Data Processing Lab 1 My Solutions > Please reference the Data Processing Lab 1 file for a description of the deliverables. Note that your script must import the data from the original text file supplied on Canvas. You can manipulate the data however you would like after it is imported to MATLAB, but you must reference the supplied file in your solution script. O 00 ~N o U B WN R N B P PR R B R R R WO ~NOOWL A WNR PLEASE READ ALL THE LOCKED COMMENTS CAREFULLY BEFORE SUBMITTING ANY ANSWERS. IF YOU DO NOT READ THESE COMMENTS, YOU ARE LIKELY TO OVERLOOK AN IMPORTANT DETAIL OF THE REQUIRED SUBMISSION STYLE. o o of Import your data and make any preliminary calculations between here and the next set of locked lines. o of 0. % Create your Boyle's Law plot between here and the next set of locked lines. F1 = figure(1) % create a figure handle called F1, for checking your answers % make sure your figure has a legend or MATLAB Grader will not grade your work correctly. % Plot your Gay-Lussac's Law figure between here and the next set of locked lines. F2 = figure(2) % create a figure handle called F2, for checking your answers % make sure your figure has a legend or MATLAB Grader will not grade your work correctly. % Perform the rest of your calculations below this locked line. Is Boyle's Law plot x data correct? Is Boyle's Law plot y data correct? Is Gay-Lussac's Law plot x data correct? Is Gay-Lussac's Law plot y data correct? Is Rair correct? Is the percent difference for Rair correct? Is the uncertainty in Rair correct?Is RCO2 correct? Is the percent difference for RCO2 correct? Is the uncertainty in RCO2 correct? Is RN2 correct? Is the percent difference for RN2 correct? Is the uncertainty in RN2 correct? Is Ru correct?Is RN2 correct? Is the percent difference for RN2 correct? Is the uncertainty in RN2 correct? Is Ru correct? Is the percent difference for Ru correct? Is the uncertainty in Ru correct?\fData Processing Lab 1: The Ideal Gas Constant Overview The purpose of this Data Processing Lab (DPL) is to help you practice your coding skills in the context of a real science/engineering problem: using data to extract some meaningful piece of information. Our lens for this first DPL will be the Ideal Gas Law, a familiar law from introductory chemistry and physics that plays an important role in many areas of engineering analysis. The hope is that this DPL doesn't just feel like a \"problem\" (e.g., from a homework set), but instead feels like an actual data analysis task. To facilitate this realistic feeling, this document is written like a lab manual from a typical physical science and/or engineering course. You will (unfortunately) not be able to carry out the experiments as described below, but you should still read through the entire document before doing anything else. There are important pieces of information throughout and skipping straight to the data processing tasks will almost assuredly leave you feeling confused. Background The Ideal Gas Law is a historically empirical law that was pieced together between the late 17" and early 19 centuries from a series of experimental observations [1]. Robert Boyle first recognized in 1662 that the pressure (P) of gases in high vacuum (i.e., low pressure) environments is inversely proportional to its volume (V), P o V1. Following Boyle's work, |. Charles and J. Gay- Lussac found that temperature (T) also played an important role in the physics, finding that V e T and P o T, respectively, for gases at low pressures in the early 1800's [1]. The final piece of the puzzle was discovered by Amedeo Avogadro who, working around the same time as Charles and Gay-Lussac, observed that the volume of a gas is directly proportional to the number of moles (n) of the gas, V o n [2]. Combining these empirical laws and introducing the proportionality constant R, gives the familiar Ideal Gas Law: PV = nR,T 1) As written, R, is the so-called universal gas constant and is the same for every ideal gas. It is also possible to write the Ideal Gas Law in terms of the mass of a gas (m) and the gas constant specific to that gas (R): PV = mRT (2) The conversion factor between these two forms of the Ideal Gas Law is the molar mass of the gas (M), typically with units of kg per kmol. The Ideal Gas Law on a per-mole basis (Eqn. 1) is commonly used in chemistry and chemical engineering, whereas the Ideal Gas Law on a per-mass basis (Eqn. 1) is commonly used in mechanical and aerospace engineering. Engineers in all fields should be familiar with both forms to facilitate efficient communication on interdisciplinary projects. The proportionality constant R,, must be known if we wish to use the Ideal Gas Law to predict the properties of an ideal gas under certain constraintscalculating the pressure of a gas given the volume, temperature, and number of moles of the gas, for example. Your primary goal in this lab is to determine R, from a large set of experimental data. You will also make plots to visually verify Boyle's Law and Gay-Lussac's Law, as well as calculate the gas constant (R) for three different gases. Page1ofs Experimental Procedure Fixed masses of 3 different gases (mass values in Appendix A) have been isolated in 3 different containers (Figure 1). The containers are well sealed, preventing any leakage, and each container is fitted with a piston that can be locked in place, allowing you to fix the container volume. Lockable Piston Pressure Tap \\ / Temperature Tap Containing Cylinder . Isolated Ideal Gas Figure 1. Experimental apparatus for studying ideal gases. A fixed mass of an ideal gas is isolated in a containing cylinder fitted with a lockable piston, providing a means for altering the volume of the ideal gas. The cylinder is also fitted with temperature and pressure taps to measure these properties, as well. Perform the following steps to collect your data. 1. Select the container labeled \"Air\". 2. Unlock the piston stop and set the container volume to 0.2 L. Lock the piston in place. Place the container on top of a hot plate. 