Transcript 01 Math skill APH015x

Topic 2: Math and Measurement
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Topic 2: Math and Measurement
1.4 Methods: Scientific method
SI metric system units
Review of Scientific Notation
Temperature scales
Density
1.5 Precision and Significant figures
Accuracy and percent error
Reading a scale
1.6 Dimensional analysis
An Example of Scientific Method
1) A sample of tin metal is heated strongly
Observation 1: The mass of the sample increases
Can you think of an hypothesis which would
explain this increase?
An Example of Scientific Method
1) A sample of tin metal is heated strongly
Observation 1: The mass of the sample increases
Adding heat energy makes objects hotter, and heavier
An Example of Scientific Method
2) A sample of copper sulfate is heated strongly
Observation 2: The mass of the sample decreases
Can you think of an hypothesis which would
explain this decrease?
An Example of Scientific Method
2) A sample of copper sulfate is heated strongly
Observation 2: The mass of the sample decreases
Revised: something is formed which leaves the crucible?
An Example of Scientific Method
1) A sample of tin metal is heated strongly
Observation 1: The mass of the sample increases
Revised?
Something is entering from outside?
An Example of Scientific Method
3) A sample of each is heated in a closed container
Observation 3: The mass of the sample remains the same
Hypothesis?
An Example of Scientific Method
3) A sample of each is heated in a closed container
Observation 3: The mass of the sample remains the same
Matter’s mass will remain the same as long as
nothing is lost to or gained from the
surroundings.
An Example of Scientific Method
Scientific Law: Mass in a closed system is always
conserved, even during chemical reactions.
Laws tell what happens or predict what will happen.
They are “observed regularities”
Laws can be “proven” by repeated experiments, however they may
need to be updated as new evidence arises.
What is needed is a “theory” – a mental picture, which can
provide an explanation for this conservation of mass.
Atomic Theory: tin, copper sulfate, and all matter is made
of tiny “atoms” …which can be gained from or lost to the
surroundings.
Scientific theories are generally accepted concepts which
we create to help explain the scientific laws that we observe.
Theories are only ideas, so they can not be proven, but we can find lots of
evidence in support.
An Example of Scientific Method
Mass in a closed system is always conserved, even
during chemical reactions.
…except during nuclear reactions. When mass can be lost.
…mass is released as large quantities of energy.
Up date: Law of conservation of mass/energy: “the combined
mass and energy of a system always remains the same.”
Types of Observations
COLD!
We make QUALITATIVE
observations of reactions
“of a quality”
Ex: changes in color and
physical state.
Types of Observations
We also make
QUANTITATIVE
observations:
MEASUREMENTS,
which involve numbers
…”AND units.
(Never write a number, without a unit)
Use SI units — based
on the metric system
5 0C
UNITS OF MEASUREMENT
Systeme International base units: aka metric units
The SI system involves units formed from base
units, combined with prefixes
Ex: Kilo-gram = a unit of 1000 grams
Kilo = 1000
Used for large quantities
Used for smaller quantities
Prefixes you should memorize are outlined
Kilo- means 1000 of that unit
ex: 1 kilometer (km) = 1000 meters (m)
1 kilo-unit = 1000 units
Centi- means 1/100 of that unit
ex: 1 meter (m) = 100 centimeters (cm)
1 unit = 100 centi-units
Milli- means 1/1000 of that unit
ex: 1 Liter (L) = 1000 milliliters (mL)
1 unit = 1000 milli-units
How many pascals (unit of air pressure) is a kilopascal?
