# Chapter 1: Measuring Up

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2 2 Discovering Design With Chemistry Now, to read how long the pencil is, we have to see how the ruler is marked off. Each number represents an inch, and there are 6 lines going from one inch mark to the next. That means the ruler is marked off in 6 th s of an inch. However, since there is space in between each 6 th -inch mark, you can actually determine whether or not the end of the pencil hits one of those marks exactly or is somewhere in between two of the marks. Let me give you the illustration again so you can do that now: If I look at the ruler, I see that the pencil goes from the -inch mark to the 6 th mark past the 8-inch mark. However, it actually goes past the 6 th mark but doesn t make it to the 7 th mark. So it is somewhere in between the mark and the mark. As a result, I would say it is at, which is halfway in between and. So I would say that the pencil is 7 inches long Usually, we like to write measurements as decimals, so I would like to get rid of the fraction. If I divide 3 by 32 (which is what the fraction means), I get So that means the pencil is inches long, right? Not quite. That measurement has too many digits in it. When a scientist reports a measurement, she has to be certain to report it to the proper precision. A ruler just isn t precise enough to measure the length of something to one hundred thousandth of an inch, which is the decimal place to which goes. A ruler is, at best, precise to 0.0 inches. So in the end, you have to round your measurement to the hundredths place. Thus, the pencil is 7.4 inches long. Note that to round the number to the hundredths place, I had to drop the 6 that is in the thousandths place. When you are forced to drop a number that ranges from 5 to 9, you add one to the number to its left. This is called rounding up. So, in order to report the length of the pencil to the proper precision, I had to round the measurement up from inches to 7.4 inches. This might sound like we are going to a lot of trouble to measure the length of a pencil, but when scientists make measurements, they have to be careful to report their answers properly. As a result, they have to know the precision of the instruments they are using. The precision of a ruler marked off in 6 th s of an inch is something you need to remember: When using a ruler that is marked off in 6 th s of an inch, report your answer to hundredths of an inch. Using Different Units Now let s suppose you want to measure the length of a pencil, but you don t live in the United States. Most likely, the ruler you have wouldn t use inches. Instead, it would probably use centimeters (abbreviated as cm ), which is another unit for measuring length. Unit A quantity that describes the measurement being made

4 4 Discovering Design With Chemistry Significant Figures Believe it or not, we aren t done with measuring length. Once again, measurement is very important in chemistry, so you have to be sure you are doing it correctly. Suppose you were measuring the length of a different pencil, as shown below: How would you report that length? As far as I can tell, the pencil lines up perfectly with the 9-cm mark. Since the pencil starts at cm and ends at 9 cm, you might be tempted to report the length as 8 cm, but that would be wrong! Remember, a ruler like this is precise enough to be reported to the hundredths of a centimeter. Thus, the correct answer for the length of the pencil is 8.00 cm. Even though the mathematical value of 8 is the same as 8.00, from a scientific point of view, there is a huge difference between a measurement of 8 cm and a measurement of 8.00 cm. When a scientist reports a measurement, he must be careful to report only the numbers he has measured. He cannot report numbers that he hasn t measured. If a scientist says, The length of the pencil is 8 cm, that means he knows the pencil is somewhere between 7.5 and 8.4 cm long. Since any number between 7.5 and 8.4 can be rounded to 8, any of those lengths would be consistent with what the scientist said. However, if the scientist says, The length of the pencil is 8.0 cm, that means he knows the pencil s length to a higher precision. He knows that it is somewhere between 7.95 and 8.04 cm long, because any of those measurements round to a length consistent with what he said. If he says, The length of the pencil is 8.00 cm, then he knows the length even more precisely. It is somewhere between and cm long. When a scientist reports a measurement, then, he reports all the digits he measured and no more. In our case, the ruler allows us to read to the hundredths of a centimeter, so we can actually measure the length of the pencil to the hundredths of a centimeter. Regardless of the actual value of the length, then, you must report it to the hundredths of a centimeter, because that tells everyone who reads the measurement how precisely the pencil was measured. When a digit in a number is actually measured, it contributes to the precision of the measurement, and we call it a significant figure. Significant figure A digit in a measurement that contributes to the measurement s precision It s important to remember that the word significant doesn t mean mathematically important. From a mathematical point of view, the two zeroes in 8.00 cm aren t important. However, from a scientific point of view, they are significant, because they have been measured. Scientists must be very careful to report all of their measurements using the proper significant figures. When you read a measurement, how can you tell whether or not a figure is significant? Here are the rules:. All non-zero figures (, 2, 3, 4, 5, 6, 7, 8, and 9) are significant. 2. A zero is significant if it is between two significant figures. 3. A zero is also significant if it s at the end of the number and to the right of the decimal point. When you read a measurement like.679 inches, then, all of the figures in that number were measured, because according to rule #, they are all significant. In a measurement like cm, however, only the 4, 5, and 6 were measured, because they are significant. Those first three zeroes are not significant

