63 experiment3 Introduction to Data Analysis LECTURE AND LAB SKILLS EMPHASIZED Determining what information is needed to answer given questions. Developing a procedure which allows you to acquire the needed information. Taking quantitative measurement on a sample. Learning how to use basic laboratory equipment. Using quantitative data to determine density mathematically and graphically. Students will work in pairs. Parts may be completed in any order. IN THE LAB Record your procedure and original data in your lab notebook along with your calculations.
64 EXPERIMENT 3: Report data collected and subsequent calculations to www.chem21labs.com. All equipment should be returned to the correct location after use. WASTE Aluminum foil can be disposed of in the trash can. If you use any liquids other than water, contact your TA for disposal information. SAFETY Safety goggles are mandatory when anyone is performing an experiment in the lab. Long pants, closed-toed shoes, and shirts with sleeves. Clothing is expected to reduce the exposure of bare skin to potential chemical splashes. Aprons are available for students wishing to have an extra layer of personal protection. Even if you are not working, you cannot anticipate what others might do. This rule is understood to be effective for the remainder of the semester, unless otherwise noted, and will not be repeated for each experiment. Additional information can be found at http://genchemlab.wordpress.com/3-mass-volume-and-units/ As your first project at the research facility, your supervisor wants you to demonstrate that you have successfully learned the skills demonstrated in your training session and can apply what you ve learned in school to answer two questions: 1) what is the density of aluminum foil, and 2) are aluminum foil sheets a cost-effective approach to their current source of aluminum? You must develop two methods or procedures to determine the density by thinking about what you can measure in the lab and what data you need to answer the questions your supervisor asks for in the data analysis section. You must have this procedure to follow in your lab notebook prior to completing the experiment in order to obtain the necessary data to complete the experiment. Here are some suggested things to think about when developing your procedure: How many times should you measure the mass of the aluminum? Should you measure samples of different sizes? What factors should be considered when determining the volume of aluminum by displacement? What shape is best for determining the volume by displacement? Will aluminum sink or float in water? What equipment is available to you in the lab? Some information that you should know: A box of foil sheets costs $9.28. The box contained 500 sheets of 12" 10¾" foil. (This is not the size you will get.) You will need at least five data points (mass volume pairs) to complete the data analysis. You should collect six or more. Dirty glassware has an effect on every experiment you do.
EXPERIMENT 3: 65 PART I MATHEMATICAL CONSIDERATIONS IN CHEMISTRY Pay Attention to Units Units play an important role in chemistry. Without units, numbers are meaningless. How important are units? For NASA, they are critical. The Mars Climate Orbiter was launched on December 11, 1998, for a mission to explore the atmosphere and climate of Mars at a cost of $327 million dollars. 1 Because of an error with the units on the calculations, the navigation of the spacecraft was incorrect and it most likely crashed. 2,3 While it won t cost you $327 million dollars if you forget a unit, it could mean the difference between passing your exams or not. Failure to use units correctly or not using them at all is a common error that students make when working problems. If the units are not correct, then the answer cannot be correct. One approach to solving problems is to see what units are given and what units are needed. Then determine the map to get from the beginning to end. For example, if you want to convert 2.5 miles to inches, you need to start with the units of miles. Write down the relationships or conversion factors that you do know that you might need. 1 mile 5280 feet 3 feet 1 yard 1 foot 12 inches It is okay if you start with more conversions than you actually use. For many problems, there is more than one way to solve it correctly. Set up the calculation first without numbers to make sure you have the units correct. miles c feet inches inches mc m feet 1 http://mars.jpl.nasa.gov/msp98/orbiter/fact.html 2 http://plus.maths.org/issue10/news/mars/ Look at the above units and see that miles will cancel with miles, feet will cancel with feet, and inches remain. If you do not have the correct units at this point, try again. Once the units are correct, then add the numbers and perform the calculation. 2.5 miles 5280 feet 12 inches c mc m= 158400 inches 1 mile 1 foot However, the calculator often gives us more numbers than we actually need so we need to determine the correct number of significant figures for the answer and decide whether it should be reported in scientific notation. Because the conversions contain exact numbers, they are not considered in determining the correct number of significant figures. The answer should have two significant figures which can be written two different ways. 160000 inches or 1.6 10 5 inches The first option can be considered as having two or six significant figures. However, it is very clear that the number written in scientific notation only has two significant figures and makes it the best answer for the problem. Significant Figures In laboratory experiments you will be making many measurements of physical quantities. The reliability of a measurement frequently depends on the quality of the instrument used to make the measurement, in addition to the skill exhibited by the experimenter in making the measurement. The number you record in your laboratory report should convey not only the value of the measured quantity, but also the reliability of the measurement. This reliability is indicated by the number of significant figures used to represent the measurement. In most volumetric measurements, you can estimate (interpolate) a reading to one more place than the actual calibrated markings. The last place is uncertain, and represents your best estimate of its value. It may be off by one unit, but is unlikely 3 http://mars.jpl.nasa.gov/msp98/news/mco990930. html
