Transcript Chapter 2: Measurements

Chapter 2
Measurements and
Calculations
1
Objectives:
 Describe the difference between a qualitative and a
quantitative measurement.
 Describe the difference between accuracy and precision.
 Write a number in scientific notation.
 State the appropriate units for measuring length, volume,
mass, density, temperature and time in the metric system.
 Determine the number of significant figures in a
measurement or calculation.
 Calculate the percent error in a measurement.
 Calculate density given the mass and volume, the mass given
the density and volume, and the volume given the density
and mass.
2
Chapter 2
Section 1
Scientific Method
3

Scientific Method is a logical approach
to solving problems by observing and
collecting data, formulating hypotheses,
testing hypotheses and formulating
theories that are supported by data.
Observations
Hypothesis
Theory
4
Experimentation
Observations
• Collecting data
• Measuring
• Communicating with other scientists
5
Measurements
Measurements are divided into two sets:
Qualitative – a descriptive measurement.
Color, hardness, shininess, physical state.
(non-numerical)
Quantitative – a numerical measurement.
Mass in grams, volume in milliliters, length
in meters.
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Hypothesis
A tentative explanation that is consistent
with the observations (educated guess).
An experiment is then designed to test the
hypothesis.
Predict the outcome from the experiments.
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Theory
Attempts to explain why something
happens.
Has experimental evidence to support
the theory.
Observations, data and facts.
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Classwork
What is the scientific theory?
What is the difference between qualitative
and quantitative measurements?
Which of the following are quantitative?
a. The liquid floats on water?
b. The metal is malleable?
c. A liquid has a temperature of 55.6 oC?
How do hypothesis and theories differ?
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Section 2
Units of
Measurement
10
Measurements represent quantities.
A quantity is something that has
magnitude, size or amount.
All measurements are a number plus a
unit (grams, teaspoon, liters).
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Number vs. Quantity
 Quantity = number + unit
12
UNITS MATTER!!
UNITS OF MEASUREMENT
Use SI units — based on the metric
system
13
Length
Meter, m
Mass
gram, g
Volume
Liter, L
Amount
mole
Temperature
kelvin, K
SI Prefix Conversions
14
Tera-
1 T(base) = 1 000 000 000 000(base) = 1012 (base)
Giga-
1 G(base) = 1 000 000 000 (base) = 109 (base)
Mega-
1 M(base) = 1 000 000 (base) = 106 (base)
Kilo-
1 k(base) = 1 000 (base) = 103 (base)
Hecto-
1 h(base) = 100 (base) = 102 (base)
Deka-
1 da(base) = 101 (base)
Base
1 (base) = 1 (base) meter, gram, liter
Deci-
1 d(base) = 10-1 (base)
Centi-
1 c(base) = 10-2 (base)
Milli-
1 m (base) = 10-3(base)
Micro-
1 µ(base) = 1 000 000 (base) = 10-6(base)
Nano-
1 n(base) = 1 000 000 000 (base) = 10-9(base)
Pico-
1 p(base) = 1 000 000 000 000(base) = 10-12(base)
SI Prefix Conversions
15
move right
move left
Prefix
Symbol
Factor
tera-
T
1012
gigamegakilohectodekaBASE UNIT
decicentimillimicronanopico-
G
M
k
h
da
--d
c
m

n
p
109
106
103
102
101
100
10-1
10-2
10-3
10-6
10-9
10-12
Learning Check
16
1. 1000 m = 1 ___
a) mm b) km c) dm
2.
0.001 g = 1 ___
a) mg
b) kg c) dg
3.
0.1 L = 1
a) mL
b) cL c) dL
4.
0.01 m = 1 ___
___
a) mm b) cm c) dm
SI Prefix Conversions
1)
17
20 cm =
______________ m
2) 0.032 L =
______________ mL
3) 45 m =
______________ m
Derived SI Units
Many SI units are combinations of the
quantities shown earlier.
Combinations of SI units form derived
units.
Derived units are produced by multiplying
or dividing standard units.
