CPChemistry
Math Tool Kit
Study and Practice Packet
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
SI Units
Derived Units
Metric Prefixes
Scientific Notation
Significant Figures
Counting Significant Figures
Calculations with Significant Figures
Dimensional Analysis
Conversion Factors
Calculating Percent Error
1
CPChemistry - Math Tool Kit - Study and Referemce Packet
I.
Units of Measurement: 7 Fundamental SI Units
Property
Length
Mass
Time
Temperature
Amount of Substance
Current
Luminous Intensity
II.
SI Unit and Standard of
Measurement
meter
kilogram
second
Kelvin
mole
ampere
candela
Symbol
m
kg
s
K
mol
A
cd
Derived Units
Property
Area
Meaning
lxw
Volume
lxwxh
Derived Unit
square meter
Symbol
m2
cubic meter
m3
mxmxm
cubic decimeter
dm3
dm x dm x dm
mxm
Volume
(liquid)
Force
mass x acceleration
newton
N
1N = 1kg-m/s2
Pressure
force/area
pascal
Pa
1Pa = 1N/m2
Energy
force x distance
joule
J
1J = 1N-m
Frequency
cycles/second
hertz
Hz
1Hz = 1wave
cycle/second
Density
mass/volume
m/s, km/hr,
m/min, etc.
Speed
kg/m3, g/cm3,
distance/time
g/mL, etc
2
III.
Table of Metric Prefixes
Prefix: Symbol: Magnitude: Meaning (multiply by):
Tera- T
1012
1 000 000 000 000
Giga- G
109
1 000 000 000
Mega- M
106
1 000 000
kilo-
k
103
1000
hecto- h
102
100
deka- da
101
10
-
-
-
-
deci-
d
10-1
0.1
centi- c
10-2
0.01
milli-
10-3
0.001
m
micro- u (mu) 10-6
0.000 001
nano- n
10-9
0.000 000 001
pico-
p
10-12
0.000 000 000 001
femto- f
10-15
0.000 000 000 000 001
3
IV.
Scientific Notation
The speed of light is approximately 300,000,000 meters per second. Working with a large
number such as this can become cumbersome so we use scientific notation to represent
very large and very small numbers. 300,000,000 m/s can also be written as: 3 x
100,000,000 or 3 x 108, where 8, the exponent, is the number of zeros.
Positive exponents
Large numbers can be written in scientific notation by moving the decimal point to the left.
For example, Avogadro's number, 602,200,000,000,000,000,000,000, is central to
chemistry. The decimal point that you don’t see is to the right of the last zero in the
measurement.
The decimal point is moved left until you have a number between 1 and 10. In the example
above, the decimal point was moved 23 places to the left. That number is now the positive
exponent of the base 10.
Negative exponents
Numbers less than 1 can be expressed in scientific notation by moving the decimal to the right.
In this instance, the decimal point needs to move to the right by 4 places to the first non-zero
number. For every place we move the decimal to the right we decrease the power of ten by one,
starting from zero. That number can be written as 7.2 x 10-4
4
Review: Power of 10 notation
For any positive whole number, n, 10n is 1 followed by n zeros. Remember a positive power of 10 means a
large number, greater than 1.
100 = 1
101 = 10
102 = 100
103 = 1000
When 10 is raised to a negative power, the exponent tells you how many places after the decimal point to
place the 1. Remember, a negative power of 10 means a small number, less than 1.
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
10-4 = 0.0001
 YOU TRY!!!  Practice
Problems: Scientific Notation
Express the following in scientific notation. Remember to retain the same significant figures.
Ordinary notation
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Scientific Notation
137,000,000
0.000290
0.00000158
738
0.020
4200
7.050 x 10-3
4.00005 x 107
2.3500 x 104
1.15 x 10-3
5
V.
Significant Figures
Some numbers are exact and some are not. For example, your family has exactly 5 people, your class has
exactly 21 students, and there are exactly 100 centimeters in one meter. The last example is a conversion
factor. There is no uncertainty in a conversion factor.
In chemistry lab this year you will be making measurements of mass, volume and temperature. Numbers
that are obtained by making measurements are not exact. There is always uncertainty as a result of the
limitations of the instrument scale and the skill of the technician reading the scale. Calculations made with
measured values must be rounded off properly to the appropriate number of significant figures. Careful
measurements together with rounding correctly make your reported measurements reliable.
The significant figures in a measurement are all of the digits known with certainty (those for which there is
a marking on the scale) plus one digit which is estimated between the smallest markings.
VI.
Counting Significant Figures
In numbers written with decimal points, count significant figures from the left beginning with the first
nonzero digit. In numbers written without decimal points, count from the right beginning with the first
nonzero digit.
Examples:
Measurements
0.0370
200
20.
20.0
400,900
0.00990
YOU TRY!!! Practice
Number of SFs
3
1
2
3
4
3
Problems: Counting SGs
measurement
1.
2.
3.
4.
5.
6.
0.23100
23100
23100.
7.203
0.00231
2000
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SFs
VII.
Calculating With Significant Figures:
☞When you use your measurements in calculations, your answer may only be as exact as your
least exact measurement!
RULE FOR ADDITION AND SUBTRACTION: Round to fewest decimal places.