1000 pascals
Atomic dimensions
A carbon atom has a radius of
91 picometers
That’s 91 x10-12meters
OR
9
of a meter
1,000,000,000,000
0.000 000 000 091 m
Table C is from your
regents reference
tables
It takes about 100,000,000,000 atoms placed side to side = 1 meter
Check your understanding
b
1. 1000 m = 1 ___
a) mm b) km c) dm
10 3
a
2. 0.001 g = 1 ___
a) mg
b) kg c) dg
b) cL c) dL
10 -3
3.
b
0.01 L = 1 ___
a) mL
4.
c
0.1 m = 1 ___
a) mm b) cm c) dm
Table 1.5 is in the textbook
But you can also
look at
regents reference table C
Review of Scientific Notation
Scientific notation is especially useful for very large or very small values
Example: Mass of the earth:
6592000000000000000000 tons
6.592 Quadrillion tons
6.592 x 10 21 tons
Scientifc notation is
split into two parts:
Notice: the “2” is the
highest level of precision
that we can be relatively
sure of. These are the
“significant” digits.
These zeros are
“placeholder”
digits. They are
not part of the
“measured” value
Review of Scientific Notation
Example:
An atom’s diameter
is an extremely small number
(ex: a carbon atom = 0.00000014 cm)
= 1.4 x 10-8 cm
Notice: the “4” is the
highest level of precision
that we can be relatively
sure of.
These zeros are
“placeholder”
digits. They are
not part of the
“measured” value
Review of Scientific Notation
Example:
An atom’s diameter
is an extremely small number
(ex: a carbon atom = 0.00000014 cm)
= 1.4 x 10-8 cm
So there will be a huge number of atoms in even
the smallest sample of matter!
12 grams of Carbon has
602000000000000000000000 atoms! = 6.02 x 10 atoms
23
Notice: 3 significant digits and 21 placeholder zero's
Converting: Normal to scientific…
Examples:
289,800,000.kg
=
Decimal moved 8 places left
(divided by 10, 8 times)
2.898 x 108 kg
Exponent multiplies
number by 10, 8 times
Notice: in changing forms you’re first dividing
… and then multiplying the number by 10,000,000.
This changes the form, but leaves the value the same.
In other words 10,000,000 was taken out of the number
and then stored in the exponent.
Notice also that the “place-holder” zero's drop out because they aren’t needed.
Converting: Normal to scientific…
Examples:
289,800,000.kg
2.898 x 108 kg
=
Decimal moved 8 places left
(divided by 10, 8 times)
0.00003050 m
=
Decimal moved 5 places right
(multiplied by 10, 5 times)
Exponent multiplies
number by 10, 8 times
3.050 x 10 - 5 m
Negative exponent divides
number by 10, 5 times
Notice: now we’re multiplying this number
…and then dividing it by 100,000 (that’s what “x10-5 “ means)
Notice again, that these “place-holder” zeros drop out because they aren’t needed
Converting: Normal to scientific…
Examples:
289,800,000.kg
=
Decimal moved 8 places left
(divided by 10, 8 times)
0.00003050 m
=
Decimal moved 5 places right
(multiplied by 10, 5 times)
2.898 x 108 kg
Exponent multiplies
number by 10, 8 times
3.050 x 10 - 5 m
Negative exponent divides
number by 10, 5 times
Notice again, that these “place-holder” zeros drop out because they aren’t needed
Notice: 10 to positive powers mean a large number (larger than 10)
While 10 to negative powers mean a decimal fraction (numbers smaller than 1)
Check your understanding
Express these numbers in Scientific
Notation:
1)
2)
3)
4)
5)
4057
0.00387
30000
2
0.4760
4.057 x 10 3
3.87 x 10 -3
3 x 10 4
2 x 10 0
4.760 x 10 -1
And back to standard…
5.093 x 106
Positive exponent: Move decimal
6 places right (x 100,000) to make
number larger
1.9060 x 10-3
negative exponent: Move decimal
3 places left (÷ 1000) to make the
number smaller
=
5093000
and we need additional 3 placeholder
zero's to carry the decimal point
=