66 EXPERIMENT 3: to be wrong by more than two units. In the laboratory, always record the proper number of significant figures for the measuring device you are using. You should record neither less nor more than this number of figures. Treating Measurements in Calculations Your measurements are treated in calculations by the same rules for determining significant figures as you were taught in lecture. 1. Nonzero digits in a reported value are always significant. Example: The number 5.4 has two significant figures, but the value of 5.40 has three significant figures. 2. Zeros are not significant when they are after a decimal point but before the first nonzero digit. If the zero comes after a decimal point and after a nonzero digit, it is considered significant. Example: In the value of 0.003, only the 3 is significant, and the number has 1 significant figure, but the value of 0.0030 has two significant figures. 3. Zeros between nonzero digits are significant. Example: The value of 5.04 has three significant figures, but the value of 5.040 has four significant figures and the value of 5.0404 has five significant figures. 4. All digits in scientific notation are significant. Example: The number of significant figures in the value of 400 is ambiguous. By writing the number in scientific notation, i.e., 4.00 10 3, we can eliminate that ambiguity and state that it has 3 significant figures. 5. Exact numbers obtained by counting or from definitions are exact numbers. Example: Addition and Subtraction To calculate the average of the values 115, 125, and 139, we sum them up and then divide the sum by 3. Since the number 3 is an exact number, the result is reported to three significant figures based on the numbers being averaged. Most exact numbers are whole numbers (i.e., there are 5 people on the elevator); that is not always the case. One inch equals exactly 2.54 cm. When adding and subtracting numbers, retain only as many significant figures in the decimal portion of the number as in the least significant of the values. The nonzero digits to the left of the decimal are all significant. Example: 76.32 5.465 0.58543 82.37 The sum of the three numbers above can have only two significant figures after the decimal point, because the number 76.32 is the number with the least number of significant figures after the decimal point (only 2). Multiplication and Division When multiplying and dividing numbers, retain in the result only as many significant figures as are contained in the number with the least number of significant figures. Example: 0.286 25.44 3.6453 26.5 The product of these three numbers can have only three significant figures, because the least accurately known number (0.286) has only three significant figures.
EXPERIMENT 3: 67 Rounding In rounding off numbers (the number of significant figures is reduced), the last digit retained is increased by 1 only if the following digit is a 5 or greater. Examples: If 6.5457 is to be rounded to 4 significant figures, it would become 6.546. If 6.5453 is to be rounded to 4 significant figures, it would become 6.545. PART II CONSIDERATIONS TO UNCERTAINIY How certain are you with your measurement? Would someone else measure the volume of water in a graduated cylinder differently? How can you express to someone else your uncertainty with your measurement? To answer these questions, scientists consider accuracy and precision of their measurements and quantify the accuracy and precision using mathematical techniques such as standard deviation and percent error (among other methods). Precision and Accuracy Many of the experiments you will perform in this course require you to measure various physical quantities, such as mass, volume, or temperature, and to use those data to calculate the value of an unknown quantity. In order to obtain good results, you must know proper techniques for collecting data, and recognize possible errors in the process. You must also be able to evaluate the quality of the data you collect, and to state your results in a meaningful way. Since an experimentally determined quantity always has some associated uncertainty, it is important to be able to give an estimate of this uncertainty in your results. In the process of evaluating your data, you will: determine the precision and accuracy of your results. subject questionable data points to data rejection methods. calculate the percent error in your answers. Precision refers to the agreement among a set of measurements that were made in exactly the same way. There are several ways to express the degree of precision. One is as deviation from the mean ; another is as relative deviation. Methods for calculating these are discussed in the next section. Accuracy refers to the agreement between a measured value and its accepted, or true, value. It is usually expressed as error, either absolute or percent. Calculation methods are discussed in the next section. Figure 3.1 illustrates the difference between precision and accuracy. Hayden-McNeil, LLC High accuracy High precision Low accuracy High precision High accuracy Low precision Low accuracy Low precision Figure 3.1.
