Observations in Science
1.4 Numbers and Measurements in Chemistry
We can observe the world in a variety of ways. For example, we may see a pro- fessional basketball player and comment that he is tall, a reasonable statement. If we consider that same person on the basketball court, however, we may want to know how tall he is. In this case, the answer to the question might be “six-ten.” To a basketball fan in the United States, this answer makes sense because that person ascribes additional meaning to the answer, specifi cally, that the numbers are in feet and inches: 6 feet 10 inches. A fan from another country might not fi nd any mean- ing in the two numbers, six and ten, because they wouldn’t make sense in metric units. After all, no one is six meters tall.
The example of the basketball player offers a good illustration of the differ- ence between qualitative and quantitative information and also points to the need to be specifi c in the way we communicate information. Science and engineering are regarded as quantitative professions, and usually this reputation is correct. But scien- tists also look at the world around us in general ways, akin to the assessment that the player is “tall.” Such general or qualitative observations can be crucial in establishing a systematic understanding of nature. As that understanding deepens, though, we usually invoke quantitative or numerical measurements and models. When discuss- ing chemistry, qualitatively or quantitatively, it is always important to communicate our observations and results as clearly as possible. In quantitative observations, this Mass is conserved in ordinary chemical
reactions. In nuclear reactions, which we will discuss in Chapter 14, mass and energy can be interconverted.
Mass is conserved in ordinary chemical reactions. In nuclear reactions, which we will discuss in Chapter 14, mass and energy can be interconverted.
Units can have consequences far more important than confusion over height.
NASA attributed the loss of a Mars probe in 1999 to errors associated with the confusion of different units used in the design.
Units can have consequences far more important than confusion over height.
NASA attributed the loss of a Mars probe in 1999 to errors associated with the confusion of different units used in the design.
usually means making sure that we carefully defi ne the terms we use. As we move through this course, you will encounter many examples of “everyday words” that take on much more specifi c meanings when used in a scientifi c context. Similarly, when we talk about numerical measurements, we need to be very careful about the ways we use numbers and units.
Units
The possible misunderstanding between basketball fans from different countries about the height of a player provides an analogy to an unfortunate reality in studying sci- ence and engineering. Science has grown over the centuries, with contributions from a number of cultures and individuals. The legacy of centuries of development is the existence of a large number of units for virtually every basic measurement in science.
Energy, for example, has long been important for human civilization and accordingly there are many units of energy. Distance and mass have also been measured for literally millennia.
The internationalization of science and engineering led to the establishment of a standard system that provides the needed fl exibility to handle a wide array of observa- tions. In this International System of Units (Système International d’Unités, or SI), care- fully defi ned units are combined with a set of prefi xes that designate powers of ten. This allows us to report and understand quantities of any size, as illustrated in Figure 1.8.
When observations are reported in this system, the base unit designates the type of quantity measured. For example, we can immediately recognize any quantity reported in meters (m) as a distance. Table 1.1 shows the base units for a number of the quantities that chemists may want to measure. These base units, however, are not always conveniently matched to the size of the quantity to be measured. When we consider the sizes of single atoms or molecules, we will see that they are on the order of 0.0000000001 meters. Dealing with so many decimal places invites confusion, so usually it is preferable to choose a unit that is better matched to the scale of the quan- tity. The standard way to do this is to use prefi xes that alter the “size” of any given base unit. A familiar example is the concept of the “kilo” unit. A kilometer is 1000 meters, and a kilobyte, familiar in computer technology, is roughly 1000 bytes. The prefi x kilo- implies that there are 1000 of whatever the base unit is. Prefi xes such as this exist for numbers large and small, and these prefi xes are provided in Table 1.2. The distance of 0.0000000001 meters could be reported as either 0.1 nm or 100 pm. Measuring time
Because powers of 2 are important in computer science, a kilobyte is actually defi ned as 1024 bytes.
Because powers of 2 are important in computer science, a kilobyte is actually defi ned as 1024 bytes.
Figure 1.8 ❚ The typical dimensions of the objects shown extend over many orders of magnitude and help point out the usefulness of the prefi xes in the SI system.
