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youCHEM 181 #0 Working With Numbers.pdf

CHEM 181 #0 Working With Numbers.pdf

Chem 181 Working With Numbers Chem 181: Working With Numbers Objectives: • Recognize the difference between exact, measured, defined, mathematical, and computed numbers • Determine the amount of significant figures in a measured or computed number • Represent large and small numbers in scientific notation • Understand the difference between precision and accuracy • Learn how to properly work with spreadsheet software to perform basic calculations and plot data Introduction: Being able to measure, calculate, manipulate, and interpret numerical data are some of the most fundamental requirements of any scientific field. Chemical experimentation is often performed and analyzed according to these quantitative measures. Recording a mass, determining the efficiency, accuracy, and precision of an experiment, finding relationships between variables, and presenting data in a clear manner all depend on a strong foundation in handling numbers. In this exercise, these topics will be discussed in more detail to provide the necessary background for working with numbers within the chemistry laboratory. Discussion: During the course of your science classes, you will likely come across two key types of measurements: those in which objects are counted and those in which characteristics are measured. If you say there are 20 students in your lab class, then that means there are exactly 20. You cannot have a fraction of a person. Therefore, when counting objects, there is no uncertainty in your measurement. Because of this idea, counted measurements are considered to be exact numbers. Now think about other measurements that may involve looking at specific characteristics such as mass, volume, or temperature. All of these measurements involve the use of a piece of equipment such as an analytical balance, a graduated cylinder, or a thermometer, respectively, and are therefore called measured numbers. Let’s say you were trying to figure out the temperature in the classroom and you had three different digital thermometers. One said the temperature was 23.4oC, the second said 23.2oC, and the third said 23.5oC. Why is there a discrepancy when all of the thermometers were in the same room? Maybe one was near a drafty window or a bright warming light. Perhaps the age or condition of the electronics produced some variation. From this example, we see a very important point: all measured numbers have some degree of uncertainty in the last digit. Every digit, including the last one, are called significant figures, which are defined as digits that are believed to be correct or nearly so. Notice that even though the last digit has some uncertainty, it still must be included in the measurement. Simply reporting the temperature as 23oC is incorrect. 1Adapted from “Handling Numbers” and “Graphing Techniques” by Andrew Weber, Georgian Court University. Chem 181 Working With Numbers Significant Figures The concept of significant figures is one of great importance in the chemistry lab. The number of significant figures tells you the precision of your instrument, and, ultimately, of your data. However, this concept does not apply to all numbers. Significant figures only apply to numbers that express measurements or computations derived from measurements. In the previous section we discussed exact and measured numbers, but there are three other types of numbers you will encounter in your chemistry course. 1. Defined (or designated) numbers 2. Mathematical numbers 3. Computed numbers Defined numbers are those that state the relationships between different units of measurement. For example, stating that there are 1000 milliliters in 1 liter, 2.54 centimeters in 1 inch, or 2.20 pounds in 1 kilogram. Mathematical numbers are numbers such as π (3.1417…) or Euler’s number, e (2.71828…). Because both of these classes of numbers are given definitively, neither type falls under the rules of significant figures. This is also true for exact numbers. Computed numbers, however, are those that are derived from measured numbers. Since measured numbers depend on the concept of significant figures, so too do computed numbers. Therefore, when dealing with either of these two classes of numbers, the following rules must be observed to determine how many significant figures a number has and how many should be reported. 1. All non-­‐zero digits are always significant. The number 6789 has four significant figures, as does 6.789. 2. When determining significant figures, count left to right starting with the first non-­‐zero digit. Therefore, all zeros preceding the first non-­‐zero digit are insignificant. 0.0123 contains 3 significant figures 0.0587359 contains 6 significant figures 0.0000004 contains 1 significant figure 3. Zeros are significant when they: a. Appear between two non-­‐zero digits b. Appear at the end of any number that includes a decimal point and after the first non-­‐zero digit 2005 contains 4 significant figures 109005 contains 6 significant figures 0.0032500 contains 5 significant figures (underlined) 37.00 contains 4 significant figures Chem 181 Working With Numbers 4. Zeros at the end of a number that doesn’t contain a decimal point are ambiguous and are not considered significant. Adding a decimal point indicates they are significant by clarifying the precision of the measurement. 