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Lesson 4
Measuring & Significant Digits
Topic 1 – Units of Measurement & Metric Prefixes
Topic 2 – Scientific Notation
Topic 3 – Accuracy, Precision and Error
Topic 4 – Significant Digits
Anything in black letters = write it in
your notes (‘knowts’)
Topic 1 – Units of Measurement & Metric Prefixes
Measurements without units are useless!
“I walked 5 today.”
“The speed of light is 186,000
“I weigh 890”
“20 of water”
All measurements need units!
SI – International System of Units
SI Base Units
We will use all of
these in this class
Quantity
SI base
unit
Symbol
Length
meter
m
Mass
kilogram
kg
Temperature
kelvin
K
Time
second
s
Number of
Things
mole
mol
Luminous
intensity
candela
cd
Electric
current
ampere
A
Commonly Used Metric Prefixes
Prefix
Symbol
Meaning
Factor
mega
M
1 million times larger than the base
106
kilo
k
1000 times larger than the base
103
BASE
Base Unit (meter, second, gram, etc)
deci
d
10 times smaller than the base
10-1
centi
c
100 times smaller than the base
10-2
milli
m
1000 times smaller than the base
10-3
micro
μ
1 million times smaller than the base
10-6
nano
n
1 billion times smaller than the base
10-9
pico
p
1 trillion times smaller than the base
10-12
Volume - Amount of space occupied by an
object (remember?)
Normal units used for volume:
Solids – m3 or cm3
Liquids & Gases – liters (L) or milliliters (ml)
1 L = 1000 mL
1 mL = 1 cm3
The volume of a material changes with
temperature, especially for gases.
Mass - Measure of inertia (remember?)
Weight - Force of gravity on a mass; measured
in pounds (lbs) or Newtons.
Weight can change with location, mass does not
Energy – Ability to do work or produce heat.
Normal units used for energy:
SI – joule (J)
non-SI – calorie (cal)
1 cal = 4.184 J
How many joules are in a kilojoule?
How many calories are in a kilocalorie?
Temperature – measure of how cold or hot an
object is.
Temperature – measure of the average
kinetic energy of molecules.
Normal units used for temp:
SI – kelvin (K)
yucky!
non-SI – celsius (°C) or Fahrenheit (°F)
Tkelvin  TCelsius  273.15
Celsius
100
divisions
100°C
Boiling point
of water
373.15 K
0°C
Freezing point
of water
273.15 K
Kelvin
100
divisions
mass
Density =
volume
Normal units for density:
g/cm3, g/mL, g/L
Densities of Some Common Materials
Solids and Liquids
Gases
Material
Density at
20°C (g/cm3)
Material
Density at
20°C (g/L)
Gold
19.3
Chlorine
2.95
Mercury
13.6
Carbon dioxide
1.83
Lead
11.3
Argon
1.66
Aluminum
2.70
Oxygen
1.33
Table sugar
1.59
Air
1.20
Corn syrup
1.35–1.38
Nitrogen
1.17
Water (4°C)
1.000
Neon
0.84
Corn oil
0.922
Ammonia
0.718
Ice (0°C)
0.917
Methane
0.665
Ethanol
0.789
Helium
0.166
Gasoline
0.66–0.69
Hydrogen
0.084
Topic 2 – Scientific Notation
We will often work with really large or
really small numbers in this class.
Standard Notation
Scientific Notation
872,000,000 grams = 8.72 x 108 grams
0.0000056 moles
= 5.6 x 10-6 moles
exponent
coefficient
6.02 x
23
10
The coefficient must be a single, nonzero digit,
exponent must be an integer.
Multiplication and Division
To multiply numbers written in scientific notation,
multiply the coefficients and add the exponents.
(3 x 104) x (2 x 102)
= (3 x 2) x 104+2
= 6 x 106
(2.1 x 103) x (4.0 x 10–7)
= (2.1 x 4.0) x 103+(–7)
= 8.4 x 10–4
To divide numbers written in scientific notation,
divide the coefficients and subtract the
exponents (top – bottom)
3.0 x 10
 3. 0 
5 2

x
10


2
6.0 x 10
 6. 0 
5
 0.5 x 10
Coefficient needs to be
2
between 1 and 10
 5.0 x 10
3
Addition and Subtraction
When adding or subtracting in Sci. Not., the
exponents must be the same.
(5.4 x 103) + (8.0 x
Not the same, need to
adjust one of the
exponents
102)
= (5.4 x 103) + (0.80 x 103)
= (5.4 + 0.80) x 103
= 6.2 x 103
Example
Solve each problem and express the answer in
scientific notation.
a. (8.0 x 10–2) x (7.0 x 10–5)
b. (7.1 x 10–2) + (5 x 10–3)
a.
Multiply the coefficients and add the exponents.
(8.0 x 10–2) x (7.0 x 10–5)
= (8.0 x 7.0) x 10–2 + (–5)
= 56 x 10–7
= 5.6 x 10–6
b.
Rewrite one of the numbers so that the
exponents match. Then add the coefficients
(7.1 x 10–2) + (5 x 10–3)
= (7.1 x 10–2) + (0.5 x 10–2)
= (7.1 + 0.5) x 10–2
= 7.6 x 10–2
Topic 3 – Accuracy, Precision and Error
Accuracy - closeness of a measurement to
the actual or accepted value.