0.0019060
and we need additional 2 placeholder
zero's to carry the decimal point
As before, we’re moving powers of 10 between the
exponent and the base coefficient)
Check your understanding
Express these numbers in normal
form
405.7
1) 4.057 x102
0.0387
2) 3.87 x10-2
3) 3.0 x102
300
4) 20 x10-2
0.020
5) 4.760 x105
476000
Click here for additional practice on this topic
Multiplying with scientific notation
Rule: Multiply coefficients, then add exponents
Sample problems:
(3.0)(6.0 x 10 3)
= (3.0)(6.0) x 103
= 18 x 103
÷10
x10
= 1.8 x 104
(3.0)(6.0 x 10 -3)
= 18 x 10 -3
÷10
x10
Notice: you’re moving
10 from the
coefficient to the power
= 1.8 x 10 -2
Multiplying with scientific notation
Rule: Multiply coefficients, add exponents
(3.0 x 10 3)(6.0 x 10 3)
= (3.0)(6.0) x 10 3 + 3
= 18 x 10 6
= 1.8 x 10 7
(3.0 x 10 2)(6.0 x 10 -5) = 18 x 10 -3 = 1.8 x 10 -2
Dividing with scientific notation
Rule: divide coefficients, subtract exponents
(6.0 x 103)
(3.0)
= (6.0) x 103
(3.0)
(3.0)
= (3.0) x 10 -3
(6.0 x 10 3)
(6.0)
= 0.50 x 10 -3
= 5.0 x 10 -4
= 2.0 x 10 3
Move to top
Switch
the
sign
Dividing with scientific notation
Rule: divide coefficients, subtract exponents
(6.0 x 103)
(3.0 x 102)
= (6.0) x 103-2
= 2.0 x 10 1
(3.0)
(3.0 x 10 2) = (3.0) x 10 2-3
(6.0 x 10 3)
(6.0)
= 0.50 x 10 -1
= 5.0 x 10 -2
Add or subtract with scientific notation
Rule: convert to like exponents,
add or subtract coefficients
Notice:
you’re adding
6 thousand + 3 hundred
(6.0 x 103) + (3.0 x 102)
÷10
x10
(6.0 x 103) + (0.3 x 103)
= (6.0 + 0.3) x 10 3
So you must
change so each
is in thousands
Your final answer
Is still in thousands
= 6.3 x 10 3
6,300 = 6.3 thousand
Check your understanding
1. (2.0)(6.3 x 10 4)
= 1.26 x 10 5
= 1.3 x 10 5
Notice the rounding: Final answer is rounded to two digits, like the starting values.
2. (6.4 x 10 4)
(2.0)
= 3.2 x 10 4
3. (2.0 x 10 -2)(6.3 x 10 4)
4. (6.4 x 10 4)
(2.0 x 10 -2)
= 1.3 x 10 3
= 3.2 x 10 6
5. (6.3 x 10 4) - (2.0 x 10 3)
0.2 x 10 4
= 6.1 x 10 4
Click here
For extra
Help and
practice
Temperature Scales
Fahrenheit is not used in science.
It is based on a frigorific (reproducible) mixture of salt and ice 0oF.
…and human body temp ~100F. Interesting huh?
Temperature Scales
Celsius is more commonly used, and is based on melting point
(0oC) and normal boiling point (100oC) of water.
Problems arise due to negative numbers (below zero). Celsius is a
relative scale. “relatively hotter, or relatively colder”
Temperature Scales
Kelvin eliminates this by reassigning numbers to the Celsius scale
so that coldest temperature possible (no molecular motion) -273oC
is assigned a value of zero on the kelvin scale.
Kelvin is an absolute scale. It is directly related to molecule speed.
Temperature Scales
Celsius is used when the change in temperature is important.
Kelvins are used when the values themselves are important.
Temperature Scales
For example: heating water from 10oC to 20oC increases its temp
by 10 degrees, but does not give it twice the heat.
This same change is 283 to 293 kelvins.
A relatively small increase in the molecules speed.
Temperature Scales
Conversions between scales are simple since a change of 1
degree on the Celsius scale is also a change of 1 degree on the
kelvin scale. One must only add 273 to the Celsius temperature to
convert to kelvins.
Celsius
is based on water
Is a relative scale
0 0C = normal melting point
100 0C = normal boiling point
Kelvin
is based on molecular speed energy
Is an absolute scale
0 K – absolute zero
- molecules stop moving
Fahrenheit
Old scale based on freezing point of
Salt water
Not a conversion you’ll have to do.