1⫻100 m 1 m
Height of human
Sheet of paper
Wedding ring
Thickness of a CD
Plant cell
Animal cell
Bacterial cell
Virus Protein molecule
Aspirin Water Atom 1⫻10–1 m
1 dm
1⫻10–2 m 1 cm
1⫻10–3 m 1 mm
1⫻10–4 m 100 µm
1⫻10–5 m 10 µm
1⫻10–6 m 1 µm
1⫻10–7 m 100 nm
1⫻10–8 m 10 nm
1⫻10–9 m 1 nm
1⫻10–10 m 100 pm
1⫻10–11 m 10 pm
1⫻10–12 m 1 pm
Optical microscope
Macroscale Microscale Nanoscale
Human eye Scanning tunneling microscope
Electron microscope
Specialized techniques are required to observe nanoscale objects.
provides another interesting example of units. Laser based experiments can measure the progress of chemical reactions on a timescale of 10−15 s. Therefore, scientific papers that report such experiments use femtoseconds, fs, to report time.
Not every quantity can be measured directly in terms of just the seven base units shown in Table 1.1, of course. Some units comprise combinations of these base units and therefore are termed derived units. The SI unit for energy, for example, is the joule (J), and 1 J is defi ned as 1 kg m2 s−2.
In principle, any quantity could be expressed in terms of appropriate combi- nations of SI base units. But in practice, many other units are so fi rmly entrenched that they remain in common use. One simple example of this is the measurement of times that are longer than seconds. Minutes, days, and years are used in these circumstances, rather than kiloseconds, and so on. In chemistry labs—and on soda containers— volumes are most often reported in liters (L) or milliliters (mL) rather than the SI unit of cubic meters (m3). Chemists also use a wide variety of units to describe concentration, which measures how much of a particular substance is present The kilogram is considered the base unit
for mass, but the names for mass units are derived by affi xing prefi xes to the gram. A kilogram is 1000 grams.
Table
❚
1.1Base quantities of the SI system of units
Property Unit, with abbreviation
Mass kilogram, kg
Time second, s
Distance meter, m
Electric current ampere, A
Temperature kelvin, K
Number of particles mole, mol Light intensity candela, cd
Table
❚
1.2Prefi xes used in the SI system
Factor Name Symbol Factor Name Symbol
1024 yotta Y 10−1 deci d
1021 zetta Z 10−2 centi c
1018 exa E 10−3 milli m
1015 peta P 10−6 micro μ
1012 tera T 10−9 nano n
109 giga G 10−12 pico p
106 mega M 10−15 femto f
103 kilo k 10−18 atto a
102 hecto h 10−21 zepto z
101 deka da 10−24 yocto y
in a mixture. Metals often contain minor impurities, and in some cases the units used are simply percentages; other units used include parts per million (ppm) and parts per billion (ppb). The ppm unit tells how many particles of a particular substance are present for every 1,000,000 particles in the sample. In ppb, the sample size is one billion particles. Impurity concentrations on the order of even a few ppm may cause problems in some applications, or they may be added intentionally to impart some desirable property. Later in this chapter, we will take up the issue of converting a measurement from one unit to another, as is frequently required in scientifi c and engineering calculations.
Although many of these SI units have found their way gradually into everyday use, the units for temperature may be the least familiar. You are probably used to see- ing temperatures in either degrees Fahrenheit or degrees Celsius, but generally, even in “fully metric” countries, the weather report does not use the Kelvin temperature scale. In general, temperature scales arise from the choice of two standard reference points that can be used to calibrate temperature with the use of a thermometer. The familiar Fahrenheit scale originally chose body temperature as one reference and set it at 100°F. (Accuracy of measurement was clearly not a priority when this temperature scale was proposed.) The second reference point was the coldest temperature that could be achieved by adding salt to ice water, a practice that lowers the melting point of ice. This established 0°F, and the temperature range between the two points was divided into 100 equal units. The scale is now defi ned by setting the freezing point of water at 32°F and the boiling point of water at 212°F.
