MATHEMATICAL
RELATIONSHIPS IN
CHEMISTRY
What You’ll Learn in this
Unit
 Significant Figures
 Scientific Notation
 Measurement
 Dimensional Analysis
 Error
 Density
 Graphical analysis
Review of Measurement Terms
 Qualitative measurements - words
 Quantitative measurements –
involves numbers (quantities)
 Depends on reliability of instrument
 Depends on care with which it is read
Precision vs. Accuracy
 Precision- the degree of agreement among
several measurements of the same quantity.
 Accuracy- the agreement of a particular value
with the true value
Uncertainty
 Basis for significant figures
 All measurements are uncertain to
some degree
 Random error - equal chance of being
high or low- addressed by averaging
measurements - expected
Significant Figures
 Meaningful digits in a measurement
 The number of significant figures in
your measurement will tell the
reader how exact the
instrumentation used
 If it is measured or estimated, it has
sig figs.
 If not it is exact (e.g. 5 apples).
Significant Figures





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All numbers 1-9 are significant.
The problem are the ZEROS.
Which ones count and which don’t?
In between numbers 1-9 does
Example:
4001……… has 4 sig figs
Now let me tell you a story…
Left handed Archer
 There once was a left handed archer who
loved to shoot decimals. Zeros could not stop
his arrow but numbers could.
 If there is a decimal in the number begin on
the left. Go through any zeros, come to the
first number then all other numbers that
follow are SIGNIFICANT!
 →0.0040
 →50.401
No decimals
 If a number has no decimals you begin on the
right hand side.
 Go through any zeros , come to the first
number.
 Then all numbers after that count
 5000←
 405,000 ←
Doing the Math
 Multiplication and division, same number of
sig figs in answer as the least in the problem
 Addition and subtraction, same number of
decimal places in answer as least in problem.
Scientific Notation
 100 = 1.0 x 102
 0.001 = 1.0 x 10-3
-- This provides a way to show significant
figures.
TOO QUICK FOR YOU!
 So here are the rules.. slowly!
1. Place decimal point after 1st real
non-zero integer. (ex) 1.0 NOT 10.0
2. Raise 10 to the exponential which
equals the number of places you
moved.
Scientific Notation
 The product of 2.3 x 10 x 10 x 10
equals 2300 (2.3 x 103)
 Note:
 Moving the decimal to the left will
increase the power of 10
 Moving the decimal to the right will
decrease the power of 10
Sample Problems
 2387
 0.00007031
 2900000000
 0.008900
 90100000
 0.00000210
Answers
 2.387 x 103
 7.031 X 10-5
 2.9 x 109
 8.900 X 10-3
 9.01 X 107
 2.10 X 10-6
Scientific Notation
 Multiplication and Division
 Use of a calculator is permitted
 use it correctly
 No calculator? Multiply the
coefficients, and add the exponents
(3 x 104) x (2 x 102) = 6 x 106
(2.1 x 103) x (4.0 x 10-7) = 8.4 x 10-4
Scientific Notation
 Multiplication and Division
• In division, divide the coefficients,
and subtract the exponent in the
denominator from the numerator
3.0 x 105
6.0 x 102
=
5 x 102
Scientific Notation
•Addition and Subtraction
•Before numbers can be added or subtracted, the
exponents must be the same
•Calculators will take care of this
•Doing it manually, you will have to make the
exponents the same- it does not matter which one
you change.
(6.6 x 10-8) + (4.0 x 10-9) = 7 x 10-8
(3.42 x 10-5) – (2.5 x 10-6) = 3.17 x 10-5
Measurement
COMMON SI UNITS
 Every
measurement has
two parts
 Number with the
correct sig - figs
 Scale (unit)
 We use the
Systeme
Internationale (SI).
Symbol
Unit Name
Quantity
Definition
m
meter
length
base unit
kg
kilogram
mass
base unit
s
second
time
base unit
K
kelvin
temperature
base unit
°C
degree
Celsius**
temperature
m3
cubic meter
volume
m3
L
liter**
volume
dm3 = 0.001 m3
N
newton
force
kg·m/s2
J
joule
energy
N·m
W
watt
power
J/s
Pa
pascal
pressure
N/m2
Hz
hertz
frequency
1/s
Metric Base Units
•Mass
- kilogram (kg)
•Length- meter (m)
•Volume- (L)
•Time - second (s)
•Temperature- Kelvin (K)
•Electric current- ampere (amp, A)
•Amount of substance- mole (mol)
•Energy – joule (j)
Prefixes
gigamega
kilo
G
- M
decicentimillimicronano-
k
d
c
m
m
n
1,000,000,000 109
1,000,000
106
103
0.1
10-1
0.01
10-2
0.001
10-3
0.000001
10-6
0.000000001 10-9
1,000
Dimensional Analysis
Using Units to solve problems
Dimensional Analysis
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

