for each set of rods, and it can be seen that the average percent errors appear to be of significant difference. While this suggests that the rods consist of two separate metals, this data suggests that the unknown rods consist of Niobium, while suggesting that the known Niobium rods do not, which is not possible. Table 10
Percent Error Table of Specific Heat Experiment
Table 10 above contains the percent errors of the niobium and unknown metal rods from the specific heat experiment. The bottom row of the table contains the average percent error of each set of metals, which are significantly different. Given this great difference in error between the sets of rods, it is likely that the unknown metal does not consist of Niobium, as the known does. 
Figure 5. Normal Probability Plot of the Niobium Linear Thermal Expansion trials Figure 5 above is a normal probability plot of the calculated linear thermal expansion coefficients for the niobium rods. It can be seen that the data points are not at all linear when compared to their expected z scores, giving evidence that running a statistical test on this data may not be reliable when used to determine whether the unknown rods consist of the same metal or not. Figure 6. Normal Probability Plot of Unknown Linear Thermal Expansion Coefficients Figure 6 above is a normal probability plot of the linear thermal expansion coefficients of the unknown rods. It can be seen that the data not at all linear, as it appears to be clustered. This suggests that a statistical test may not be a reliable way to determine whether or not the unknown metal consists of the same metal as the known niobium samples since the data is not normally distributed.
Figure 7 shows the normal probability plot for the specific heat values for the niobium rod. The graph shows the specific heat values compared to their expected z scores. It can be seen that the values are linear and fairly close to the line of best fit, meaning that the data is normally distributed. 
Figure 8. Normal Probability Plot for the Unknown Rod Specific Heat Trials Figure 8 shows the normal probability plot for the specific heat of the unknown metal rod. From the graph, it can be observed that the data points are fairly linear when compared to their expected z score. The values also cluster along the line of best fit which means that the data is normally distributed. By looking at the normal probability plots of the calculated coefficients of linear thermal expansion, it can be seen that the data collected is not normally distributed for either of the sample populations. This suggests that any statistical test that is used to compare these data could be potentially unreliable since it is assumed that if there are less than thirty trials in each sample, the data is normally distributed. In this case, each sample contains fifteen data points and is not normally distributed. It is also assumed that the samples are independent of each other. In this experiment, the trials are completely independent of each other; therefore, that assumption is met. In the specific heat experiment, it can be seen by looking at its normal probability plots that the data is fairly normally distributed with no significant outliers, which meets the assumption of normal data. In this experiment, the trials were also completely independent of other trials, which constitutes that the data is valid to run a reliable statistical test. Since the purpose for this research was to determine whether or not the unknown metal rods consisted of Niobium as the known rods did, a two sample t-test was used to compare the linear thermal coefficients and the specific heats of the known and unknown rods. The formula to run a two sample t test is: 
In this formula, the known mean is the mean of the Niobium trials for each experiment, the unknown mean is the mean of the unknown metals for each experiment, standdevkn is the sample standard deviation of the Niobium trials, standdevuk is the sample standard deviation from the mean of the unknown metal trials, trialkn is the number of Niobium trials, and trialuk is the number of unknown metal trials. A t-test is used to compare separate samples to determine whether or not they have the same population mean, which in turn will determine if they are likely to have come from the same population or not, which in this case was used to determine if the unknown rods consisted of Niobium or not. In the linear thermal expansion experiment, the two metals are being tested against the following hypothesis: 
In this hypothesis, µkn represents the population mean of the Niobium linear thermal expansion coefficients and µuk represents the population mean of the unknown metal’s linear expansion coefficients.  Figure 9. 2-Sample t Test for Linear Thermal Expansion
Figure 9 above is of the completed 2-sample t test along with the calculated means and standard deviation for each of the samples. The null hypothesis was rejected on the alpha level of 0.10 due to the p value being much less than 0.10. These results suggest that the unknown metal rods are not composed of the Niobium that the known rods are composed of. The p value of 0.009 means that if the null hypothesis were true, there would only be a 0.9% chance of collecting data this extreme, which is highly unlikely. However, because the assumptions for this test were not met, the results of this test may not be reliable.  
Figure 10. 2-Sample t Test for Specific Heat Figure 10 shows the completed 2-sample t test and the calculated values for the means and standard deviations for each of the samples. The null hypothesis was rejected because the p value is less than the standard alpha value of 0.05. This evidence suggests that the two values have different identities, meaning that the unknown metal rods are not made up of Niobium as the known rods are. The p value of 9.8811*10-12 means that if the trials were repeated, there would be a 9.88*10-10 percent chance of gathering data as extreme as this if the null hypothesis is considered to be true.
