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Measurement and Significant
Figures
Precision and Accuracy
• What is the difference between precision
and accuracy in chemical measurements?
• Accuracy refers to how close you are to
the true value.
• Precision refers to how close several
measurements are to each other.
Precision and Instruments
• Your measurement will only be as good as
the instrument you use.
• Precision is limited by the gradations -or
markings -on your instrument.
• We can typically estimate to one-tenth of a
gradation mark when using graduated
instruments in Chemistry.
Measurement Practice
Accurate
(the average),
not precise
BEAKER
47 +/- 1 mL
Not accurate
but is precise
CYLINDER
36.5 +/- 0.1 mL
Accurate
and precise
BURET
20.38 +/- 0.01 mL
Uncertainty in Measurement
• Any measurement will have some degree of
uncertainty associated with it.
• For example, if you are 1.6 meters tall, we
know that you are exactly ONE meter tall
(not 0 or 2) but the second digit is an
estimate and contains some uncertainty (it
could be .58 rounded up, or .62 rounded
down).
• Scientific measurements are rounded off so
that the last digit is the only one that is
uncertain. Preceding digits are known with
certainty.
Significant figures
• What is the difference between the
measurements 25.00 mL and 25 mL?
• The first measurement is known to a greater
degree of precision and contains more
significant figures –it could be 25.01 or 24.99
whereas the second measurement lies between
24 and 26.
• The number of significant figures tells us how
well we know a measurement.
• The known numbers PLUS the last uncertain
number in a measurement are significant.
Rules for determining significant figures
• Any non-zero number is significant.
example: 762 has 3, and 2500 has 2
• Zeros: (a) leading zeros are not significant, they are just place
holders. Ex: 006471 has 4, and 0.00284 has 3
(b) “Captive” zeros between nonzeros are significant. Ex: 1.008
has 4 and 12046 has 5
(c) Trailing zeros are significant ONLY if the number contains a
decimal point. Ex: 1.0 x102 has 2, and 3000. has 4
• Exact numbers are numbers that are determined by counting
(not measurement) or by definition are assumed to have an
infinite number of significant figures.
example: 1 minute equals 60 seconds
15 students are in class today
Practice Problem
•
•
•
•
•
•
•
•
How many sig figs are in each?
4.5090
0.00607
6.7 x103
200.
250
698,000.1
2.0000 x106
Addition and Subtraction Using
Significant Figures
• The answer must have the same number of
decimal places as the least precise
measurement used in the calculation.
• For example, consider the sum
12.11
18.0
+ 1.013
31.123
• The answer is 31.1 since 18.0 only has one
decimal place.
Multiplication and division using
sig. figs.
• The number of significant figures in the
answer is the same as the least precise
measurement (lowest number of sig. figs.)
used in the calculation.
• For example, consider the calculation
4.56 x 1.4 = 6.38
• The correct answer is 6.4 (it should only
have two sig figs since 1.4 has only two)
Dimensional Analysis
• We will often need to convert form one unit to
another when solving problems in Chemistry.
• The best way to do this is by a method called
dimensional analysis (a.k.a. factor-label
method).
• For example, consider a pin measuring 2.85 cm
in length. Given that one inch is equal to 2.54
cm, what is its length in inches?
• 2.85 cm x
1 in
= 1.12 in
2.54 cm
Using the method…
• In math you use numbers, in chemistry we use
quantities. A quantity is described by a number and a
unit.
• 100 is a number : 100 Kg is a quantity (notice that in
chemistry we give meaning to the numbers). In
science we solve a lot of the "math" by watching the
units of the quantities
• There are two main rules to solving science
problems with the factor-label method:
• 1. Always carry along your units with any
measurement you use.
• 2. You need to form the appropriate labeled ratios
(equalities).
Unit Conversion Practice
• A pencil is 7.00 inches long. How long is it
in cm?
• ANSWER: 17.8 cm
• A student has entered a 10.0 km race.
How long is this in miles?
• ANSWER: 6.22 mi
• The speed limit on many highways in the
U.S. is 55 mi/hr. What is this in km/hr?
• ANSWER: 88 km/hr
Linking conversion ratios
• Sometimes you will need to multiply by
more than one ratio to get to your desired
units, you can do this by using linking
units. Your setup will look like this:
• Example: How many inches are in 1.00
meter given the equality 1 inch = 2.54 cm
and 1 meter = 100 cm?
1.00 m x 100 cm x 1 inch = 39.4 in
1m
2.54 cm
Advanced Problem
• A Japanese car is advertised as having a gas
mileage of 15 km/L. Convert this rating to
mi/gal. (Given conversion factors 1L=1.06 qt
and 4 qt= 1 gal)
• ANSWER:
15 km x 1mi x 1 L x 4 qt = 35.38 mi/gal
L
1.6 km 1.06 qt 1 gal
With correct sig figs this rounds to 35 mi/gal
More Examples:
• Convert 50.0 mL to liters:
• How many seconds are in two years?
Scientific Notation
• The primary reason for converting numbers into
scientific notation is to make calculations with
unusually large or small numbers less
cumbersome.
• Because zeros are no longer used to set the
decimal point, all of the digits in a number in
scientific notation are significant, as shown by
the following examples:
• 2.4 x 1022 has 2 significant figures
9.80 x 10-4 has 3 significant figures
1.055 x 10-22 has 4 significant figures
Converting to Sci. Not.
•The following rule can be used to convert numbers into scientific
notation:
The exponent in scientific notation is equal to the number of
times the decimal point must be moved to produce a number
between 1 and 10.
•Example: In 1990 the population of Chicago was 6,070,000. To
convert this number to scientific notation we move the decimal
point to the left six times.
6,070,000 = 6.070 x 106
•To convert numbers smaller than 1 into scientific notation, we
have to move the decimal point to the right. The decimal point in
0.000985, for example, must be moved to the right four times.
0.000985 = 9.85 x 10-4
Exponent Review
• Some of the basics of exponential mathematics
are given below:
– Any number raised to the zero power is equal to 1.
ex: 10 = 1 and 100= 1
– Any number raised to the first power is equal to itself.
ex: 11 = 1 and 101 = 10
– Any number raised to the nth power is equal to the
product of that number times itself n-1 times.
ex: 22 = 2 x 2 = 4 and 105 = 10 x 10 x 10 x 10
x 10 = 100,000
– Dividing by a number raised to an exponent is the
same as multiplying by that number raised to an
exponent of the opposite sign.
ex: 5 ÷ 102 = 5 x 10-2 = 0.05
Practice Problem
• Convert the following numbers into
scientific notation:
• (a) 0.004694 (b) 1.98 (c) 4,679,000
ANSWER: (a) 4.694 x 10-3
(b) 1.98 x 101
(c) 4.679 x 106
THE END