Unit 2- Measurements, Math, and the Mole

Unit 2- Measurements, Math,
and the Mole
“No human endeavor can be called
science if it cannot be demonstrated
mathematically”
- Leonardo Da Vinci
Modified from sciencegeek.net

Nature of Measurement
Measurement - quantitative observation
consisting of 2 parts
Part 1 - number
Part 2 - scale (unit)
Examples:
20 grams
6.63 x 10 -34 Joule seconds

Significant Figures

Measuring and Significant Figures
Significant figures = indicate precision in a measurement.
How to: record all digits in a measurement that are known
with certainty on the device plus one final digit, which is
an estimation.
Ruler
Units are in Centimeters (the arrow is the end of the object)
2.83 cm
Known (because of markings on
the ruler)
Estimated (you must include)

Ex. 1
Practice- What are the best
measurements?
Ex. 2
6
a. 3 cm
b. 3.0 cm
c. 3.00 cm
Answer: 3.0 cm
a. 5.29 mL
b. 5.290 mL
5
Answer: 5.29 mL

Thermometer
Measuring Practice
graduated cylinder
Answer: 21.7 °C
Answer: 30.0 mL

Rules for identifying Significant
Figures
#1= Non-zero integers always count
as significant figures.
ex: 3456 has 4 sig figs
ex: 300
has 1 sig fig
Zeros (placeholders vs. Accuracy)
#2 =sandwiched zeros always count as
significant figures.
ex: 16.07 has 4 sig figs.

Rules for Counting Significant
Zeros
Figures - Details
#3- Leading zeros NEVER count as
significant figures (they are
placeholders).
Ex:
0.0486 has 3 sig figs.

Rules for Counting Significant
Figures - Details
Zeros
#4= Trailing zeros are significant
only if the number contains a
decimal point.
Ex: 9.300 has 4 sig figs.
Ex: 2000 has 1 sig fig (no decimal)
Ex: 2000. has 4 sig figs (decimal)
Ex: 0.02020 has 4 sig figs (decimal)

Sig Fig Practice #1
How many significant figures in each of the following?
1.0070 m 5 sig figs
17.00 kg 4 sig figs
100,890 L 5 sig figs
3.29 x 10 3 s 3 sig figs
0.0054 cm 2 sig figs
320 cm 2 sig figs

Multiplying & Dividing with Sig Figs
Multiplication and Division: # sig figs in the
result equals the number in the least precise
measurement used in the calculation.
Ex: 6.38 x 2.0 =
12.76 13 (2 sig figs)
Ex 2: 570 ÷ 2.5 =
228 230 (2 sig Figs)

Sig Fig Practice #2
Calculation Calculator says: Answer
3.24 m x 7.0 m 22.68 m 2 23 m 2
100.0 g ÷ 23.7 cm 3 4.219409283 g/cm 3 4.22 g/cm 3
0.02 cm x 2.371 cm 0.04742 cm 2 0.05 cm 2
710 m ÷ 3.0 s 236.6666667 m/s 240 m/s
1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft
1.030 g ÷ 2.87 mL 0.35888 g/mL 0.359 g/mL

Adding and Subtracting with Sig
Figs (fyi only…don’t need to know)
Addition and Subtraction: The number
of decimal places in the result equals the
number of decimal places in the least
precise measurement.
6.8 + 11.934 =
18.734 18.7 (3 sig figs)

Metrics

Commonly Used (SI) Metric Units
Quantity Name Abbreviation
Mass Gram g
Length meter m
Temperature Kelvin or Celsius K or °C
Volume Liters or centimeter 3 L or cm 3
Density Grams per centimeter 3 g/ cm 3
Amount of
Substance
mole
mol
3 countries don’t use metrics: Burma,
Liberia, and the USA

A List of the Metric Prefixes
Multiplier
Prefix Symbol Numerical Exponential
yotta Y 1,000,000,000,000,000,000,000,000 10 24
zetta Z 1,000,000,000,000,000,000,000 10 21
exa E 1,000,000,000,000,000,000 10 18
peta P 1,000,000,000,000,000 10 15
tera T 1,000,000,000,000 10 12
giga G 1,000,000,000 10 9
mega M 1,000,000 10 6
kilo k 1,000 10 3
hecto h 100 10 2
deca da 10 10 1
no prefix means:
deci d 0.1 10¯1
centi c 0.01 10¯2
milli m 0.001 10¯3
micro µ 0.000001 10¯6
nano n 0.000000001 10¯9
pico p 0.000000000001 10¯12
femto f 0.000000000000001 10¯15
atto a 0.000000000000000001 10¯18
zepto z 0.000000000000000000001 10¯21
yocto y 0.000000000000000000000001 10¯24

Scientific Notation

Scientific Notation… why??
In science, we deal with some very
LARGE numbers:
1 mole = 602000000000000000000000
In science, we deal with some very
SMALL numbers:
Mass of an electron =
0.000000000000000000000000000000091 kg

Imagine the difficulty of calculating
the mass of 1 mole of electrons!
0.000000000000000000000000000000091 kg
x 602000000000000000000000
???????????????????????????????????