68 EXPERIMENT 3: Standard Deviation In most real experiments, the true value of a quantity is not known. Therefore, we must find a way to use our data to get the best possible estimate of the true value for the quantity being determined. One common estimate of the true value is the mean (X). The mean is simply the arithmetic average of all the data points: / Xi... X x1 x2 x3 = = x n + + n + + n The usefulness of the standard deviation is that it is expressed in units of the original measurement, and can be used to describe the position of any observation relative to the mean. It can be shown mathematically that, for a distribution with an infinite number of replicate measurements, 68.3% of the observed values will fall within ± 1s of the mean; 95.5% will fall within ± 2s of the mean; and 99.7% within ± 3s of the mean. where X = the mean value (or average), Σ = the sum of, X i = the individual data points (i = 1, 2, 3,, n), and n = the total number of data points. One way to express precision is by means of the standard deviation. To discuss this, we must first discuss the normal distribution. Frequency 68% 95% 99% -3s -2s -1s +1s +2s +3s Measured characteristic Figure 3.2. If a very large number of determinations of a quantity are done, all of the values will not be exactly the same, due to random errors. On these graphs, X represents the mean, which is the best estimate of the true value. The width of the curve indicates the precision of the measurements. A tall, thin curve would indicate good precision, while a broad, flat curve would show poor precision. The standard deviation can be used to measure the width of a normal distribution. The standard deviation is defined as: Example Suppose that a density determination of a liquid is done in the laboratory, and the following data are obtained: Experiment Number Density (g/ml) 1 2.60 2 2.90 3 2.70 4 2.90 5 2.50 s = / ^Xi - Xh ^n - 1h 2 From this data, calculate the average and standard deviation for the results. where s the standard deviation, n the number of observations. Step 1: Calculate the sum of the data: Σ X i = 13.60 Step 2: Calculate the average of the values: / X Xi = = 13.60 = 2.720 n 5
EXPERIMENT 3: 69 Step 3: Calculate the absolute value of the deviation of each result (d X i X ), the sum of the deviations (Σ d ), the square of d values ( d 2 ), and the sum of the square of the d values (Σ d 2 ). Tabulate these values in a new table: Experiment Number X i X i X X i X 2 1 2.60 2.60 2.720 = 0.12 0.014 2 2.90 2.90 2.720 = 0.18 0.032 3 2.70 2.70 2.720 = 0.020 0.00040 4 2.90 2.90 2.720 = 0.18 0.032 5 2.50 2.50 2.720 = 0.22 0.048 n 5 Σ X i 13.60 Σ (X i X) 2 0.13 Step 4: The standard deviation can then be calculated from the formula: s = 0.13 0.177 ^5-1h = Thus, we could state that the result of the density determination together with its standard deviation is 2.72 ± 0.18 g/ml. Note that the average cannot have more significant figures than the measurements that make up the average and that the standard deviation has the same number of decimal places as the average. As the value of the standard deviation tells us about the variation or spread of our data points within our measurements and its uncertainty, it does not make sense to include more than one significant figure in the standard deviation. Our measurement is correctly reported as 2.7 ± 0.2 g/ml. This indicates that our measurements were variable or not precise. Percent Error When doing an experiment that has been done before, it is useful to evaluate the quality of the results because it gives you an indication of how well you did the experiment. Percent error can be used to determine the accuracy of the results. Accuracy is how close your experimental value is to the accepted value. Precision is how close your values are to one another for multiple trials of the same experiment. One way to evaluate the accuracy of your results is to determine the percent error in your experimental value using the equation shown. % error = accepted value - experimental value # 100% accepted value Note the absolute value sign in the formula which results in the percent error always being a positive value. While having a low percent error is important, in this course, the focus will be on learning how to calculate the percent error and understanding why, if you have a high percent error. Regardless of the value of the percent error, you should always include the calculation in your results section and include it in the summary of results presented in your discussion. Additionally, you should also discuss potential sources of error in the discussion even if you have a low value for the percent error. A low percent error doesn t necessarily mean that everything was done perfectly, since you could have had two sources of error that offset one another. One way to come up with potential sources of error is to look at each step of the procedure and ask yourself, What could have gone wrong in this step? See the information on lab reports for more detailed information about what to include in your discussion including information about sources of error. It is very important to realize at this stage that you can have a very small deviation in your data (indicating high precision) but your result may be significantly off from the true value (if the accuracy is low).