In this research, the experimenters set out to determine whether or not a set of unknown metal rods consisted of the same metal as the set of known rods, niobium. It has been hypothesized that the unknown rods would be made up of niobium if the percent errors of the specific heat and the coefficient of linear thermal expansion of each was less than five percent. Through experimentation, it has been determined to accept the hypothesis. The decision to accept the hypothesis was based off of physical observations as well as the data collected. By looking at the two sets of metals, the niobium rods had a different shade of gray than the unknown rod, and was much more lustrous than the unknown rods. A magnet was also placed against the known and unknown rods, and when the unknown rods both attracted a magnet while the niobium rods did not, it was determined to be highly unlikely that they were the same. Once the experiments were completed, the data was analyzed and the specific heats and linear thermal expansion coefficients were compared between the niobium rods and the unknown rods. This was achieved by using a two sample t test, comparing the sample means of each set of rods. For each experiment, the t test returned a p-value of less than 0.01, suggesting that if the metals were the same, there is a less than one percent chance of catching data this extreme, providing strong evidence that the two metals were not the same. Though each experiment suggested significant difference between the metals, the linear thermal expansion experiment showed results that were antithetical to what was known about the metals. While the significance test suggested that the metals were different, when looking at the data collected along with the percent error of each data point, it can be seen that the niobium rods have consistently larger percent errors than those of the unknown rods when compared to the linear thermal expansion coefficient of niobium. Reasons for this are unknown; however, it has been theorized that due to the smaller mass of the niobium rods along with their lower specific heat than the unknown rods, the niobium rods could have contained less energy inside of them when taken from the water, allowing for them to retract faster, as what heat energy was contained in the rods would escape faster than the unknown rods. This could have affected the trials because if the niobium rods retracted faster, the total change in length would not be caught by the linear thermal expansion jig, since there was a short time between the metal leaving the water and entering the jig. If the total change in length was not accurately recorded because of this, the coefficient of linear thermal expansion would be greatly affected, causing the niobium rods to have a higher percent error than the unknown rods. One problem that was encountered by the researchers was the effectiveness of the wooden jigs used in testing linear thermal expansion. Although the jigs did register some movement, they were not very sturdy. This meant that if they were moved and the bottom of the jig was disturbed, the results would not be accurate. Newer jigs made of metal would fix this problem, as the old wooden jigs would often bend and that would cause them to be inconsistent. Another obstacle the experimenters encountered was the inability to effectively transfer the metal rods into the linear thermal expansion jig or the calorimeter. This was mainly because the tongs that were used were neither large enough nor strong enough to completely pick up the heavier unknown metal rods. Due to this, many trials were re-executed with a new set of tongs so that the results were more accurate. Another possible flaw in this research is that the experimenters assumed that the metal had risen to the same temperature of the boiling water. After being left in the boiling water for only five minutes, it is highly unlikely that the temperature of a metal with a high specific heat would have completely risen to the recorded value, but due to time constraints, the time allowed for the metal to warm up was set to five minutes. Because of this flawed assumption, the specific heat and linear thermal expansion results may have been inaccurate due to the fact that the metal may not have been the same temperature throughout. To expand on this research, other intensive properties such as malleability, ductility, conductivity, and hardness could be used to determine or verify the results gathered in this experiment. These properties could be measured using other tools, which were not at the exposal of the researchers, such as a metal stretcher to measure ductility, metal formers to test the malleability, an ohmmeter to measure conductivity, etc. Even though specific heat and linear thermal expansion are properties that are unique to each substance, the methods of calculating these values are quite tedious, and may not be the most accurate way of identifying metals, unless more accurate equipment, such as digital jigs to measure the change in length of a material, is used. One could also expand onto this research by further testing the unknown metals with the intention of identifying the element or alloy of which it is made up. The properties used in this experiment are useful for not only determining the identity of a metal, but they can also determine how a metal will react when exposed to energy, or even how much a given metal will expand when exposed to a certain amount of energy is applied to it. Knowing this information is important in such industries as road building and designing cooling systems. When building a road, the engineer designing the road must know the linear thermal expansion of the concrete that is to be used, as it will tell if it will crack when the weather changes, causing it to expand and contract. As it pertains to cooling systems, the specific heat of the coolant must be known in order to build the system. This is because when building the cooling system, the designer needs to know how much heat energy the system can absorb, and how that will affect the temperature of the coolant. In the end, the researchers used the intensive properties of linear thermal expansion and specific heat in order to identify the unknown metals either as being the same or different than the niobium rods that they used as a reference, and were able to correctly identify the unknown rods as being different.
Since Niobium is often used as a strengthening agent in steels, a nail was produced from it. Due to niobium’s tendency to strengthen an alloy, it is a great material to use when making nails, for the added strength is needed in order for the nail to become lodged into another material without it bending or breaking. 
Figure 1. Isometric View of a Nail Composed of Niobium Figure 1 shows a nail composed of niobium, which is used for a wide variety of things including construction. Niobium is suitable for this material because it used as a strengthening agent for steel, which many nails are made of. The cost of this nail would be twelve cents if it was made of pure niobium, which was determined by taking the cost of niobium per gram and multiplying that value by the mass of the nail.
Figure 2. Drawing of Niobium Nail Figure 2 shows the drawing of the nail made of niobium along with all of its dimensions.