The Celsius scale was developed in a similar way, but with the freezing point of pure water set at 0°C and the boiling point of water at 100°C. Figure 1.9 shows the relationship between these scales. Conversions between the two scales are given by the following expressions:
°F = (1.8 × °C) + 32 (1.1)
°C = (°F − 32) /1.8 (1.2)
Scientifi c uses of temperature require yet another temperature scale. The choice of the kelvin as the standard refl ects mathematical convenience more than familiarity.
The Kelvin scale is similar to the Celsius scale but draws its utility from the fact that the lowest temperature theoretically possible is zero kelvin. It violates the laws of nature to go below 0 K, as we will see in Chapter 10. The mathematical importance of this defi nition is that we are assured we will not divide by zero when we use a formula
Gabriel Daniel Fahrenheit invented the alcohol thermometer in 1709 and the mercury thermometer in 1714.
Gabriel Daniel Fahrenheit invented the alcohol thermometer in 1709 and the mercury thermometer in 1714.
Boiling point of water
Fahrenheit Celsius
212°
32° 0° 273
180° F 100° C 100 K
100° 373
Kelvin (or absolute)
Freezing point of water
Figure 1.9 ❚ The Fahrenheit, Celsius, and Kelvin temperature scales are compared. The freezing point of water can be expressed as 32°F, 0°C, or 273 K. The boiling point of water is 212°F, 100°C, or 373 K.
that has temperature in the denominator of an expression. Conversions between Cel- sius degrees and kelvins are common in science and are also more straightforward.
K = °C + 273.15 (1.3)
°C = K − 273.15 (1.4)
Engineers in some disciplines use the Rankine temperature scale (°R), which is an absolute scale whose degrees are the same size as those of the Fahrenheit scale.
Numbers and Signifi cant Figures
We often encounter very small and very large numbers in chemistry problems. For example, pesticide production in the world exceeds millions of tons, whereas pesticide residues that may harm animals or humans can have masses as small as nanograms.
For either type of number, scientifi c notation is useful. Numbers written using sci- entifi c notation factor out all powers of ten and write them separately. Thus the num- ber 54,000 is written as 5.4 × 104. This notation is equivalent to 5.4 × 10,000, which clearly is 54,000. Small numbers can also be written in scientifi c notation using nega- tive powers of ten because 10−x is identical to 1/10x. The number 0.000042 is 4.2 × 10−5 in scientifi c notation.
When numbers are derived from observations of nature, we need to report them with the correct number of signifi cant fi gures. Signifi cant fi gures are used to indicate the amount of information that is reliable when discussing a measurement.
“Pure” numbers can be manipulated in a mathematical sense without accounting for how much information is reliable. When we divide the integer 5 by the integer 8, for example, the answer is exactly 0.625, and we would not round this to 0.6. When numbers associated with an observation are used, however, we must be more careful about reporting digits. To understand why, consider whether or not to accept a wager about some measurement.
For example, an almanac lists the population of Canada as 33,507,506. Suppose a study concluded that 24% of the people who live in Canada speak French. Based on this information alone, the wager is that 8,041,801 Canadians speak French. Should we accept the bet? What if the wager was that roughly 8.0 million Canadians speak French? Is this a better bet? This scenario shows the importance of signifi cant fi gures.
The fi gure of 8,041,801 is not believable because we don’t really know if 24% is ex- actly 24.000000%. It could just as well be 23.95% and the correct answer would then be 8,025,048. Both answers might qualify as “roughly 8 million,” an answer with a more reasonable number of signifi cant fi gures.
This type of reasoning has been formalized into rules for signifi cant fi gures (digits) in numbers reported from scientifi c observations. When a measurement is reported nu- merically, generally we consider that each digit given is known accurately, with one im- portant exception. There are special rules for zero such that it is sometimes signifi cant and sometimes not. When a zero establishes the place for the decimal in the number, it is not signifi cant. Thus the measurement 51,300 m has two zeros that are not sig- nifi cant, as does the measurement 0.043 g. A zero is signifi cant when it is the fi nal digit after a decimal place or when it is between other signifi cant digits. Thus the zeros in both 4.30 mL and 304.2 kg are signifi cant. When numbers are written properly in scientifi c notation, all of the digits provided are signifi cant. Example Problem 1.2 provides some additional practice in determining the number of signifi cant fi gures in measurements.