4. Connect a thermocouple reader to the thermocouple wire protruding from the side of the container. Also connect the pressure-sensing tube to a diaphragm-type pressure gauge. 5. Set the hot plate to 40C. Wait ~3 minutes for the gas temperature to stabilize. The gas temperature will likely stabilize in the range of 36 - 38C, depending on the temperature in the lab on the day you are running your experiment. 6. Increase the set temperature on the hot plate by the difference between the target temperature and the stable gas temperature. 7. Wait for the temperature to stabilize then repeat step 6 if necessary. 8. Record the pressure at the fixed volume and temperature. Note that the pressure gauge measures the gauge pressure of the gas: Pyauge = Pabsolute Patm, in kPa, where the subscript 'atm' represents the ambient pressure. 9. Repeatsteps 5 -8 for {T | T [40C, 80C]} in increments of 2C. 10. Repeatsteps 2 -9 for {V | V [0.2 L,1L]} in increments of 0.05 L. 1. Repeat this experiment for CO, and N, with the following adjustments: a. ForCO,, use {V |V [0.3L, 1 L]} and the same temperature range. b. ForN,, use {V |V [0.1L,1L]} and the same temperature range. Pagez2of 5 Analysis Use the raw data provided in the Excel file \"Raw_Data.xlxs\" to perform the analyses outlined below. This data file contains pressure measurements at each pair of temperature and volume values, P(V,T), described in the Experimental Procedure section above. 1 Visually verify Boyle's Law by plotting P vs V for air, where P is the absolute pressure in kPa and V is in m* Your plot should contain 3 different lines representing T = {50C, 60C, 70C). Visually verify Gay-Lussac's Law by plotting P vs T for CO,, where P is the absolute pressure in kPa and T in in C. Your plot should contain 4 different lines representing V = {0.4 L, 0.6L, 0.8L, 1L}. Use the measured pressure data to calculate the average value of the specific gas constant (R) 3 qset kPa: for each gas, in units of [ k::: ] Don't forget that you must use absolute temperature and pressure in the Ideal Gas Law. The measured ambient pressure in the lab while you performed you experiments can be found in Appendix A. Call these variables Rair, RCOz, and RNz, respectively. Calculate the percent difference between your calculated values of R a (Appendix B). Call these variables diffRair, diffRCO2, and difi int difference as defined by . accepted value calculated value % diff = | | + 100 accepted value Calculate the statistical uncertainty in the specific gas constant for each gas. Call these variables uRair, uRCOz, and uRNz, respectively. See Appendix C for a short discussion on making this calculation. Repeat the average, percent difference, and statistical uncertainty calculations for the universal gas constant, in units of [. using the data across all 3 gases. Call these variables Ru, diffRu, and uRu. Note: You will have to account for the fact that there are a different number of samples (data points) for each of the gases in your analysis. To see why this matters, consider the two arrays x; and x,: x=[3333333333333], x=1[2] The mean of each of these arrays is = 3 and X; = 2, respectively. We might naively presume that we can calculate the mean value of the entire set of data, x; U x,, where U represents the \"union\" of the two sets, as s R X Ux; = 12 2 =125 We can clearly see that this is wrong, however, since X; U x; should be much closer to 3 than to 2. You must think carefully about how to properly incorporate all the data when performing these universal gas constant calculations. Page3ofs Appendix A This appendix contains information about the isolated masses of each gas and the atmospheric pressure during the experimental measurements. Table 1. Isolated masses of each ideal gas contained in their respective piston-cylinder devices. Property Measured Value Mass of air, mg;; 525 mg Mass of CO,, mco, 495 mg Mass of N, my, 575 mg Ambient pressure, pyym 1.02 bar Appendix B This appendix contains nominal values for the universal gas constant, the specific gas constant for each gas, and the molar mass of each gas. These values were sourced from the Cengal and Boles introductory text on thermodynamics [1]. Table 2. Nominal values of important properties for the ideal gases of interest. Property Nominal Value ; ] U rsal tant, R 8314 niversal gas constan T : J Gas constant for air, m,;, 287.0 _kg X J Gas constant for CO,, mco, 188.9 m J Gas constant for N,, my, 296.8 m Molar mass for air, Mg, 28.97 ke kmol Molar mass for CO,, Mo, 44.01 . kmol kg Molar mass for N, My, 28.013 r Appendix C This appendix provides a brief discussion on calculating the statistical uncertainty of a sample data set. There are several assumptions associated with this calculation, which you will learn more about in your statistics course. Page 40of 5 No physical measurement is complete without an uncertainty. For example, when you use a ruler, you can never say with 100% confidence that you've measured something to be exactly 2.34 inches; you must always acknowledge that your measurement has some uncertainty. This uncertainty is typically reported using the + symbol. For the ruler measurement this might look like 2.34 1 0.05 in. The + value generally represents the upper and lower bounds of the so-called 95% confidence interval [3]. When you analyze a data set you generally calculate the mean value (u) of some parameter of interest and its corresponding standard deviation (o). If your data set has more than N = 30 samples (data points), the sample mean and the 95% confidence interval are reported as M + 1.96- VN (A. 1) where the factor after the +, 1.96-, is commonly called the uncertainty. Note: The std function in MATLAB allows you to normalize by N - 1 or N. Please use the default function setting, which normalizes by N - 1. Works Cited [1] Y. A. Cengal, M. A. Boles and M. Kanoglu, Thermodynamics: An Engineering Approach, New York, NY: McGraw-Hill, 2024. [2] M. S. Silberberg, Chemistry: The Molecular Nature of Matter and Change, New York, NY: McGraw-Hill, 2009. [3] P. Lisa Sulliva [Online]. Available: https://sphweb.bumc.bu.edu/otIt/mph- modules/bs/bs704_confidence_intervals/bs704_confidence_intervals_print.html. Page 5 of 5