Download Chapter 2 Measurement & Problem Solving

Chapter 2
Measurement & Problem Solving
Uncertainty
• There is a certain amount of doubt in
every measurement
– It is important to know the uncertainty
when measurements are recorded or read
– Why is this important in science?
• This helps to communicate the amount of
accuracy and precision of the
measurement
Accuracy
• The closeness between a
measurement and its true value.
– The true value is rarely known, so
accuracy is based on how close
independent studies agree
– The closer they agree, the more
confidence scientists have in the
accuracy of their results
Accuracy
• Accuracy is affected by determinate
errors
– Determinate errors are errors due to
1) poor technique
2) incorrectly calibrated instruments
Which graduated cylinder is
more accurate?
• 50 mL
up by 1 mL
• 25 mL
up by 0.5 mL
• 10 mL
**Has a smaller scale so measurement will be
closer to true value
up by 0.1 mL
Precision
• The reproducibility of an experiment
(how close are the repeated measurements to each other)
• Precision is affected by indeterminate errors
– Indeterminate errors- errors due to
estimating the last uncertain digit of a
measurement
• Random
• Cannot be eliminated
Graduated Cylinder
3.0 mL
5.72 mL
0.33 mL
Precision
What are other
examples of when
precision is
important to you?
Which graduated cylinder is
more precise?
• 50 mL
up by 1 mL
• 25 mL
up by 0.5 mL
• 10 mL
**Has a smaller scale so measurement will be
more precise
up by 0.1 mL
Is it possible to be precisely
inaccurate?
YES!!
Image from: http://celebrating200years.noaa.gov/magazine/tct/tct_side1.html
What is the volume?
• Up front, two of the graduated cylinders
have water in them
• 10 students to “read” the volumes
– Write down your measurements on a scratch
piece of paper:
Volumes of water in
50mL grad. cylinder =
10mL grad. cylinder=
• 1 student to write measurements on the
board and tally them if there are
multiples
Is the variation in those
measurements due to
determinate or indeterminate
errors?
BOTH!
Determinate errors- may have had poor technique
(didn’t read from bottom of meniscus, didn’t lower
eyes to the level of the meniscus)
Indeterminate errors- estimating the last digit (when
the level of the meniscus is between two distinct
lines)
**This illustrates that there is always uncertainty
in measurements!
Uncertainty
• Instruments are not always calibrated
perfectly (remember, humans do the calibrating and
many times the measuring!)
• It is very important that in science we
can communicate how accurate or
precise our measurements are
Communicating Uncertainty
• In science, it is understood that the
last digit of a number is “uncertain”
or the estimated digit
(if another person were to make the same measurement,
the number may be different)
*The estimate is made between
the smallest divisions on the scale
of the instrument
(i.e. ruler, graduated cylinder, thermometer)
**Generally, the more numbers after the
decimal point, the more precise the
instrument is
General Rule for Measuring
Uncertainty
• Unless otherwise indicated, assume that the
uncertainty in a measurement is one-half the
smallest division
• Examples:
________________
_________________ _______________
Which is more precise?
1002 cm
or
Instrument measures every 10
cm, so estimated to the ones
place
1002.39 cm
Instrument measures every 0.1
cm, so estimated to the
hundredths place
In both numbers,
which
This
number is measured to more
decimal places which indicates
numbers are uncertain?
smaller increments of measurement
Things to keep in mind…
• When using an electric device, the last digit
reported will be considered the uncertain
digit
• “2” and “2.0” are not the same number!
– By saying “2” we are saying the number is not 1
or 3
– By saying “2.0” we are saying the number is not
1.9 or 2.1
**It is the responsibility of the experimenter to
write numbers in a way to reflect the
precision of the instrument being measured
Standard Deviation
• Values reported as results from experiments are in the
form of the “mean plus-or-minus one standard
deviation” ( X ± s.d.)
– This gives a range over which we have confidence
that the true value will fall
• Used in scientific journals
• Requires a sample of at least three
• Most calculators are able to compute the mean and
standard deviation
– The standard deviation indicates how much each
value differs from the average
– The size of the deviation would indicate precision in
running a sequence of experiments (the smaller the
deviation, the more precise the work was)
Scientific Notation
• Scientific Notation is often used to
write very large or very small
numbers
– The numbers are expressed as a
coefficient number multiplied by an
exponent of 10
exponent
coefficient
4.32 x 105
Scientific Notation
• The coefficient is a number greater
than 1 but less than 10
• The exponent is:
– Positive if the number is greater than 1
Ex: 1,200 = 1.2 x 103
– Negative if the number is less than 1
Ex: 0.023 = 2.3 x 10-2
– Zero if the number is equal to 1
Ex: 1 = 100
Addition & Subtraction with
Scientific Notation
1) Exponents must be made equal
before you can perform the operation
Ex: 5.4x103 + 6.0x102 = 5.4x103 + 0.60x103
2) Then add or subtract the coefficients
& the exponent will stay the same
Ex: 5.4x103 + 0.60x103 = 6.0x103
Multiplication and Division with
Scientific Notation
1) Multiply the coefficients and add the
exponents
Ex: (3.0x104) x (5.0x102) = 15.0x106 = 1.50x107
2) Divide the coefficients and subtract
the exponents – be careful when
subtracting a negative exponent!!
