Chapter 5
Measurements and
Calculations
1
5.1 Scientific Notation
Objective:
To show how very large or very small
numbers can be expressed as the
product of a number between 1 and
10 and a power of 10.
2
Scientific Measurements
In science we have two types of
observations:
Qualitative: observations of color,
odor, appearance, etc.
Quantitative: observations of
measurement.
3
Scientific Measurements
Scientific measurements must always be
represented as a number and a unit.
During a lab, if units are not included, or
improperly included, it could result in
drastically different results.
In science we often make measurements that
are very large or small. To make things
easier to write we use scientific notation.
4
Scientific Notation
When using scientific notation for very large
numbers:
1. Move the decimal to the left until there is
only one digit between 1 and 10.
2. Count the number of places you move
the decimal and make this the power of 10.
3. Rewrite your number as a decimal times
10 to the power of x. X= the number of
places you moved the decimal.
5
Scientific Notation
EXAMPLE:
930
1. Move the decimal to make the
number between 1 and 10.
930 => 9.30
2. Multiply by 10 raised to a power of 2,
because the decimal was moved 2
places.
9.3x 102
6
Scientific Notation
When a very small number is involved, the
decimal must be moved to the right instead
of the left, in this case we make the power
of 10 a negative number.
0.0000093 => move the decimal 6 places
to the right.
9.3 x 10-6
7
Scientific Notation
EXAMPLES:
Convert to scientific notation:
1. 8,000
2. 75,600,000,000
3. 0.000000546
4. 0.0004876
8
Scientific Notation
Multiplication and Division of Scientific
Notation:
When multiplying 2 numbers with exponents,
you must add the exponents.
EXAMPLE: (3.2 x 104)(2.8 x 103)=
(3.2 x 2.8) x 104+3= 9.0 x 107
When dividing you subtract the exponents.
EXAMPLE: 6.4 x 103 = 6.4/8.3 x 103-5
8.3 x 105
=0.77 x 10-2 = 7.7 x 10-3
9
Scientific Notation
Addition and subtraction of scientific
notation requires that all the numbers
are raised to the same power of 10
EXAMPLE:
1.31 x 105 + 4.2 x 104 =
13.1 x 104 + 4.2 x 104 =(13.1 + 4.2) x 104
= 17.3 x 104 = 1.73 x 105
10
5.2 Units
Objective: To learn the English, metric,
and SI systems of measurements.
The units part of a measurement tells us
what scale or standard is being used
to represent the results of the
measurement.
The need for common units is necessary
so that scientists may have a universal
language.
11
Units
The English system of units is the system used in
the United States and includes inches, feet,
miles.
The metric system is used internationally and is
known as the International System or SI Units.
Below are some basic SI Units.
Mass
kilogram
kg
Length
meter
m
Time
second
s
Temperature
Kelvin
K
12
Units
Below are the prefixes used for the metric system.
13
5.3 Measurements of
Length, Volume, and Mass
Objective: To use the metric system to
measure length, volume, and mass.
To measure the length or distance, we
commonly use a ruler. The unit used
for length is the meter.
14
Measurements
15
Measurements
Volume is the amount of
3-dimensional space occupied by a
substance.
If you have a cube that is 1m x 1m
x 1m, the volume of this cube
would be 1m3 or 1 cubic meter.
1 cm3 = 1 mL.
1000 mL = 1 L
16
Measurements
What this instrument is
called?
Graduated cylinder
What would you measure
with this?
Liquid volume
17
Measurements
When measuring mass we use the unit,
kilogram.
In this class we will be measuring in
grams.
How many grams are in a kilogram?
What type of instrument is used to
measure mass?
18
5.4 Uncertainty in
Measurement
Objectives:
To learn how uncertainty in a
measurement arises.
To learn to indicate a measurement’s
uncertainty by using significant
figures.
19
Uncertainty
Whenever a measurement is made with
instruments such as a ruler, graduated
cylinder or thermometer some estimation
must be made.
If you and your group must measure the
length of a pin, you may get results such as
the following:
2.84 cm, 2.85 cm, 2.86 cm, 2.85 cm, 2.86 cm
Why aren’t these results all exactly the same?
20
Uncertainty
If we look at fig. a we
see that the pin is
between 2.8 and
2.9 cm.
So if we enlarge that
distance and break
the space between
2.8 and 2.9 into 10
even pieces, we
can see that it
lands on precisely
5.
This means that the
distance is 2.85.
You must always
estimate a digit
beyond the marked
reading.
21
Uncertainty
So if we look back at your group’s readings
we see that the first 2 digits are all the
same, it is the estimated third digit is where
things vary.
