Transcript CHEM 10: CHAPTER TWO

CHEM 10: CHAPTER TWO
MEASUREMENT & PROBLEM
SOLVING
1
Scientific Notation
Very large and very small numbers are often
encountered in science.
Large: 602210000000000000000000
And small: 0.00000000000000000000625
Very large and very small numbers like these
are awkward and difficult to work with.
A method for representing these numbers in a
simpler form is called scientific notation.
Large: 6.022 x 1023
And small: 6.25 x 10-21
2
Scientific Notation
To write a number as a power of 10
- Move the decimal point in the original
number so that it is located after the first
nonzero digit.
- Follow the new number by a
multiplication sign and 10 with an
exponent (power).
- The exponent is equal to the number of
places that the decimal point was shifted.
3
Write 6419 in scientific notation.
decimal after
first nonzero
digit
power of 10
1
2
3
6.419
641.9x10
64.19x10
6419.
6419
x 10
4
Write 0.000654 in scientific
notation.
decimal after
first nonzero
digit
6.54 x
0.000654
0.00654
0.0654
0.654
power of 10
-4
-2
-1
-3
10
5
Scientific Notation
Adding and subtracting in exponential or
scientific notation:
- must be in same power of ten
Try adding:
3.47x102 + 5.93267x105
0.00347x105 + 5.93267x105 =
5.93614x105
6
Scientific Notation
Multiplying and dividing:
10a * 10b = 10a+b
10a/10b = 10a-b
Your calculator will do all this for you, if you enter the
numbers correctly!
Group practice:
A. 3.47 x 102 * 1.20 x 10-3 =
B. 0.0012 + 1.3 x 10-2 =
C. 3.47 x 102 / 1.20 x 10-3 =
A. 4.16 x 10-1
B. 1.4 x 10-2
C. 2.89 x 105
Measurements
Experiments are performed.
Numerical values or data are obtained from
these measurements.
The values are recorded to the most
significant digits provided by the
measuring device.
The units (labels) are recorded with the
values.
Form of a Measurement
numerical value
70.0 kilograms = 154 pounds
unit
Significant Figures
The number of digits that are known plus
one estimated digit are considered
significant in a measured quantity
known
5.16143estimated
Significant Figures
The number of digits that are known plus
one estimated digit are considered
significant in a measured quantity
known
6.06320 estimated
The
temperature
Temperature
is
oC is expressed
21.2
estimated
to be
oC. The last 2 is
to
3 significant
21.2
figures.
uncertain.
The
temperature
Temperature
is
oC is expressed
22.0
estimated
to be
to
3 osignificant
22.0
C. The last 0 is
figures.
uncertain.
The
temperature
Temperature
is
oC isto
22.11
expressed
estimated
be
oC. The last 1
to
4 significant
22.11
figures.
is uncertain.
Exact Numbers
Exact numbers have an infinite number of
significant figures.
Exact numbers occur in simple counting
operations
12345
• Defined numbers are exact.
12 inches
100
centimeters
= 1 foot
= 1 meter
Significant Figures
All nonzero numbers are significant.
461
Significant Figures
All nonzero numbers are significant.
461
Significant Figures
All nonzero numbers are significant.
461
Significant Figures
All nonzero numbers are significant.
3 Significant
Figures
461
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
401
Significant Figures
A zero is significant when it is between
nonzero digits.
5 Significant
Figures
93 . 006
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
9 . 03
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
55 . 000
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
2 . 1930
Significant Figures
A zero is not significant when it is before the
first nonzero digit.
1 Significant
Figure
0 . 006
Significant Figures
A zero is not significant when it is before the
first nonzero digit.
3 Significant
Figures
0 . 709
Significant Figures
A zero is not significant when it is at the end
of a number without a decimal point.
1 Significant
Figure
50000
Significant Figures
A zero is not significant when it is at the end
of a number without a decimal point.
4 Significant
Figures
68710
Rounding Off Numbers
Often when calculations are performed extra digits
are present in the results.