0F = 9/5(0C) + 32
Only used in USA and Belize
Conversions involving
Fahrenheit are unlikely,
but you should be aware
of the relationships
between the scales
Calculations of
Temperature:
Using a formula
K = ˚C + 273.15
• Body temp = 37 ˚C + 273 = 310 K
• Liquid nitrogen = -196 ˚C + 273 = 77 K
Or
˚C = K - 273.15
Check your understanding
Room temperature is 298 Kelvins.
What is this expressed in Celsius?
˚C = K - 273
˚C = 298 - 273
˚C = 25
1) Write down the relevant formula
2) substitute given values
3) Calculate
4) Evaluate: is answer reasonable?
Since 273 Kelvins = 0 celsius, then 298 kelvins should be about 25 higher also
Some units are “Derived”
ex: Cubic length = Volume
Notice: 1 cm x 1cm x 1cm = 1cm3
that’s 1 “cubic” centimeter
Also, 10cm x 10 cm x 10 cm = 1000 cm3
Which is
1 dm x 1 dm x 1 dm = 1 dm3
that’s a cubic decimeter
Notice 1 dm3 is a.k.a. 1 liter
So: 1 cm3 is also 1 milliliter
DERIVED UNITS:
Cubic length = Volume
Oh yeah, since the density of
water is 1 gram per cm3,
1 cm3 H2O weighs 1 gram
and 1 liter of water weighs
1000 grams or 1 kilogram.
Its good to remember that.
DENSITY - an intensive physical property
(specific to a substance)
Compactness of its matter (mass)
Ratio of mass compared to volume
Density 
mass (g)
volume (cm3)
Mercury
Platinum
Aluminum
13.6 g/cm3
21.5 g/cm3
2.7 g/cm3
DENSITY - an intensive physical property
(specific to a substance)
Compactness of its matter (mass)
Ratio of mass compared to volume
Same 1
cm3
of volume.
Different masses.
Which sample is most compact (dense)?
The same amount of mass
that is in the water sample
is compacted into a much
smaller space in a sample
of mercury. That’s dense!
Same 13.6 grams of mass.
Different volumes.
Which sample is less dense?
Problem A piece of copper has a mass
of 57.54 g. It displaces 6.4 cm3 .
Calculate its density (g/cm3).
Using a formula: ACE the problem!
1. Analyze (and Plan) : List knowns and unknowns:
Mass = 57.54 grams Volume = 6.4 cm3
Density?
D =
Plan: Write the formula:
M
V
2. Calculate (set up) problem by substituting values
D = 57.54 grams
6.4 cm3
Don’t forget
to include units!
D = 8.9953125
3. Evaluate (round, and label) answer: is it reasonable?
What should be the mass of one of the 6 cm3’s?
D = 9.0 g/cm3
I rounded to two digits – the least number of digits in the original values!
Check your understanding
Osmium has a density of 22.5 g/cm3. What is
the volume of a 50.00 g sample?
1) 0.45 cm3
2) 2.22 cm3
3) 11.3 cm3
Rearrange
Formula
Using
algebra
Substitute
And solve
Evaluate?
If 22.5 g is 1 cm3, how large is 50 grams?
V ___
D =
D
V =
M V
V D
M
D
V = _50.00 g_
22.5 g/cm3
Notice that given units
Cancel to give desired unit
Be sure to show complete setups in your notes!
1.5 Uncertainty in Measurement
Can you hit the bull's-eye?
Three targets
with three
arrows each to
shoot.
How do
they
compare?
Both
accurate
and precise
What is accuracy?
What is precision?
Precise
but not
accurate
accurate
but not as
precise
“near the bullseye”
Data which is close to a standard
“a grouping”
Data which is reproducible
(though not necessarily accurate)
Accuracy and percent error
Accuracy is represented by percent error
Absolute error – how far a measured value is
away from an accepted value
Percentage – value per 100 units
Percentage error:
% error = measured value – accepted value x 100
Accepted value
A student calculates the density of mercury to be
13.2 g/cm3. If the accepted value is 13.6 g/cm3,
what is the percentage error?