2700 contains 2 significant figures 2700. contains 4 significant figures These rules are important to follow whenever taking measurements in the lab. Additionally, it should be noted that whenever recording data from a digital device such as an analytical balance or a digital thermometer, all digits from the device must be recorded and considered significant. When recording data from analog devices such as alcohol thermometers or graduated cylinders, you must include one digit more than the smallest increment. For example, a 10-­‐mL graduated cylinder has lines measuring tenths of a milliliter (one decimal place), so the volumes should be recorded to the hundredths of a milliliter (two decimal places). Oftentimes, measured data is only obtained in order to perform calculations where additional rules regarding significant figures will also be necessary, the most important of which is that results calculated from measurements can only be as certain as the measurements themselves; therefore calculated values should never be stated with fewer or more significant figures than the least significant number in the calculation. In addition and subtraction, this means the result must be rounded to the placeholder of the least accurately known measurement. For example: 18.1 the result must be rounded to 294.8 since the 273.497 least significant placeholder is the tenths + 3.25 position in the number 18.1. 294.847 In subtraction: 723.19 the result must be rounded to 703.54 since the -­‐ 19.651 least significant placeholder is the hundredths 703.539 position in the number 723.19. In other words, when adding or subtracting, your answer should contain the same number of decimal places as your measurement with the fewest. The procedure for multiplication and division is slightly different in that instead of looking at only decimal places, you must look at the total number of significant figures in each of your measurements. The number of significant figures in your calculated number will be the same as the number of significant figures in your measurement with the fewest. For example: (1.234) x (5.67) = 6.99678 In this example, the result must contain 3 significant figures to match the 3 in 5.67. Therefore it must be rounded to 7.00. Notice that the zeros are kept in the answer since they are significant! . Chem 181 Working With Numbers 22.001/7.0436 = 3.12354478 This answer must contain 5 significant figures since each measurement contains 5. Therefore, the answer must be rounded to 3.1235. Scientific Notation At times in chemistry, you will come across problems that involve very small or very large numbers. For example the diameter of a hydrogen atom is roughly 0.000000000106 meters, or there are about 602,200,000,000,000,000,000,000 molecules in one mole (a unit you will learn much more about in the near future!). It is very cumbersome to write out all of these zeros, and trying to carry them through a calculation greatly increases the likelihood of making a mistake. Therefore, you will frequently see a shorthand expression for these numbers to make them easier to work with. This shorthand is called scientific notation and has the form: N x 10exponent Where N is a decimal number with a value 1 ≤ N < 10 and the exponent is either a positive or negative whole number. The exponent tells the location of the decimal place when the number is written out. When the exponent is positive it means the decimal was moved that many places to the left compared to the original number (i.e. to go back to the original measurement you would need to move the decimal place back that many places to the right). When the exponent is negative it means the decimal was moved that many places to the right compared to the original number (so you would move it left to go back). For example 0.000000000106 meters = 1.06 x 10–10 meters 602,200,000,000,000,000,000,000 = 6.022 x 1023 Note in the first example that the decimal was moved to the right so the exponent is negative and in the second example that the decimal was moved to the left so the exponent is positive. Also note that both factors (N) are given as decimal numbers between 1 and 10 and that the number of significant figures didn’t change. Calculations in the Laboratory As previously discussed, the reason significant figures are so important in the laboratory is that they help the experimenter analyze the precision and accuracy of his or her data. The precision of an experiment is a measure of how reproducible data is, or how close the data is to itself. Recall that a calculation can only be as precise as its measurements. In order to analyze the precision of a dataset, the experimenter should calculate the standard deviation. Standard deviation is a measure of precision with the form: Chem 181 Working With Numbers !
