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UNCERTAINTIES IN MEASUREMENTS
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UNCERTAINTY IN MEASUREMENTS
Different measuring devices have different uses
and different degrees of precision
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ESTIMATING UNCERTAINTY
• To estimate a measuring devices
uncertainty, read to ½ the
smallest scale division.
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INDICATING UNCERTAINTY IN A MEASUREMENT
• Indicate using + after the recorded measurement
stating the ½ beyond the smallest scale division
followed by the units
•
Example:
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INDICATING UNCERTAINTY IN A MEASUREMENT
For digital devices, uncertainty is
always +/- .1 of the smallest
scale
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Significant figures - Counting
• Counting Significant Figures – Atlantic Pacific Method
• Decimal point absent (Atlantic) – begin on right side
of number and cross out all zero digits until first
non – zero digit, remaining digits are significant
• Decimal point present (Pacific) - begin on left side
of number and cross out all zero digits until first
non – zero digit, remaining digits are significant
• Exact numbers – have infinite number of significant
figures
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Examples: Counting Sig. Figs
•
•
•
•
•
•
•
234 cm
67000 cm
45000 cm
560. cm
0.5630 cm
1.0034 cm
0.00467 cm
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SCIENTIFIC NOTATION
• Used to shorten large or small numbers
• Standard Form
Base - one number to the left of the decimal
place, with all the significant figures shown
Exponent – is the number of places the decimal
point must be shifted to give the number in
standard form
Negative Exponent – value less than 1
Positive Exponent – value greater than 1
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EXAMPLES: SCIENTIFIC NOTATION
124300 =
0.00362 =
1300000 =
1.23E-5 =
5.61 x 10^6 =
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SIGNIFICANT FIGURES IN MATHEMATICAL
OPERATIONS
• Addition/Subtraction – the answer will have the
same number of decimal places as the
measurement with the fewest decimal places
• Multiplication/Division – the answer will have the
same number of significant figures as the
measurement with the fewest significant figures
• Mixed operations – follow the order of operations,
determining s.f. for each step, do not round until
the end!
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EXAMPLE OF SIG. FIGS IN CALCULATIONS
33.5 cm + 7.88 cm + 0.977 cm =
23000 km + 8.7 km =
67.23 cm x 9.22 cm =
(200 cm x 3.333) + (300 x 1.35) cm =
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CALCULATORS AND SCIENTIFIC NOTATION
• Calculators handle scientific notation by only
inputting the exponent, using an EXP or EE key
enter the base as you would a regular
number, then press EXP or EE, then enter the
exponent
• Display – the calculator used E to show
exponent ( E means x 10 )
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RANDOM ERRORS - PRECISION
• Random errors - Precision
•A random error makes the measured value
both smaller and larger than the true value
(this happens by chance alone)
•Reduce random errors- repeat the
experiment
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SYSTEMATIC ERRORS
• Systematic error - Accuracy
• Errors due to "incorrect" use of
equipment or poor experimental design
• Makes the measured value always smaller or
larger than the true value, but not both.
• An experiment may involve more than one
systematic error and these errors may nullify
one another
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CATEGORIES OF SYSTEMATIC ERRORS
• Personal errors – the result of ignorance,
carelessness, prejudices, or physical limitations on
the experimenter.
• Instrumental Errors - attributed to imperfections in
the tools with which the analyst works.
• Method Errors - results when you do not consider
how to control an experiment.
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DETERMINATION OF ERROR
• Accuracy - how close a
measurement is to the true value of
a quantity.
• Precision - how close several
measurements are to each other.
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PRECISION VS. ACCURACY
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EVALUATING ACCURACY AND PRECISION IN DATA
• When evaluating whether data is accurate or precise
you could look at
Accuracy
Precision
Single Data Point
A single data point is accurate if
it is close the literature value
A single data point is
precise if it has many
decimal places/significant
figures. Meaning, you
used a very precise tool or
method for measuring the
value.
Set of Data Points
A set of data points are accurate
if the average or mean is close
to the literature value. This is
the reason we do multiple trials.
Any random errors cancel out
when the average is taken, thus
random error is reduced.
