Download Use of Significant Figures

Measurement
Uncertainty in Measurement
Significant Figures
2-1
Measurement
Observation can be both
QUALITATIVE and QUANTITIVE
A qualitative observation
is a description in words.
A quantitative observation
is a description with numbers and units.
A measurement is a comparison to a standard.
2-2
Units are important
45 000
has little meaning, just a number
$45,000
has some meaning - money
$45,000/yr more meaning - person’s salary
2-3
Uncertainty in Measurement
Use of Significant Figures
It is important to realize that a measurement
always has some degree of uncertainty,
which depends on the precision of the
measuring device.
Therefore, it is important to indicate the
uncertainty in any measurement. This is done
by using significant figures.
2-4
Uncertainty in Measurement
• Every measurement has an uncertainty
associated with it, unless it is an exact,
counted integer, such as the count of trials
performed or a definition.
2-5
Uncertainty in Measurement
• Every calculated result also has an
uncertainty, related to the uncertainty in the
measured data used to calculate it. This
uncertainty should be reported either as an
explicit ± value or as an implicit uncertainty,
by using the appropriate number of
significant figures.
2-6
Uncertainty in Measurement
• The numerical value of a "plus or minus" (±)
uncertainty value tells you the range of the
result.
• When significant figures are used as an
implicit way of indicating uncertainty, the last
digit is considered uncertain.
2-7
Uncertainty in Measurement
. A significant figure is one that has been
measured with certainty or has been
'properly' estimated.
The significant figures in a number includes all
certain digits as read from the instrument
plus one estimate digit.
2-8
Uncertainty in Measurement
Significant digits or significant figures
- are digits read from the measuring
instrument plus one doubtful digit
estimated by the observer. This doubtful
estimate will be a fractional part of the
least count of the instrument.
2-9
Uncertainty in Measurement
All measurements contain some uncertainty.
• Limit of the skill and carefulness
of person measuring
• Limit of the measuring tool/equipment
being used
Uncertainty is measured with
Accuracy
How close to the true value
Precision
How close to each other
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Precision
How well our values agree with each other.
Here the numbers
are close together
so we have good
precision.
xx
x
• Poor accuracy.
• Large systematic
error.
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Accuracy
How close our values agree with the true value.
Here the
average value
would give a
accurate number
but the numbers
don’t agree, are
not precise.
x
x
x
Large random error
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Accuracy and precision
Our goal!
Good precision
and accuracy.
xx
x
These are
values we
can trust.
2 - 13
Accuracy and precision
Predict the effect on accuracy and precision.
• Instrument not ‘zeroed’ properly
• Reagents made at wrong concentration
• Temperature in room varies ‘wildly’
• Person running test is not properly trained
2 - 14
Types of errors
Systematic
Instrument not ‘zeroed’ properly
Reagents made at wrong concentration
Random
Temperature in room varies ‘wildly’
Person running test is not properly trained
2 - 15
Errors
Systematic
• Errors in a single direction (high or low).
• Can be corrected by proper calibration or
running controls and blanks.
Random
• Errors in any direction.
• Can’t be corrected.
Can only be
accounted for by using statistics.
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Errors
Systematic: ACCURACY
• Errors in a single direction (high or low).
• Can be corrected by proper calibration
or running controls and blanks.
Random: PRECISION
• Errors in any direction.
• Can’t be corrected.
Can only be
accounted for by using statistics.
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Significant figures
Method used to express precision.
You can’t report numbers better than the method
used to measure them.
67.2 units = three significant figures
Certain
Digits
Uncertain
Digit
ONLY ONE
UNCERTAIN DIGIT
IS REPORTED
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Significant figures
The number of significant digits is
independent of the decimal point.
These numbers
All have three
significant figures!
255
25.5
2.55
0.255
0.0255
2 - 19
Significant figures:
Rules for zeros
Leading zeros are not significant.
0.421 - three significant figures
Leading zero
Captive zeros are significant.
4012 - four significant figures
Captive zero
Trailing zeros are significant.
114.20 - five significant figures
Trailing zero
2 - 20
Significant figures
Zeros are what will give you a headache!
They are used/misused all of the time.
Example
The press might report that the federal
deficit is three trillion dollars. What did they
mean?
$3 x 1012 meaning +/- a trillion dollars
or
$3,000,000,000,000.00 meaning +/- a penny
2 - 21
Significant figures
In science, all of our numbers are either
measured or exact.
•
Exact - Infinite number of significant figures.
•
Measured - the tool used will tell you the level
of significance. Varies based on the tool.
Example
Ruler with lines at 1/16” intervals.
A balance might be able to measure to the
nearest 0.1 grams.
