EXERCISE 1. SCIENTIFIC MEASUREMENTS, DATA TREATMENT AND CALCULATIONS

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1 EXERCISE 1. SCIENTIFIC MEASUREMENTS, DATA TREATMENT AND CALCULATIONS NOTE: YOU ONLY HAVE TO PRINT OUT AND TURN IN PAGES This is a review of some important techniques that you need to handle the data you measure in chemistry: scientific notation, the metric system, dimensional analysis, use of significant figures and plotting data. Refer to your textbook for more details on subjects that are not clear to you. Scientific Notation: Scientific notation is a method used to simplify the expression and handling of very large or very small numbers. For example, the velocity of light in a vacuum is 29,979,000,000 cm/s and the distance between the centers of two hydrogen atoms in an H 2 molecule is cm. How can we express these numbers in a more convenient manner? In scientific notation, a number is expressed in the form a 10 n where a is a number greater than 0 and less than 10, and n is an exponent of 10 (which can be a positive or negative integer, or zero). A number is converted scientific notation in the following way. First, move the decimal point of a number until it is in the ones place. For each place the decimal point is moved to the left, increase n by one. For each place the decimal point is moved to the right, decrease n by one. Therefore, in the above example, More examples are given in Table 1. 29,979,000,000 cm/s = cm/s cm = cm You can perform simple mathematical operations on numbers expressed in scientific notation: 1. Multiplication. Multiply the decimal parts and add the exponents. ( )( ) = ( ) = ( )( ) = ( ) 10 7+(-3) = =

2 Table 1. Examples of Numbers Converted into Scientific Notation Number Number Expressed in Scientific Notation 1,000, , , (10 0 = 1) Division. Divide the decimal parts, and subtract the exponent in the denominator from the exponent in the numerator ( 7) ( + 3) 10 = 10 = Addition and subtraction. Before the numbers can be added or subtracted, first express the numbers so that they have the same exponent. Then add or subtract the decimal parts. The exponent remains constant. ( ) ) = ( ) + ( ) = See the Appendix in your text for more examples. The Metric System: The metric system is a way of expressing measurements that uses units that are all based on factors of ten. This makes interconversion of metric units much simpler than our conventional English system. For this reason, we often use the metric system to report scientific data. Table 2 displays the most commonly used metric units and their conversion into English units. Other conversions can be found your text. 2

3 Table 2. Metric and English Units Property Metric Unit English Unit Conversion Factor length meter (m) inch (in) m/in mass gram (g) pound (lb) g/lb volume liter (L) quart (qt) L/qt Multiples or fractions of the base units are indicated by the use of prefixes. The most frequently used prefixes are given in Table 3. Table 3. Metric Prefixes Prefix Factor Symbol kilo 10 3 k centi 10-2 c milli 10-3 m micro 10-6 µ nano 10-9 n Some examples using metric prefixes are: 1.0 milliliter (ml) equals liters. 2,400 grams (g) equals 2.4 kilograms (kg). Temperature: There are three common temperature scales: Fahrenheit, Celsius (centigrade) and Kelvin (absolute). Of these, the latter two are most frequently used by chemists and almost all calculations that use temperature require it be expressed in Kelvins. The relationships between the temperature scales are given in Table 4. Table 4. Common Temperature Scales Scale Symbol Freezing Point of Boiling Point Water of Water Conversion Factor Fahrenheit F Celsius C C = 5/9 ( F -32) Kelvin K K = C

4 Following are some examples to illustrate the units described above: (a) 23.5 inches to meters and centimeters (23.5 in)( m/in) = m, (0.597 m)(100 cm/m) = 59.7 cm (b) 245 pounds to kilograms (245 lb)(453.6 g/lb)(1 kg/1000 g) = 111 kg (c) 72 F to C 5/9 (72 32) = 0.55(40) = 22 C Dimensional Analysis: Units are an important part of all measurements you make. It makes no sense to say the length of an object is 5.0. Does this mean 5.0 cm? 5.0 in.? 5.0 ft.? How do we convert between different units? Dimensional analysis is a useful way of interconverting numbers that have dimensions or units. The most effective way to solve these problems is by using conversion factors. Whenever you begin to solve a problem, start with the information that you have. Then set up each conversion carefully, being sure not to invert the factor and/or the units. Cancel the units that appear in the numerator and denominator to be sure you ve done the conversion correctly. A few examples are given below: (a) Find the number of meters in yards. 36 in ( yd) 1 yd 2.54 cm 1 in 1 m 100 cm = 91.4m (b) The speed limit is 55 MPH. Express this in m/s. (Notice the two conversions going on simultaneously the conversion from miles to meters, and the conversion from hours to seconds.) 55 mi 1 hr 5280 ft 1 mi 12 in 1 ft 2.54 cm 1 in 1 m 100 cm 1 hr 3600 s = m s (c) What is the weight of 1.0 gal of water? (given 1 gal = 231 in 3, and the density of water is 1 g/cm 3 ) ( 1.0 gal) 231 in3 1 gal 2.54 cm 1 in g H O 2 1 kg 1 cm g = 3.8 kg 4