Precision - closeness of repeated
measurements to each other
Accuracy and Precision
Darts on a dartboard illustrate the difference between
accuracy and precision.
Good Accuracy,
Good Precision
Poor Accuracy,
Good Precision
Poor Accuracy,
Poor Precision
The closeness of a dart to the bull’s-eye corresponds to the degree of
accuracy. The closeness of several darts to one another corresponds
to the degree of precision.
Error
Suppose you measured the melting point of a
compound to be 78°C
Suppose also, that the actual melting point value
(from reference books) is 76°C.
The error in your measurement would be 2°C.
error  accepted  experiment al
Error is always a positive value
How far off you are in a measurement doesn’t
tell you much.
For example, lets say you have $1,000,000 in
your checking account. When you balance your
checkbook at the end of the month, you find that
you are off by $175; error = $175
Now, lets be more realistic, you have $225 in
your checking account and after balancing you
are off by $175!
In both cases, there is an error of $175.
But in the first, the error is such a small portion of
the total that it doesn’t matter as much as the
second.
So, instead of error, percent error is more
valuable.
error
percent error 
x 100
accepted
Percent error compares the error to the
size of the measurements.
Topic 4 – Significant Digits
In any measurement, the last digit is estimated
30.2°C
The 2 is estimated (uncertain) by the
experimenter, another person may say
30.1 or 30.3
9.3 mL
0.72 cm
Increasing Precision
The significant figures in a measurement are
the numbers that are part of the measurement.
Zeros that are NOT significant are called
placeholders.
Rules for determining Significant Figures
1. Every nonzero digit in a reported measurement is assumed to be significant.
2. Zeros appearing between nonzero digits are significant.
3. Leftmost zeros appearing in front of nonzero digits are not significant. They act
as placeholders. By writing the measurements in scientific notation, you can
eliminate such placeholding zeros.
4. Zeros at the end of a number and to the right of a decimal point are always
significant.
5. Zeros at the rightmost end of a measurement that lie to the left of an understood
decimal point are not significant if they serve as placeholders to show the
magnitude of the number.
5 (continued). If such zeros were known measured values, then they would be
significant. Writing the value in scientific notation makes it clear that these zeros are
significant.
6. There are two situations in which numbers have an unlimited number of
significant figures. The first involves counting. A number that is counted is exact.
6 (continued). The second situation involves exactly defined quantities such as
those found within a system of measurement.
“Line Through” Method for Counting Sig Figs
1. If there is a decimal, start from the left and draw
a line through any zeros, the numbers remaining
are significant.
2. If there is no decimal, start the line from the
right.
A Shorter Method
SOURCE: Skylar Morben, 2014 MRHS Graduate
How many significant digits?
100
1.00
0.23
0.0034
1.01
1005.4
0.10
100.0
54.0
How many significant digits are in the following measurements?
a) 150.31 grams
b) 10.03 mL
c) 0.045 cm
d) 4.00 lbs
e) 0.01040 m
f) 100.10 cm
g) 100 grams
h) 1.00 x 102 grams
i) 11 cars
j) 2 molecules
An answer can’t be more accurate
than the measurements it was
calculated from
Rules for Add/Subtracting Sig Figs
The answer to an +/- calculation should be
rounded to the same number of decimal places
as the measurement with the least number of
decimal places.
2.05 cm
+ 1.1 cm
3.15 cm
3.2 cm
32.10
g
+ 5.0012 g
37.1012
37.10 g
g
Rules for Mult/Division Sig Figs
The answer to a x/÷ calculation should be
rounded to the same number of sig figs as the
measurement with the least number of sig figs.
2.0 cm
x 1.89 cm
3.78 cm
3.8
cm2
2
8.19 g
 8.1 mL
1.01111111... g
g
1.0
mL
mL
Always round your final answer off to
the correct number of significant digits.
Draw a box around the significant digits in the
following measurements.
a) 2.2000
b) 0.0350
c) 0.0006
d) 0.0089
e) 24,000
f) 4.360 x 104
g) 0.0708
h) 1200
i) 0.6070
k) 21.0400
l) 0.007
m) 5.80 x 10-3
Round off each of the following numbers to two
significant figures.
a) 86.048
b) 29.974
c) 6.1275
d) 0.008230
e) 800.7
f) 0.07864
g) 0.06995
h) 7.096
i) 8000.10
Express each of the following numbers in standard scientific
notation with the correct number of significant digits.
a)
b)
c)
d)
e)
f)
g)
h)
0.00000070
25.3
825,000
826.7
43,500
65.0
0.000320
0.0432
Perform the following arithmetic. Round the answers to the proper
number of sig. figs. Box in your final answer and don’t forget units!
a)2.41 cm x 3.2 cm
c) 81.4 g  104.2 cm3
b) 4.025 m x 18.2 m
Perform the following arithmetic. Round the answers to the proper
number of sig. figs. Box in your final answer and don’t forget units!
d) 822 mi  0.028 hr
e) 10.89 g / 1 mL