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

Part 1 - numberPart 2 - scale (unit)

Examples:20 grams

6.63 x 10-34 Joule seconds

Measurement - quantitative observation consisting of 2 parts

Significant Figures

Measuring and Significant Figures

Units are in Centimeters (the arrow is the end of the object)

Ruler

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.

2.83 cm

Known (because of markings on the ruler)

Estimated (you must include)

Practice- What are the best measurements?

a. 3 cm

b. 3.0 cm

c. 3.00 cm5

6Ex. 2

a. 5.29 mL

b. 5.290 mLAnswer: 3.0 cm

Answer: 5.29 mL

Ex. 1

Thermometer

Measuring Practice

graduated cylinder

Answer: 30.0 mL

Answer: 21.7 °C

Rules for identifying Significant Figures

#1= Non-zero integers always count as significant figures.

ex: 3456 has 4 sig figsex: 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 Figures - Details

Zeros#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 #1How 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 103 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

3.24 m x 7.0 m

Calculation Calculator says: Answer

22.68 m2 23 m2

100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3

0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2

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 centimeter3 L or cm3

Density Grams per centimeter3 g/ cm3

Amount of

Substance

mole mol

3 countries don’t use metrics: Burma, Liberia, and the USA

A List of the Metric PrefixesMultiplier

Prefix Symbol Numerical Exponential yotta Y 1,000,000,000,000,000,000,000,000 1024

zetta Z 1,000,000,000,000,000,000,000 1021

exa E 1,000,000,000,000,000,000 1018

peta P 1,000,000,000,000,000 1015

tera T 1,000,000,000,000 1012

giga G 1,000,000,000 109

mega M 1,000,000 106

kilo k 1,000 103

hecto h 100 102

deca da 10 101

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

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

Scientific Notation… why??

Imagine the difficulty of calculating the mass of 1 mole of electrons!

0.000000000000000000000000000000091 kgx 602000000000000000000000

???????????????????????????????????

A method of representing very large or very small numbers in the form:

M x 10n

M is a number between 1 and 10 (can be a decimal too)

n is an integerEx: 6.3 x 103

How to Write in Scientific Notation

2 500 000 000

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

123456789

Step #4: Re-write in the form M x 10n

2.5 x 109

The exponent is the number of places we moved the decimal.

0.0000579

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 10n

1 2 3 4 5

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 108

4. 5, 234, 000

5. 0. 00120

6. 7.49 x 10-4

7. 80, 080, 100, 000

1. 6 x 109

2. 6 x 10-9

3. 600, 000, 000

4. 5.234 x 106

5. 1.20 x 10-3

6. 0.000749

7. 8.00801 x 1010

PERFORMING CALCULATIONS IN SCIENTIFIC

NOTATION

Multiplying

Multiply the M factors together and the exponents are addedtogether.

EX: (4 x 106)X (3 x 105)

12 x 1011

Best answer with Sig Figs : 1 x 10 12

Change to correct Scientific notation format: 1.2 x 1012

Practice Problem

(8 x 106m)X (3 x 104 m)

Answer: 24 x 1010 m2 … but it’s not quite correct.

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 1011 m2

Dividing

EX: (4 x 106)÷ (2 x 105)

Divide the M factors together and the exponents are subtracted from each other.

2 x 101

Practice Problem

(8 x 106 g)

÷ (2 x 104 mL)

4 x 102 g/mL

Another way of seeing it written

6 X 10 8 m x 2 x 10 3 m

3 X 10 4 s

= 12 X 10 11 m2

3 X 10 4 s = 4 x 10 7 m2 / s

Multiply top and divide by bottom

How to use your calculator with Scientific Notation

1. Practice (6.0 x 105) x (4.0 x 103) 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 10x buttons!!

4. Enter the exponent number. Use the +/-button to change its sign.

(some calculators will show this for the answer: 2.49 -you have to know it is 2.4 x 109)

Answer= 2.4 x 109

Practice with Calculator

(5.23 X106)

X (7.1 x 10-2)

3.7133 x 105 , but there are 2 sig figs3.7133 x 105 , but there are 2 sig figs

3.7133 x 105 , but there are 2 sig figs

Best answer: 3.7 x 105

Adding and Subtracting with Scientific Notation

(FYI ONLY… don’t really use Chem.)

(4 x 106)+ (3 x 106)

IF the exponents are the same, we simply add or subtract the numbers in front and bring the exponent down unchanged.

7 x 106

(4 x 106)+ (3 x 105)

If the exponents are NOT the same, we must move a decimal to make them the same.