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19
Volume
Volume (m3) is the amount of space
occupied by an object.
length x width x height
Also expressed as cubic centimeter (cm3).
When measuring volumes in the laboratory
a chemist typically uses milliliters (mL).
1 mL = 1 cm3
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Density
Density – the ratio of mass to volume, or
mass divided by volume.
mass
Density =
volume
m
D=
v
Density is often expressed in
grams/milliliter or g/mL
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Density
Density is a characteristic physical property
of a substance.
It does not depend on the size of the
sample.
As the sample’s mass increases, its
volume increases proportionally.
The ratio of mass to volume is constant.
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Density
Calculating density is pretty
straightforward.
You measure the mass of an object by
using a balance and then determine the
volume.
For a liquid the volume is easily measured
using for example a graduated cylinder.
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Density
For a solid the volume can be a little more
difficult.
If the object is a regular solid, like a cube,
you can measure its three dimensions and
calculate the volume.
Volume = length x width x height
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Density
If the object is an irregular solid, like a rock,
determining the volume is more difficult.
Archimedes’ Principle – states that the
volume of a solid is equal to the volume of
water it displaces.
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Density
Put some water in a graduated cylinder
and read the volume. Next, put the
object in the graduated cylinder and read
the volume again.
The difference in volume of the graduated
cylinder is the volume of the object.
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Volume Displacement
A solid displaces a matching volume of
water when the solid is placed in water.
33 mL
25 mL
27
Learning Check
What is the density (g/cm3) of 48 g of a metal if
the metal raises the level of water in a graduated
cylinder from 25 mL to 33 mL?
1) 0.2 g/cm3
2) 6 g/cm3 3) 252 g/cm3
33 mL
25 mL
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PROBLEM: Mercury (Hg) has a density
of 13.6 g/cm3. What is the mass of 95 mL
of Hg in grams?
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PROBLEM: Mercury (Hg) has a density of
13.6 g/cm3. What is the mass of 95 mL of Hg?
First, note that
1 cm3 = 1 mL
Strategy
Use density to calc. mass (g) from
volume.
mass
(g)
Density =
volume(ml)
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PROBLEM: Mercury (Hg) has a density of
13.6 g/cm3. What is the mass of 95 mL of Hg?
)
(g
mass
Density =
volume (ml)
13.6 g/mL =
mass ( g )
95 (ml)
Mass = 1,292 grams
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Learning Check
Osmium is a very dense metal. What is its
density in g/cm3 if 50.00 g of the metal
occupies a volume of 2.22cm3?
1) 2.25 g/cm3
2) 22.5 g/cm3
3) 111 g/cm3
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Solution
Placing the mass and volume of the osmium
metal into the density setup, we obtain
D = mass = 50.00 g =
volume
2.22 cm3
= 22.522522 g/cm3 = 22.5 g/cm3
33
Learning Check
The density of octane, a component of
gasoline, is 0.702 g/mL. What is the
mass, in kg, of 875 mL of octane?
1) 0.614 kg
2) 614 kg
3) 1.25 kg
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Learning Check
The density of octane, a component of
gasoline, is 0.702 g/mL. What is the
mass, in kg, of 875 mL of octane?
1) 0.614 kg
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Densities of Common Materials
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Material
Density (g/mL)
Water
Ice
Table sugar
Copper
Gasoline
Mercury
1.00
0.92
1.59
8.92
0.67
13.6
Classwork
Textbook: page 42, questions 3 and 5
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Conversion Factors
Conversion factor – Used to convert from
one unit to the other. A ratio of units.
Example: the conversion between
quarters and dollars:
4 quarters
1 dollar
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or
1 dollar
4 quarters
Conversion Factors
Example:
Determine the number of quarters in 12
dollars?
Number of quarters = 12 dollars x conversion factor
4 quarters
? Quarters = 12 dollars x
= 48 quarters
1 dollar
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How many minutes are in 2.5 hours?