2.84 cm, 2.85 cm, 2.86 cm, 2.85 cm, 2.86 cm
The third digit is called an uncertain number.
When making a measurement you must
always record the certain digits (those
marked) plus the first uncertain number.
22
Uncertainty
Any measurement always has some
degree of uncertainty.
The uncertainty of a measurement
depends of the measuring device.
The ruler has uncertainty that occurs in
the hundredths place.
A beaker would have uncertainty that
occurs in the 10s or ones place.
23
Uncertainty
The numbers recorded in a measurement are
called significant figures.
The number of significant figures is
determined by the uncertainty of the
measuring device.
With the ruler for measuring the pin, the
uncertainty fell in the hundredths place.
The uncertainty is usually assumed to be + 1
unless otherwise indicated.
24
5.5 Significant Figures
Objective:
To learn to determine the number of
significant figures in a calculated
result.
25
Significant Figures
Chemistry requires doing many types of
calculations. With each calculation we must
consider what happens when arithmetic is
done with numbers that contain uncertainties.
Mathematicians have come up with some rules
for uncertainty.
26
Significant Figures
Rules for Counting Significant Figures:
1.
Nonzero integers. Nonzero integers always count
as significant figures.
For example, the number 1457 has four nonzero
integers, all of which count as significant figures.
2. Zeros. There are three classes of zeros.
a. Leading zeros are zeros that precede all of the
nonzero digits. They never count as significant
figures.
For example, in the number 0.0025, the 3 zeros simply
indicate the position of the decimal point. The number
27
only has 2 significant figures, the 2 and the 5.
Significant Figures
2. b. Captive zeros are zeros that fall between
the nonzero digits. They always count as
significant figures.
For example, the number 1.008 has 4
significant figures.
c. Trailing zeros are zeros at the right end of
the number. They are significant only if the
number is written with a decimal point.
The number one hundred written as 100 has
only one significant figure, but written as
100. has 3 significant figures.
28
Significant Figures
Rules for Counting Significant Figures cont.:
3. Exact numbers. Often calculations involve
numbers that were not obtained using
measuring devices but were determined by
counting: 10 experiments, 3 apples, 8
molecules. Such numbers are called exact
numbers. They can be assumed to have an
unlimited number of significant figures.
Exact numbers can also arise from definitions.
For example, 1 inch is defined as exactly 2.54 cm.
Thus, in the statement 1in. = 2.54 cm, neither
2.54 nor 1 limits the number of significant
figures when it is used in a calculation.
29
Significant Figures
Use your rules to determine the number
of significant figures:
1. 0.0108
2. 0.0050060
3. 5.030 x 103
4. 480
30
Significant Figures
On your calculator you will find you have several
extra numbers and you must round off.
Rules for Rounding Off:
1.
If the digit to be removed
a. is less than 5, the preceding digit does not
change. Example: 1.33 = 1.3
b. Is equal to or greater than 5, the preceding
digit is increased by 1. Example: 1.36 = 1.4
2. In a series of calculations, carry the extra digits
through to the final result and then round off.
31
Significant Figures
Rules for Using Significant Figures in Calculations:
1.
For multiplication or division, the number of
significant figures in the result is the same as that
in the measurement with the smallest number of
significant figures. We say this measurement is
limiting, because it limits the number of
significant figures in the result.
For example: 4.56 x 1.4 = 6.384 => 6.4 2 sig figs
You try: 8.315 = ?
298
32
Significant Figures
For addition or subtraction, the limiting term is
the one with the smallest number of decimal
places.
For example: 12.11+18.0+1.013 = 31.123
The correct answer would be 31.1
You try: 0.6875 – 0.1 = ?
2.
NOTE: For multiplication/division the significant
figures are counted. For addition/subtraction the
decimal places are counted.
33
5.6 Problem Solving and
Dimensional Analysis
Objective:
To learn how dimensional analysis can be
used to solve various types of problems.
The problem:
Your mother asks you to buy 3 dozen cookies.
The cookies come in packages of 6 cookies.
How many packages of cookies must you
buy?
34
Dimensional Analysis
In order to solve this problem and many
others in chemistry we are going to use a
process called dimensional analysis. We
are going to analyze the dimensions or units
involved here.
First we must figure out the conversion
factors. These are the numbers that help
us convert from one unit to another.
1 dozen = ? cookies
1 package = ? cookies
35
Dimensional Analysis
So the conversion factor for the first is
1 dozen = 12 cookies, also if we read the problem
we can learn that
1 package = 6 cookies
We can now use this information to calculate how
many packages we need.
