Unit 2: MEASUREMENT

1. Scientific Notation
2. Metric System
3. Accuracy and Precision
4. Measuring & Counting Significant Figures
5. Calculations with Significant Figures
6. Density
7. Graphing Density

1
1. Scientific
Notation

2
 http://www.youtube.com/watch?v=7T77G5XwTnE

Scientific Notation
• A shorthand method of displaying very large (distance to
the sun) or very small numbers (lengths of atoms).
• Consists of a coefficient, a base 10, and an exponent
• e.g. 3.95 x 103
• The coefficient must be between 1 and 10 or it is not in
scientific notation.
• If the exponent is positive (such as above), the number will
be large (greater than 1).
• If the exponent is negative, the number will be small (less
than 1).

3
 Express in Scientific Notation
• Ex. 3756 = 3.756 x 103

• 0.000493 = 4.93 x 10 ‐4

4
 Express in Standard Notation

• E.g. 5.21 x 104 =

The exponent is positive, so make the coefficient a
large number (move the decimal to the right)
5.21 x 104 = 52100

• 2.694 x 10‐5
The exponent is negative, so make the coefficient a
small number (move decimal to the left).
2.694 x 10‐5 = 0.00002694

5
 Practice
• Put in scientific notation
• 1. 8720000 =
• 2. 0.0000513 =
• 3. 5302 =
• 4. 0.00117 =
• Put in standard notation
• 5. 7.03 x 10‐2 =
• 6. 1.38 x 104 =
• 7. 3.99 x 10‐5 =
• 8. 2.781 x 107 =

6
 Practice ‐ Answers
• 1. 8720000 = 8.72 x 106
• 2. 0.0000513 = 5.13 x 10‐5
• 3. 5302 = 5.302 x 103
• 4. 0.00117 = 1.17 x 10‐3
• 5. 7.03 x 10‐2 = 0.0703
• 6. 1.38 x 104 = 13800
• 7. 3.99 x 10‐5 = 0.0000399
• 8. 2.781 x 107 = 27810000

7
 Write in Scientific Notation:

• 1. 34.79 x 103 =

• 2. 0.497 x 106 =

• 3. 19.5 x 10‐2 =

• 4. 0.837 x 10‐4 =

8
Write in Scientific Notation ­ Answers

• 1. 34.79 x 103 = 3.479 x 104

• 2. 0.497 x 106 = 4.97 x 105

• 3. 19.5 x 10‐2 = 1.95 x 10‐1

• 4. 0.837 x 10‐4 = 8.37 x 10‐5

9
To add, subtract, multiply, and divide
numbers that are in scientific notation, use
your calculator.

10
How to do Sci Not. On the Calc.

11
1. Type in the coefficient

2. Hit the EXP (or EE or 10x button)

3. Type in the exponent.

If the exponent is negative, use the
negative button (­) or +/­. Don't use
the subtract button for negatives!

12
Give the answer in scientific notation
Ex. (2.56 x 10­4)(3.87 x 103)

Answer: 9.91 x 10­1

13
Homework:

Scientific
Notation
Worksheet

14
2. Metric System

15
• A measurement must include some form of
unit (otherwise it’s just a number).
• E.g.
• 10.5 cm A length
• 247.93 g A mass
• 0.25 mL A volume

16
 Base Units
• Common BASIC units of metric
Written
Quantity Symbol
Unit

length meter m
mass gram g
time second s
volume liter L
Degrees C
temperature Celsius

amount mole mol

17
 Units Derived from Base Units
Examples of Derived Units
Quantity Written Unit Symbol
meter per
speed m/s
second
density gram per liter g/L
concentration moles per liter M

area meters squared m2

volume meters cubed m3

18
 Common Multiples
of Base Units
(prefixes)

http://www.youtube.com/watch?v=KfrCaKyhwZk

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 http://www.media.pearson.com.au/schools/cw/au_sch_derry_ibcsl_1/int/SIPrefix/0106.html

Equivalent
Written Prefix Prefix Symbol
multiplier

mega M 106 = 1 million

kilo k 103 = 1 thousand

hecto h 102 = 1 hundred

Move Move
deca da 101 = ten
Decimal Decimal
Right Left
(base) no prefix 100 = one

deci d 10­1 = 1 tenth

10­2 = 1
centi C
hundredth
10­3 = 1
milli m thousandth
10­6 = 1
micro u millionth

20
 Single Unit Conversions
• 27.6 kg = ____________
2760000 cg ?
We are moving down on our conversion table, therefore we
move the decimal to the right. In this case, 5 jumps to the right!
The new number is larger because the new unit is smaller.

