3. DESARROLLO DEL PROYECTO
3.1 ESTUDIO DEL ENTORNO
3.1.5 Análisis de las Cinco (5) Fuerzas de Porter
When a calculation combines operations (multiplication, division, addition, and subtraction), apply the rules for figure significance before and after every step involving addition or subtraction. Table 3-3 H
shows how these rules affect rounding.
Examples 1 and 2 in Table 3-3 are straightforward. The operations involve only addition or subtrac- tion, and the values of x and y force the final result to stop in the third decimal place. In examples 3 and 4, the operations involve only multiplication and division, which means that the final result can have no more significant figures than does x, which has only two.
Example 5, however, combines the operations of multiplication and addition. The first step is mul- tiplication of x by y, giving 0.314578. Because x has only two significant figures, this product must be rounded to 0.31 before the next step. It is then added to z to give 92.1142, which must be rounded to 92.11 because “0.31” has only two decimal places.
Example 6 is similar to 5. The first step is multiplication of y by z, giving 365.56432, which must be rounded to 365.6 because the value of y has four significant figures. This number is then added to x to give 365.7, which has only one decimal place because “365.6” has only one.
Examples 7 and 8 follow logic similar to that of examples 5 and 6.
The Exception of Repeated Measurements
Suppose that you quantify the drug methotrexate in the same patient specimen five times. The results, in “μmol/L,” are 62.33, 62.69, 61.56, 63.02, and 61.79. The average of these five values is
62.33 + 62.69 + 61.56 + 63.02 + 61.79
5 = 62.278
But remember that, in the addition step, the last significant figure in the sum must occupy the same place as the last significant figure in the measurement that has the greatest uncertainty, which in this case is the hundredths place:
62.33 62.69 61.56 63.02 +61.79 311.39 Next, we divide the sum by 5 to get the average:
311.39
5 = 62.278
The number “5” in the denominator is an exact count—not an estimation or a measurement; therefore, it does not bear on the number of significant figures in the final answer. But the sum in the
x y z operations
Raw Result from calculator (without rounding at any step)
Final Result (after proper rounding
at each step) 0.079 3.982 91.8042 1 x + y + z 95.8652 95.865 2 x + y - z -87.7432 -87.743 3 x
#
y#
z 28.879581 29 4 x , (y#
z) 0.000216104 0.00022 5 (x#
y) + z 92.118778 92.11 6 x + (y#
z) 365.64332 365.7 7 (x, y) + z 91.824039 91.824 8 x+ (y , z) 0.1223749 0.122numerator has five significant figures, and the rule for multiplication and division requires, in this case, that the result of the calculation also have five significant figures. If we obey this rule, then the average is 62.278. However, because each concentration has only four significant figures, the average itself should have no more than four; it cannot be more certain than any of the measurements that went into it. It would seem, then, that the correctly reported average is 62.28 μmol/L. But is it really?
Notice that the five concentrations, which range from 61.79 to 63.02, vary not only in the tenths and hundredths places but also in the ones place. It is not defensible to claim certainty to the hundredths place (62.28 μmol/L) when the first uncertain digit in the concentration is in the ones place. The value “62.28” means that the real concentration lies between 62.275 and 62.284, but there is no way to justify this conclusion with such a wide range of individual concentrations. As stated earlier in this chapter, we customarily report measurements such that only the last digit is uncertain. Therefore, the correctly reported average for our five concentrations must terminate in the ones place: 62 𝛍mol/L.
This exception for repeated measurements reflects a flaw in the rules of figure significance. The final section of this chapter addresses this issue.
CheCkpoint 3-3
1. Specify the number of significant figures that should be present in the result of each calculation.
(a) 2.445 * 0.921 (b) 451 , 9.3 (c) 0.0022435 * 66.2778 (d) 2378 , 1.09880 2. With the correct number of significant figures, give the result of each calculation.
(a) 101 + 33.5 (b) 0.02775- 0.0104 (c) 2.0046 + 0.11708 (d) 55.66 - 2.189 3. With the correct number of significant figures, give the result of each calculation.
(a) (2.33 + 0.988) * 66 (b) 1190.7558+ 83 (c) (18 * 0.500) - 2.3 (d) 3445.9 + 0.8850.919 (e) 0.08475 + 0.40(106) (f) 1.0556 - 0.00664 - 0.802 (g) 0.033000(1.229845 + 0.0009443 - 0.0730721) 1. (a) 3 (b) 2 (c) 5 (d) 4 2. (a) 135 (b) 0.0174 (c) 2.1217 (d) 53.47 3. (a) 220 (b) 267 (c) 6.7 (d) 3446.9 (e) 40 (f) 0.247 (g) 0.038205
SigniFicAnT FiguReS in exponenTiAl
expReSSionS And logARiThmS
As Chapter 2 explains, an exponential expression, such as 8.332 * 106, has two parts: the significand
(8.332) and the exponential term (106). All the significant figures in an exponential expression are in the
significand; there are none in the exponential term.