It is necessary to drop these extra digits so as to
express the answer to the correct number of
significant figures.
When digits are dropped the value of the last digit
retained is determined by a process known as
rounding off numbers.
Rounding Off Numbers
Rule 1. When the first digit after those you want to
retain is 4 or less, that digit and all others to its
right are dropped. The last digit retained is not
changed.
4 or less
80.873
Rounding Off Numbers
Rule 1. When the first digit after those you want to
retain is 4 or less, that digit and all others to its
right are dropped. The last digit retained is not
changed.
4 or less
1.875377
Rounding Off Numbers
Rule 2. When the first digit after those you
want to retain is 5 or greater, that digit and
all others to its right are dropped. The last
digit retained is increased by 1.
drop
5 or
these
greater
figures
increase by 1
6
5.459672
CALCULATIONS AND
SIGNIFICANT FIGURES
The results of a calculation cannot be more precise than
the least precise measurement.
Learn how to determine number of sig figs in answers
after performing calculations, including multiply/divide
and add/subtract.
In multiplication or division, the answer must contain the
same number of significant figures as in the
measurement that has the least number of significant
figures.
CALCULATIONS AND
SIGNIFICANT FIGURES
The results of an addition or a subtraction must
be expressed to the same precision as the
least precise measurement.
- The result must be rounded to the same
number of decimal places as the value with
the fewest decimal places.
2.3 has two significant
figures.
(190.6)(2.3) = 438.38
190.6 has four
significant figures.
Answer given
by calculator.
The answer should have two significant
figures because 2.3 is the number with
the fewest significant figures.
Round off this
digit to four.
Drop these three
digits.
438.38
The correct answer is 440 or 4.4 x 102
Add 125.17, 129 and 52.2
Least precise number.
Answer given
by calculator.
Round off to the
Correct answer.
nearest unit.
306.37
125.17
129.
52.2
306.37
1.039 - 1.020
Calculate
1.039
Answer given
by calculator.
1.039 - 1.020
= 0.018286814
Two significant
1.039
figures.
1.039 - 1.020 = 0.019
0.019
= 0.018286814
1.039
0.018286814
0.018
286814
The answer should have two significant
Drop these
Correct
answer.
figures because 0.019 is the number
6 digits.
with the fewest significant figures.
The Metric System
The metric or International System (SI,
Systeme International) is a decimal system of
units.
It is built around standard units.
It uses prefixes representing powers of 10 to
express quantities that are larger or smaller
than the standard units.
International System’s
Standard Units of Measurement
Quantity
Name of Unit
Abbreviation
Length
Mass
meter
kilogram
m
kg
Temperature
Kelvin
K
Time
second
Amount of substance mole
s
mol
Prefixes and Numerical Values for SI Units
Numerical Value
Power of 10
Equivalent
Prefix
Symbol
exa
peta
E
P
1,000,000,000,000,000,000 1018
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
—
—
1
100
Prefixes and Numerical Values for SI Units
Numerical Value
Power of 10
Equivalent
Prefix
Symbol
deci
d
0.1
10-1
centi
c
0.01
10-2
milli
m
0.001
10-3
micro

0.000001
10-6
nano
n
0.000000001
10-9
pico
p
0.000000000001
10-12
femto
f
0.00000000000001
10-15
atto
a
0.000000000000000001
10-18
GROUP RACE FOR ANSWERS IN METRIC
a.
b.
c.
d.
e.
one millionth of a scope = ____scope
0.01 mental = ____mental
1,000,000 phones = ___phones
2000 mockingbird =___bird
1/1000 tary =___tary
MEMORIZE THESE ENGLISH/METRIC
CONVERSIONS: (table 2.3 plus others)
1 lb = 453.59 grams
1 inch = 2.54 cm
(exactly)
1 mile = 1.609 km
1.0567 qt = 1 L
1000 mL = 1 L
1 mL = 1 cm3
1 cal = 4.184 Joule 1 atm = 760.00 torr
oF = 1.8oC + 32
K = oC + 273.15
Dimensional Analysis
Dimensional analysis converts one unit to
another by using conversion factors.
unit1 x conversion factor = unit2
Basic Steps
1. Read the problem carefully. Determine what is
to be solved for and write it down.
