7# FE!q;yyy :zx V* V/'VVVVVVVVExperiment I Mass Measurement PRE-LAB ASSIGNMENT Reading: Before coming to your discussion section read the following: 1. The Introduction in this lab manual. 2. Sections 1-5 and 1-6 in Olmstead and Williams. 3. The remainder of this experiment in this lab manual. Pre-lab assignment to hand in: Prepare written responses to the following questions and submit your work to your lab instructor at the beginning of your discussion section. 1. Outline the steps you would take to weigh a 15 g sample of metal on our analytical balances (see Experiment I Appendix 2). The analytical balances have a weighing accuracy of 0.0002 g. Express the weighing accuracy of the balances as a percentage of the 15 g sample weight. 2. The method of least squares provides equations to determine the slope and intercept of the "best straight line" to fit experimental data. In Part II, which of these values gives a good approximation of the weight of the bolt? Briefly explain the logic of this extrapolation method for determining the weight of the bolt (two or three sentences). Introduction This experiment is an introduction to several important methods routinely used by scientists in their everyday work. You will learn how to measure the mass of objects using analytical and centigram balances. In Part I, you will calculate the density of aluminum. In Part II, you will learn some standard methods for graphical data analysis including linear curve fitting and extrapolation. PART I--DENSITY INTRODUCTION Density is a fundamental property of matter and is often used to describe substances. It is simply a number which relates the mass of an object to its volume. Density must be measured and expressed as a property at a given temperature and pressure. The mass of a substance is invariant with changing temperature or pressure, but the volume does change. The most common units for a liquid or solid are g/cm3 (also written g cm-3), or g/mL. The mass and volume of a liquid are easily measured experimentally. For large solid objects the mass can be determined with a balance and the volume can be obtained either by measuring dimensions and computing, or more accurately, from the volume of liquid displaced when the object is submerged. In this experiment you will determine the density of aluminum. PROCEDURE Determine the mass of a piece of aluminum on a digital centigram balance. These balances have an accuracy of 0.01 g. Record your weight to that accuracy. Determine the volume of the piece of aluminum either by using a ruler or by the volume of water displaced when the aluminum is submerged in a graduated cylinder (see Experiment I-Appendix 3 for the proper way to read the volume in a graduated cylinder). Calculate the density. Repeat this procedure twice using a different piece of aluminum each time. Average the three values for the density of aluminum. PART II--LINEAR CURVE FITTING AND EXTRAPOLATION INTRODUCTION This portion of the lab exercise has three purposes: a) to introduce you to a graphical extrapolation technique. b) to consider some simple statistics and the method of least squares for finding the best straight line through a set of data points. c) to give you practice with weighing procedures and our analytical balances. Extrapolation: Often it is impossible to measure quantities directly. Thus, methods must be devised to determine quantities indirectly. One of these indirect methods is graphical extrapolation. In this exercise you will determine the mass of a bolt by graphical extrapolation. The experiment is somewhat contrived, but it introduces many important techniques without having to worry about time consuming experimental details. You will use these techniques next week to study a chemical reaction. Next to your balance you will find a bottle containing five loose nuts and a bolt with a nut firmly attached to it. The objective of the experiment is to determine the mass of the bolt without removing the fixed nut. To do this, you will weigh (to the nearest 0.0002 g) the bolt-nut combination together. Then you will repeat the weighing five times adding another loose nut to the balance pan with each successive weighing. If masses of the nuts are nearly identical, a plot of aggregate mass (on the y-axis) versus the number of nuts (on the x-axis) should be a straight line. The corresponding mass where the line intersects the y-axis (number of nuts = 0) should be a good estimate of the mass of the bolt alone. An example plot is shown in Figure 1. A straight line has been fit to the data using the least squares technique, which is described below. The equation for the straight line is printed on the plot. This straight line has been extended until it intersects the y-axis at the x value for zero nuts. The extension of the line to a given point is called extrapolation. The mass of the bolt alone would equal the y intercept or 1.2589g.  Figure 1. The determination of the mass of a bolt by extrapolation The method of Least Squares: The example above required that a straight line be drawn through a series of experimental points. Because of the experimental error in doing laboratory measurements, all of the plotted points will usually not fall on a perfect straight line, so the line which best represents all of the data points, the best straight line, must be approximated. The term best straight line implies that the line chosen lies the closest to the experimental points.  