• 6542mm = ____________
0.06542 hm ?
We are moving up our conversion table, therefore we move the
decimal to the left. In this case, 5 jumps to the left. The new
number is smaller because the new unit is larger.

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3 jumps

3 jumps

22
 More Single Unit Conversions
• 0.0723 g = ___________ Mg
• 32.4 daL = ___________ dL
• 2.43 x 10‐2 hm = ________ m
• 6.65 x 103 dg = _________ ug

23
 Answers
• 0.0723 g = 7.23 x 10‐8 Mg
• 32.4 daL = 3240 dL
• 2.43 x 10‐2 hm = 2.43 m
• 6.65 x 103 dg = 6.65 x 108 ug

24
 Double Unit Conversions
• e.g. 2.46 cL/s = __________ mL/ms

• Step 1: To convert the numerator (top) unit, do exactly the

same as you would for a single unit conversion.
e.g. 2.46 cL/s = 24.6 mL/s
• Step 2: To convert the denominator (bottom) unit, do exactly
as you would a single conversion, but reverse the direction of
your decimal jump.
e.g. 24.6 mL/s = (three jumps LEFT) 0.0246 mL/ms

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Double Unit Con. Examples

• 16.5 cg/hL = ___________ug/L

• 2.34 x 10­1 m/s = __________km/ds

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 Answers

• 16.5 cg/hL = 1650 ug/L

• 2.34 x 10­1 m/s = 2.34 x 10­5 km/ds

27
HOMEWORK:

Metric
Worksheet

28
3. Accuracy & Precision

29
 Accuracy

• Depicts how close a measured value is

to the actual value

• EX. If you weigh 150 lbs. But the

scale reads 130 lbs., the scale is not
accurate.

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 Precision of a single
measurement

• A term used to depict how many decimal places

(place values) you can acquire from a measuring
device

• If scale 1 reads 142.6lbs and scale 2 reads

143lbs, then scale 1 is more precise – it has
smaller measuring increments (tenths compared
to ones)

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 Precision of a single
measurement
• Which measurement is more precise?
a. 2.3 cm b. 2.32 cm

Which glassware is more precise?

OR

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 Precision of multiple
measurements
• Precision can be used to describe
reproducible measurements as well

• Ex. You weigh the same piece of zinc on

a scale 3 times, and you get 7.60, 7.61, and
7.59 grams. the scale is precise (it gives
reproducible results).

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THE TWO MEANINGS OF PRECISION
1. To how many decimal places a
measuring device can measure to (the
more decimal places, the more precise)

2. The reproducibility of a measurement. If

you measure something three times, will the
measuring device give the same result every
time.

34
A scale that is giving incorrect masses (is
not accurate), but does give reproducible
results (is precise) needs to be
CALIBRATED (kind of like setting your
watch to the correct time).

A scale that is not precise could mean

two things:
­either it doesn't measure to many
decimal places OR
­it doesn't give reproducible results
­OR both

35
 Suppose you are shooting for the bullseye!

Results not Results Results Accurate

Accurate or Inaccurate Accurate And
reproducible But But not reproducible
reproducible reproducible

http://www.youtube.com/watch?v=_LL0uiOgh1E

36
4. Measuring & Counting
Significant Figures
http://www.dlt.ncssm.edu/core/c1.htm up to 3:50 on video

37
 Significant Figures
• When counting objects we can find an exact
number that's undisputable (100% accurate)
• eg numbers of students in class
• number of books on a shelf

• When measuring quantities you are always

limited by the precision of the measuring device
(what place value it can measure to). The
precision can always be better (to infinity), thus
you can never get a completely accurate,
undisputable measurement
• eg length of classroom, mass of person
• When making a measurement, there is a limit
to the amount of meaningful digits

38
For example, measure the length of the arrow
using the ruler:

You couldn't say the measurement was

28.7492743. There's a limit to the amount of
digits in your measurement that are meaningful.

39
 Significant Figures
• A significant figure (or significant digit) is a
measured or meaningful digit.
• Significant figures (or “Sig figs”) are the digits
known to be exact plus one more that may have
some uncertainty but is an educated guess
• The following examples show how many digits
can be determined in different cases.