How to Write in Scientific Notation
A method of representing very large or
very small numbers in the form:
M x 10 n
‣M is a number between 1 and 10 (can
be a decimal too)
‣ n is an integer
Ex: 6.3 x 10 3

2 500 000 000
9
8
7
.
Step #1: Insert an understood decimal point
Step #2: Decide where the decimal must end
up so that one number is to its left
Step #3: Count how many places you bounce
the decimal point
Step #4: Re-write in the form M x 10 n
6
5
4
3
2
1

2.5 x 10 9
The exponent is the
number of places we
moved the decimal.

0.0000579
1 2 3 4 5
Step #2: Decide where the decimal must end
up so that one number is to its left
Step #3: Count how many places you bounce
the decimal point
Step #4: Re-write in the form M x 10 n

5.79 x 10 -5
The exponent is negative
because the number we
started with was less
than 1.

Practice: Convert the following
into or out of scientific notation.
1. 6, 000, 000, 000
2. 0. 000 000 006
3. 6 x 10 8
4. 5, 234, 000
5. 0. 00120
6. 7.49 x 10 - 4
7. 80, 080, 100,
000
1. 6 x 10 9
2. 6 x 10 - 9
3. 600, 000, 000
4. 5.234 x 10 6
5. 1.20 x 10 - 3
6. 0.000749
7. 8.00801 x 10 10

PERFORMING
CALCULATIONS
IN SCIENTIFIC
NOTATION

Multiplying
EX: (4 x 10 6 )
X (3 x 10 5 )
12 x 10 11
Multiply the M factors
together and the
exponents are added
together.
Change to correct Scientific
notation format: 1.2 x 10 12
Best answer with Sig Figs : 1 x 10 12

Practice Problem
(8 x 10 6 m)
X (3 x 10 4 m)
Answer: 24 x 10 10 m 2
correct.
… but it’s not quite
M must be a number between 1-10, right now
it’s 24. So, you must make 24 into 2.4 and
move the decimal one more place.
Best answer: 2.4 x 10 11 m 2

Dividing
EX: (4 x 10 6 )
÷ (2 x 10 5 )
2 x 10 1
Divide the M factors
together and the exponents
are subtracted from each
other.

Practice Problem
(8 x 10 6 g)
÷ (2 x 10 4 mL)
4 x 10 2 g/mL

Another way of seeing it written
Multiply top and divide by bottom
6 X 10 8 m x 2 x 10 3 m
3 X 10 4 s
= 12 X 10 11 m 2
3 X 10 4 s = 4 x 10 7 m 2 / s

How to use your calculator with
Scientific Notation
1. Practice (6.0 x 10 5 ) x (4.0 x 10 3 ) on your
calculator
2. Punch the number (the digit number) into
your calculator.
3. Push the EE or EXP button. Do NOT type
in x10 or use the 10 x buttons!!
4. Enter the exponent number. Use the +/-
button to change its sign.
Answer= 2.4 x 10 9
(some calculators will show this for the answer:
2.4 9 -you have to know it is 2.4 x 10 9 )

Practice with Calculator
(5.23 X10 6 )
X (7.1 x 10 -2 )
3.7133 x 10 5 , but
3.7133 x 10 5 , but there
are 2 sig figs
there are 2 sig figs
Best answer: 3.7 x 10 5

Adding and Subtracting with
Scientific Notation
(FYI ONLY… don’t really use Chem.)

(4 x 10 6 )
+ (3 x 10 6 )
7 x 10 6
IF the exponents are
the same, we simply
add or subtract the
numbers in front and
bring the exponent
down unchanged.

(4 x 10 6 )
+ (3 x 10 5 )
If the exponents
are NOT the
same, we must
move a decimal
to make them
the same.

4.00 x 10 6
+4.30
.30 x 10 6
YES!
x 10 6

A Problem for you…
Scientific Notation
(2.37 x 10 -6 )
+ (3.48 x 10 -4 )

Solution…
0.0237 x 10 -4
+ 3.48 x 10 -4
3.5037 x 10 -4

Calculating Density

Mass = amount of matter
something contains
Weight Vs. Mass
» Doesn’t change in outer
space
Weight = measurement of the
pull of gravity on an
object.
» Changes in outer space.