70 EXPERIMENT 3: Example Refer to our previous standard deviation calculation. We determined the density of our liquid to be 2.72 g/ml. Suppose that the accepted value in the literature for the density is 2.43 g/ml. The relative or % error of the result is: % error = accepted value - experimental value accepted value 243. - 272. = 243. = 11.9% # 100% # 100% The precision of the experiment is given by the average value ± standard deviation, while the accuracy of the experiment is given by the percent error. In our case: precision: 2.72 ± 0.18 accuracy: 11.9% error It is important to note that you cannot comment on the accuracy of the experiment unless you know the actual value of the unknown that you are investigating. Density is an example of a physical property of a substance and can be used to identify an unknown because the density of a sample at a given temperature is constant. It is important to note that the density is constant at a given temperature because as the temperature changes, so will the density. Look in Table 3.1 to see how the density of water changes with temperature. Why does the density of water change when it freezes? Does the mass of the sample change with temperature? Does the volume of the sample change with temperature? The mass of the water stays the same, but the volume increases as the water temperature decreases which leads to a lower value of density. Water is fairly unique in this respect since most substances experience a decrease in volume with a decrease in temperature. Table 3.1. Density of water as a function of temperature. 4 Temperature ( C) Density (g/ml) 0 0.99984 20 0.99821 40 0.99222 60 0.98320 80 0.97182 100 0.95840 PART III DENSITY CONSIDERATIONS Density (d) is a mathematical combination of mass (m) and volume (V) which all states of matter have as shown in the following equation. d = m V Many of the solutions used in this course will be aqueous (water-based) solutions. Water has a density of 1.00 g/ml at 20 C. Substances that have a density greater than 1.0 g/ml will sink when placed in water while substances with a density less than 1.0 g/ml will float. Ice floats because it has a density of 0.98 g/ml. While there can be a variety of units for density, it is typically reported as g/ml for most solutions and solids. 4 Properties of Water in the Range 0 100 C in CRC Handbook of Chemistry and Physics, ed. David R. Lide, 6 10. Boca Raton: CRC Press, 1993.
EXPERIMENT 3: 71 Graphically Determining Density The density of a sample can be determined using the previous equation but with several data points, graphical methods are more effective. The equation for a line is expressed as y mx b where m is the value of the slope and b is the y- intercept. A math review can be found in the appendix of your chemistry textbook. The density equation can be related to the equation for a line. The graph of mass versus volume will have an intercept at zero (b 0) because if a sample has no volume, then it must have no mass and vice versa. The density equation can be rearranged to get m dv y mx (with b 0) Linear regression line m = dv through the data Using several pairs of mass volume data, a graph can be constructed whose slope is the value of the density for that substance at that temperature. We must assume a constant temperature throughout the experiment in order to use this technique because density changes with temperature. Volume by Displacement For objects whose dimensions cannot be easily measured, the easiest way to determine the volume is by displacement of a liquid. If a known volume of liquid is in a graduated cylinder and a solid is added to it, then the solid will displace an amount of liquid equal to its volume. This is known as measuring volume by displacement. It accounts for small holes or for odd-shaped objects since the liquid can easily surround the shape completely. There are two conditions that are necessary for this to work correctly and they are 1. The liquid must be one that will not react with the solid in the time frame of the experiment. 2. The density of the solid must be greater than that of the liquid so that it will sink and displace its entire volume. Mass Slope of the line gives the value of the density PART IV MATERIALS CONSIDERATIONS The following items are available for your use in the lab: Volume Figure 3.3. Graphical method of determining density. Standard laboratory equipment and supplies including balances, graduated cylinders, burets, beakers, etc. Pieces of aluminum foil Rulers If you believe that you need additional equipment, please contact your TA.
72 EXPERIMENT 3: Data Analysis Make sure to show all of your calculations in your lab notebook as a record of how you completed your calculations. Don t forget to include your units and correct number of significant figures! Then, go onto Chem21 and report your results. 1. Determine the cost per square inch of the aluminum foil. Report answer in units of dollar/ in. 2 2. Determine the grams per square inch using the data in Trial 1. 3. Determine the cost per gram for the aluminum foil. Report answer in dollar/grams. 4 8. Determine the density of the aluminum foil for each trial (1 5). 10. Determine the standard deviation for the density of the aluminum foil. 11. Determine the density of the aluminum foil using the graphical method. 12. Using an online or print source, find the accepted value for the density of aluminum, including the temperature at which is it accurate. Include a reference to the source in your lab report. 13. Determine the percent error in the density based on your average calculated density value. 14. Determine the percent error in the density based on your graphical density value. 9. Determine the average density of the aluminum foil.