5 Chapter : Measuring Up 5 because none of them are between significant figures, and none of them are at the end of the number. See if you can understand how to use these rules in the following examples. Example. How many significant figures are in 02.0 inches? Rule # tells us that the and 2 are significant. Rule #2 tells us that the zero in between the and 2 is also significant. The last zero is both at the end of the number and right of the decimal, so Rule #3 tells us it is significant. In the end, then, all four of the figures in 02.0 inches are significant. How many significant figures are in cm? The first two zeroes are not between significant figures. They are also not at the end of the number. Thus, they are not significant. Rule # tells us the 4 and 5 are significant, and that makes the 0 in between them significant, according to Rule #2. As a result, there are three significant figures. How many significant figures are in 2,000 inches? Rule # tells us that the and 2 are significant. The zeroes are all at the end of the number, but that s not enough. Rule #3 says they must be at the end of the number and to the right of the decimal. Thus, even though they are mathematically important, they are not significant. That means there are two significant figures. Don t think of this as some useless exercise about following rules. It is very important to know which figures in a measurement are significant and which are not, because that tells you what was actually measured. Make sure you understand this by answering the following question. Comprehension Check 2. How many significant figures are in each of the numbers below? a in b. 2,007 cm c. 609,000 cm d in The concept of significant figures allows me to make an important distinction. People often talk about precision and accuracy as if they are the same thing, but to a scientist, they are completely different. Precision is determined by the instrument you are using to make your measurement. If you have a very fine measuring instrument, you can report your measurement with a lot of decimal places in it. Accuracy, on the other hand, indicates how correct your measurement is. You can use a very precise instrument, but if you use it wrongly, you will end up with an answer that is precise but not accurate. For example, we know that a U.S. dollar bill has a length of 5.6 cm. Suppose someone uses a very precise measuring device wrongly and measures the length of a dollar bill to be cm. That s a very precise answer, because it has a lot of decimal places in it, but it isn t anywhere close to the correct answer, so it is not accurate. Suppose someone else uses an imprecise instrument and reports that the dollar bill is 6 cm long. Even though the measurement is not very precise (it doesn t even report the answer to tenths of a centimeter), it is very accurate, because the correct length of a dollar bill, reported to two significant figures, is 6 cm. Accuracy and precision, then, are different things. Precision is based on the significant figures involved, while accuracy depends on how close the measurement is to being correct.