Example
4.1cm + 0.07cm
Unrounded answer
4.17cm
Rounded answer
4.2cm
18.3m – 11m
7.3m
7m
8.120g-7.090g
1.03g
1.030g
Explanation
4.1 has one decimal so
answer rounded to tenths
place
11 has no decimals so
answer rounded to ones
place
Both measurements have
three decimals so answer
should have three decimals
RULE FOR MULTIPLICATION AND DIVISION: Round to fewest significant figures.
(abbreviated SFs)
Example
Unrounded answer
4.1cm x 0.07cm
0.287cm2
Rounded answer
7.079𝑐𝑚
0.53𝑠
13.356603774𝑐𝑚
𝑠
13𝑐𝑚
𝑠
0.53 has two SFs, answer
rounded to two SFs
8.120m x 7.090m
57.5708m2
57.57 m2
Both measurements have
four SFs so answer should
have four SFs
0.3
cm2
Explanation
0.07 has one SF, answer
rounded to one SF
☞ Notice that units of measurement are carried through the calculations and shown in all results.
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YOU TRY!!! Practice
Problems: Calculations with SFs
Practice Problems: Using Significant Figures in Calculations
Example
1
Unrounded
answer
45.71cm x 0.20cm
2
10.2𝑠
0.4
3
100. mm – 1.6 mm
4
4302g + 0.837g
5
87.3cm – 1.655cm
6
2.099g + 0.05681g
7
2.4𝑔
𝑥 15.82𝑚𝐿
𝑚𝐿
8
105.725𝑔
39.1𝑚𝐿
8
Rounded answer
Explanation
VIII. Dimensional Analysis:
Units in science are sometimes called dimensions. Keeping track of units in calculations is called
dimensional analysis.
When you multiply or divide numbers with units (measurements) you also multiply or divide the units.
Examples:
1. Area = length x width = 3cm x 2cm = 6cm2
2.
32 𝑠
4𝑠
= 8 (The time units, seconds, have divided out.)
3. Density =
𝑚𝑎𝑠𝑠
𝑣𝑜𝑙𝑢𝑚𝑒
=
853.76𝑔
310.1𝑚𝐿
= 2.7531763947
𝑔
𝑚𝐿
= 2.753
𝑔
𝑚𝐿
(The units haven’t changed in the calculation and so are brought out in the result. The answer is rounded
to the same number of SFs as the measurement with the fewest, the volume, which has four SFs.)
4. Convert 3.72 hours to seconds:
?s = 3.72h x
IX.
60𝑚𝑖𝑛
1ℎ
x
60𝑠
1𝑚𝑖𝑛
= 13,392s = 13,400s (Hours & minutes divided out)
Conversion Factors
Conversion Factors can be used to convert from one unit of measurement to another. The ability
to convert between units of measurement with confidence is essential in the lab, research,
industry, and hospital settings. Another name for a conversion factor is Unit Equality. That’s
because a conversion factor is equal to ONE; the original quantity will not be changed when you
multiply it by a Unit Equality. The reciprocal of a unit equality is also equal to one! Conversion
factors are written like fractions.
Example:
Convert the height of a student from inches to centimeters:
? cm = 63𝑖𝑛 𝑥
2.54𝑐𝑚
1𝑖𝑛
= 160𝑐𝑚
? 𝑐𝑚 = 63𝑖𝑛 𝑥
9
1𝑐𝑚
0.3937𝑖𝑛
= 160𝑐𝑚
X.
Calculating Percent Error (Percent Difference)
Use the equation: 𝑷𝒆𝒓𝒄𝒆𝒏𝒕 𝑬𝒓𝒓𝒐𝒓 =
𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝒗𝒂𝒍𝒖𝒆−𝒂𝒄𝒄𝒆𝒑𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆
𝒂𝒄𝒄𝒆𝒑𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆
𝒙 𝟏𝟎𝟎
Example:
Your teacher asks you to demonstrate your skill on the balance by massing an object in the lab. You
measure its mass 6.878g but the teacher’s measurement was 6.085g. If the teacher’s value is the accepted
value, what is the percent error in your mass measurement?
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐸𝑟𝑟𝑜𝑟 =
6.878−6.085
6.085
=
0.793
6.085
= 13.0% 𝑤𝑖𝑡ℎ 3𝑆𝐹𝑠
It’s important to show the intermediate step above because this step determines the number of SFs in your
answer. If your answer is a negative value it simply means that your result is lower than the true value.
 YOU TRY!!!  Practice
Problems: Percent Error
1. What is the percent error of a length measurement of 0.229cm if the correct value
is 0.225cm?
2. A handbook gives the density of calcium as 1.54
in a density of 1.25
𝑔
𝑚𝐿
. What is the percent error?
10
𝑔
𝑚𝐿
. Lab measurements resulted
YOUR NAME:
Practice Problems: Unit conversions using dimensional analysis
Convert 0.000830m to cm.
Convert 7.56kg to g.
Convert 4.02 hours to seconds.
Add 9.78m to 245cm. Express your answer in cm.
Convert 10km to miles.
Convert 36.7miles to kilometers.
The distance from wing to wing is 0.75 mi. How many cm is this?
A Motrin tablet is 200. mg. How many ounces is this?
The price of gas in Germany is $2.118 per liter. What is the price per gallon?
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