!!!(! − )! −1 Where xi is the individual trial data, is the average of the trial data, and N is the number of trials. The closer the standard deviation is to zero, the more precise the data. For example, if we were to calculate the standard deviation of our three temperatures from before (23.2oC, 23.4oC, 23.5oC), we would do it as follows: = = 23.2 − 23.367 ! + 23.4 − 23.367
3−1 ! + 23.5 − 23.367 ! = 0.153 where 23.367 is the average of 23.2, 23.4, and 23.5 (taken by adding all of the numbers together and dividing by the number of trials, in this case, 3). Since the standard deviation here is relatively low, the precision of the measurements is relatively high. In addition to looking at the precision of data, it is also useful to know the accuracy of your measurements. Accuracy refers to how close experimental data is to the theoretical (or accepted) value and is typically measured by percent error. Percent error is calculated as: − ℎ
% = × 100 ℎ The lower the percent error, the more accurate your data. For example, the density of water is accepted to be 1.00 g/mL; if an experimenter calculates it to be 0.97 g/mL, then the percent error would be: 0.97 − 1.00
% = × 100 = 3% 1.00 It should be noted that even though precision and accuracy are often used interchangeably in colloquial language, they are used for different purposes in the chemistry lab and should not be mixed up. It is also important to note that data can be precise without being accurate, and vice versa. If a student calculates the density of water over a series of tests to be 0.723 g/mL, 0.724 g/mL, and 0.725 g/mL, then the data would be precise (have a low standard deviation) but it would not be accurate (have a high percent error). Similarly, if another student obtained densities of 0.75 g/mL, 1.00 g/mL, and 1.25 g/mL then the average of these values would give very accurate data (low percent error) but the values themselves are very imprecise (high standard deviation). Using Spreadsheets and Graphing Data Finally, obtaining measurements and performing calculations is clearly an important aspect of experimentation, but that data won’t mean much if it is not presented clearly. For this reason, in addition to properly working with numbers, a good Chem 181 Working With Numbers experimenter will need to learn how to manipulate and present his or her data. Here, it is recommended you familiarize yourself with a spreadsheet program such as Microsoft Excel. Spreadsheets are very powerful computer programs that allow users to enter and manipulate numerical data and produce high quality graphs to visualize relationships between datasets. Learning how to use spreadsheets can seem like a daunting task, but they are incredibly useful tools with just a basic level of understanding. It should be noted that your version of Excel may not look exactly the same as the figures shown in this section, but the general procedures outlined should still be very similar and just as applicable. Figure 1. Basic Excel spreadsheet All spreadsheet work begins with the cell. Cells are the array of boxes that appear within the worksheet that can be identified by their column letter and row number. For example, when you open your spreadsheet, the box in the upper left corner would be cell A1, with the one immediately below being A2 and the one immediately to the right being B1. Just as you could use these location names to quickly find a particular cell, so too can Excel use these names as shortcuts. This is what makes calculations possible. When you want to start entering data into a spreadsheet, you simply left click a cell and type your information. In order to move onto another cell, you could click a different box, press Enter, or use your directional arrows on your keyboard. In order to make the most out of a spreadsheet it is a good idea to limit the amount of data you put into each cell. The real power of Excel comes from its ability not only to hold data but also to actually perform functions with that data. Excel can do massive amounts of calculations on very large datasets in extremely short amounts of time. The way of telling Excel that you want it to calculate something rather than just holding the data is by starting the cell with “=”. For example, if you typed =2+2 into a cell and pressed enter, the cell would display 4. Not only can Excel handle basic calculations using the symbols +, -­‐, *, and / for addition, subtraction, multiplication, and division, respectively, but it can also perform more difficult or time-­‐consuming math. This frequently involves referring back to other cells and will be discussed shortly. Some handy functions you might find useful include: • =SUM which can add a large amount of data together • =AVERAGE which finds the mathematical mean of a dataset • =STDEV which takes the standard deviation of a dataset • =SQRT which takes the square root of a number • =(some number)^2 which squares a number • =LN which takes the natural logarithm of a number Chem 181 Working With Numbers =LOG which takes the logarithm (base 10) of a number =MEDIAN which finds the median number in a dataset When performing calculations such as these it is possible to just enter all of the numerical data, but it is more practical to use the callback shortcuts to data already in the spreadsheet. For example, let’s say you wanted to find the average density of water over six different experiments. In the lab you found the density to be 0.96 g/mL, 0.98 g/mL, 1.01 g/mL, 1.02 g/mL, 1.03 g/mL, and 1.06 g/mL. It is certainly fine to find the average by typing in “=AVERAGE(0.96, 0.98, 1.01, 1.02, 1.03, 1.06)” but if the data is already in the spreadsheet, the much more efficient method would be to have Excel refer back to the cells containing the data. If your six densities were in cells A1-­‐ A6, instead of retyping all of the numbers, you could simply find the average by typing Figure 2. Calculations. “=AVERAGE(A1:A6)” or by typing “=AVERAGE(“ and then highlighting the data you want Excel to average and closing the parentheses. For small datasets this may not seem that useful, but for larger experiments that involve tens or hundreds of numbers, Excel can be more helpful than a calculator. Additionally, you’ll find Excel to be a time-­‐saver if you want to perform many calculations at once. For example, let’s say for a different experiment you need to take the logarithm of each measurement recorded in lab. You could calculate this using a calculator or using your new Excel knowledge and typing in =LOG(number) but again this gets tedious and time-­‐consuming if you have a large amount of data. If you organize your data appropriately in your spreadsheet, you can actually perform all of the calculations at once. To do this, type in the function you want to perform into a new cell. After you click on a cell, you should notice a small, solid black, blue, or grey square in the bottom right corner of the cell. Once the function has been typed in, click and drag that box in the same direction as all of your other data and Excel will perform that same calculation for every other number in that group. This same procedure can be applied when performing calculations involving more than one cell. For example, if you were trying to calculate a density, you would divide mass by volume, two numbers you could find in a lab. It is common to organize data in columns in Excel, so you may put all of your masses in column A and all of your corresponding volumes in column B. Then in column C you could type in =A1/B1 to calculate the density. By dragging that small box down, Excel would now calculate A2/B2, A3/B3, A4/B4, etc. You can also type in functions where you always calculate with the same cell. For example, if you obtain volume in milliliters but need it in liters, you may type all of your measured volumes in column A, and then the number 1000 in cell B1 since milliliters/1000 = liters. If you typed in =A1/B1 into C1 then the calculation would happen as normal, but if you went to drag down that small box to carry the function across all of your measurements, you would see error messages. This is because Excel is trying to calculate A2/B2, A3/B3, etc., but now there is no number in B2, B3, etc. In order to get Excel to always just divide all of your numbers by cell B1 (in this case, 1000), you would enter your calculation as =A1/$B$1 instead. Using the two dollar signs tells Excel to lock that cell in for •
• Chem 181 Working With Numbers all other calculations. Now dragging that box will give you all of your volumes in liters as desired. Figure 3. (Left) Performing calculations involving more than one cell; (Right) How to lock a cell for future calculations. Note the solid black square in the lower right corner of cell C3 (Left) and C2 (Right) used for performing multiple calculations. Once you have your data and calculations properly organized, you may want to provide a graph as part of your experimental analysis. Excel can generate many different types of charts and graphs but for this experiment we will focus on linear plots since that is what you will encounter most frequently in the chemistry lab. In order to create a graph on Excel, you will want to put the data for your independent variable in column A. The independent variable is the experimental data that is under the control of the experimenter and it belongs on the x-­‐axis (horizontal axis) on a graph. You then need to put your dependent variable in column B. The dependent variable is the condition that depends on the other variable and gets placed on the y (or vertical) axis. With your data correctly placed, click and drag over the whole dataset to highlight the two columns. Then, depending on your version of Excel, you will either click the Insert tab or the Charts tab. Within the Charts submenu, find the option labeled “Scatter” and select “Marked