% error = measured value – accepted value
Accepted value
% error = 13.2 - 13.6
13.6
x 100
x 100
= - 2.94 PERCENT
NEGATIVE ERROR = MEASURED VALUE BELOW ACCEPTED!
Precision and “deviation”
Various types of “deviations” are used to express precision
From a set of data we find the mean
then find the deviation…
Our “average” could be
expressed as 9 ± 2.75
This is technically the
“average” deviation
On our data deviates from the
mean by an average of 2.75
You may not encounter this in
chemistry, but likely in other
applications throughout life!
Taking a measurement (reading a scale)
. l2. . . . I . . . . I3 . . . .I . . . . I4. .
cm
First digit (known)
=2
2.?? cm
Second digit (known)
= 0.7
2.7? cm
Third digit (estimated) between 0.04- 0.06
Length reported
=
2.75 cm
(or 2.74 or 2.76 OK too)
These are the significant digits (or significant figures)
 All the digits we’re sure of, plus 1 estimated digit.
Taking a measurement
. l2. . . . I . . . . I3 . . . .I . . . . I4. .
Length reported
=
cm
2.75 cm
What digit comes next? After the the 5 in the value 2.75?
Its not a zero! (not necessarily)
It could be any digit (0-9)
We can’t be more precise with the scale.
All measured values have uncertainty.
There is a limit to the precision that can be achieved with
any measuring device.
(Its important to recognize that you can’t simply tack zeros onto numbers)
Notice on this scale we must estimate
the one’s place.
We obviously can’t estimate the
tenth’s place.
We could NOT record this as 27.0
since the 7 itself is estimated.
Does this make sense to you?
Zero as a measured value
. l3. . . . I . . . . I4 . . . . I . . . . I5. .
cm
What is the length of the line?
First digit
Second digit
Last (estimated) digit is
4.?? cm
4.7? cm
4.70 cm
Zero in this example means “exact or “on the line”
What digit comes after the zero?
We don’t know (it could be any digit)
zero is commonly a placeholder
Mass of the earth:
4
Significant
digits
6592000000000000000000 tons
6.592 x 10 21 tons
18
Placeholder
zeros
We can assume that this is not a very exact estimation. The zeros at the end are
holding the place for the actual digits which are unknown to us.
zeros as placeholders
Radius of a carbon atom:
2
Sig
digs
Also, by
convention, a zero
is placed in front
of the decimal in
order to help us
spot the decimal.
Its not a real digit.
77 picometers
77 x
10-12
m
0.0000000000077 m
11
placeholder
zeros
These zeros must
be used to show
us the placement
of the decimal.
They don’t even
represent real
digits like in the
last example!
zeros as placeholders
Placeholders left after rounding:
Ex:734567
Rounded to nearest thousand:
735000
3 placeholder zeros
3 sig digs
Note: the zeros left after rounding are only placeholders
They drop out when not needed – as in scientific notation
7.35
x 10 5
Counting Significant Digits
RULE 1. All non-zero digits in a measured number are significant.
(Only a zero can be used as a placeholder)
38.15 cm
5.6 ft
65.6 lb
122.55 m
Number of Significant digits
4
Remember: sig digs are
2
the values written down
3
___
from a measurement.
5
___
RULE 2. In exact or defined quantities there is an infinite level of
precision – an infinite number of significant figures
ex: 25 students
or
5 beakers
or 1 inch = 2.54 centimeters
All of these values are considered to have infinite significant figures!
This means there is no limit to the certainty of the value.
Leading Zeros
RULE 3. Leading zeros in decimal fractions are
NOT significant (they’re placeholders).
Recall that leading zeros will drop out once we change to scientific notation?
Number of Significant Figures
0.008 mm
1
0.0156 oz
3
0.0042 lb
2
____
0.000262 mL
3
____
Try to focus on the
significant digits in
quantities, and ignore the
placeholder zero's.
Sandwiched Zeros
RULE 4. Zeros between nonzero numbers are significant.