9 Chapter : Measuring Up 9 inches and divided the two. This should give you the number of centimeters in an inch. If you did your measurements correctly, all three of your numbers should be consistent with a value of 2.54 cm/in, which is the accepted value for how many centimeters are in an inch. What does it mean to be consistent with? You have to think about significant figures. This is why they are so important. When you made your measurement, you estimated the last significant figure, didn t you? After all, the metric ruler was marked off in tenths of a centimeter. You estimated where the edge of the book fell in between the marks to report your answer to the hundredths place. This tells us something pretty important: There is always some error in the last significant figure of a measurement. When scientists compare measurements, they take this into account. For example, suppose you got an answer of 2.52 centimeters per inch. That s consistent with the correct value. After all, there is error in the last significant figure. Because we know that, two measurements can be different in their last significant figure and still be consistent with one another. In the same way, if you got an answer of 2.57 centimeters per inch, that s still consistent with 2.54 centimeters per inch. How much error is acceptable in the last significant figure of a measurement? Would 2.59 centimeters per inch be consistent with 2.54 centimeters per inch? In some ways, it s a judgment call that varies from scientist to scientist. For the purposes of this course, as long as one measurement s last significant figure is no more than 3 different from the last significant figure of the other measurement, the two are consistent with one another. Converting Between Units Now that you know a relationship between centimeters and inches, you can switch between the two rather easily. This process is called unit conversion, and you will use it over and over again in chemistry. There are several ways you can do a unit conversion, but I am going to teach you a method that applies to many different situations in chemistry. As a result, even if you are already comfortable converting inches into centimeters and vice-versa, you still need to learn this method, as it will apply to a lot of what you will study in this course. The method you will use is called the factor-label method, and it makes use of a fact that you learned when you did arithmetic. When multiplying fractions, you were probably taught to look for numbers that appear in both the numerator and denominator of the fractions you were multiplying. For example, in the following multiplication: 7 25 You should have been taught that the 25 s cancel, because in a fraction, you are simply dividing the top number by the bottom number. Since =, the only numbers left are the 7 and 2. Thus: While you are probably used to recognizing this fact when it comes to numbers, remember that the same thing applies to units! After all, what happened in Example.3 when we divided centimeters by centimeters? They cancelled, leaving no unit! 25 2 = 7 2

10 0 Discovering Design With Chemistry How does this help? Well, let s go back to the results of your experiment. You learned that there are 2.54 centimeters per inch. Even if you didn t get that exact answer, it s the accepted value, so it s the one we are going to use. What does that mean? It means that a measurement of 2.54 centimeters is the same as a measurement of one inch. In other words: 2.54 cm = in This is called a conversion relationship, because it tells you the relationship between two different units. Since the two measurements in a conversion relationship are equal, think about what the following fractions represent: 2.54 cm in OR in 2.54 cm When I take two equal things and put them over each other in a fraction, what happens? They cancel each other out, equaling. What does that tell you? It tells you that you can multiply anything by either of the fractions above, and you aren t changing its value, because you are just multiplying by something that equals, and multiplying by doesn t change the value of anything. So, let s suppose I have a measurement like 2.0 inches. How many centimeters would that be? Well, I can take that measurement and make it into a fraction by putting it over. After all, you can make any number a fraction by just putting it over : 2.0 in Now I can multiply this by either of the fractions I showed you before, because they both equal. However, I will be very careful about how I do that. I will multiply by the fraction on the left. Why? Look what happens: 2.0 in 2.54 cm = 30.5 cm in Notice that the unit of inches appears in the numerator of the first fraction and the denominator of the second. Because of this, it cancels. That leaves us with only one unit centimeters. That s the unit we wanted! So we now know that 2.0 inches is the same as 30.5 cm. This is the way you will use the factor-label method when converting units. You will put the measurement you want to convert over so it becomes a fraction. Then, you will multiply by a fraction you build from the conversion relationship. The fraction you use should have the unit you want to get rid of on the bottom and the unit you want to keep on the top. That way, the old unit cancels, and the new unit is what s left. I will show you a couple of examples in a moment, but first, let s worry about the significant figures. Most conversion relationships don t bother to list all the significant figures. This is because most of the conversion relationships you will see, including the one between inches and centimeters, are exact. That means inch is exactly centimeters. In other words, both numbers in the conversion relationship have an infinite number of significant figures. This means when you do the conversions in this chapter, you don t have to worry about the significant figures in the conversion relationship. You only have to count the significant figures in the original measurement