You should recall that a line can be described by the general equation y = mx+b, where m is the slope of the line and b is the point where the line meets the y-axis, the y-intercept. The vertical distance from an experimental point to the line is often referred to as the deviation for that point. The best line will produce the smallest sum of the deviations. In the derivation of the expression for the best line (see Experiment I-Appendix 1), the deviations are squared before they are summed so that negative deviations do not cancel positive deviations. This is the reason for the name "method of least squares". The slope and intercept can be calculated from the following equations (see Experiment I-Appendix 1 for the derivation of these equations.): Slope = F(nxy -xy,n(x2) - (x)2) (n is the number of experimental points) Intercept = F(y(x2) - xyx,n(x2) - (x)2) Although it appears to be a lot of work at first, these calculations can be performed rather easily using a calculator or a computer spreadsheet program, such as Microsoft Excel. An example of the calculations for six experimental data points is shown in the table below. DATA: xyx2xy16.726516.726527.8374415.674839.0257927.0771410.11811640.4724511.00722555.0360612.23533673.4118x = 21y = 56.9502(x2) = 91xy = 218.3986 Slope = F((6)(218.3986) - (21)(56.9502),6(91) - (21)2) = 1.0899 Intercept = F((56.9502)(91) - (218.3986)(21),(6)(91) - (21)2) = 5.6771 One estimate of the error in the intercept value is the average deviation for the experimental points. The average deviation, AVS0(,d), can be calculated from the following equation: AVS0(,d) = F(ABS[ y - (mx + b) ],n) where m and b are the calculated slope and intercept, respectively, n is the number of experimental points, and x and y are the coordinates of each point. ABS is an abbreviation for the operation of taking the absolute value of the term inside the brackets. The Analytical Balance and Weighing Techniques: Accurate weighing is a fundamental operation in modern chemical work. It is very important that you learn good weighing technique in order to perform well in the lab and to maintain the high quality of our delicate analytical balances. Please follow the weighing instructions given in Experiment I-Appendix 2. Experimental Procedure Graphical Determination of the Mass of a Bolt Next to the balance you will find a bottle containing five loose nuts and a bolt with a nut firmly attached to it. The objective of the experiment is to determine the mass of the bolt without removing the fixed nut. To do this, weigh (to the nearest 0.0002 g) the bolt-nut combination together. Then repeat the weighing five times adding another loose nut to the balance pan with each successive weighing. Make a table of the number of nuts used in each step and the corresponding aggregate mass of the nuts and bolt. Plot the data using Cricket Graph. To get the most precision out of your graph, choose your scale so that your line will fill as much of the page as possible. You do not need to start at zero mass on the y-axis; just be sure that the line will intercept the y-axis on the page and that all of your data points fit on the page (see the example plot on page 3 for an experiment using just 5 nuts). The nuts do not have to be exactly the same mass for the method to give a reasonably accurate value for the mass of the bolt. If the nuts do vary in mass, it is better that they are not added in any systematic way with respect to their masses, such as from lightest to heaviest. If some trend were introduced, the points would appear to define a curved rather than a straight line plot. Variations in the mass of the nuts do limit the precision to which we can determine the mass of the bolt. there is a way of estimating the reliability of our result. In fact, you can find the slope and intercept for the best straight line fit of your data and get an estimate of the uncertainty (or probable error) in the mass of the bolt by using the Method of Least Squares. A least squares "spreadsheet template" for Microsoft Excel is available on the lab Macintosh computers. If you are unfamiliar with the use of the computers or the program (Microsoft Excel), ask your lab instructor for assistance. Using the template provided, enter your data to create your least squares table similar to the one on page 4 and calculate the slope and intercept of the least squares line, and the average deviation of the y-values. Print your data table and include a copy of the table with your report. The original table should be firmly attached in your notebook. If you wish to retain a disk copy of your table, you may copy your data into the CH141Lab folder on the Chemistry Departments file server. NOTE: If you have experience with Microsoft Excel and wish to create your own data table you may do so. However, be sure to clearly label your columns and rows to aid instructors in evaluating your work. Also, if you need assistance with the Macintosh graphics program Cricket Graph, ask your lab instructor or an assistant at the Mac Lab to help you. REPORT To be submitted at the beginning of the next lab period.Your report should include the following information, clearly labeled and in this order: Part I 1. The masses of three pieces of aluminum. 2. The length measurements of the pieces and their calculated volumes, or the volumes of displaced water in your graduated cylinder. 3. The density calculation. Calculate the density for each trial, and average the three values. Remember, for this and future reports, always show calculations and include units. 4. Your discussion including evaluations of your precision and the likely accuracy of your density. Look up the density of aluminum and compare it to your calculated density (you can use the Handbook of Chemistry and Physics). Since this discussion should be in your lab notebook, you may simply submit a photocopy of your notebook page. Part II 1. Your analytical balance number 2. Your least squares data table (computer print out); i.e., the Excel spreadsheet. 3. The graph of your data. Please review the comments on graph preparation in the Introduction to this lab manual. 