Scatter” or “Scatter with only Markers”. Figure 4. Standard look of the insert tab with several chart options. Chem 181 Working With Numbers Figure 5. Scatter chart menu; notice the “Marked Scatter” option in the upper-­‐left. A graph featuring nothing but data points should now appear next to your data, but it is missing many key features. As previously mentioned, we are assuming that we are looking for a linear relationship. This means we want to find the best-­‐fit line to see how the data points are connected. To add the best-­‐fit line, right-­‐click on one of the data points themselves. Click the option that says “Add Trendline”; then within the trendline menu, under Options, check the boxes labeled “Display equation on chart” and “Display R-­‐squared value on chart”. Recall from algebra that the equation of a line is given by y = mx + b, so the equation of the line gives the slope (m) and y-­‐intercept (b), which may be useful in finding key relationships. The R-­‐squared value is called the coefficient of determination and is a measure of how well the data points fall on the best-­‐fit line, where an R2 value of 1 shows perfectly linear data. Figure 6. Data point menu. Notice the “Add Trendline” option second from the bottom. Chem 181 Working With Numbers Current, mA Figure 7. Format Trendline options. Notice the “Display equation on chart” and “Display R-­‐
squared value on chart” options checked on the bottom. You should also always include axes and chart titles so that anyone looking at your graph can immediately identify what is being compared and make their own conclusions about the data. To add titles, go to the Chart Layout tab and use the appropriate dropdown arrows to add titles as necessary. You may also want to delete the legend since you only have one series of data and change or delete the gridlines to further clean up the graph, but these are more aesthetic fixes that fall to personal preferences. Overall, a lot of this business with handling numbers and utilizing spreadsheets seems complicated. However, these are essential topics when working in the chemistry lab. Though it may be daunting at first, a bit of practice goes a long way towards stronger levels of understanding. As you become more comfortable you will hopefully find these topics to be useful even outside of the laboratory. Change in Current with Increasing Concentration of Known Conductivity 45.00 40.00 35.00 30.00 y = 4.4365x + 1.9167 25.00 R² = 0.98353 20.00 15.00 10.00 5.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 Conductivity, mS Figure 8. Final plot with axes and graph titles, line equation, and R2 value. Chem 181 Working With Numbers Name _________________________________________ Date _________________________ Sec. No. ___________ Working With Numbers Pre-­‐Laboratory Assignment 1. How do exact numbers differ from measured numbers? 2. How many significant figures does the measurement 1000 cm have in it? Explain your answer. 3. What does the exponent tell you when a number is written in scientific notation? 4. If an experiment was expected to produce 1253 J of heat but only produced 1197 J, what was the percent error? Show your work. Chem 181 Working With Numbers 5. An experiment was performed to look at how density changes when the ratio of water and ethanol was changed. The following data was obtained: % Mass 1.00 2.50 3.50 4.00 6.00 7.50 9.50 Ethanol Density, 0.9963 0.9936 0.9918 0.9910 0.9878 0.9855 0.9826 g/mL Use Excel or a similar spreadsheet program to plot these six data points and answer the following questions. a. What is the independent variable? b. What is the dependent variable? c. What is the equation of the best-­‐fit line? d. What is the value of the coefficient of determination? e. According to the best-­‐fit line, what should the density of pure water be (i.e. what is the density when % mass ethanol = 0.00)? Show your work. Chem 181 Working With Numbers Name _________________________________________ Date _________________________ Sec. No. ___________ Working With Numbers Post-­‐Laboratory Assignment 1. Identify the following numbers as exact, measured, calculated, mathematical, or defined. a. The number of centimeters in one meter _______________________________ b. The number of students in a classroom _______________________________ c. The number π _______________________________ d. The circumference of the Earth _______________________________ e. The weight of a bag of apples _______________________________ f. The number of apples in that bag _______________________________ 2. Determine the number of significant figures in each of the following numbers. a. 1932.75 __________ e. 0.42930000 __________ b. 1004 __________ f. 12743500. __________ c. 75432000 __________ g. 0.0000000062 __________ d. 0.0093392 __________ h. 23490001243 __________ 3. Convert the following numbers to scientific no...

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