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How long is each line?

In Figure 1, the line is 1.6cm, therefore 2 sig figs

In Figure 2, the line is 1.63cm (or 1.62 or 1.64), so 3 s.f.
The number of sig figs consists of certain digits + one
uncertain (educated guess) digit.
The precision of the measuring device determines the
number of sig figs. Fig. 2 has a higher precision

41
• On the centimetre ruler above we know the length at
the arrow is between 2 cm and 3 cm
• If the smaller divisions are 0.1 cm we know the length
is between 2.8 cm and 2.9 cm
• We can’t read another digit, but we can estimate how
many tenths of a division past 2.8 to the arrow
• We can estimate 2 tenths of a division which gives a
measurement of 2.82 cm

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• We state the measurement as 2.82 cm.
• We are certain about the first 2 digits and
have some certainty about the third
• eg ‐ we know the third digit is not 0 or 9, (but
it might be 1 or 3)
• This measurement has 3 sig figs
• We cannot give the measurement of 2.8275
because we cannot be that precise with this
ruler

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• More than 12, less than 13
• More than 12.3, less than 12.4
• Estimated length = 12.33 cm
• (4 significant figures)
• Note it could also be estimated as 12.32 cm or
12.34 cm ‐ be as accurate as you can
• Any of these last 3 would be an acceptable
measurement

44
Length is between 4 and 5 cm. Arrow is right at the 0.5cm mark

Our guess digit will be a 0 as the measurement is right on the line.

Length can be reported as 4.50 cm. The 0 is the one guess digit.

45
How many degrees Celsius?
• Decide what each marked division
represents
• Estimate between marked divisions
• Estimated temperature
• Between 21 and 22 degrees C
• Best estimate 21.8 degrees C
• 3 sig figs

46
 Graduated Cylinder

• Estimated volume is between 20 and 30

(read at bottom of meniscus curve) mL
• Large division is 5 mL, each small one is 1
mL
• Estimate between 27 and 28 mL
• Volume = 27.5 mL
• 3 sig figs

47
 Graduated Cylinder

• Large division is 0.5 mL, each small one is

0.1 mL
• Volume is 5 mL, but we know it more
precisely. We can read 5.0 using marked
divisions and estimate one more decimal
place
• Volume = 5.00 mL (3 sig figs)

48
If the increments on a measuring device
don't change by 1 (or 0.1 or 0.01 etc), use
educated guessing when making a
measurement. If the last guess digit is a
random guess, it should not be a sig fig.

Measure:

332mL

49
 Measure:

640mL

3.48mL

mL mL

50
What if the measurement has been
made by someone else, and then you
are to work with it. How do you count
the number of sig figs in the
measurement?

51
Rules for Counting Significant Figures
• A) all non‐zero digits are significant
• B) zeros are significant if:
• They are at at the end of a number to the right of the decimal
point. ex. 2.50 (3 sig figs as the 0 counts)
• They are sandwiched by non‐zero numbers.
ex. 2002 (4 sig figs) or 10.003 (5 sf)
• C) zeros that help define the number are not
significant (zeros at the end of a number to the left of
the decimal). ex. 100 has only 1 sig fig but 100.0 has 4
sig figs
• OR zeros leading off a number ex. 0.00034 has 2sf

52
 Examples
• 34.500
• 5 significant figures
• 0.0087
• 2 significant figures
• 350.007
• 6 significant figures
• 1500
• 2 significant figures
• 120.0
• 4 significant figures

53
 Scientific Notation & Sig Figs
• What if you measure 100 to three sig figs?
How would you show this?
• Use scientific notation…1.00 x 102 is 100
expressed with three sig figs
• Sig figs for scientific notation:
• The number of digits in the coefficient IS the
number of sig figs!