E X A M P L E P RO B L E M 1. 2
An alloy contains 2.05% of some impurity. How many signifi cant fi gures are reported in this value?
Strategy Use the general rule for signifi cant fi gures: digits reported are signifi cant unless they are zeros whose sole purpose is to position the decimal place.
Scientifi c notation provides an advantage here because any digits shown in the number are signifi cant. There are no zeros needed for placing the decimal.
Scientifi c notation provides an advantage here because any digits shown in the number are signifi cant. There are no zeros needed for placing the decimal.
Solution In this case, all digits reported are signifi cant, so there are three signifi cant fi gures in the number.
Check Your Understanding How many signifi cant fi gures are reported in each of the following measurements? (a) 0.000403 s, (b) 200,000 g
We also need to account for signifi cant fi gures in assessing values that we obtain from calculations. The general principle is that a calculated value should be reported with a number of signifi cant fi gures that is consistent with the data used in the cal- culation. Our earlier story about the number of French-speaking Canadians provides some insight into this issue as well. The wager that roughly 8.0 million such people live in Canada was the more attractive. The reason: 8.0 million has the same number of signifi cant fi gures as the value 24%, so it provides a similar quality of information.
Three key rules will be required to determine the number of signifi cant fi gures in the results of calculations.
Rule 1: For multiplication and division, the number of signifi cant fi gures in a result must be the same as the number of significant figures in the factor with the fewest signifi cant fi gures. When 0.24 kg is multiplied by 4621 m, the result on a calculator reads 1109.04, but if signifi cant fi gures are correctly reported, the result will be 1100, or 1.1 × 103 kg m. The value 0.24 kg has only two signifi cant fi gures, and so the result should also have just two sig- nifi cant fi gures.
Rule 2: For addition and subtraction, the rules for signifi cant fi gures center on the posi- tion of the fi rst doubtful digit rather than on the number of signifi cant digits. The result (a sum or difference) should be rounded so that the last digit retained is the fi rst uncertain digit. If the numbers added or subtracted are in scientifi c notation with the same power of 10, this means that the result must have the same number of digits to the right of the decimal point as in the measurement that has the few- est digits to the right of the decimal point. The number 0.3 m added to 4.882 m yields 5.182 on a calculator, but should be reported as 5.2 m. The fi rst uncertain digit in the values added is the ‘3’ in 0.3 m, so the result should be rounded to one decimal place. In this case, the last allowed digit was “rounded up” because the fi rst nonsignifi cant digit, 8, was greater than 5. There are two possible conven- tions for rounding in such cases. In this text, we will round down for numbers 4 and smaller and round up for numbers 5 and larger. Example Problem 1.3 pro- vides some practice in working with signifi cant fi gures.
E X A M P L E P RO B L E M 1. 3
Report the result for the indicated arithmetic operations using the correct number of signifi cant fi gures. Assume that all values are measurements.
(a) 4.30 × 0.31 (b) 4.033 + 88.1 (c) 5.6/1.732 3 104
Strategy Check the rules for signifi cant fi gures as needed and determine which rules apply. Carry out the calculation and express the result in the correct number of signifi cant fi gures.
Solution
(a) 4.30 × 0.31 = 1.3 (b) 4.033 + 88.1 = 92.1 (c) 5.6/(1.732 × 104) = 3.2 × 10−4 Check Your Understanding Determine the value of the following expressions using the correct number of signifi cant fi gures: (a) 7.10 m + 9.003 m, (b) 0.004 g × 1.13 g
The rules above apply to any numbers that result from most of the measurements that we might make. But in the special case of countable objects, we must also consider one additional rule.
Rule 3: When we count discrete objects, the result has no ambiguity. Such measure- ments use exact numbers, so effectively they have infi nite signifi cant fi gures.
Thus if we need to use information such as four quarts in a gallon or two hydrogen atoms in a water molecule, there is no limitation on significant fi gures. Note that this rule also applies when we work with the various prefi xes in the SI system. There are exactly 100 centimeters in a meter, so the factor of 100 would never limit the number of signifi cant fi gures in a calculation.