Density= Mass / Volume
• How much “stuff” per unit
volume
•The density of a substance is unique and never
changes regardless of it’s size or shape.
Ex: Gold’s density = 19.3 g/ mL
(same for the ring & gold bar)

• Is a physical property
that is unique to each
material
• Can be used to identify
a substance
Density
Density of ice= .92g/cm 3

Can find by:
Volume- amount of space an object
occupies
1) Using a ruler
• only if it’s a certain shape
(ex: cube)
• Measured in : cm 3
2) Water displacement
• Use for odd shaped objects
• Measure in: mL
*1 cm 3 = 1 mL (volume is the
same regardless of which
way you measure it)

Ex: Muscle is more dense than fat
Muscle density= 1.06 g/ml
Fat density= 0.9 g/ml
This is why
people who
start
exercising
don’t always
lose weight,
but clothes
fit
differently

Dimensional Analysis

Sample Problems to Solve
A) How many seconds has a 16 year old been
alive?
B) How many liters of water will Kate consume in
1 year? Kate drinks 8 glasses of water in 1 day
& 1 glass = 300 mL. There are 1000 mL in 1
L.

The Math behind Dimensional Analysis
6 zooms
x
4 zetas
3 zooms
=
6 x 4 Zetas
3
=
24 Zetas
3
= 8 Zetas
•This is the level of math you will need to know in this unit.
•You would cancel out everything on the top and bottom
that are the same (including words).
• In the end, multiply everything on the top and divide it by
everything that has been multiplied on the bottom.

Dimensional Analysis
• It is a method to solve conversion type problems
• Uses Conversion Factors (relate 2 things to each other)
• These are either given to you or assumed to be known
– Ex: 4 laps = 1 mile (they are equal to each other)
Can also be written as a fraction:
4 Laps
1 mile = Or 1 mile • Can be written in
1
4 laps either direction
When solving problems use this method:
GIVEN X CONVERSION FACTOR = Find

Sample Dimensional Analysis Problem
Ex: Convert 15 cm into meters
Given x conversion factor = Find
Ex: 15 cm x 1 meter =
100 cm
.15 meter

Dimensional Analysis
Sample Problem #2
•You're throwing a pizza party for 15 people.
•each person might eat 4 slices.
•each pizza will cost you $14.78
•1 pizza will be cut into 12 slices.
•How much is the pizza going to cost you?
Starting x conversion factors
= find
15 people 4 slices 1 pizza 14.78 dollars
1 person 12 slices 1 pizza
=
$73.90
$ 74
dollars

Good use of D.A. - performed by a
senior planning a trip
A group of friends are going to Disneyland for
graduation. Driver wants to know how much to
charge his friends the trip to cover his gas costs?
It’s 876.8 miles round trip.
-Car gets 18 miles/ gallon
$ 4.30/ 1 gallon gas
-6 people

DIMENSIONAL ANALYSIS REVIEW
If an amoeba needs to eat 4 cyanobytes to be full.
How many amoebas can you feed if you have 6.0
cyanobytes?
6 cyanobytes 1 amoeba = 1.5 amoebas
4 cyanobytes

DIMENSIONAL ANALYSIS REVIEW
If you have 3 drimples, how many blobs could you
make if it takes 1 drimple to make 4 clomps and
6 clomps to make a blob?
3 drimples 4 clomps 1 blob = 2 blobs
1 drimple 6 clomps

DIMENSIONAL ANALYSIS REVIEW
If you have 8.0 grams of a substance, how many
moles would you have if 1 mole has a mass of 32
grams?
8.0 grams 1 mole = 0.25 moles
32 grams

DIMENSIONAL ANALYSIS REVIEW
If 3 amoebas need 5 x 10 -2 meters 2 of living space.
How many amoebas can you have if you have a
petri dish that is 20. x 10 -2 meters 2 ?
20. x 10 -2 m 2 3 ameobas = 12 amoebas
5 x 10 -2 m 2

DIMENSIONAL ANALYSIS REVIEW
If you have 10. grams of a substance, how many
atoms would you have if 1 mole has a mass of 40
grams and 1 mole contains 6 x 10 23 atoms?
10. g 1 mol 6 x 10 23 atoms = 1.5 x 1023 atoms
40 g 1 mol

DIMENSIONAL ANALYSIS REVIEW
What would be the mass of 1.5 x 10 23 molecules, if
6 x 10 23 molecules equals 1 mole and 1 mole is 80
grams?

DIMENSIONAL ANALYSIS REVIEW
How many moles are in 116 grams of magnesium
hydroxide, Mg(OH) 2 ?

DIMENSIONAL ANALYSIS REVIEW
How many atoms are in 44 grams of boron?

The Mole
naked mole rat

What is the Mole?
It’s an amount, like a dozen (1 dozen = 12)
1 mole = 6.02 x 10 23 particles
(a.k.a. Avogadro’s Number)
Amedeo
Avogadro
‣Experiments are performed with moles of atoms
(not individual atoms).
‣A mole is defined as: amount of a substance
that contains as many particles as there are atoms
in 12 g or Carbon-12.