Conversion factor
2.5 hr x
cancel
40
60 min
1 hr
= 150 min
Sample Problem
• You have $7.25 in your pocket in
quarters. How many quarters do you
have?
7.25 dollars
X
41
4 quarters
1 dollar
= 29 quarters
Learning Check
A rattlesnake is 2.44 m long. How
long is the snake in cm?
a) 2440 cm
b) 244 cm
c) 24.4 cm
42
Solution
A rattlesnake is 2.44 m long. How
long is the snake in cm?
b) 244 cm
2.44 m x 100 cm
1m
43
= 244 cm
Homework
Textbook: page 59 and 60
Questions: 1, 2, 7,14, 28 and 30
Due:
44
Section 3
Using Scientific
Measurements
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Accuracy and
Precision
Accuracy – refers to how well the
measurements agree with the accepted
or true value.
Precision – refers to how well a set of
measurements agree with each other.
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ACCURATE = CORRECT
PRECISE = CONSISTENT
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Accuracy and Precision
Three targets
with arrows.
How do they
compare?
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Both
accurate
and precise
Precise
but not
accurate
Neither
accurate
nor precise
Accuracy and Precision
Student 1
Student 2
Student 3
Student 4
Trial 1
27.77 cm
27.30 cm
27.55 cm
27.30 cm
Trial 2
27.30 cm
27.60 cm
27.55 cm
27.29 cm
Trial 3
27.56 cm
27.97 cm
27.53 cm
27.31 cm
Average
27.54 cm
27.62 cm
27.54 cm
27.30 cm
The accepted length of the object is 27.55 cm.
Based on the average values of the
measurements which students had the best
accuracy?
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Accuracy and Precision
Student 1
Student 2
Student 3
Student 4
Trial 1
27.77 cm
27.30 cm
27.55 cm
27.30 cm
Trial 2
27.30 cm
27.60 cm
27.55 cm
27.29 cm
Trial 3
27.56 cm
27.97 cm
27.53 cm
27.31 cm
Average
27.54 cm
27.62 cm
27.54 cm
27.30 cm
The accepted length of the object is 27.55 cm.
Based on the individual trials of the
measurements which students had the best
precision?
50
Percent Error
The accuracy of an individual value can be
compared with the correct or accepted value
by calculating the percent error.
Percent error is calculated by subtracting the
accepted value from the experimental value,
dividing the difference by the accepted value,
and then multiplying by 100.
Percent error =
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Value (experimental) – Value (accepted)
Value (accepted)
x 100
Percent Error
 Indicates accuracy of a measurement
% error =
experimental  accepted
accepted
 100
your value
given value
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Percent Error
 A student determines the density of a
substance to be 1.40 g/mL. Find the %
error if the accepted value of the density
is 1.36 g/mL.
% error =
1.40 g/mL  1.36 g/mL
1.36 g/mL
% error = 2.9%
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 100
Classwork
Textbook: page 45, question 2
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Percent Error Worksheet
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Significant Figures
Significant figures – a measurement consists
of all the digits known with certainty plus
one final digit, which is uncertain or
estimated.
Significant figures are the number of digits
that you report in your final answer of a
mathematical problem.
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Significant Figures
Indicates precision of a measurement.
Recording Sig Figs
Sig figs in a measurement include the
known digits plus a final estimated digit
2.31 cm
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Determining Significant Figures
RULE 1. All non-zero digits (1-9) in a
measured number are significant.
RULE 2. Leading zeros in decimal numbers
are NOT significant.
RULE 3. Zeros between non-zero numbers
are significant.
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Counting Significant Figures
RULE 4. Trailing zeros in numbers without
decimals are NOT significant.
RULE 5. Trailing zeros in numbers with
decimals are significant
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Counting Significant Figures
RULE 1. All non-zero digits (1-9) in a measured
number are significant. Only a zero could
indicate that rounding occurred.
Number of Significant Figures
38.15 cm
5.6 ft
65.6 lb
122.55 m
60
4
2
___
___
Leading Zeros
RULE 2. Leading zeros in decimal numbers
are NOT significant.