54
 Same number with different
amounts of Sig. Figs.
• 1200
• 1200.0
• 1.2 x 103
• 1.20 x 103
• 1.200 x 103

55
 Perfect Numbers
• REMEMBER: Counting numbers or defined
values are considered to be exact or perfect
numbers and are exempt from rules of sig.
figs.
• 7 cheers for chemistry (counting #)
• 100cm = 1m (conversions: defined values)

56
 Practice ‐ How many Sig Figs
• 13.0 mm
• 48.07 g
• 0.050 cm
• 1001 L
• 5 students
• 15000 g
• 1 L = 1000 mL
• 3.00 x 10 ‐3 kg

57
 Practice ‐ How many Sig Figs
• 13.0 mm 3 sig figs
• 48.07 g 4 sig figs
• 0.050cm 2 sig figs
• 1001 L 4 sig figs
• 5 students perfect number
• 15000 g 2 sig figs
• 1 L = 1000 mL perfect number
• 3.00 x 10 ‐3 3 sig figs

58
 Homework:
Counting Sig Figs
Worksheet AND
Measuring with Sig
Figs Worksheet

http://www.youtube.com/watch?v=ZuVPkBb­z2I

59
5. Calculations with Significant Figures

60
 Significant Figures and
Multiplication/Division
• The answer can have no more SIG FIGS than
the LEAST significant of all terms
• E.g. 10.4 x 4.1 x 0.03963
• (3 sig figs) (2 sig figs) (4 sig figs)
• Answer can only be given to 2 sig figs
• Calculator gives = 1.6898232
• Answer = 1.7 ( 2 sig figs)

61
• E.g. (3.428 x 105) x ( 5.98 x 10‐2)
• 8.7615 x 104
• Least number of sig figs is 3 so answer
must be rounded to 3 sig figs
• Calculator gives 0.233971808483
• Answer is 0.234 or, in sci. notation, the
answer is 2.34 x 10‐1

62
Sig Figs and Addition/Subtraction
• The answer goes to the same PLACE VALUE
of the term that is least precise

• E.g. 3.4893 0.35 is least precise of the three

• + 0.35 measurements (only to hundredths)
• + 0.17938
4.01868 = 4.02
• Answer can only have 2 decimal places

63
• E.g. 583.4 ‐ 1.256 =

• 583.4 Line up
• 1.256 decimal
places
582.144

= 582.1 (One decimal place)

64
• When adding or subtracting in scientific
notation, change the number to standard form
first, and then calculate answer using sig figs.
• 5.43 x 10‐4 + 6.235 x 10‐1
• 0.000543
• + 0.6235
• 0.624043 Answer: 6.240 x 10­1

http://www.youtube.com/watch?v=kB2szfcwu1A

65
 Calculating Tips
• Always carry a couple extra unsignificant
digits when doing calculations, so you don`t
have rounding errors
• Round off only the FINAL ANSWER to the
correct number of significant digits.
ex. find 34.3 x 0.3455 answer when answer when
rounding after carrying extra
Then, square the answer
each step: digits:
and divide by 3.00
47.3 46.8
8

66
 Mixed Calculations
• Evaluate to the correct amount of sigfigs:
• 50.35 x 0.106 – 25.37 x 0.176 =
• Order of operations à BEDMAS
• Multiply 1st…but show sigfigs in bold:
• 50.35 x 0.106 = 5.3371
• 25.37 x 0.176 = 4.46512

67
• Now…perform the subtraction operation.
• Since both of the multiplications are ‘legal’
to the 2nd decimal place, the result of the
subtraction operation must be rounded to
the 2nd decimal place as follows:
• 5.3371 – 4.46512 = 0.87192
• The answer is then rounded to 0.87
• Again, always carry more digits through to
the end, and round to the proper number of
sigfigs at the end only.
10

68
 HOMEWORK:

Sig Fig Calculation

Worksheet

11
http://www.wwnorton.com/college/chemistry/gilbert2/chemtours.asp

69
http://www.youtube.com/watch?v=J8bRciuMLqQ

6. Density

70
What's more dense?
A big box of OR A little piece of
wooden chairs iron

http://www.youtube.com/watch?v=H2Rlt3YM1To

71
 Density
• The mass contained in a given volume of
space
m
Formula: d=m/v
d v

Which is more dense??

2

72
 Units

• Mass: mg, g, kg

• Volume: m3 , cm3, L, mL (1cm3 = 1mL)

• Density: g/cm3, g/mL, kg/L etc.