Ex: (8.0x104) ÷ (2.0x10-2) = 4.0x106
Significant Figures
• In science, numbers are only
significant if they were measured.
The last number reported is the
“estimated” number and therefore is
the uncertain number
Significant Figures
• All non-zero digits are significant
• What to do with the zeros:
– Zeros between significant digits are
significant
(Sandwich Rule)
Ex: 307cm (3 sig figs)
– Zeros at the end of a number and to the
RIGHT of a decimal are significant
Ex: 5.20cm (3 sig figs)
Significant Figures
• Zeros at the end of a number and to
the LEFT of an assumed decimal may
or may not be significant
**The presence of a decimal indicates
significance, while the absence of the
decimal point makes it insignificant
Ex: 750.cm (3 sig figs)
750cm (2 sigs figs)
Significant Figures
• “Cosmetic” zeros written to the LEFT
of the first non-zero digit are NOT
significant
Ex: 0.146cm (3 sig figs)
09 cm (1 sig fig)
• Zeros that are place holders are NOT
significant
Ex: .006cm (1 sig fig)
0.014cm (2 sig figs)
Significant Figures
• The difference between zeros used for accuracy and
those being used as place holders can only be
addressed using scientific notation
All zeros are placeholders (no decimal) 1 sig fig
– 1,000,000
The first two zeros are significant 3 sig figs
– 1.00x106
6 sig figs
– 1.00000x106 All the zeros are significant
Sig Fig Practice
1.
2.
3.
4.
5.
6.
7.
8.
9.
23.46 mL 4 sig figs
0.0036 s 2 sig figs
854.236 g 6 sig figs
6.02x1023 molecules 3 sig figs
0.98 mol 2 sig figs
2 sig figs
0023 m
4 sig figs
2.000 J
1.00026x10-3 cm 6 sig figs
824 mg 3 sig figs
Exact Values
• Some numbers are exact values which involve no
uncertainty.
•
For example, there is an exact number of people in the classroom. If
there are 20 people, the zero is automatically significant.
• Chemistry examples:
– There are 4.184 joules in a calorie
– There are exactly 2 hydrogen atoms in a water molecule
– Stoichiometric coefficients and subscripts
6CO2
– Defined quantities
• (i.e. If we know 1 liter = 1000 mL, the number 1000 actually has
an infinite number of significant digits)
Significant Figures & Rounding
• Round down if last (or leftmost) digit
dropped is 4 or less
Ex: 56.3231 cm
56.32cm
• Round up if the last digit dropped is 5
or more
Ex: 4.5272 cm
4.53cm
**Make sure you only use the last digit being
dropped to decide which direction to round
(ignore all digits to the right of it)
Addition & Subtraction of
Significant Figures
• The number of places to the right of the
decimal point (degree of accuracy),
determines the number of sig figs reported
Add the decimals as
you would normally
(add in placement
zeros if it helps
you, but do not
consider them
when you decide on
the # of sig figs
15.340
15.34 cm
4.100
4.1
cm **This number has the
+ 23.584 cm least number of digits
43.024 cm to the right of the
decimal – Answer
should be rounded to
the tenth place
Correct Answer:
43.0 cm
Multiplication & Division of
Significant Figures
• The number or reported digits is determined by the
lease accurate value (the one with the fewest
significant figures)
23.5 cm x 63.215 cm x 2. cm = 2,994.46 cm
Correct answer: 3000 cm or 3 x 103cm
This number has the least
amount of sig figs, so the
answer will need to have only 1
sig fig
Keep in mind…
• For calculations involving multiple
steps, round only the final answer
• In calculations involving both
multiplication/division and
addition/subtraction, do the steps in
parenthesis first
Determining Sig Figs in
Measurements
• The last significant figure in your measurement
needs to be one that you estimate
• The digits preceding this last digit, are considered
“certain” because the instrument will have specific
“marks” that should be exact measurements.
• The final “estimated” digit will be located between
the specific “marks” (depending on the location of
the measurement, but you may also decide that
the estimated digit is a zero)
How many centimeters is
this pencil?
Your answer should have
3 sig figs!
The ruler measures 0.1cm
(1mm) so your measurement
should have an estimated
digit in the hundredths
position
~6.12 cm
How many mL are in this
graduated cylinder?
Your answer should have 4
sig figs.
The graduated cylinder
measures to every 0.1 mL
so your estimated digit will
be in the hundreths
position
~40.30 mL
http://www.proprofs.com/quiz-school/story.php?title=science-benchmark-1-review-test
How long is this pen?
The tip of the pen is
between 4.7 cm and
4.8 cm. You will need
to estimate the digit
between the two
points
http://www.concord.org/~ddamelin/chemsite/b_measurement/sig_fig.html