(zeros can not be rounded or dropped, unless they are
on an end of a number.)
Number of Significant Figures
50.8 mm
3
2001 min
4
0.702 lb
3
____
0.00405 m
3
____
Trailing Zeros – the tricky ones
RULE 5a. Trailing zeros in numbers without decimals are
assumed to be placeholders. They would take the place of other
digits after rounding.
Number of Significant Figures
25,000 in.
2
rounded to thousands
200 yr
1
rounded to hundred
48,600 gal
25,005,000 g
3
____
5
____
In other words we assume
numbers like this are
rounded off unless we
know otherwise.
Trailing Zeros – the tricky ones
RULE 5b. Trailing zeros in numbers with decimals are significant.
We leave the decimal at the end to show this.
Number of Significant Figures
25,000. in.
5
200. yr
3
48,600. gal
5
___
25,005,000. g
8
___
Check your understanding
A. Which answers contain 3 significant figures?
1) 0.4760
2) 0.00476
Zero as part of measurement
3) 4760
Zeroes are only placeholders here
B. All the zeros are significant in
1) 0.00307
2) 25.300
3) 2.050 x 103
C. 534,675 rounded to 3 significant figures is
1) 535
2) 535,000
3) 5.35 x 105
Rounded to nearest thousand
After rounding, leave zeroes in
place of missing digits
Rounding in calculations: Addition/subtraction rule
The answer has the same number of “tens places” as the
measurement with the fewest tens places.
25.2 ? nearest tenth (1 decimal place) Unknown digit after the “2”
+ 1.35 nearest hundredth (2 dp’s)
26.55 Round answer to nearest tenth since we can’t be sure of hundredth's place
answer 26.6 one decimal place
Quick check: round each to the correct number of significant figures.
A. 235.05 + 19.6 + 2.1 =
1) 256.75
2) 256.8
3) 257
Round to tenth’s place
B.
58.925 - 18.2
1) 40.725
=
2) 40.73
3) 40.7
Round to tenth’s place
Rounding: Multiply/divide rule
A quick example: Let’s calculate an area from the following dimensions and see
how precision affects the calculated values:
25.4 cm x 3.2 cm = 81.28 cm2
But how many digits should be kept when rounding?
Remember that we don’t know what the next digit is.
25.4? cm x 3.2? cm
Now lets see…the largest numbers that could round down would be 4’s
25.44 cm x 3.24 cm
=
82.4256 cm2
….While the smallest numbers that could round up would be
25.35 cm x 3.15 cm
=
79.8525 cm2
How many digits are still at least close in agreement? Only the first two!
So: 25.4 cm x 3.2 cm = 81.28 cm2 = 81 cm2
3 sig figs x 2 sig figs = keep 2 sig figs
The rule: Round the answer to the number of digits, as the measurement
with the fewest significant digits.
Choose the answers that are rounded
correctly:
A. 2.19 X 4.2 =
1) 9
B.
C.
3 sf’s x 2 sf’s = keep 2 sf’s in final answer
2) 9.2
3) 9.198
4sf’s x 1 sf’s = keep 1 sf’s in final answer
4.311 ÷ 0.07 =
1) 61.58
2) 62
2.54 X 0.0028
=
0.0105 X 0.060
1) 11.3
2) 11
3) 60
Least is 2 sf’s
3) 0.041
Hint: Your desired unit is meters per second:
m/s
So, Divide meters by seconds
Do you need more help with
significant figures?
Click here for a tutorial
1.6 Dimensional Analysis
Dimensional analysis is the term given to a system used to solve complex
math problems. It is THE accepted method used in science.
It looks complicated, but once you get the technique down, you won’t
believe how much easier problem solving becomes. So, keep an open mind.
Dimensional analysis uses conversion factors
Conversions using the known equalities between units
Ex: 1 inch = 2.54 centimeters
We build conversion factors (fractions) in which the numerator and
denominator are EQUAL quantities expressed in different units
ex Factors:
1 in.
2.54 cm
or
2.54 cm
1 in.