A set of data points are
precise if they are all very
close together. This
definition refers to the
consistency of the data.
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PERCENTAGE UNCERTAINTIES AND
PROPAGATION OF ERRORS
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Percent Uncertainties and
errors
• Percent Error
•
(accepted value – experimental value) x 100
•
accepted value
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MEASUREMENT
Temperature Conversions
A. Temperature
• Temperature
• measure of the average KE of
the particles in a sample of
matter
Kelvin  oC  273.15
9o
Fahrenheit 
C  32
5
5 o
Celsius  ( F  32)
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A. Temperature
•
Convert these temperatures:
1)
25oC = ______________K
2)
-15oF = ______________ K
3)
315K = ______________ oC
4)
288K = ______________ oF
CH. 3 - MEASUREMENT
Dimensional Analysis
Conversion Factors
Problems
A. Problem-Solving Steps
1. Analyze
2. Plan
3. Compute
4. Evaluate
B. Dimensional Analysis
• Dimensional Analysis
• A tool often used in science for converting units
within a measurement system
• Conversion Factor
• A numerical factor by which a quantity expressed in
one system of units may be converted to another
system
B. Dimensional Analysis
• The “Factor-Label” Method
• Units, or “labels” are canceled, or “factored” out
g
g
cm 

3
cm
3
B. Dimensional Analysis
• Steps to solving problems:
1. Identify starting & ending units.
2. Line up conversion factors so units cancel.
3. Multiply all top numbers & divide by each
bottom number.
4. Check units & answer.
C.Fractions
Conversion
Factors
in which the numerator and
denominator are EQUAL quantities expressed
in different units
Example:
1 in. = 2.54 cm
Factors: 1 in.
and
2.54 cm
2.54 cm
1 in.
How many minutes are in 2.5 hours?
Conversion factor
2.5 hr x
1
cancel
60 min
1 hr
= 150 min
By using dimensional analysis / factor-label method,
the UNITS ensure that you have the conversion right
side up, and the UNITS are calculated as well as the
numbers!
C. Conversion Factors
Learning Check:
Write conversion factors that relate
each of the following pairs of units:
1. Liters and mL
2. Hours and minutes
3. Meters and kilometers
E. Dimensional Analysis Practice
• You have $7.25 in your pocket in quarters.
How many quarters do you have?
$7.25
1
X
4 quarters
1 dollar
E. Dimensional Analysis Practice
How many seconds are in 1.4
days?
1.4
24 hr 60
days
min
60 s
1
day
1
min
1 hr
= 12000 s
E. Dimensional Analysis Practice
• How many milliliters are in 1.00 quart of milk?
E. Dimensional Analysis Practice
• You have 1.5 pounds of gold. Find its volume in cm3 if
the density of gold is 19.3 g/cm3.
E. Dimensional Analysis Practice
Your European hairdresser wants to cut your hair 8.0 cm
shorter. How many inches will he be cutting off?
E. Dimensional Analysis Practice
Roswell football needs 550 cm for a 1st down.
How many yards is this?
E. Dimensional Analysis Practice
A piece of wire is 1.3 m long. How many 1.5-cm
pieces can be cut from this wire?
E. Dimensional Analysis Practice
• How many liters of water would fill a container
that measures 75.0 in3?
Base Units
• In the SI (Le Système International
d'Unités) system of measurement there
are seven base units
• These units are used in the measurement
of different quantities and independent of
each other
Base Unit
Quantity
Unit
Symbol
Mass
Grams
g
Length
Meter
m
Amount of
Substance
Mole
mol
Time
Second
s
Electric Current
Ampere
A
Kelvin
K
candela
cd
Temperature
Luminous Intensity
D. SI Prefix Conversions
•Memorize the following chart. (next slide)
•Find the conversion factor(s).
•Insert the conversion factor(s) to get to the correct
units.