Number of Significant Figures
61
0.008 mm
1
0.0156 oz
3
0.0042 lb
____
0.000262 mL
____
Sandwiched Zeros
RULE 3. Zeros between nonzero numbers are
significant.
Number of Significant Figures
62
50.8 mm
3
2001 min
4
0.702 lb
____
0.00405 m
____
Trailing Zeros
RULE 4. Trailing zeros in numbers without decimals
are NOT significant. They are only serving as place
holders.
Number of Significant Figures
63
25,000 in.
2
200 yr
1
48,600 gal
____
25,005,000 g
____
Trailing Zeros
RULE 5. Trailing zeros in numbers with
decimals are significant.
Number of Significant Figures
3030.0
0.000230340
50.0
25,005,000.0
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5
6
3
9
Significant Figures
Counting Sig Fig Examples
1. 23.50
2. 402
3. 5,280
4. 0.080
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Learning Check
A. Which answers contain 3 significant
figures?
1) 0.4760
2) 0.00476
3) 4760
B. All the zeros are significant in
1) 0.00307
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2) 25.300
3) 2.050 x 103
Learning Check
In which set(s) do both numbers
contain the same number of
significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150,000
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Learning Check
State the number of significant figures in each of
the following:
A. 0.030 m
1
2
3
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B. 4.050 L
2
3
4
C. 0.0008 g
1
2
4
D. 3.00 m
1
2
3
E. 2,080,000 bees
3
5
7
Rounding Figures
Rule 1: If the first number to be dropped
is 5 or greater, drop it and increase the last
retained number by 1.
Rule 2: If the first number to be dropped is
Less than 5, drop it and leave the last
retained number unchanged.
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Significant Numbers in Calculations
 Please learn how to calculate with significant
figures on your own.
 This includes: adding, subtracting,
multiplying and dividing.
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Significant Numbers in Calculations
The number of significant figures you can report
in your answer is based on the type of calculation
performed.
Significant figures are needed for final answers
from
1) adding or subtracting
2) multiplying or dividing
71
Adding and Subtracting
The answer must have the same number of
decimal places as the measurement with the
fewest decimal places.
25.2
one decimal place
+ 1.34 two decimal places
26.54
answer 26.5 one decimal place
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Learning Check
In each calculation, round the answer to the
correct number of significant figures.
A. 235.05 + 19.6 + 2.1 =
1) 256.75
2) 256.8
B.
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58.925 - 18.2 =
1) 40.725
2) 40.73
3) 257
3) 40.7
Multiplying and Dividing
For multiplication or division, the answer
can have no more significant figures than
are in the measurement with the fewest
number of significant figures.
74
Multiplying and Dividing
You calculate the density of an object that
has a mass of 3.05 g and a volume of
8.47 mL. The following division on a
calculator will give a value of
0.360094451.
Density =
Mass
Volume=
0.360094452 g/mL
75
3.05 g
8.47 mL =
The answer must be rounded to the
correct number of significant figures.
The values of the mass and volume used
only have 3 significant figures.
Therefore, the answer can only have 3
significant figures.
0.360094452 g/mL
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0.360 g/mL
Learning Check
A. 2.19 X 4.2 =
1) 9
B.
C.
77
4.311 ÷ 0.07 =
1) 61.58
2) 9.2
2) 62
2.54 X 0.0028 =
0.0105 X 0.060
1) 11.3
2) 11
3) 9.198
3) 60
3) 0.0413
Significant Figures in Conversion Factors
Conversion factors are considered exact
numbers. 100 cm/m
Therefore they are not considered when
determining the correct number of
significant figures.
4.608 m x 100 cm = 460.8 cm
m
4 sig. fig.
78
1 sig. fig.
4 sig. fig.
Classwork
Textbook: page 50, questions 1-4
79
Homework
Significant Figures
Worksheet
Due:
80
Scientific Notation
Please learn scientific notation on your
own.
81
Skip
Proceed to slide 102
82
Scientific Notation
Scientific notation is a way of
expressing really big numbers or really
small numbers.