Use conversion factors to change the units
Conversion factors = 1
1 foot = 12 inches (equivalence statement)
12 in = 1 = 1 ft.
1 ft.
12 in
 2 conversion factors
 multiply by the one that will give you the
correct units in your answer.
Example Problem
 The speed of light is 3.00 x 108 m/s. How far
will a beam of light travel in 1.00 ns?
 Well, we know that 1.00 ns = 10-9 seconds
 (3.00 x 108 m) X (10-9 s) =
s
(1.00 ns)
3.00 x 10-9 m/ns
Example Problems


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11 yards = 2 rod
40 rods = 1 furlong
8 furlongs = 1 mile
The Kentucky Derby race is 1.25 miles. How long
is the race in rods, furlongs, meters, and
kilometers?
 A marathon race is 26 miles, 385 yards. What is
this distance in rods, furlongs, meters, and
kilometers?
Volume
 The space occupied by any sample of
matter
 Calculated for a solid by multiplying the
length x width x height
 SI unit = cubic meter (m3)
 Everyday unit = Liter (L), which is non-SI
Units of Mass
 Mass is a measure of the quantity of matter
 Weight is a force that measures the pull by
gravity- it changes with location
 Mass is constant, regardless of location.
 The SI unit of mass is the kilogram (kg), even
though a more convenient unit is the gram
 Measuring instrument is the balance scale
Density
 Which is heavier- lead or feathers?
 It depends upon the amount of the
material
 A truckload of feathers is heavier
than a small pellet of lead
 The relationship here is between
mass and volume- called Density
Density
 Ratio of mass to volume
 D = m/V
 Common units are g/mL, or possibly
g/cm3, (or g/L for gas)
 Useful for identifying a compound
 Useful for predicting weight
 An intensive property- does not depend on
what the material is
Things related to density
 density of corn oil is 0.89 g/mL and
water is 1.00 g/mL
 What happens when corn oil and
water are mixed?
 Why?
 Will lead float?
Example Problem
 An empty container weighs 121.3 g. Filled with
carbon tetrachloride (density 1.53 g/cm3 ) the
container weighs 283.2 g. What is the volume of
the container?
Density and Temperature
 What happens to density as the
temperature increases?
 Mass remains the same
 Most substances increase in volume
as temperature increases
 Thus, density generally decreases as
the temperature increases
Density and water
 Water is an important exception
 Over certain temperatures, the
volume of water increases as the
temperature decreases
 Does ice float in liquid
water?
 Why?
Specific Gravity
 A comparison of the density of an
object to a reference standard (which
is usually water) at the same
temperature
 Water density at 4 oC = 1 g/cm3
Formula
Specific gravity =
D of substance (g/cm3)
D of water (g/cm3)
• Note there are no units left, since they
cancel each other
• Measured with a hydrometer
• Uses?
• Gem purity
• differentiating between different types of crude
oils/gasoline
• urine tests for concentration of all chemicals in
your urine
Temperature
 Heat moves from warmer object to
the cooler object
 Glass of iced tea gets colder?
 Remember that most substances
expand with a temperature increase?
 Basis for thermometers
Temperature scales
 Celsius scale- named after a Swedish
astronomer
 Uses the freezing point (0 oC) and
boiling point (100 oC) of water as
references
 Divided into 100 equal intervals, or
degrees Celsius
Temperature scales
 Kelvin scale (or absolute scale)
 Named after Lord Kelvin
 K = oC + 273
 A change of one degree Kelvin is the
same as a change of one degree
Celsius
 No degree sign is used
Temperature scales
 Water freezes at 273 K
 Water boils at 373 K
 0 K is called absolute zero, and equals
–273 oC
Temperature
 A measure of the average kinetic energy
 Different temperature scales, all are talking
about the same height of mercury.
 In lab take the reading in ºC then convert to
our SI unit Kelvin
 ºC + 273 = K
100ºC = 212ºF
0ºC = 32ºF
100ºC = 180ºF
1ºC = (180/100)ºF
1ºC = 9/5ºF
Example problem
 A 55.0 gal drum weighs 75.0 lbs. when empty.
What will the total mass be when filled with
ethanol?
density 0.789 g/cm3
3.78 L
 1 lb = 454 g
1 gal =
Error Calculations
Error =
Experimental value- accepted value
% error = [error]
accepted value
X 100
Graphing
 The relationship between two variables is
often determined by graphing
 A graph is a “picture” of the data
Graphing Rules –
10 items
1. Plot the independent variable on the x-axis
(abscissa) – the horizontal axis. Generally
controlled by the experimenter
2. Plot the dependent variable on the y-axis
(ordinate) – the vertical axis.
3. Label the axis.
 Quantities (temperature, length, etc.) and
also the proper units (cm, oC, etc.)
4. Choose a range that includes all the results
of the data
Graphing Rules
5. Calibrate the axis (all marks equal)
6. Enclose the dot in a circle (point protector)
7.Give the graph a title (telling what it is about)
8. Make the graph large – use the full piece of
paper
9. Indent your graph from the left and bottom
edges of the page
10. Use a best fit line, do not connect points