The researchers would like to thank Mrs. Jamie Hilliard for her assistance in designing the experiments that were used in this research, as well as providing the necessary materials and lab space that was needed to complete the experiments. They would also like to thank Mr. Mark Supal for providing the jigs that measured the change in length of the metal rods in the linear thermal expansion experiment. Finally, Mrs. Christine Kincaid Dewey is thanked on behalf of the researchers for her criticism of the data analysis and interpretation section because without her criticism, the interpretation of the data would not have taken the same shape that it did in the paper. Appendix A: Calorimeter Instructions
¾ inch PVC pipe (20 cm long) ¾ inch PVC caps (2) PVC sealant PVC primer High density Styrofoam (2 x 3 x 7 in.) Clamp
7/8 inch diameter forstner drill bit 1/8 inch diameter drill bit 50 grit sandpaper Ruler
Drill Press Procedure: 1. Using a ruler, find the center of the 2 x 3 inch side of the Styrofoam block. Mark the center. 2. Place the forstner drill bit inside of the drill press and clamp the Styrofoam to the drill press with the marked side face up. 3. Drill a hole through the center of the 2 x 3 inch side of the Styrofoam block to the bottom of the block. 4. Using the sandpaper, scratch one end of the PVC pipe until the end is fairly rough. 5. Apply the primer over the scratched end. 6. After the primer has dried, which takes approximately five seconds, apply the sealant over the primer, and inside of the PVC cap, quickly placing one cap over the end of the pipe. Allow the sealant to dry. 7. When the sealant has dried, push the pipe into the Styrofoam block with the uncapped end face up. Caution: Since the Styrofoam is incredibly fragile, it is advised that the experimenter takes immense care when pushing the pipe into the block, as it may take a bit of force to fit it into the Styrofoam. 8. Clamp the other PVC cap down to the drill press, and, using the 1/8 inch drill bit, drill a hole through the top of the cap. 9. Run water through the inside of the PVC pipe to remove any Styrofoam or glue that may be leftover. Diagram: 
Figure 11. Isometric Drawing of Calorimeter Figure 11 above is an isometric model of the calorimeter that was used to complete the specific heat experiments, along with a drawing that is labeled with all of its dimensions. Appendix B: Sample Calculations In order to determine the specific heat of the metal rod tested, an equation was utilized where massm was the mass of the metal, massw was the mass of the water, tempw was the initial temperature of the water, tempm was the initial temperature of the metal, tempe was the equilibrium temperature of the metal and the water, and cf was the correction factor for the calorimeter.
Figure 12 below shows a sample calculation using the equation for specific heat, using rod B from trial 2 of the known metals. 
The correction factor used in the specific heat formula was calculated by using the mean calculated specific heat from the pre-trials and subtracting it from the known specific heat. 
Figure 13 below shows the calculation of the correction factor for the specific heat trials.
To determine the linear thermal expansion coefficient of the metal rods, an equation was needed where lengthi is the initial length, lengthc is the change in length, tempf is the final temperature of the metal, tempi is the initial temperature, and α is the linear thermal expansion coefficient. 
Figure 14 below is a sample calculation using the linear thermal expansion equation.
The following equation was used to calculate the percent error of the metal rods which was then used to analyze the data. The experimental value was computed in the lab while the true value was the published value for that property. 
Shown in Figure 15 below is a sample calculation using the percent error equation using trial 2 rod B from the known metals.
To analyze the data using a two sample t test, the following equation was used to determine the t value. This value was then used to determine a p value which gave the probability of receiving these results if the two metal rods had the same identity. The standdevkn and trialkn are the known standard deviation and number of trials respectively. Similarly, the standdevuk and trialuk are the unknown standard deviation and number of trials respectively.
Shown in Figure 16 is a sample calculation using the equation to find the t value.
Below is the equation used to calculate the standard deviations for both the known and the unknown data. These values were then utilized in the equation to determine the t-value which was then used to conclude if the metals were identical. 
Works Cited: "223 Physics Lab: Linear Thermal Expansion." 223 Physics Lab: Linear Thermal Expansion. Clemson University, n.d. Web. 25 Mar. 2014. "Coefficient of Thermal Linear Expansion." St. Louis Community College. N.p., n.d. Web. 26 Mar. 2014. "Measuring the Coefficient of Linear Expansion of Copper, Steel, and Aluminum." University of Nevada, Reno. N.p., 04 Mar. 2013. Web. 26 Mar. 2014. Nave, Carl R. "Light Water Reactors." Light Water Nuclear Reactors. Georgia State University, 17 Jan. 2014. Web. 26 Mar. 2014. Nave, Carl R. "Thermal Expansion." Thermal Expansion. N.p., n.d. Web. 25 Mar. 2014. "Niobium." - Element Information, Properties and Uses. Royal Society of Chemistry, 08 May 2014. Web. 08 May 2014. "Specific Heat." Specific Heat. Ohlone College, 12 Nov. 2008. Web. 26 Mar. 2014. "Thermal Expansion." - Summary. Glenn Elert, n.d. Web. 26 Mar. 2014. Yüklə 399,2 Kb. Dostları ilə paylaş:
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