2. Tabulate the data given in the problem.
Label all factors and measurements with the proper
units.
Dimensional Analysis
Basic steps – continued:
3. Determine which principles are involved
and which unit relationships are needed
to solve the problem.
You may need to refer to tables for needed
data.
4. Set up the problem in a neat, organized
and logical fashion.
Make sure unwanted units cancel.
Use sample problems in the text as guides
for setting up the problem.
Dimensional Analysis
Basic steps continued:
5. Proceed with the necessary mathematical
operations.
Make certain that your answer contains
the proper number of significant figures.
6. Check the answer to make sure it is
reasonable.
Metric Units of Length
Unit
Abbreviation Metric Equivalent
Exponential
Equivalent
kilometer
meter
km
m
1,000 m
1m
103 m
100 m
decimeter
dm
0.1 m
10-1 m
centimeter
cm
0.01 m
10-2 m
millimeter
mm
0.001 m
10-3 m
micrometer
m
0.000001 m
10-6 m
nanometer
nm
0.000000001 m
10-9 m
How many millimeters are there in 2.5 meters?
Use the conversion factor with millimeters in the
numerator and meters in the denominator.
1000 mm
= 2500 mm
2.5 m x
1m
3
2.5 x 10 mm
Convert 3.7 x 103 cm to micrometers.
Centimeters can be converted to micrometers by
writing down conversion factors in succession.
cm  m  meters
1m
10 μm
7
x
3.7 x 10 cm x
= 3.7 x 10 μm
100 cm
1m
6
3
Convert 3.7 x 103 cm to micrometers.
Centimeters can be converted to micrometers by
two stepwise conversions.
cm  m  meters
1m
1
3.7 x 10 cm x
= 3.7 x 10 m
100 cm
3
10 μm
7
3.7 x 10 m x
= 3.7 x 10 μm
1m
6
1
Convert 16.0 inches to centimeters.
Use this
conversio
n factor
2.54 cm
1 in
2.54 cm
16.0 in x
= 40.6 cm
1 in
Metric Units of mass
Unit
Abbreviation Gram Equivalent
Exponential
Equivalent
kilogram
gram
kg
g
1,000 g
1g
103 g
100 g
decigram
dg
0.1 g
10-1 g
centigram
cg
0.01 g
10-2 g
milligram
mg
0.001 g
10-3 g
microgram
g
0.000001 g
10-6 g
Convert 45 decigrams to grams.
1 g = 10 dg
1g
45 dg x
= 4.5 g
10 dg
An atom of hydrogen weighs 1.674 x 10-24 g. How
many ounces does the atom weigh?
Grams can be converted to ounces by a series of
stepwise conversions.
1 lb = 454 g
-24
1.674 x 10
1 lb
-27
gx
 3.69 x 10 lb
454 g
16 oz = 1 lb
-27
3.69 x 10
16 oz
-26
x
 5.90 x 10 oz
lb
1 lb
An atom of hydrogen weighs 1.674 x 10-24 g. How
many ounces does the atom weigh?
Grams can be converted to ounces using a linear
expression by writing down conversion factors
in succession.
1 lb
16 oz
-26
x
x
 5.90 x 10 oz
1.674 x 10 g
454 g
1 lb
-24
Derived Units: Area & Volume
Area: Measure of the amount of two-dimensional
space occupied. It is a derived unit from the two
dimensions of area: length x width.
Volume: Measure of the amount of threedimensional space occupied. It is a derived unit
from the three dimensions of volume: length x width
x height.