Since the value on the top
and bottom are equivalent
They both carry a value of “one”
Quick Check: Write conversion factors that relate each of the following pairs of units:
1. Liters and mL
1 liter = 1000 milliliters
2. Hours and minutes
1 hour = 60 minutes
3. Meters and kilometers
1000 meters = 1 kilometer
1L
1000 mL
1 hr
60 min
1 Km
1000 m
OR
OR
OR
1000 mL
1L
60 min
1 hr
1000 m
1 Km
Setting up conversions
A sample problem 1 : How many minutes are in 2.5 hours?
We begin by writing the original value with its given unit
2.5 hr x
60 min =
1 hr
(2.5)(60) min
1
= 150 min
We will multiply our original value by a conversion factor
We build the conversion factor with Given unit (hours) on the bottom and
desired unit (minutes) on the top, so units can be “cancelled out”
Finish by multiplying across as shown by the setup
The generic setup is
Given unit
x
desired unit
given unit
Sample Problem 2
You have $7.25 in your pocket in quarters.
How many quarters do you have?
Equality: How many quarters = 1 dollar?
4 quarters = 1 dollars
What is your given unit? What is your desired unit?
Desired unit
Given unit
7.25 dollars
4 quarters
1 dollar
4 quarters
X
1 dollar
= 29 quarters
Learning Check
A rattlesnake is 2.44 m long. How
long is the snake in cm?
1 meter = 100 centimeters
Don’t think, first write your setup:
a) 2440 cm
b) 244 cm
c) 24.4 cm
2.44 m x 100 cm
1m
The setup tells us to multiply 2.44 x 100
Is that reasonable?
A practical problem: Driving in Canada, the speed limit
says 90 km per hour, how fast is that in miles per
hour? (Don’t want to get a speeding ticket, do I?)
90 km x 6.2 miles = (90)(6.2) = 55.8 miles
10 km
(10)
Notice that the value on the bottom gets divided.
km to miles
Step 1: ID Starting and ending unit
Step 2: Do I know an equality?
Is the answer
Reasonable?
10 km = 6.2 miles
Miles are larger,
So value should
Be smaller!
Step 3: Write down starting value and unit
Step 4: Write X conversion fraction
Step 5: cancel units and calculate
Ending unit
Starting unit
Plus it seems
A familiar
Speed
limit
Help: Calculating fractions review
Notice that in this technique, fractions are calculated.
3
2
x
4 x
____
1
18 in
x
4
3
4
2
= 12
6
=
1 ft
12 in
16
2
=
=2
= 8
18
12
Simply multiply across.
If it helps
put the starting
value over 1
= 1.5 ft
Adding the units allows us to check our setup for correctness.
PROBLEM: Mercury (Hg) has a density
of 13.6 g/cm3. What is the mass of 95 mL
of Hg in grams?
Solve the problem using DIMENSIONAL
ANALYSIS.
PROBLEM: Mercury (Hg) has a density of
13.6 g/cm3. What is the mass of 95 mL of Hg?
1. Analyze: knowns V = 95 mL
D = 13.6 g/cm3 or
13.6 g = 1 cm3
1 cm3 = 1 mL
unknowns - mass
2. Setup, calculate, round
95 cm3
x 13.6 g =
1 cm3
1292 g
=1300 g
3. Evalute: Is our answer reasonable?
Volume Displacement
A solid displaces a matching volume of
water when the solid is placed in water.
Check your understanding
What is the density (g/cm3) of 48 g of a metal if
the metal raises the level of water in a graduated
cylinder from 25 mL to 33 mL?
1) 0.2 g/ cm3
2) 6 g/cm3 3) 252 g/cm3
33 mL
25 mL
33 mL
25 mL
Using metric prefixes
45 nm = _____pm
x 10-9
45 nm
Substitute 10-9
for nano in nm
X
3)
1 pm
45
(10
=
-12
10
m 10-3
Switch to top, ÷10
-3
becomes x 10
-3
+3
Convert meters
to picometers
= 45000 pm
(Easier: if you recognize that pico is 1000 times smaller than nano, move decimal right 3 places
to make number larger. As long as its stated on AP exam, its OK)
Try these: Convert the following into new units. Show labeled factor label setup.
