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The following study guides are linked to Anne McKenna's website (http://employees.csbsju.edu/amckenna/CH123), with permission by the author.Significant Figures and Uncertainty
All measurements have some degree of uncertainty; how great the
uncertainty is depends on both the precision of the measuring device and the
skill of its operator. Significant figures
are those digits in a measured number (or result of a calculation
with measured numbers) believed to be correct by the person making the
measurement. The correct use of
significant figures is an attempt to indicate the precision of a measured
number.
A portion of a typical mercury thermometer is shown in Figure 1.
Figure 1. Typical mercury
thermometer. What temperature is
represented by the thermometer shown in the figure?
It is obvious that the temperature is somewhere between 20°C and 30°C.
Inspection of the figure reveals that there are 10 equal divisions
between the 20°C and 30°C markings--thus the smallest graduation represents 1°C.
As a result, we can easily determine that the temperature is between 26°C
and 27°C. Close inspection of the
figure reveals that the mercury in the thermometer is closer to the line
representing 27°C than 26°C. By
careful inspection and estimation, as a "best guess," the temperature
might be reported as 26.8°C. All
three of the digits in this temperature are believed to be correct although the
last digit in the temperature was estimated. Rules
for Determining the Number of Significant Figures
The following rules may be used to determine the number of significant
figures in measured quantities: In
measured quantities:
1. All
digits that are non-zero are significant.
[The number 28.1 contains three significant figures; 11,387 contains five
significant figures.]
2. Captive
Zeros--Zeros between non-zero digits are significant.
[The number 1004 contains four significant figures; 5091.08 contains six
significant figures.]
3. Leading
Zeros--Zeros to the left of the first non-zero digit are not
significant; they simply indicate the position of the decimal point.
[The number 0.0086 contains two significant figures; 0.02088 contains
four significant figures. The
following example demonstrates why leading zeros are not significant.
Consider, for instance, a length of 2.3 mm which is the same as 0.0023 m.
Since we are dealing with the same measured value, its number of
significant figures cannot change when we change the units.
Therefore, both quantities have two significant figures and in the 0.0023
m we don’t count the zeros that come before the 2 as being significant since
they are only needed to locate the position of the decimal point.]
4. Trailing
(Terminal) Zeros--Zeros to the right of the decimal point are significant if
they are at the end of the number (or, of course, if they are followed by
non-zero digits).
[The number 0.0120 contains three significant figures; 820.0 contains
four significant figures; 90.00380 contains seven significant figures.
NOTE:
These zeros would not be written unless those digits were known to be
zeros.]
Trailing zeros in a number without an explicit decimal point may or may
not be significant. A mass of 1300
grams may indicate a mass closer to 1300 grams than to 1200 or 1400 grams. In this case the zeros serve only to locate the decimal point
and are not significant. If the
mass is closer to 1300 grams than 1290 or 1310 grams, the zero to the right of
the 3 is significant. If the mass
is closer to 1300 grams than 1299 or 1301 grams, both zeros are significant.
For convenience (in most cases), we will assume that when a measured
number ends in zeros that are not to the right of a decimal point, these
trailing zeros are not significant unless otherwise specified.
One way to indicate that trailing zeros are significant is to write the
decimal point. If 1300. grams (note
the decimal point) is written, it would indicate four significant figures. Unfortunately, using this technique it is not possible to
indicate the number 1300 having three significant figures.
The ambiguity can best be avoided by using scientific notation:
1.3 x 103 (two significant
figures); 1.30 x 103 (three
significant figures); 1.300 x 103
(four significant figures). NOTE:
In using scientific notation, the exponent only locates the decimal point
and therefore does not affect the number of significant figures. How
many significant figures do each of the following numbers have?
1. 897
2. 0.0093
3. 67.02
4. 2.440 x
103
5. 400
Rounding
Rounding
is the procedure of dropping nonsignificant digits in a calculation result and
adjusting the last digit reported. In
rounding off numbers, look at the leftmost digit to be dropped.
If this digit is less than 5, simply drop it and all digits farther to
the right; 8.72 rounded off to two significant figures rounds off to 8.7.