•When converting to or from a base unit, there will only
be one step. To convert to or from any other units,
there will be two steps.
move right
move left
A. SI Prefix
Conversions
Prefix
Symbol
Factor
tera-
T
1012
gigamegakilohectodekaBASE UNIT
decicentimillimicronanopico-
G
M
k
h
da
--d
c
m

n
p
109
106
103
102
101
100
10-1
10-2
10-3
10-6
10-9
10-12
D. SI Prefix Conversions
Tera-
1 T(base) = 1 000 000 000 000(base) = 1012 (base)
Giga-
1 G(base) = 1 000 000 000 (base) = 109 (base)
Mega-
1 M(base) = 1 000 000 (base) = 106 (base)
Kilo-
1 k(base) = 1 000 (base) = 103 (base)
Hecto-
1 h(base) = 100 (base) = 102 (base)
Deka-
1 da(base) = 101 (base)
Base
1 (base) = 1 (base)
Deci-
10 d(base) = 1(base)
Centi-
100 c(base) = 1 (base)
Milli-
1000 m (base) = 1(base)
Micro-
1 (base) = 1 000 000 µ = 10-6(base)
Nano-
1 (base) = 1 000 000 000 n = 10-9(base)
Pico-
1 (base) = 1 000 000 000 000 p = 10-12(base)
D. SI Prefix Conversions
a. cm to m
b. m to µm
c. ns to s
d. kg to g
D. SI Prefix Conversions
1)
20 cm =
______________ m
2) 0.032 L =
______________ mL
3) 45 m =
______________ m
D. SI Prefix Conversions
4) 805 Tb = ______________ b
Terabytes
bytes
D. SI Prefix Conversions
1) 400. g = ______________ kg
1) 57 Mm = ______________ nm
• 1 cm3 = _______________ dm3
• 1 kL2 = _____________ L2
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DENSITY FUN
Objective
• Student will demonstrate an
understanding of density by using
it to perform different unit
conversions.
A. Derived Units
• Combination of base units
• Volume – length  length  length
1 cm3 = 1 mL
1 dm3 = 1 L
• Density – mass per unit volume (g/cm3)
M
D=
V
Broken
Heart
M
D V
Density
• Density is a physical property of matter, as each element and
compound has a unique density associated with it.
• The density of any sample of a substance at the same
temperature will always be the same.
Density
• If a substance is more dense than a liquid it will sink in it, if it is
less dense than it will float
• The density of water is 1.00 g/mL, however this varies slightly
with temperature
B. Density Calculations
• An object has a volume of 825 cm3 and a density of 13.6
g/cm3. Find its mass.
GIVEN:
WORK:
V = 825 cm3
D = 13.6 g/cm3
M=?
M = DV
M
D V
M = (13.6 g/cm3)(825cm3)
M = 11,220 g
M = 11,200 g
B. Density Calculations
A liquid has a density of 0.87 g/mL. What volume is
occupied by 25 g of the liquid?
GIVEN:
WORK:
D = 0.87 g/mL
V=?
M = 25 g
V=M
D
M
D V
V=
25 g
0.87 g/mL
V = 28.7 mL = 29 mL
B. Density Calculations
You have a sample with a mass of 620 g and a volume of 753
cm3. Find its density.
GIVEN:
WORK:
M = 620 g
V = 753 cm3
D=?
D=M
V
M
D V
D=
620 g
753 cm3
D = 0.82 g/cm3
C. Density Calculations with DA
• Used when units do not agree
• Conversions must be made before using formula
M
D=
V
g
D=
3
cm
C. Density Calculations with DA
•
You have 3.10 pounds of gold. Find its volume in cm3 if the density
of gold is 19.3 g/cm3.
cm3
lb
3.10 lb 1 kg 1000 g 1 cm3
2.2 lb
1 kg
19.3 g
= 73.0 cm3
C. Density Calculations with DA
•
You have 0.500 L of water. Find its mass in ounces if
the density of water is 1.00 g/cm3.
L
oz
0.500 L 1000 mL 1 cm3 1.00g 1 kg 2.2 lbs 16 oz
1L
1 mL 1 cm3 1000 g 1kg 1lb
= 17.6 oz
• I threw a plastic ball in the pool for my dog to fetch. The mass
of the ball was 125 grams.
