Chemistry 65 Chapter 2 1 SCIENTIFIC NOTATION Scientific Notation is a convenient way to express very large or very small quantities. General form: n A x 10 1 A <10 n= integer Using scientific notations: Changing between conventional and scientific notation: 75,000,000 changes to 7.5 x 10 7 (7 to the left) 0.00642 changes to 6.42 x 10 -3 ( 3 to the right) Addition and subtraction (NOT COVERED) Multiplication and division : 1. Change numbers to exponential form. 2. Multiply or divide coefficients. 3. Add exponents if multiplying, or subtract exponents if dividing. 4. If needed, reconstruct answer in standard exponential notation. Examples: 1. Multiply 30,000 x 200,000 9 5) (4 5 4 6x10 6x10 ) )(2x10 (3x10 2. Divide 60,000 by 0.003 7 (-3)] - [4 3 - 4 2x10 x10 3 6 3x10 6x10
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ERRORS IN MEASUREMENT - profpaz.commatter. For measurements to be useful, a measurement standard must be used. A standard is an exact quantity that people agree to use for comparison.
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Chemistry 65 Chapter 2
1
SCIENTIFIC NOTATION
Scientific Notation is a convenient way to express very large or very small quantities.
General form:
nA x 10 1 A <10 n= integer
Using scientific notations:
Changing between conventional and scientific notation:
75,000,000 changes to 7.5 x 107 (7 to the left)
0.00642 changes to 6.42 x 10 -3
( 3 to the right)
Addition and subtraction (NOT COVERED)
Multiplication and division :
1. Change numbers to exponential form.
2. Multiply or divide coefficients.
3. Add exponents if multiplying, or subtract exponents if dividing.
4. If needed, reconstruct answer in standard exponential notation.
Examples:
1. Multiply 30,000 x 200,000
95)(454 6x106x10))(2x10(3x10
2. Divide 60,000 by 0.003
7(-3)]-[4
3-
4
2x10x103
6
3x10
6x10
Chemistry 65 Chapter 2
2
Follow-up Problems:
1. (5.5 x 10
3) (3.1 x 10
5) =
2. (9.7 x1014
)(4.3 x10-20
) =
3.
6
2
2.6x10=
5.8x10
4.
-5
-8
1.7x10=
8.2x10
5. (3.7x 10 -6
) x (4.0 x 10 8) =
6. (8.75x1014
)(3.6x108) =
7.
-28
13
1.48x10=
7.25x10
Chemistry 65 Chapter 2
3
ACCURACY & PRECISION
For measurements to be useful, it is important that they be precise and accurate.
Accuracy is closeness of a measurement to an external standard.
Precision is closeness of a measurement to another similarly obtained measurement.
Two types of error can affect measurements:
Systematic errors: those errors that are controllable, and cause measurements to be either
higher or lower than the actual value.
Random errors: those errors that are uncontrollable, and cause measurements to be both
higher and lower than the average value.
Evaluate the accuracy and precision of each set of data shown below:
Chemistry 65 Chapter 2
4
ERRORS IN MEASUREMENTS
Two kinds of quantities are used in science:
Counted or Defined: exact numbers; no uncertainty (error)
Measured: are subject to error; have uncertainty (error)
Uncertainty in Measurements:
Every measurement has uncertainty because of instrument limitations, human error,
and number of measurements.
The uncertainty in a measurement appears in the last recorded digit.
15 ± 1 cm (14 cm or 16 cm)
15.3 ± 0.1cm (15.2 cm or 15.4 cm)
An uncertainty of one unit is assumed in all measurements, unless otherwise specified.
In reading a measurement scale, it is wrong to record more than one estimated digit.
The last digit is the estimated one.
8.65 cm
8.6 cm
Chemistry 65 Chapter 2
5
RECORDING MEASUREMENTS TO THE
PROPER NO. OF DIGITS
What is the correct value for each measurement shown above?
a) 28 mL (1 certain, 1 uncertain)
b) 28.2 mL (2 certain, 1 uncertain)
c) 28.31 mL (3 certain, 1 uncertain)
Chemistry 65 Chapter 2
6
SIGNIFICANT FIGURES
Scientists use significant figures to express the precision of a measurement.
Significant figures are the number of certain and uncertain digits
Not significant
Leading zeros Significant
All zeros between
non-zero digits
0 . 0 0 4 0 0 4 5 0 0
Significant:
all non-zero
integers
Significant:
zeros to the end of a
number and right of
decimal point Examples:
Determine the number of significant figures in each of the following measurements:
0.05082 in 4 significant figures
41.0 C 3 significant figures
14.303 m 5 significant figures
0.00025 L 2 significant figures
150000 mg ambiguous (should be written in scientific notation)
1.5 x 105 mg 2 significant figures
1.50 x 105 mg 3 significant figures
1.500 x 105 mg 4 significant figures
Chemistry 65 Chapter 2
7
SIGNIFICANT FIGURES IN CALCULATIONS
Multiplication and Division:
The measurement with the least certainty limits the certainty of the results; or
The answer must contain the same number of significant figures as in the measurement with
the least number of significant figures.
Examples:
5.02 x 89.665 x 0.10 = 45.0118 = 45 (3 sf) (5 sf) (2 sf) (calculator answer) (2 sf)