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Significant Figures and Measurement I

VIEWS: 10 PAGES: 6

  • pg 1
									                                                                   Revised 1/10/2005 WAB       1



                      Significant Figures and Measurement: I

Purpose: Introduce the concept of significant figures, including 1) determination of the
number of significant figures in a number, 2) determination of the number of significant
figures which should be included in the result of a calculation, and 3) determination of the
number of significant figures which should be recorded when making a measurement.
Students will practice using standard laboratory equipment used to measure temperature,
mass, and volume. The use of volumetric glassware (i.e. volumetric pipet, and buret) is
introduced.

Measurement and significant figures are closely related topics which are extremely important
when performing any lab work. Temperature, mass, and volume measurements routinely will
be made in general chemistry lab. Volume and mass measurements will involve a number of
different devices, the use of which needs to be explained. However, before discussing
various measuring devices, the concept of significant figures will be covered.

Significant Figures

In general chemistry there are three basic concepts which need to be understood in order to
master the concept of significant figures

       I. Determining the number of significant figures within a given number.
       II. Determining the number of significant figures which should be included in the
            result of a calculation.
       III. Determining the number of digits which should be recorded when making a
            measurement.

The first two concepts may have been discussed in general chemistry lecture. The following
five rules can be used to determine the number of significant figures present within a number
(concept I).


               1. All nonzero digits are significant. Example: there are three significant
                  figures in the number 25.3.
               2. Zeros between nonzero digits are significant. Example: the number
                  12.03 contains four significant figures. By rule 1, the 1, 2, and 3 are
                  significant. The 0 is significant by this rule.
               3. Zeros to the left of the first nonzero digit are not significant. Example:
                  the number 0.00206 contains three significant figures. By rule 1, the 2 and
                  6 are significant. By rule 2 the zero between the 2 and 6 is significant. By
                  this rule the zeros to the left of the 2 are not significant. You can think of
                  these zeros as place holders. Alternatively, if this number is expressed in
                  scientific notation (2.06*10-3) it is easy to see there are only three
                  significant figures. In scientific notation only the number (i.e. 2.06) is
                  important in determining the number of significant figures.
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               4. Zeros to the right of the decimal point, which are at the end of a
                  number are significant. Example: the number 0.000200 contains three
                  significant figures. The 2 is significant by rule 1, and the two zeros
                  following the 2 are significant by this rule. Rule 3 indicates the zeros to
                  the left of the 2 are not significant. Again, expressing this number in
                  scientific notation (2.00*10-4) may help to demonstrate there are three
                  significant figures.
               5. Zeros to the left of the decimal point, which are at the end of a
                  number may or may not be significant. Example: 130 may have two or
                  three significant figures, depending on the interpretation of the person who
                  wrote the number. The ambiguity is removed by expressing the number in
                  scientific notation. If the number is expressed as 1.30*102, then it has
                  three significant figures. If it is expressed as 1.3*102, then it has two
                  significant figures. I will not purposely try to trick you with this rule, but
                  instead will display such numbers in scientific notation so as to remove the
                  ambiguity. Some people will include a decimal point (130.) when
                  implying the zero is significant, and leave the decimal point off (130)
                  when implying the zero is not significant.
               6. Exact conversion factors and counting numbers are considered to
                  have an infinite number of significant figures. Example: note Table 1-
                  7 on page 19 of wdp. Many of the equivalencies given in this table are
                  exact conversion factors (i.e. it is defined 1 inch is equivalent to 2.54
                  centimeters, and 1Å is exactly 1*10-10 meters). Such exact conversion
                  factors are considered to have an infinite number of significant figures.
                  Examples of counting numbers include, 37 pennies (you cannot have a
                  fraction of a penny), or having 3 sisters.

The rules used to determine the number of significant figures which should be included in the
result of a calculation (concept II) are presented below. Examples are also presented in the
text (wdp, 24-25). Multiplication/division calculations are treated differently than
addition/subtraction calculations.

       1. Multiplication and/or Division When two or more numbers are multiplied
          and/or divided, the resulting quantity should have as many significant figures as
          the number with the fewest significant figures.

For example consider the following:

                                       (4.1 * 10 −2 )(1.031 * 10 2 )
                                                 0.0108

The number of significant figures in each of the numbers involved in the calculation is given
in Table 1.
                                                                       Revised 1/10/2005 WAB    3

                                               Table 1
                                    Number      Significant Figures
                                    4.1*10-2             2
                                   1.031*102             4
                                     0.0108              3


According to the rule, the result can only have 2 significant figures.

(4.1 * 10 −2 )(1031 * 10 2 ) 4.2271
                .
                            =        = 39139815 = 3.9139815 * 10 2 = 3.9 * 10 2
                                          .
          0.0108              0.0108

The calculator displayed the value 391.39815. The result was switched from standard
notation to scientific notation in order to avoid the ambiguity discussed in rule 5 of
significant figures. The final result is displayed with two significant figures as required.