83
Scientific Notation
Scientific notation – numbers are written in
the form M x 10n, where M is a number
greater than or equal to 1 but less than ten
and n is a whole number.
Example: 65,000 = 6.5 x 104
When numbers are written in scientific
notation, only the significant figures are
shown.
84
Small numbers are handled in a similar
way; the decimal point is moved to the
right:
0.00012 = 1.2 x 10-4
The decimal place is moved 4 places to the right.
There should be only one digit to the left
of the decimal place.
85
Examples
• Given: 289,800,000
• Use: 2.898 (moved 8 places)
• Answer: 2.898 x 108
• Given: 0.000567
• Use: 5.67 (moved 4 places)
• Answer: 5.67 x 10-4
86
To change scientific notation
to standard form…
• Simply move the decimal point to
the right for positive exponent.
• Move the decimal point to the left
for negative exponent.
(Use zeros to fill in places.)
87
Example
• Given: 5.093 x 106
• Answer: 5,093,000 (moved 6 places
to the right)
• Given: 1.976 x 10-4
• Answer: 0.0001976 (moved 4 places
to the left)
88
Learning Check
• Express these numbers in
Scientific Notation:
1)
2)
3)
4)
5)
89
405789
0.003872
3000000000
2
0.478260
Calculations with Scientific Notation
Addition and Subtraction
These operations can only be performed
if the values have the same exponent (n).
If they do not, adjustments must be made
to the values so that the exponents are
equal.
90
Once the exponents are equal, the M
coefficients can be added or subtracted.
The exponents of the answer remain the
same.
Example:
4.2 x 104 + 7.9 x 103 kg
One of these figures needs to be changed
to make the exponents equal.
91
4.2 x 104
+ 7.9 x 103
4.2 x 104
+ 0.79 x 104
4.99 x 104
Round to correct number
of sig. fig. - 2
5.0 x 104
92
Alternate method:
4.2 x 104
+ 7.9 x 103
42. x 103
+ 7.9 x 103
49.9 x 103
Round to correct number
of sig. fig. - 2
5.0 x 104
93
Calculations with Scientific Notation
Multiplication and Division
To multiply numbers, you multiply the
coefficients (M) and add the exponents
(n).
To divide numbers, you divide the
coefficients (M) and subtract the
exponents (n).
94
Multiplication
Example:
9.25 x 10-2 x 1.37 x 10-5 kg
(9.25 x 1.37) x (10-2 + -5) = 12.7 x 10-7
1.27 x 10-6
95
Multiplication
Example:
5.23 x 106 x 7.1 x 10-2 kg
(5.23 x 7.1) x (106-2) = 37.133 x 104
Round to correct number
of sig. fig. - 2
37 x 104
3.7 x 105
96
Division
Example:
8.27 x 105 ÷ 3.25 x 103 kg
(8.27 ÷ 3.25) x (105-3) = 2.54 x 102
97
Division
Example:
5.44 x 107 ÷ 8.1 x 104 kg
(5.44 ÷ 8.1) x (107-4) = 0.6716 x 103
Round to correct number
of sig. fig. - 2
0.67 x 103
6.7 x 102
98
Scientific Notation
 Calculating with Sci. Notation
(5.44 × 107 g) ÷ (8.1 × 104 mol) =
Type on your calculator:
5.44
EXP
EE
7
÷
8.1
EXP
EE
4
= 671.6049383 = 6.7 × 102 g/mol
99
EXE
ENTER
Practice Problems
a) (4 x 102 cm) x (1 x 108cm)
b) (2.1 x 10-4kg) x (3.3 x 102 kg)
c) (6.25 x 102) ÷ (5.5 x 108)
d) (8.15 x 104) ÷ (4.39 x 101)
e) (6.02 x 1023) ÷ (1.201 x 101)
100
Classwork
Textbook: page 54, questions 1-4
101
Homework
Textbook: page 59 and 60
Questions: 16, 21, 35, 36, 38,
39, 43 and 44
102