The expression above is the same as 8,332,000, a number that consists of four significant figures. Thus, when written exponentially, all four of those figures appear in the significand.
8,332,000 = 8.332 * 106
Four significant figures
The significand: contains all four significant figures
f f
The exponential term: has no significant figures
The exponential term tells us nothing about the number of significant figures; it only specifies the order of magnitude. After all, if the expression were 8.332 * 1013, there would still be four significant figures,
even though the value has seven more zeros: 83,320,000,000,000.
How does one identify the significant figures in a logarithm? A logarithm comprises two parts: the characteristic, which is the set of numbers to the left of the decimal point, and the mantissa, which is the set of numbers to the right of the decimal point. Consider, for example, the logarithm of 150,000, or 1.5 * 105:
5.17609 characteristic mantissa
f
The characteristic merely locates the decimal point, telling us only the power of 10, which in this case is 5. It is not a significant figure. The mantissa, however, does contain significant figures and must contain the same number of them as does the argument. In this case, because there are two significant figures in the argument (150,000), there must also be two in the mantissa. Calculated to two significant figures, therefore, the logarithm of 150,000 is 5.18. To convince yourself of this rule, consider the following table.
Regardless of the power of 10, every argument in this table has two significant figures. In fact, they are the same two: “1” and “5.” Clearly, the characteristic of the logarithm changes from one argument to the next, whereas the mantissa stays the same. Thus, the characteristic says nothing about the two significant figures, instead showing only the location of the decimal point. The mantissa is what captures the “1” and “5,” no matter how many zeros follow them in the argument.
Therefore, in counting significant figures in a logarithm, ignore the characteristic. Only the mantissa con- tains significant figures, and it contains the same number of them as does the argument. For example, if the argument is 365, there are three significant figures, and the logarithm is reported as 2.562. If the argument is 2.3374 * 109, there are five significant figures, and the logarithm is reported as 9.36873.
CheCkpoint 3-4
1. Count the significant figures in each of these expressions:
(a) 3.772 * 10-5 (b) 0.110* 105 (c) 9.6 * 1017 (d) 3.09400 * 105 2. For each number, write its base-10 logarithm with the correct number of significant
figures.
(a) 1.22760 * 109 (b) 0.00330 (c) 600 (d) 10,460
3. Write each number in exponential notation with the correct number of significant figures. (a) 5,003,000,000 (b) 25,010 (c) 0.0000116 (d) 0.00070070 1. (a) 4 (b) 3 (c) 2 (d) 6 2. (a) 9.089057 (b) -2.481 (c) 2.8 (d) 4.0195 3. (a) 5.003 * 109 (b) 2.501 * 104 (c) 1.16 * 10-5 (d) 7.0070 * 10-4 Argument log10 1.5 0.17609 15 1.17609 150 2.17609 1500 3.17609 15,000 4.17609 150,000 5.17609 1,500,000 6.17609
AbSoluTe And RelATive unceRTAinTy
There are more-rigorous ways to quantify uncertainty than relying on the number of significant figures. Practicing scientists usually express measurements in the form “x { y,” where x is the value of the measurement and y is the uncertainty. The volume in Figure 3-1, for example, might be reported as “31.72 mL { 0.02 mL.”
Suppose you weigh a substance on an analytical balance and it gives a mass of 4.38 g; the balance’s stated uncertainty is 0.05 g. You record the mass as “4.38 g { 0.05 g,” which implies that the true mass is between 4.33 and 4.43 g. The “0.05 g” is the absolute uncertainty, which simply represents the raw amount of uncertainty in the measurement.
The importance of that uncertainty, however, is captured in the relative uncertainty, which is the fraction of the measurement’s value represented by the absolute uncertainty:
relative uncertainty = measurement’s valueabsolute uncertainty
In other words, it tells us how large the uncertainty is in relation to the measurement. It is usually expressed as a percentage. In our example, the relative uncertainty is
0.05 g
4.38 g = 0.011 = 1.1%
Note that the relative uncertainty has no units because the ratio cancels them out. Therefore, the measure- ment may be expressed as “4.38 { 1.1%.”
If someone reports to you the measurement “22 mg,” you have no indication of uncertainty apart from the number of significant figures. Because the measurement was rounded to the nearest whole number, the true value is presumably between 21.5 and 22.5, making the uncertainty “{ 0.5.” There- fore, the implied relative uncertainty is 0.5 out of 22:
0.5
22 = 0.023 = 2.3%
Consequently, the result of any calculation involving this value (22 mg) can have an implied relative uncertainty no smaller than 2.3%.
CheCkpoint 3-5
1. Calculate the relative uncertainty for each of the following measurements. (a) 23.66 g { 0.03 g
(b) 0.00557 moles{ 0.00005 moles (c) 469 mL { 10 mL
2. Calculate the implied relative uncertainty for each of the following measurements. (a) 0.571 g/L (b) 145 mg (c) 4 g
1. (a) 0.1% (b) 0.9% (c) 2%
2. (a) 0.09% (b) 0.3% (c) 13%