If the leftmost digit to be dropped is 5 or greater, increase the
preceding number by 1 and drop all digits farther to the right; 3.847 rounded
off to three significant figures rounds off to 3.85. NOTE:
This technique for rounding is slightly different than that presented in
the textbook. [McMurray and Fay,
3rd ed. p. 22. "If the digit you remove is a 5 with nothing following,
round down.] Using the procedure
described in this handout, 5.664525 becomes 5.66453 when rounded to six
significant figures (rather than 5.66452 as in the textbook).
The procedure presented in this handout is the same procedure used by
calculators and is more widely used than the one in the textbook.
Although rounding is generally straightforward, one point requires
special emphasis. Suppose that the
number 9.348 needs to be rounded to two significant figures. Look only at the first number to the right of the 3 (i.e.,
4). Since this number is less than
5 the number is rounded to 9.3 because 4 is less than 5.
It is incorrect to round sequentially.
For example, do not round the 4 (because the 8 is greater than 5) to 5 to
give 9.35 and then round the 3 to 4 to give 9.4.
When rounding, use only the first number to the right of the last
significant figure. Round
off these numbers to the indicated places.
6. Round
21.246 to the tenths place
7. Round
0.0986 to the hundredths place
8. Round
7,392 to three significant figures
9. Round 498
to two significant figures
10. Round
349 to one significant figure
Rules
for Use of Significant Figures in Calculations A full explanation of
the procedure to determine the number of significant figures in a calculation
involves a mathematical treatment of data known as error analysis.
For most purposes, a simplified procedure using two easy-to-remember
rules is sufficient. These rules only give an approximate value of the actual
error, but that approximation is often good enough.
1. When
adding or subtracting measured quantities, the result should be reported with
the same number of decimal places as that of the number with the least number of
decimal places (or, an answer cannot be more precise than the least precise
measurement used to arrive at the answer).
[Thus 3.45 + 2.1124 = 5.56; the number 3.45 has the least number of
decimal places (2) and thus limits the final answer to two decimal places.
This is really just common sense because if the number 3.45 is ± 0.01
(it could be 3.44, 3.45, or 3.46), it would be illogical to expect that the
answer would have a lower uncertainty than the least certain number.]
2. When
multiplying or dividing measured quantities, the product (or quotient) should
contain no more digits than the measurement with the least number of significant
figures involved in the calculation.
[If you drove your car 315 miles on 11.33 gallons of gasoline, you can
calculate your gas mileage by dividing the number of miles by the number of
gallons of gasoline; 315 miles/11.33 gallons = 27.80229479 mpg.
Although this answer on a ten-digit pocket calculator, the answer is not
really as precise as it appears and should properly be rounded off to 27.8 mpg.
The final answer has three significant figures because 315 contains fewer
significant figures (3) than 11.33 (4) and this limits the precision of the
final answer.] Perform
each of the following operations and write the answer with the correct number of
significant figures:
11. 28.14 +
10.2
12. 3.45 +
7.95
13. 76 -
0.600
14. 13.444/6.7
15. 720 x
0.08
16. 6,300 +
180 -3,200
17. (1.000 x
3.61)/0.86
18. 1.71 x
103 - 5.0 x 102
19. (77.5 -
77.2)/77.5
_________ Exact Numbers It
is important to distinguish nonintegral numbers (i.e., measured numbers) from exact, integral ones, those that are defined or that result
from counting. For example, there
are exactly 3 ft in a yard, exactly four people in my immediate family, exactly
1000 grams in a kilogram, exactly seven days in one week, and exactly 12 eggs in
a dozen eggs. One inch is defined
as exactly 2.54 cm. Thus in the
statement 1 in = 2.54 cm, both the 1 and the 2.54 are considered exact.
In addition, in the formula for the circumference of a circle, 2πr,
the 2 is exact because it is defined as part of the formula.
Exact numbers are numbers with no associated uncertainty.
They can therefore be considered to have an infinite number of
significant figures. As a result,
exact numbers never limit the number of significant figures in a
calculation--the number of significant figures in a calculation result depends
only on the number of significant figures in quantities having uncertainties.
Most of the numbers you will be dealing with in chemistry are not exact
numbers--they represent numbers which have been measured. |
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