Using the “mole” by weighing
Ex: a scientist wants 1 mole of Na and 1 mole of Cl
to make NaCl.
Is he or she going to count out 6.02 x 10 23 Na and Cl
atoms? NO WAY!
Chemists can "count" atoms or molecules by knowing
how much 1 mole of every substance weighs…
The molar mass! It’s on the periodic table

How big is Avogadro’s #?
• An Avogadro's number of soft drink cans would
cover the surface of the earth to a depth of over
200 miles.
• If you spread Avogadro's number of unpopped
popcorn kernels across the USA, the entire
country would be covered in popcorn to a depth of
over 9 miles.
• If we were able to count atoms at the rate of 10
million per second, it would take about 2 billion
years to count the atoms in one mole.

Molar Mass
Atomic mass
Mass of 1 atom
of C= 12.01 amu
Molar Mass
Mass of 1 mole of C atoms (6.02 x 10 23 atoms)
12.01 g/ mol
Here’s How do we know?
It weighs 12.01 grams on a balance

Practice with the Mole Concept
1
H
1.01
8
O
16.00
1. What weighs more, a mole of Hydrogen or
a mole of Oxygen? Oxygen (16.00 grams /mole)
(Because a Hydrogen atom weighs more than
an Oxygen atom) Note: only units change
2. Which has more atoms, a mole of Oxygen or
a mole of Hydrogen? Both have 6.02x 10
23
atoms
3. What’s the molar mass of H 2 O? 18.02 g/ mol (see demo)
4. Mass of 1 molecule of H 2 O? = 18.02 amu

The Mole vs. a Dozen
1 dozen cookies = 12 cookies
1 mole of cookies = 6.02 X 10 23 cookies
1 dozen cars = 12 cars
1 mole of cars = 6.02 X 10 23 cars
1 dozen Al atoms = 12 Al atoms
1 mole of Al atoms = 6.02 X 10 23 atoms
Note that the NUMBER is always the same, but the MASS is
very different!
Mole is abbreviated mol (gee, that’s a lot quicker to write,
huh?)

Using the Mole in Chemistry

The Mole- a conversion factor
**1 mole = 6.02 x 10 23 particles **
1 mole
6.02 x 10 23 particles**
OR
6.02 x 10 23 particles**
1 mole
**Particles = atoms or molecules **

Molar Mass- a conversion factor
Since 6.02 X 10 23 is impossible to count, chemists know the mass of a
mole (the molar mass)
1 mole
x grams
OR
X grams
1 mole
use periodic table to determine the mass because every substance is different.
Ex: Write the molar mass of Ca as a conversion factor
40.08 grams
1 mole Ca
Or
1 mole Ca
40.08 grams
**Can also be written 40.08 g/ mol**

Molar mass of Compounds
Ex: Write the molar mass of H 2 O as a conversion factor
Ex : H 2 O = 18.02 grams
(you need to add)
2 Hydrogen’s : (1.01) 2 = 2.02
1 Oxygen: (16.00) 1 = 16.00
= 18.02 g
Molar mass of H 2 O=
1 mole
18.02 g

“Mole Map” for Calculations
use molar mass use Avogadro’s number
Grams Moles particles
g/ mol
1 mol/ 6.02 x 10 23
(use the periodic table)
Everything must go through Moles!!!

Converting Moles and Atoms
What is the # of moles of S in 1.8 x 10 24 S atoms?
GIVEN X
CONVERSION FACTOR = Find**
1.8 X 10 24 Atoms x 1 mole
6.02 X 10 23 Atoms
= # of Moles
= 1.8 X 10 24 mole
6.02 X 10 23 = 3.0 mole

Converting Moles and Grams
How many grams of Al are in 3.00 moles of Al?
You will need:
1. Molar mass of Al … 1 mole Al = 27.00 g Al
2. Conversion factors for Al
27.00g Al or 1 mol Al
1 mol Al 27.00 g Al
3. GIVEN X CONVERSION FACTOR= Find
3.00 moles Al x 27.00 g Al
1 mole Al
= 81.00 g Al

Sample of Atoms/Molecules and
Grams
How many atoms of Cu are present in 35.4 g of
Cu?
35.4 g Cu 1 mol Cu 6.02 X 10 23 atoms Cu
63.5 g Cu 1 mol Cu
= 3.4 X 10 23 atoms Cu

Dimensional Analysis Fun…
from Lewis Caroll’s Through the looking Glass
If there are 5 frumious Bandersnatches,
how many Jabberwocks are there?
There are 20 tumtum trees in the tulgey wood.
In each tulgey wood is one frumious Bandersnatch.
There are 5 slithy toves in 2 borogoves.
There are 2 mome raths per Jabberwock.
There are 2 Jubjub birds in 200 tumtum trees.
There are 200 mome raths in each borogove.
There are 5 Jubjub birds per slithy tove.