3 dozen x 12 cookies x 1 package = 6 pkgs.
1
1 dozen
6 cookies
We can cancel units that are the top and bottom of
the equation because this equals 1.
36
Dimensional Analysis
If you divide 2 this equals 1. The same rule
2
applies when dividing units. Cookies = 1
cookies
Or elephants = 1
elephants
It does not matter what the unit is, if it is on
the top and bottom of the equation, they
equal 1. And anything times one is
anything. 2x1= 2 or cookies x 1 = cookies
37
Dimensional Analysis
So let’s look at our problem again.
3 dozen x 12 cookies x 1 package = 6 pkgs.
1
1 dozen
6 cookies
The dozen cancels, the cookies cancel and we
are left with only packages as a unit.
IT IS VERY IMPORTANT TO ALWAYS CLEARLY
WRITE DOWN YOUR UNITS!!!
If you do not, you WILL lose track and your
answer will become incorrect.
38
Dimensional Analysis
Earlier we measured the length of a pin. We
found this pin to be 2.85 cm, how many
inches is this? There are 2.54 cm in 1 inch.
1. Before we start the problem, read carefully
and find what we piece of information we
are trying to calculate here.
How many inches is the pin?
2. Then write down the information you are
given.
The pin is 2.85 cm and 2.54 cm = 1 in.
39
Dimensional Analysis
Now we can set up our problem.
3. Always start with the given value, in this
case it would be 2.85 cm.
2.85 cm x 1 inch
=?
1
2.54 cm
4. Next write in your conversion factor(s).
5. Now check your units and see if they
cancel.
If they don’t….then you need to find the
missing units.
40
Dimensional Analysis
6. Draw a light line through your canceled
units.
2.85 cm x 1 inch
= 1.12 inches
1
2.54 cm
7. Multiply and divide you answer and add
your final units to your answer.
8. Check your significant figures.
9. Ask whether your answer makes
sense.
41
Dimensional Analysis
The length of a marathon race is
approximately 26.2 miles. What is
this distance in kilometers?
HINT: 1 mi = 1760 yd and 1 m = 1.094
yd
1.
42
Dimensional Analysis
Your set up:
26.2 mi x 1760 yd x 1 m =
1
1 mi
1.094 yd
Do the units cancel?
What units is the answer supposed to be
in?
43
Dimensional Analysis
26.2 mi x 1760 yd x 1 m
x 1km =
1
1 mi
1.094 yd 1000 m
Do your units all cancel?
How many significant figures?
What is the correct answer?
Does this seem reasonable?
44
5.7 Temperature
Conversions: An Approach
to Problem Solving
Objective:
To learn the three temperature scale.
To learn to convert from one scale to
another.
To continue to develop problem-solving
skills.
45
Temperature Conversions
There are three scales used for
measuring temperature:
Fahrenheit – Part of the English
System
Celsius – Used in the metric system
Kelvin – The base unit in the SI units
and also known as the absolute scale.
46
Temperature Conversions
The thermometers
indicate the
freezing pts and
boiling pts on
each scale.
Notice there is an
equal distance
between the 2
pts on the
Celsius and
Kelvin scales,
but not the
Fahrenheit.
47
Temperature Conversions
There are simple equations
you can use to convert
temperatures from one
scale to another.
Celsius -> Kelvin
Because these units are
equal, all we have to do is
add 273 to the Celsius
temperature.
T C + 273 = TK
Kelvin -> Celsius
TK - 273 = T C
48
Temperature Conversions
When converting with
the Fahrenheit scale
we must do more
math because the
scales are not equal
units.
Celsius ->Fahrenheit
T F=1.80(T C)+32
Fahrenheit -> Celsius
T C= T F – 32
1.80
49
Temperature Conversions
Convert the following:
1. 38 C =
F
2. 145 C =
K
3. 373 K =
C
4. 25 C =
K
5. 85 F =
K
50
5.8 Density
Objective: To define density and its units.
Density: the amount of matter present in a
given volume of substance or is mass per
unit volume.
In mathematical terms this means:
Density = mass
or is the ratio of mass
volume
to volume
51
Density
What type of units would you use for
density?
Mass => ? Units
Volume => ? Units
Mass = grams (g)
Volume = milliliters (ml) = cubic cm =
cm3/cc
So density units = g/ml or g/cm3
52
Density
So what does the term density really mean?
If something is very dense...what does that
mean?
What if something is not very dense?
You know how to calculate density, but how
would you measure density?
What type of instruments would you need?
53