73
 Example
1.0mL of water at 4°C has a
mass of 1.0g.
• Therefore, the density of
water is…

• 1.0 g/mL
http://www.wiredchemist.com/anim­density

74
Density determines whether an object
sinks or floats!
• Is the density of ice greater than
or less than water?
• LESS THAN (it floats!)
• Is the density of ice greater than
or less than isobutanol?
• GREATER THAN (it sinks!)
5

75
• Will the following sink or float in
water?
• Aluminum 2.70 g/cm3 SINK
• Cork 0.72 g/cm3 FLOAT
• Cooking Oil 0.920 g/mL FLOAT

• How do you know?

• Compare with the density of water.
http://www.kentchemistry.com/links/Measurements/Density.htm
both videos
6

76
 • Density of Some Common Substances
Density (g/cm3) or (g/mL)
• Air 0.0012
• Feathers 0.0025
• Wood(Oak) 0.6 ­ 0.9
• Ice 0.92
• Water 1.00
• Bricks 1.84
• Aluminum 2.70
• Steel 7.80
• Silver 10.50
• Gold 19.30

Density is an intensive property of a substance, 7

meaning it's the same for a substance no

matter the size or mass of the substance (at a
specific temperature).

77
Density and Temperature
• Is density temperature dependent?

Yes!

78
• As a substance’s temperature increases,
the molecules move faster and space out,
therefore density decreases (less mass in a
given volume).
• As a substance’s temperature decreases,
the molecules move slower and pack in,
therefore density increases (more mass in a
given volume).
• For most substances, density is ranked as
follows
solid phase > liquid phase > gas phase
9

79
 So why does ice float?
• When water solidifies to make ice,
H2O molecules are kept a certain
distance from one another due to
hydrogen bonds between molecules.
These H­bonds hold the molecules
further apart compared to liquid phase

• Liquid H2O molecules still have

hydrogen bonding, but the increased kinetic
energy of the molecules allows for more
movement and thus molecules are able to be
closer together (H­bonds form and re­form
with great frequency due to the high KE).
http://preparatorychemistry.com/water_flash.htm
*very end of animation

80
 Examples of Density Problems
• An iron bar has a mass of 19600g and a volume
of 2.50L. Find the density of the bar in g/L.

d v

density = 19 600g = 7840 g/L

2.50L
14

81
• Mercury has a density of 13600g/L. What
volume (in mL) is occupied by 425g of Mercury?

v = m = 425g = 0.0313L = 31.3mL

d 13 600L
m

d v

82
• A 25.0g piece of steel (denisty 7.80g/cm3) is
dropped into a graduated cylinder that has
16.00mL of water. What will the water level
be after the steel has been added?
REMEMBER: 1cm3 = 1mL m

v = m = 25.0g = 3.21mL d v
d 7.80g/mL

16.00 + 3.21mL = 19.21mL

83
 Homework:

Density Worksheet

http://www.youtube.com/watch?v=fE2KQzLUVA4&feature=results_main&playnext=1
&list=PLD2AE3B1A10707D42
15

84
7. Determining Density
from Graphs

85
• Density is defined as the mass of a certain
volume of a substance.
• Therefore, we can determine the density of a
substance from experimental measurements of
its mass and its volume

86
 Graphing Terms
• Independent Variable ‐ the variable that is
manipulated as part of the experiment
• placed on the x‐axis of the graph
• In the following example, we manipulate the
VOLUME of an unknown liquid.
• Dependent Variable ‐ the variable that changes as a
result of the independent variable changing
• placed on the y‐axis (in this case, MASS)
• as we change the volume (x value) of the unknown
liquid, the mass (y value) will change

87
• Slope ‐ mathematical relationship
between the x and y variables on a
linear graph
• “steepness” of the graph

(change in y value)
(change in x value)

88
• If volume is the independent variable (x
value) and mass is the dependent variable (y
value), then the slope of the graph will be the
density of the substance.

89
 Example
• The volume of several samples of
an unknown liquid are measured by
graduated cylinder.
• The mass of each sample is then
found using a mass balance

90
 Volume (x) ­ mL Mass (y) ­ g
0.0 0.0
5.0 4.4
10.0 8.8
15.0 13.1
20.0 17.5
25.0 21.9

• Graph the results with volume on the x axis and

mass on the y axis.

91
 Rise = 17.5 g
Run = 20.0 mL

92
 Density (from slope﴿
• From the graph, the density of the unknown
liquid would be:

• d = 0.875 g/mL

• liquid air (air cooled down to liquid state) has

a density of 0.875g/mL at ‐200 degrees C

93
Homework:

Density Graphing
Worksheet

94