       2. Addition and/or Subtraction When two or more numbers are added and/or
          subtracted, the resulting quantity should have the same number of places past the
          decimal point as the number with the fewest places past the decimal point.
          Remember, you are interested in determining the correct number of significant
          figures which should be reported in the resulting quantity. This is determined by
          noting the number of places past the decimal point in each number involved in
          the calculation. Consider the following example:

                                                      .
                                         1775.280 − 151

                                               Table 2
                    Number      Significant Figures    Places Past Decimal Point
                    1775.280             7                         3
                      15.1               3                         1


                               1775.280 − 15.1 = 1760.18 = 1760.2

As the rule states, the result should have one place past the decimal point. When the result is
displayed with one place past the decimal point, it has five significant figures. Notice the
number of significant figures in the result is not the same as the number of significant figures
in either of the numbers involved in the calculation!

In addition to determining the correct number of significant figures for the result of a series of
calculations, the result must be rounded to this correct number of digits. The following three
rules of rounding will be used in general chemistry lab:

       1. If the first digit past the last significant digit is less than 5, then leave the last
          significant digit unchanged and drop all nonsignificant digits. For example, if
          64.24565 is to be displayed with 3 significant figures, then it should be rounded to
          64.2.
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       2. If the first digit past the last significant digit is greater than 5, or is a 5
          followed by other nonzero digits, then add 1 to the last significant digit and drop
          all remaining nonsignificant digits. If 64.24565 is to be displayed with 4
          significant figures, then it should be rounded to 64.25.
       3. If the only digit past the last significant digit is a 5, leave the last significant
          digit unchanged and drop the 5 when the last significant digit is even or zero
          (64.24565 rounded to 6 significant digits is 64.2456).
          Add 1 to the last significant digit and drop the 5 when the last significant digit is
          odd (3.8675 rounded to 4 significant digits is 3.868).
          This rule always results in the last significant digit being even or zero.

One final comment concerning rounding; round once at the end of a multi-step calculation.
Do not round at each intermediate step of a calculation.

Concepts I and II are vitally important, and will be used in practically every lab. In addition
to these concepts, one must be able to determine the number of digits which should be
recorded when a measurement is made (concept III). The definition of the phrase significant
figure can be appreciated when it is considered from the point of making a measurement.
Significant figure is defined as the number of digits in a physical quantity that are known to
be certain, plus the first uncertain digit. This definition does not make much sense without
an example.

Consider the ruler shown in Figure 1A. Each tick mark on the ruler corresponds to 1
centimeter, so it can be seen the line is longer than 8 centimeters and less than 9 centimeters.
Thus, one digit (the 8) is known to be certain. The definition of significant figures indicates
the first uncertain digit must also be included. This digit is uncertain because it is described
with a unit (0.1cm) which is smaller than the smallest unit indicated on the measuring device.
Each individual must guess the first uncertain digit, and different people will guess different
values. One student may guess this first uncertain digit to be 7, and therefore record the
length of the line as 8.7 cm. This measurement has 2 significant figures (8 is known with
certainty, the 7 is the first uncertain digit). Someone else may say the first uncertain digit is
6. This student would record the length of the line as 8.6 cm. The measurement would still
have 2 significant figures. Different people estimate the uncertain digit differently.
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                 1    2    3    4    5    6    7    8    9 10 11 12 13


                                         Figure 1A




                 1    2    3    4    5    6    7    8    9 10 11 12 13


                                          Figure 1B

In Figure 1B the same line is measured with a ruler marked off in smallest increments of 0.1
centimeter (1 millimeter). In this case, one is certain the line is greater than 8.7 cm but less
than 8.8 cm. One student may estimate the first uncertain digit to be a 6, and would report a
length of 8.76 cm. This measurement has three significant figures; the 8 and 7 are certain
(can be read directly off the measuring device), the 6 is uncertain. Another student may think
the first uncertain digit is other than 6. Regardless of the estimated value of the first
uncertain digit, the measurement will have three significant figures. This discussion and
figures 1A and 1B indicate the number of significant figures which should be recorded
when making a measurement depends on the scale of the measuring device.

In order to apply the idea of significant digits to measurements one must differentiate
between analog and digital measuring devices. Digital measuring devices output the
measurement on a digital display (similar to a calculator display). Measuring devices which
digitally display a measurement are designed so all displayed digits are significant (those
known to be certain plus the first uncertain digit). Analog measuring devices, which include
rulers, thermometers, beam balances, graduated cylinders, and burets require you to make the
measurement. In this case it is up to you to record all digits known to be certain and the first
uncertain digit. There are a few digital measuring devices which will be used in general
chemistry lab, the most notable being the electronic balance. However, the majority of
measuring devices are of the analog variety, and you must be able to determine the correct
number of significant figures which should be recorded.

         Two terms often encountered when making measurements or analyzing data include
accuracy and precision. Accuracy refers to how close a measured quantity is to the “true” or
“accepted” value. For example, suppose the mass of metal rod is known to be exactly 15.453
g (i.e. the “true” value). You weigh this rod twice, obtaining a mass of 15.782 g and 15.436
g. The measured value 15.436 g is more accurate than 15.782 g because it is closer to the
                                                                  Revised 1/10/2005 WAB       6


“accepted” value. Precision refers to the agreement among a series of measurements. A
series of measurements with a small range (i.e. all measurements are very similar) are more
precise than a series of measurements with a larger range.

								
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