2 - 22
Significant figures:
Rules for zeros
Scientific notation - can be used to clearly
express significant figures.
A properly written number in scientific
notation always has the the proper number
of significant figures.
0.00321
=
3.21 x 10-3
Three Significant
Figures
2 - 23
Scientific notation
• Method to express really big or small
numbers.
Format is
Mantissa x Base Power
Decimal part of
original number
Decimals
you moved
We just move the decimal point around.
2 - 24
Scientific notation
If a number is larger than 1
• The original decimal point is moved X
places to the left.
• The resulting number is multiplied by 10X.
• The exponent is the number of places you
moved the decimal point.
1 2 3 0 0 0 0 0 0. = 1.23 x 108
2 - 25
Scientific notation
If a number is smaller than 1
• The original decimal point is moved X
places to the right.
• The resulting number is multiplied by 10-X.
• The exponent is the number of places you
moved the decimal point.
0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7
2 - 26
Scientific notation
Most calculators use scientific notation when
the numbers get very large or small.
How scientific notation is
displayed can vary.
It may use x10n
or may be displayed
using an E.
They usually have an Exp or EE
This is to enter in the exponent.
1.44939 E-2
cos tan
CE
ln
7
8
9
/
log
4
5
6
x
1/x
1
2
3
-
x2
EE
0
.
+
2 - 27
Examples
378 000
3.78 x 10 5
8931.5
8.9315 x 10 3
0.000 593
5.93 x 10 - 4
0.000 000 4
4 x 10 - 7
2 - 28
Significant figures
and calculations
An answer can’t have more significant figures
than the quantities used to produce it.
Example
How fast did you run if you
went 1.0 km in 3.0 minutes?
speed
= 1.0 km / 3.0 min
= 0.33 km / min
0.333333333
cos tan
CE
ln
7
8
9
/
log
4
5
6
x
1/x
1
2
3
-
x2
EE
0
.
+
2 - 29
Significant figures and calculations
Addition and subtraction
Report your answer with the same number
of digits to the right of the decimal point as
the number having the fewest to start with.
123.45987 g
+ 234.11 g
357.57 g
805.4 g
- 721.67912 g
83.7 g
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Significant figures and calculations
Multiplication and division.
Report your answer with the same number
of digits as the quantity have the smallest
number of significant figures.
Example. Density of a rectangular solid.
25.12 kg / [ (18.5 m) ( 0.2351 m) (2.1m) ]
= 2.8 kg / m3
(2.1 m - only has two significant figures)
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Example
257 mg
\__ 3 significant figures
102 miles
\__ 3 significant figures
0.002 30 kg
\__ 3 significant figures
23,600.01 $/yr
\__ 7 significant figures
2 - 32
Rounding off numbers
After calculations, you may need to round off.
2 - 33
Rounding off
If a set of calculations gave you the following
numbers and you knew each was supposed to
have four significant figures then -
2.5795035 becomes 2.580
1st uncertain digit
34.204221 becomes 34.20
2 - 34
Converting units
Factor label method
• Regardless of conversion, keeping track
of units makes thing come out right
• Must use conversion factors
- The relationship between two units
• Canceling out units is a way of checking
that your calculation is set up right!
2 - 35
Common conversion factors
SomeEnglish/ Metric conversions
1 liter
1 kilogram
1 meter
1 inch
= 1.057 quarts
= 2.2 pounds
= 1.094 yards
= 2.54 cm
Factor
1.057 qt/L
2.2 lb/kg
1.094 yd/m
2.54 cm/inch
2 - 36
Example
A nerve impulse in the body can travel as fast
as 400 feet/second.
What is its speed in meters/min ?
Conversions Needed
1 meter
1 minute
=
=
3.3 feet
60 seconds
2 - 37
Example
?
m
400 ft
=
min
1 sec
?
m
400 ft
=
min
1 sec
7273
m
min
x
1m
3.3 ft
60 sec
x
1 min
x
1m
3.3 ft
60 sec
x
1 min
....Fast
2 - 38
Extensive and intensive properties
Extensive properties
Depend on the quantity of sample measured.
Example - mass and volume of a sample.
Intensive properties
Independent of the sample size.
Properties that are often characteristic of the
substance being measured.
Examples - density, melting and boiling points.
2 - 39
Density
Density is an intensive property of a substance
based on two extensive properties.
Density =
Mass
Volume
Common units are g / cm3 or g / mL.
g / cm3
Air
Water
Gold
0.0013
1.0
19.3
cm3 = mL
g / cm3
Bone
Urine
Gasoline
1.7 - 2.0
1.01 - 1.03
0.66 - 0.69
2 - 40
Example.