5 (d) Gold costs about $400/ounce. How much would 1.00 ml cost if the density of gold is 19.3 g/ml? ( 1.00 ml) 19.3 g 1 ml 1 lb g 16 oz 1 lb $400 1 oz = $272 Significant Figures: In science, there are two terms that are regularly used (and confused!) to describe the worth of experimental data: Precision and Accuracy. Precision is described as the reproducibility of measurements obtained in an experiment; in short, how close are the individual measurements to one another. It is assumed that the methods used to obtain the individual measurements were exactly the same. Accuracy, on the other hand, is defined as how close the results are to the true (or scientifically accepted) result. What this means is that you can say nothing about the accuracy of your data unless you already know what the answer is! In almost all work you will do in chem lab you will deal primarily with the precision of your results and not the accuracy (that will be something that the TA or Professor will assess). There is some degree of uncertainty in every measured number because every measuring device has limited precision. For instance, consider two bathroom scales: one with a dial that tell you that you weigh roughly 150 pounds, and a second with a digital screen that tell you that you weigh pounds. The digital scale has a higher precision. When you report your data, you should reflect the precision of the instrument you used in your measurements. But what if you need to convert your weight (150.3 pounds) into kilograms? Multiplying by the conversion factor gives you kilograms, an answer with much higher precision (more decimal places) than you measured. This higher precision is artificial. How can you decide what sort of precision is significant? Unfortunately, most calculators do not understand how many significant figures to report, so it is up to you to decide this. But fortunately, it is not a difficult concept! The rules for significant figures are relatively simple; they are summarized below. Table 5 illustrates the number significant figures in a some measured numbers. Rules for Significant Figure Determination 1. All non-zero integers ALWAYS count as SigFigs. example: the numbers and both have FIVE SigFigs 2. Exact numbers NEVER limit the number of SigFigs in a calculation and as a result are assumed to have an UNLIMITED number of SigFigs. Exact numbers are those that are determined by counting rather than measuring; they also arise from definitions of quantities (such as a reported density, a molar mass, a conversion factor, etc.). examples: a) 102 people were in the room indicates that EXACTLY 102 people were in the room, not 101 or 105 or people. The number 102 is an EXACT number and carries with it an unlimited number of SigFigs. b) 1 lb = 16 oz or 1 mol of atoms = x atoms indicate EXACT quantities by definition and as a result both carry an unlimited number of SigFigs. 5

6 3. Treatment of zeros (note: if you have a problem with SigFigs it will be here). There are three types of zeros: i) Leading zeros (zeros that precede all of the non-zero digits [i.e. are to the left]) NEVER count as SigFigs. example: Say you have a counter on a turnstile that reads The zeros here are NOT significant (think about it...they simply tell you that less than 1100 people have passed through your turnstile). This number has TWO SigFigs. Another example is ; here this number has four leading zeros, none of which are significant...all they do is fix the decimal point. This number has TWO SigFigs. ii) Captive zeros (zeros that fall between non-zero digits) ALWAYS count as SigFigs. example: has three captive zeros ALL of which are significant. This number therefore has FIVE SigFigs. iii) Trailing zeros (zeros at the right end of a number) are ONLY SIGNIFICANT IF THE NUMBER CONTAINS A DECIMAL POINT. You will be able to determine the difference between a trailing zero and a zero in an exact number from the context of a problem. example: 120 has two significant figures, while 120. has three; the decimal point indicates that the zero IS significant. What about ? If you said it has FIVE SigFigs, you d be correct! By the way, if I said that I counted 120 oranges, now that number has THREE SigFigs, since it is now an exact number and should be best represented by writing Keep in mind the following rules when performing mathematical operations on your data: 1. When multiplying or dividing, the result can have no more significant figures than the least precise factor. The "least precise" factor is that number in which you have the least confidence, or that has the most error. 2. When adding or subtracting, the result has no more significant decimal places than the least precise piece of data. See your text for a good discussion of significant figures. 6