12 2 Discovering Design With Chemistry yards. If I wanted to measure the distance between cities, I would probably use miles. Each unit is good for measuring a specific kind of distance. In the metric system, there are also different units, but unlike the English system, all these units are defined relative to a base unit. The base unit for measuring distance in the metric system is the meter (abbreviated as m ). It is about the same length as a yard, so whatever you might measure in yards in the English system can be easily measured in meters in the metric system. An American football field, for example, is 00.0 yards from goal line to goal line. That s the same as 9.44 meters. So what s a centimeter? It s one hundredth (0.0) of a meter. In the metric system, you modify the base unit to come up with a unit that is appropriate for what you are measuring. When you measure the length of something like a pencil, using centimeters makes sense, because a pencil is much, much shorter than a meter. So you modify the base unit of the metric system with the prefix centi, which means 0.0. If you want to measure something even smaller, say the length of a grain of rice, you might want to use an even smaller unit. The millimeter is one thousandth (0.00) of a meter, because the prefix milli means A long grain of rice is about 7 millimeters long, while a short grain is about 4 millimeters long. On the other hand, if you want to measure something really long, like the distance between two cities, you might use the kilometer, which is,000 meters. The driving distance between Washington, DC and New York, NY, for example, is 365 kilometers. There are a lot more prefixes in the metric system, which makes it very flexible. The table below summarizes a few of them. Please note that you do not need to know the meaning of all these prefixes. The ones I expect you to know by heart (kilo, centi, and milli) are put in boldface type. The others are there just to give you a feel of how flexible the metric system is. Metric Prefixes and Their Meanings Prefix Abbreviation Meaning Prefix Abbreviation Meaning mega M,000,000 centi c 0.0 kilo k,000 milli m 0.00 hecto H 00 micro µ deca Da 0 nano n When you are measuring distance, then, you can choose a prefix to put in front of the base unit. The prefix you use will determine the size of the unit you are using. Now, of course, because we know the meanings of these prefixes, it is rather easy to convert between one unit and another in the metric system. We can use the factor-label method just as we did before. See how this works in an example. Example.5 An Olympic pool is 0.05 kilometers in length. How long is that in meters? In order to do a conversion, we need a conversion relationship. The prefix kilo means,000, so we just write km = and then replace the prefix with its meaning:

13 Chapter : Measuring Up 3 km =,000 m Notice how the left-hand side of this conversion relationship has the unit with the prefix (km), while the right-hand side has the prefix s meaning (,000) followed by the base unit (m). Now we just put the original measurement over to make it a fraction and multiply by a fraction made from the conversion relationship, making sure km cancels: 0.05 km,000 m = 50 m km There is only one significant figure in 0.05, so there can be only one significant figure in the answer, which is 50 m. Comprehension Check 6. The diameter of a U.S. dime is 7.9 millimeters. How many meters is that? Scientific Notation Before we move on and start learning about measuring area and volume, I want to bring up an important point related to significant figures. Sometimes, we have to change the way we write down numbers in order to make sure the significant figures are communicated properly. To see what I mean, consider the Olympic swimming pool mentioned in Example.5. Suppose you measure it with a measuring tape marked off in meters. You can estimate between the marks, but the pool s edge falls right on the 50-meter mark. That means you would write its length as 50.0 meters. How many significant figures does that measurement have? It has three. Now, suppose I asked you to convert that into centimeters. What would you do? Well, centi means 0.0, so you would write down the conversion relationship, cm = 0.0 m. You would then multiply by a fraction made from that conversion relationship so that meters cancel: 50.0 m cm 0.0 m = 5,000 cm There is a problem with that answer. Can you see what it is? The original measurement has three significant figures, so the answer needs to have three. How many significant figures are in 5,000 cm? Only one! The zeroes are at the end of the number, but they are not to the right of the decimal place. Thus, they are not significant. In order to make two of them significant so that the entire number can have three significant figures, I need to get two of the zeroes to be right of the decimal. How can I do that? The answer is to use scientific notation. When you write a number in scientific notation, there is only one figure to the left of the decimal. All other figures are written to the right of the decimal. To keep that from changing the actual value of the number, you then multiply the number by 0 raised to some exponent. The proper answer to the equation above, for example, is cm. Notice that there are now three significant figures in the measurement, because the two zeroes are at the end of the number and to the right of the decimal. In addition, the value is the same, because 0 3 is,000, and 5.00,000 = 5,000.