4. Your discussion, including evaluation of your precision and the likely accuracy of your estimate of the weight of the bolt. Also comment on any large variations in your data and give possible explanations. Since this discussion should be in your lab notebook, you may simply submit a photocopy of your notebook page. References W.E. Harris and B. Kratochvil, "An Introduction to Chemical Analysis," Saunders, 1981. D.G. Peters, J.M. Hayes, and G.M. Hieftje, "Chemical Separations and Measurements," Saunders, 1974. Experiment I - Appendix 1 Derivation of the Equations for Slope and Intercept in the Method of Least Squares m = slope of the least squares line b = intercept of least squares line y = experimental y-values x = experimental x-values n = number of data points used d = deviation of an experimental y-value from the y-value predicted by the least squares line mx + b = least squares line prediction of a y-value associated with an experimental x-value From the definitions above: d = y - (mx + b) (1) and d2 = [y - (mx + b)]2 = y2 - 2mxy - 2by +m2x2 + 2mbx + b2 (2) We wish to obtain the slope and intercept of the line which will produce the smallest value for the sum of the squares of the deviations. Thus we must derive the equation for the sum of the deviations over all of the points (n). This equation is found by summing each term in equation (2) over n data points: d2 = y2 - 2mxy - 2by +m2(x2) + 2mbx + nb2 (3) Now we will allow the slope and intercept to vary until we reach values which produce the minimum value for equation (3). This is done by taking the partial derivative of equation (3) with respect to m and then separately with respect to b. In general, the minimum value for an equation is obtained with variable values which make the derivative of the equation equal zero. Thus, the slope and intercept for the Least Squares Line will be those derived by setting both derivative equations equal to zero and solving for m and b simultaneously. Partial derivative of (3) with respect to m: F((d)2,m) = -2xy + 2m(x2) + 2bx = 0 (4) Partial derivative of (3) with respect to b: F((d)2,b) = -2y + 2mx + 2nb = 0 (5) Solving equation (5) for m we obtain: m = F(y - nb,x) (6) Substituting m from equation (6) into equation (4) we obtain: -2xy + 2 F((y - nb)(x2),x) + 2bx = 0 (7) Equation (7) is solved for b, yielding equation (10): -xyx + y(x2) - nb(x2) + b(x)2 = 0 (8) b[n(x2) - (x)2] = y(x2) - xyx (9) b = F(y(x2) - xyx,n(x2) - (x)2) (10) Substituting b from equation (10) into equation (6) and simplifying, we finally obtain equation(14): m = F(y - n B(F(y(x2) - xyx,n(x2) - (x)2)) ,x) (11) m = F(ny(x2) - y(x)2 - ny(x2) + nxxy,x[n(x2) - (x)2]) (12) m = F(nxxy - (x)2y,x[n(x2) - (x)2]) (13) m = F(nxy - xy,n(x2) - (x)2) (14) Experiment I Appendix 2 WEIGHING INSTRUCTIONS Mettler College 150 Analytical Balances  Figure 1. Diagram of Mettler College 150 balance with draft shield removed. 1. BEFORE WEIGHING, BE SURE THAT: a. The pan is clean and the weighing-chamber doors are closed. b. The display reads zero. Press the tare bar to zero the display. 2. LOADING THE PAN: a. Never place the sample to be weighed directly on the balance pan. Use a container or weighing paper. b. Place the sample on the center of the balance pan with tongs or forceps. c. Close the weighing-chamber doors. 3. WEIGHING THE SAMPLE. a. Allow the digital display to reach a constant reading. A small circle will disappear in the upper left side of the display when the balance has stabilized. 4. COMPLETING THE WEIGHING a. Record the result. b. Remove the sample from the balance pan. REMEMBER: Use a container or weighing paper when weighing samples. Basic Operating Principle of the Analytical Balance Figure 2 is a schematic of the weighing mechanism of the analytical balance. The base of the balance pan sits on top of an electromagnet which is positioned inside of a permanent magnet. A weight (i.e. the sample) placed on the balance pan depresses the pan. The pan movement is detected by the null detector causing electrical current to flow through the electromagnet. The magnetic field of the electromagnet is attracted or repelled by the permanent magnets restoring the balance pan to its original position. The current needed to restore the balance pan to its original position is proportional to the mass of the sample on the pan. The instrument is calibrated to display in units of mass (grams). The operating principle of the balance is really pretty simple. You only realize the degree of sophistication of the balance when you consider that the balances can accurately differentiate between two samples having masses differing by one part in a million!  Figure 2. Schematic of the electronics of the Mettler College 150 balance. Since the weighing mechanism of the balance relies on complicated electric circuits, each balance must be calibrated periodically to correct for drift in the electronics. Once a week we put the balances into a calibration mode and calibrate the balances using a standard 100.0000 g weight. The computer in each balance then adjusts the balance electronics to correct for any drift. We do not want you to calibrate your balance yourself since incorrect calibration will result in improper measurements for an entire week. However, standard weights are available in the weighing room if you would like to check the calibration of your balance. Be careful not to touch the standard weights with your hands since the oils from your fingers will add a significant amount of mass to the weight.  This experiment was adapted from one created by Dr. Richard Ramette, Carlton College, Northfield, Minnesota. CH 141 Lab: Expt. I -- -- CH 141 Lab: Expt. 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