4.00 x 106

.30 x 106

4.30 x 106

YES!+

A Problem for you…

(2.37 x 10-6)+ (3.48 x 10-4)

Scientific Notation

+ 3.48 x 10-4

Solution…0.0237 x 10-4

3.5037 x 10-4

Calculating Density

Weight Vs. Mass

Mass = amount of matter something contains

» 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

Ex: Gold’s density = 19.3 g/ mL

(same for the ring & gold bar)

•The density of a substance is unique and neverchanges regardless of it’s size or shape.

Density

• Is a physical property that is unique to each material

• Can be used to identify a substance

Density of ice= .92g/cm3

Volume- amount of space an object occupies

Can find by:

1) Using a ruler

• only if it’s a certain shape (ex: cube)

• Measured in : cm3

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/mlFat 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 zoomsx

4 zetas

3 zooms=

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.

6 x 4 Zetas

3= 24 Zetas

3= 8 Zetas

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 = 1Or 1 mile

4 laps

• Can be written in

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?

15 people 4 slices 1 pizza 14.78 dollars

1 person 12 slices 1 pizza

Starting x conversion factors = find

= $73.90$ 74dollars

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 meters2 of living space.

How many amoebas can you have if you have a

petri dish that is 20. x 10-2 meters2?

20. x 10-2 m2 3 ameobas = 12 amoebas

5 x 10-2 m2

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 1023 atoms?

10. g 1 mol 6 x 1023 atoms = 1.5 x 1023 atoms

40 g 1 mol

DIMENSIONAL ANALYSIS REVIEW

What would be the mass of 1.5 x 1023 molecules, if

6 x 1023 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?

1 mole = 6.02 x 1023 particles(a.k.a. Avogadro’s Number)

Experiments are performed with moles of atoms (not individual atoms).

Amedeo Avogadro

A mole is defined as: amount of a substance that contains as many particles as there are atoms in 12 g or Carbon-12.

It’s an amount, like a dozen (1 dozen = 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 1023 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

Here’s How do we know?

Mass of 1 atom of C= 12.01 amu

Molar Mass

Mass of 1 mole of C atoms (6.02 x 1023 atoms)

12.01 g/ mol

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 H2O? 18.02 g/ mol (see demo)

4. Mass of 1 molecule of H2O? = 18.02 amu

The Mole vs. a Dozen

1 dozen cookies = 12 cookies1 mole of cookies = 6.02 X 1023 cookies

1 dozen cars = 12 cars1 mole of cars = 6.02 X 1023 cars

1 dozen Al atoms = 12 Al atoms1 mole of Al atoms = 6.02 X 1023 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**

6.02 x 10 23 particles**

1 moleOR

**Particles = atoms or molecules **

Molar Mass- a conversion factorSince 6.02 X 10 23 is impossible to count, chemists know the mass of a

mole (the molar mass)

use periodic table to determine the mass because every substance is different.

1 mole

x gramsOR

X grams

1 mole

Ex: Write the molar mass of Ca as a conversion factor

40.08 grams

1 mole Ca

Or1 mole Ca

40.08 grams

**Can also be written 40.08 g/ mol**

Molar mass of Compounds

Ex: Write the molar mass of H2O as a conversion factor

Ex : H2O = 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

1 mole18.02 g

Molar mass of H2O=

use molar mass use Avogadro’s numberGrams Moles particles

Everything must go through Moles!!!

“Mole Map” for Calculations

g/ mol

(use the periodic table)

1 mol/ 6.02 x 10 23

Converting Moles and Atoms

What is the # of moles of S in 1.8 x 1024 S atoms?

GIVEN X CONVERSION FACTOR = Find**

= # of Moles1.8 X 10 24 Atoms x 1 mole

6.02 X 10 23 Atoms

= 1.8 X 10 24 mole

6.02 X 10 23= 3.0 mole

How many grams of Al are in 3.00 moles of Al?

Converting Moles and Grams

3. GIVEN X CONVERSION FACTOR= Find

3.00 moles Al x 27.00 g Al1 mole Al

1. Molar mass of Al … 1 mole Al = 27.00 g Al

You will need:

2. Conversion factors for Al27.00g Al or 1 mol Al1 mol Al 27.00 g 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 1023 atoms Cu

63.5 g Cu 1 mol Cu

= 3.4 X 1023 atoms Cu

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.

Dimensional Analysis Fun…

from Lewis Caroll’s Through the looking Glass