7 Table 5. Significant Figure Examples Number Number of Significant Figures Comments All three figures are significant Note 0 s before a number are not significant Note 0 s after a decimal are significant 4,000 1 or 4 This can be ambiguous and is one reason why scientific notation is better See? This way we know this measurement has only one significant figure and this measurement has four significant figures. 12 in/ft Not relevant Defined conversion factors are exact g/kg Not relevant Defined conversion factors are exact. Practice: (a) Go back to the earlier examples and be sure they all reflect the proper number of significant figures. (b) What is sum of 624.3, 0.007, 3.75 and 14? Answer: 642 (the result is limited by the number 14, which has no significant digits after the decimal point). Notice that, because we are adding the numbers, we are not using least precise number in terms of significant figures to limit our answer (0.007 has only one s.f. while our answer has three!) but instead are using the least precise number in terms of decimal places. This is the major difference between addition/subtraction and multiplication/division (next example) and one that you must keep straight! (c) What is the product of 62.4, , and 54? Answer: 49,000 (the result is limited by the number 54, which only has two significant figures). Notice that even though one of the number is extremely precise ( has 6 significant figures) it is the the least precise number that limits the precision of the result! Graphing: A graph is a pictorial presentation of a collection of data. A graph allows you to display your experimental data clearly and can reveal trends or relationships in the data that may not be clear in a table. 7

8 The first step in designing a graph is to organize the data into a table. Not only is it easier to collect and record in a table, it is easier to keep track of your data as you graph it. Choose graph paper that has divisions with the same level of precision as your data. For instance, if your data is very precise, select graph paper that has many small divisions. Equipped with your data and graph paper, you should now decide what each axis represents. Conventionally, the horizontal axis (x) is used for plotting the independent variable in an experiment. The independent variable is that quantity which has been varied by the experimenter. The vertical axis (y) is used for plotting the dependent variable, or observable quantity. This is the quantity that is measured as a function of the other variable. For instance, if we were measuring the growth of trees over time, the x-axis would be time (passing independently) while the y-axis would represent the height of the trees (which varies with time). Remember to label the axes with the names and units of the variables you are plotting. Label your graph as well!! Try to use the full sheet of graph paper and use the longer edge with the variable that has the larger range of values. The scale divisions on your graph should be spaced at regular intervals, for example powers of ten (10, 100, etc.) Use a sharp pencil to plot the actual data points on the graph. To show the relationship between the data points, draw the best straight line or smooth curve through the points. Do not simply connect the dots! If you are plotting more than one set of data on the same graph, small squares or triangles can be used as alternate symbols, so as not to confuse the different sets of data. Whether a graph is linear or curved will usually be obvious once the data has been plotted. Slope and Intercept The steepness of a straight line is called its slope. The slope of a line is calculated from any two points on the line (not necessarily data points; in fact it is preferable that you do NOT use data points). Choose two points on the line that are fairly far apart (see Figure 1), find the difference between the Y values (the rise ) and divide by the difference in the X values (the run ; slope = rise run). slope = y y 2 1 = rise x 2 x 1 run It is important to include the units as well as the sign of the calculated slope. A line will have a negative slope if it is angled down and will have a positive slope if it is angled up. The line in Figure 1 has a positive slope. 8

9 Figure 1. The intercept of a straight line is the point or value on the Y axis when X = 0. Once you know the slope of the line, you can find the intercept by using the equation for a line: Y = (slope) X + (y-intercept) By plugging in any data point values for X and Y, and the slope, you can calculate the intercept. The Appendix of your text should have a good discussion of graphing techniques. NOTE: YOU ONLY HAVE TO PRINT OUT AND TURN IN PAGES

10 Name Date Exercise 1: Treatment of Data Exercises Remember to include the proper number of SIGNIFICANT FIGURES in the following exercises. Express each of the following number in scientific notation. (1) 137,000,000 (2) (3) (4) (5) -98,500 (6) 9,200 (2,000)(850)(0.032) (7) ( )(42,000) = (8) = 46,700 18,000,000 (9) = ( )(24,300)(950) (10) = (64)(2,000)(35.45) 10

11 Convert the following: (11) 17 pounds to grams (12) in to mm (13) 450 F to C (14) 13 pounds water to liters of water. (The density of water is 1.00 g/ml.) (15) 27.0 ml to liters (16) 155 F to K (17) 98.6 F to C (18) 450 nm to cm Solve the following using dimensional analysis: (19) An auto engine has a displacement of 427 in 3. What is this in cm 3? (hint: in 3 = in. x in. x in.) 11

12 (20) Light travels at mi/s. How far does light travel km/yr? (21) Iron has a density of 7.20 g/ml. Calculate the mass of 3.6 x 10 3 cm 3 of iron. (22) A block has dimensions of 2.54 in x 7.0 in x 3 ft. What is the volume of the block in ml? (23) The density of alcohol is 0.8 g/ml. What is the weight of 1.0 liter of alcohol? (24) A solution contains 4.52 g of sugar per liter. How much solution would be required to supply 2.0 kg of sugar? 12

13 (25) Plot the following data on the graph paper provided: (Y) Mass of liquid (g) (X) Volume of liquid (cm 3 ) Determine the slope and y-intercept of your graph. Watch your significant figures! (26) Graph these data on the graph paper provided: Temperature of Water ( C) Vapor Pressure (kpa) (27) Make sure that your answers to (1) - (26) have the proper number of significant figures. 13

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