14 4 Discovering Design With Chemistry So, when you have a large number, you can change it into scientific notation by moving the decimal place to the left until there is only one figure to the left of it. Then, you multiply by 0 raised to the same power as the number of places you moved the decimal. Converting a number like 4,000,000 to scientific notation, then, looks like this: move the decimal to the left seven places. Multiply the result by 0 7. If you need some of the zeroes to be significant in order to properly follow the rules of significant figures, you can just add as many as you need after the 4, because any zero at the end of the number and to the right of the decimal is significant. Now, of course, you can go the other way as well. Suppose you have a really small number, like You might find it annoying to write all those zeroes. Since they aren t significant, we can get rid of them with scientific notation. In this case, however, you have to move the decimal place to the right to get the to the left of it, so you have to multiply by 0 raised to a negative exponent: move the decimal to the right seven places. Multiply the result by 0-7. Of course, if you have a number in scientific notation, you can always convert it back to standard decimal notation by doing the reverse. If the number is multiplied by 0 to a positive exponent, it s a large number, so you move the decimal to the right by the number of places given in the exponent. If the number is multiplied by 0 to a negative exponent, you know the number is small, so you move the decimal to the left by the number of places given in the exponent. If this seems a bit confusing, the following examples should clear things up. Example.6 Convert 60,000 into scientific notation so that it has four significant figures. To get it into scientific notation, we need to move the decimal place to in between the 6 and the. That means moving it five places. Since the number is big, the exponent on the ten will be positive 5. To have four significant figures, we will need to keep two of the zeroes. That means the answer is Convert into scientific notation. To get it into scientific notation, we need to move the decimal place to in between the 4 and the 5. That means moving it six places. Since the number is small, the exponent on the ten will be -6. That means the answer is Convert into decimal notation. In this case, we need to get out of scientific notation. Since the exponent is negative, we know the number is small. That means moving the decimal place to the left a total of 2 places (the value of the exponent). Thus, the answer is

15 Chapter : Measuring Up 5 Convert into decimal notation. In this case, we need to get out of scientific notation. Since the exponent is positive, we know the number is large. That means moving the decimal place to the right a total of 9 places (the value of the exponent). Thus, the answer is 3,400,000,000. Comprehension Check 7. Convert the following numbers to scientific notation so they each have five significant figures: a. 53,000,000,000 b Convert the following numbers into decimal notation: a b Now that you understand scientific notation, you might be wondering when you should use it. In general, whether or not you use scientific notation is up to you. A number like 5,467 is identical in every way to , so you can report it using either decimal or scientific notation. The only time you are required to use scientific notation is when the significant figures can only be properly expressed with scientific notation. For example, suppose you do some math on a few measurements and end up with an answer of 700. However, suppose the rules of significant figures require you to report that answer to two significant figures. There is no way to do it with decimal notation. The only way to properly report your answer is to use scientific notation so that one of the zeroes can be made significant. In this case, then, you would have to use scientific notation and report your answer as Measuring Area and Volume: Derived Units Now that you know the ins and outs of measuring distance, it s time to learn about a couple of measurements that can result from measuring distance and then doing some math. Suppose, for example, you wanted to put carpet in a room. One of the first things you will probably do is go shop for the kind of carpet you want. When you find the right carpet, the price you are given will be based on square feet, square yards, or square meters. What does that mean? A squared distance unit is a measurement of area. If the room that you are carpeting has a length of 3.00 meters and a width of 3.50 meters, you can calculate the area of the room by multiplying the length times the width. When you multiply the numbers, you get 0.5, and when you multiply the units, you have meters meters, or meters 2 (abbreviated as m 2 ). That unit is called square meters, which is a metric unit for area. Units like this are called derived units, because you get them by doing math with units. While area is an important thing to measure when you are laying carpet or trying to get a feel for how much room is in a house, a more important thing for chemists to measure is volume. After all, matter is defined as something that has mass and takes up space. If you measure an object s volume, you are measuring how much space it takes up. Consider, for example, a box. How would you determine how much space it takes up? If you measure its length, width, and height and then multiply the three together, you will have measured the volume of the box, because volume = lengthwidthheight. What unit will you get when you do that? Well, let s suppose you measured the length to be.20 m, the width to be

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