SI unit = cubic meter (m3)
Usually measure liquid or gas volume in milliliters
(mL)
1 L is slightly larger than 1 quart (1 L = 1.0567 qt)
1 L = 1000 mL = 103 mL
1 mL = 0.001 L = 10-3 L
1 mL = 1 cm3
Convert 4.61 x 102 microliters to
milliliters.
Microliters can be converted to milliliters by a
series of stepwise conversions.
L  L  mL
1L
-4
x

4.61x10
L
4.61x10 μL
6
10 μL
2
1000 mL
-1
4.61x10 L x
= 4.61 x 10 mL
1L
-4
Convert 4.61 x 102 microliters to
milliliters.
Microliters can be converted to milliliters using
a linear expression by writing down conversion
factors in succession.
L  L  mL
1L
1000 mL
-1
4.61x10 μL x 6
x
= 4.61 x 10 mL
10 μL
1L
2
Heat
A form of energy that is associated with the
motion of small particles of matter.
Heat refers to the quantity of this energy
associated with the system.
The system is the entity that is being heated
or cooled.
Temperature
A measure of the intensity of heat.
It does not depend on the size of the
system.
Heat always flows from a region of higher
temperature to a region of lower
temperature.
Temperature Measurement
The SI unit of temperature is the Kelvin.
There are three temperature scales: Kelvin,
Celsius and Fahrenheit.
In the laboratory temperature is commonly
measured with a thermometer.
180 Farenheit Degrees
= 100 Celcius degrees
180
=1.8
100
It is not uncommon for temperatures in the
Canadian plains to reach –60.oF and below
during the winter. What is this temperature in oC
and K?
o
o
o
F - 32
C=
1.8
60. - 32
o
C=
= -51 C
1.8
It is not uncommon for temperatures in the
Canadian planes to reach –60.oF and below
during the winter. What is this temperature in oC
and K?
o
K = C + 273.15
o
K = -51 C + 273.15 = 222 K
Density is the ratio
of the mass of a
substance to the
volume occupied by
that substance.
mass
d=
volume
Density varies with temperature
d
d
4oC
H 2O
o
80 C
H 2O
1.0000 g
g
=
= 1.0000
1.0000 mL
mL
1.0000 g
g
=
= 0.97182
1.0290 mL
mL
Density examples
Try to rank: air, lead, feathers, water,
gasoline
Air < feathers < gasoline < water < lead
1.28 g/L <0.5 g/mL <0.7 g/mL< 1.0 g/mL < 11.3 g/mL
gas
(solid)
liquid
liquid
solid
The density of ether is 0.714 g/mL. What is the
mass of 25.0 milliliters of ether?
Method 1
(a) Solve the density equation for mass.
mass
d=
volume
mass = density x volume
(b) Substitute the data and calculate.
0.714 g
25.0 mL x
= 17.9 g
mL
The density of ether is 0.714 g/mL. What is the
mass of 25.0 milliliters of ether?
Method 2 Dimensional Analysis. Use density as
a conversion factor. Convert:
mL → g
g
=g
The conversion of units is mL x
mL
0.714 g
25.0 ml x
= 17.9 g
mL
The density of oxygen at 0oC is 1.429 g/L. What is
the volume of 32.00 grams of oxygen at this
temperature?
Method 1
(a) Solve the density equation for volume.
mass
d=
volume
mass
volume =
density
(b) Substitute the data and calculate.
32.00 g O2
volume =
= 22.40 L
1.429 g O2 /L
The density of oxygen at 0oC is 1.429 g/L. What is
the volume of 32.00 grams of oxygen at this
temperature?
Method 2 Dimensional Analysis. Use density as
a conversion factor. Convert:
g→L
The conversion of units is
L
gx =L
g
1L
32.00 g O 2 x
= 22.40 L O 2
1.429 g O 2
SPECIFIC GRAVITY
• An older concept still used in many places.
• Specific gravity of a liquid or solid is
always its density referenced to water at
the same temperature.
• Specific gravity of a gas is always
referenced to air at the same temperature
and pressure.