• What must the volume be to have a density of 0.500 g/mL. ( I
want it to float of course!)
• A little aluminum boat (mass of 14.50 g) has a volume of
450.00 cm3. The boat is place in a small pool of water and
carefully filled with pennies. If each penny has a mass of 2.50
g, how many pennies can be added to the boat before it sinks?
Matter
• Matter is defined as anything that has mass and takes up
space.
• Volume is the amount of space matter occupies.
• The smallest building block of matter is the atom.
States of Matter
• Solid- definite
volume and definite
shape; vibrate at
fixed points
• Liquid- has a definite
volume but an
indefinite shape;
particles can move
past one another and
will take the shape of
its container
• Gas- Neither a
definite volume nor a
definite shape;
particles move
rapidly and will take
the shape of its
container
• Plasma- high
temperature state in
which atoms lose
electrons (Sun and
stars)
Pure Substances
• Element- The simplest type of a pure substance
made of only one kind of atom. The smallest
particle of an element is an atom.
Elements are list on the periodic table
• Compounds- Pure substances that are made of
two or more elements that are chemically
bonded. Examples are sugar, carbon dioxide,
ammonia, baking soda, and vinegar.
Identify the following as an
Element or Compound
• Na
Elements
Compound
• MgO2
Elements
• As
Elements
• Cl2
Compound
• H2O
• Fe(NO3)2 Compound
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Compounds
• Compounds can be broken down into simpler
substances while elements cannot. Heat and
electricity are often used to break apart
compounds.
• The properties of a compound are very different
from the elements that make up the compound.
Ex.- Salt is a compound made from sodium ( a
silvery metal) and chlorine ( a greenish
poisonous gas)
Compounds
• The smallest particle of a compound is a
molecule. Molecules of a compound are all
alike. Water in the ocean is like water that
comes from the faucet.
• Atoms are represented by symbols and
molecules are represented by formulas.
• Every element and compound has chemical
properties that are used for classification.
Classifying Matter
• Matter is classified into one of two
groups: Pure Substances or Mixtures
• Pure substances include compounds
and elements and they have a definite
composition.
• Mixtures contain more than one
substance; therefore, the composition
varies.
Mixtures
• A mixture is a blend of two or more kinds of
matter which retains its own identity and
properties.
• Mixtures that have a uniform composition are
called homogeneous mixtures or solutions.
• Mixtures that do not have uniform compositions
are called heterogeneous mixtures.
• In Chemistry, the amount of substance present in
a mixture is referred to by percent by mass of
the substance.
Properties
• Physical properties
can be observed
without changing the
identity of the
substance.
•
•
•
•
•
Melting Point
Boiling Point
Freezing Point
Color
Mass
• Chemical Properties
can be observed
when new
substances are
produced.
• Ability to Burn
• Ability to Tarnish
• Ability to Rust
Types of Physical Properties
• Intensive properties
do not depend on
• Extensive properties
the amount of
depend on the amount
matter present.
of matter present.
• Volume
• Mass
• Amount of Energy
•
•
•
•
Melting Point
Boiling Point
Density
Conductivity
Physical Change
• A change is a substance that does not involve a
change in the identity of the substance
• Examples include
•
•
•
•
•
Breaking
Tearing
Changes of state
Grinding
Cutting
Phase Changes
• When a substance undergoes a phase
change it is a physical change
• Melting: Solid to liquid
• Freezing: Liquid to solid
• Vaporizing: Liquid to gas
• Condensing: Gas to liquid
• Sublimation: Solid to gas
• Deposition: Gas to Solid
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Chemical Change
• Chemical Reactions- changing of a
substance into something else
• Examples include
• Burning, Rusting, Tarnishing, Reacting
• Reactants- substances that react
• Products- Substances that are produced
Chemical or Physical Change
• Dissolving salt in water Physical Change
• Burning oil
Chemical Change
• Melting solid iron Physical Change
• Sugar ferments
Chemical Change
• Copper sulfate crystals are broken Physical Change
• Paper is torn in half Physical Change
• Iron rusts
Chemical Change
• An egg is cooked Chemical Change
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