Density calculation
What is the density of 5.00 mL of a fluid if it
has a mass of 5.23 grams?
d = mass / volume
d = 5.23 g / 5.00 mL
d = 1.05 g / mL
What would be the mass of 1.00 liters of this
sample?
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Example.
Density calculation
What would be the mass of 1.00 liters of the
fluid sample?
The density was 1.05 g/mL.
density = mass / volume
so
mass
= volume x density
mass
= 1.00 L x 1000 ml x 1.05 g
L
mL
= 1.05 x 103 g
2 - 42
Specific gravity
The density of a substance compared to a
reference substance.
density of substance
Specific Gravity =
density of reference
• Specific Gravity is unitless.
• Reference is commonly water at 4oC.
• At 4oC, density = specific gravity.
• Commonly used to test urine.
2 - 43
Specific gravity measurement
Hydrometer
Float height will
be based on
Specific
Gravity.
2 - 44
Measuring time
The SI unit is the second (s).
For longer time periods, we can
use SI prefixes or units such
as minutes (min), hours (h),
days (day) and years.
Months are never used they vary in size.
2 - 45
The mole
Number of atoms in 12.000 grams of 12C
1 mol
mol
=
=
6.022 x 1023 atoms
grams / formula weight
Atoms, ions and molecules are too small to
directly measure - measured in u.
Using moles gives us a practical unit.
We can then relate atoms, ions and
molecules, using an easy to measure unit the gram.
2 - 46
The mole
If we had one mole of water and one mole of
hydrogen, we would have the name number
of molecules of each.
1 mol H2O
1 mol H2
= 6.022 x 1023 molecules
= 6.022 x 1023 molecules
We can’t weigh out moles -- we use grams.
We would need to weigh out a different
number of grams to have the same number
of molecules
2 - 47
Moles and masses
Atoms come in different sizes and masses.
A mole of atoms of one type would have a
different mass than a mole of atoms of
another type.
H
- 1.008 u or grams/mol
O
- 16.00 u or grams/mol
Mo - 95.94 u or grams/mol
Pb - 207.2 u or grams/mol
We rely on a straight forward system to
relate mass and moles.
2 - 48
Masses of atoms
and molecules
Atomic mass
• The average, relative mass of an atom in
an element.
Atomic mass unit (u)
• Arbitrary mass unit used for atoms.
• Relative to one type of carbon.
Molecular or formula mass
• The total mass for all atoms in a compound.
2 - 49
Molar masses
Once you know the mass of an atom, ion, or
molecule, just remember:
Mass of one unit
- use u
Mass of one mole of units - use grams/mole
The numbers DON’T change -- just the units.
2 - 50
Masses of atoms
and molecules
H2O - water
2 hydrogen
1 oxygen
mass of molecule
2 x 1.008 u
1 x 16.00 u
18.02 u
18.02 g/mol
Rounded off based
on significant figures
2 - 51
Another example
CH3CH2OH - ethyl alcohol
2 carbon
6 hydrogen
1 oxygen
mass of molecule
2x
6x
1x
12.01 u
1.008 u
16.00 u
46.07 u
46.07 g/mol
2 - 52
Molecular mass vs. formula mass
Formula mass
Add the masses of all the atoms in formula
- for molecular and ionic compounds.
Molecular mass
Calculated the same as formula mass
- only valid for molecules.
Both have units of either u or grams/mole.
2 - 53
Formula mass, FM
The sum of the atomic masses of all elements
in a compound based on the chemical
formula.
You must use the atomic masses of the
elements listed in the periodic table.
CO2
1 atom of C and 2 atoms of O
1 atom C x 12.011 u
2 atoms O x 15.9994 u
Formula mass
=
=
=
12.011 u
31.9988 u
44.010 u
or
g/mol
2 - 54
Example - (NH4)2SO4
OK, this example is a little more complicated.
The formula is in a format to show you how
the various atoms are hooked up.
( N H 4) 2S O 4
We have two (NH4+) units and one SO42- unit.
Now we can determine the number of atoms.
2 - 55
Example - (NH4)2SO4
Ammonium sulfate contains
2 nitrogen, 8 hydrogen, 1 sulfur & 4
oxygen.
2N
8H
1S
4O
x
x
x
x
14.01
1.008
32.06
16.00
=
=
=
=
28.02
8.064
32.06
64.00
Formula mass
=
132.14
The units are either u or grams / mol.
2 - 56
Example - (NH4)2SO4
How many atoms are in 20.0 grams of
ammonium sulfate?
Formula weight
grams/mol
Atoms in formula
= 132.14
= 15 atoms / unit
1 mol
moles = 20.0 g x
= 0.151 mol
132.14 g
atoms
units
23
atoms = 0.151 mol x 15 unit x 6.02 x10
mol
atoms = 1.36 x1024
2 - 57