(a) 340 ml =
L
340 ml x 10-3 liter = 0.34 L
1 ml
(c) 1.4 x 103 m =
cm
1.4 x 103 m x 1 cm
10 -2 m
(e) 3.5 dm3 =
3.5 dm3 x
(
(b) 235 g =
235 g x 1 kg
103 g
(d) 4.7 Kilocalories =
= 1.4 x 105 cm
cm3
10 cm
1 dm
(g) 2.3 x 10-5 mm =
2.3 x 10-5 (x 10-3)m x
Kg
= 0.235 kg
4.7 kcal x 103 cal
1 kcal
(f) 32 Joules =
)
3 = 3.5 x 10 3 cm3
pm
1 pm =
10-9-1 m
= 4.7 x 103 cal
calories
32 J x 1 cal =
4.2 J
(h) 1.0 oz =
23 pm
calories
grams
1 oz x 1 lb x 1 kg x 103 g =
16 oz
2.2 lb
1 kg
Equalities: 1 calorie = 4.2 joules / 1 lb = 16 oz. / 2.2 lb = 1 kg
28 g
AP
AP: Sample Problem – multi-step problems
How many seconds are in 1.4 days?
Unit plan: days  hr  min  seconds
1.4 days x 24 hr
1 day
x 60 min x 60 sec =
1 hr
1 min
= 120,000 sec
Check
ending
unit
AP: Sample Problem – multi-step problems
How many feet is a meter?
Unit plan: 1 m  cm  in  feet
1 meter
x 100 cm x 1 inch x 1 foot =
2.5 cm 12 inches
1m
How do we plug this into a calculator?
1 x 100 ÷ 2.5 ÷ 12 =
3.3 ft
Multiply tops, divide by bottoms
Check
ending
unit
AP: What about Square and Cubic units?
• Use the conversion factors you already
know, but when you square or cube the
unit, don’t forget to cube the number
also!
• Best way: Square or cube the ENTIRE
conversion factor
• Example: Convert 4.3 cm3 to mm3
4.3 cm3 10 mm
(
1 cm
3
=
)
4.3 cm3 1000 mm3
1 cm3
= 4300 mm3
Dealing with Two Units
AP
If your pace on a treadmill is 65 meters per
minute, how many seconds will it take for
you to walk a distance of 8450 feet?
Analyze (and plan):
Known
Starting: distance 8450 ft
Rate (equality): 65 m / 1 min
Unit plan:
First convert:
Unknown
Ending unit :
seconds
ft  inches  cm  meters
Since rate is 65 meters per minute
Then 65 m = 1 minute (for this problem only)
Then convert:
Meters  minutes  seconds
Calculate your setup:
8450 ft x 12 in x 2.54 cm
1 ft
x 1m
100 cm
1 in
x 1 min x 60 sec
65 m
1 min
= 2400 sec
In using your calculator simply work across multiplying tops and dividing bottoms
(Its one of those math rules: you can multiply and divide in any order. skip the ones of course)
8450
x
12
x
2.54
÷
100 ÷ 65
x
60
Evaluate: Is your answer reasonable?
Lets look at estimating next 
=
2377
AP
Estimating answers:
AP exam part A, is multiple choice and requires calculating without a calculator
Lets look at how to simplify a setup and estimate a numerical answer!
First: Lets round off all values to one significant digit
8000
10
3
8450 ft x 12 in x 2.54 cm
x 1m
100 cm
1 ft
1 in
x 1 min x 60 sec
= 2377 sec
65 m
1 min
70
Lets simplify:
By canceling 10’s
(8000)(10)(3)(60)
(100)(70)
Cancel out number from top and bottom that are close
= (80)(10)(3)
= 2400
Wow!
AP
Need more help with dimensional
analysis? Click here for the
chemistry solutions tutorial
Khan academy tutorial on unit conversions