Ch 2:
Measurements and Problem Solving
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Chapter 2: Measurements and Problem Solving
Bonus Problems: 29, 31, 35, 41, 43, 47, 49, 57, 63, 69, 73, 85, 87, 91, 95, 101, 103, 111
Measurements:
Scientific measurements need to be understood by many and must include
significant number values with units.
If I told you the temperature outdoors is 32, would you consider that hot or
cold? That depends on the unit: 32˚C is equivalent to 90 ˚F which is a “hot”
day, while 32˚F is the freezing temperature of water to ice.
Significant numbers include all the numbers that are know with certainty plus
one more estimated number. If you want to solve for the density
(mass/volume) of a liquid that you measure to have a mass of 2.0 g while
occupying 3.0 ml how many numbers should the answer have?
a) 0.7 g/ml, b) 0.67 g/ml, c) 0.667 g/ml, d) 0.66666666666666667 g/ml
Scientific Notation: Writing Large and Small Numbers
Scientific Notation removes the place holding zeros.
Move the decimal to obtain a number between 1 to 10 (one digit) followed by
a decimal and include only the significant numbers
Multiply the number by 10 raised to an exponent.
#.### x 10#
141,000 sec = 1.41 x 105 sec
000 000 000 0561 m = 5.61 x 10-11 m
Uncertainty in Measurement:
Inexact (has uncertainty, last digit is estimated)
Exact (definitions i.e. 1 ft = 12 in, or whole items i.e. number of students enrolled)
Precision (measurements agree with each other)
Accuracy (measurement agrees with true value)
Significant Figures:
Significant Figures-only the last digit is uncertain, others known with certainty.
When recording experimental measurements you always want to include all known
numbers plus one estimated digit. Do not round off known measured values.
Ch 2:
Measurements and Problem Solving
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Rules for significant figures:
Unlimited significance 1000mm = 1 m
Exact numbers
Numbers 1 through 9 (nonzero) Significant
Never significant
Leading zero
25.223 g
0.00402 kg
Captive zero between nonzero
Significant
2.005 x 108 atoms
Trailing zero with decimal
Significant
3.200 x 104 s
Trailing zero without decimal
Uncertain
1400 miles
Significant Figures in Calculations:
1. Addition and Subtraction-minimum significant to the right taken
2. Multiplication and Division-least number of significant numbers
3. Follow order of operations you have learned in math courses
Rounding Numbers:
Remove all nonsignificant numbers. Less than 5 round down (drop it), when 5 or
more round up.
Units of Measurement:
SI Units (Systeme International d’Unites) chosen by international agreement in
1960 based on the metric system. This has become the standard system for
scientific measurements
Physical Quantity
Unit
Abbreviation
kilogram
kg
Mass
Length
meter
m
Time
second
s
Electric current
Luminous intensity
Temperature
ampere
candela
Kelvin
A
cd
K
Amount
mole
mol
Volume (derived unit), dm3
liter
L
Density (derived unit), mass/V
grams/cm3
Energy (derived unit), kgm2s2
joule
Solid, liquid: g/cm3=g/ml,
for gases: g/L
J
Ch 2:
Measurements and Problem Solving
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Prefixes used in the metric system. (memorize the bolded prefixes)
Prefix
Abbrev. Meaning
Example
12
T
10
1 Terabyte = 1 x 1012 bytes
Tera
Giga
G
109
1 Gigabyte = 1 x 109 bytes
Mega
M
106
1 Megameter = 1 x 106 meter
kilo
k
103
1 km = 1 x 103 m
unit
-
1
deci
d
10-1
10 dm = 1 m
centi
c
10-2
102 cm = 1 m
milli
m
10-3
103 mm = 1 m or 1 mm = 0.001 m
10-6
106 m = 1 m or 1 m = 1 x10-6 m
micro
nano
n
10-9
109 nm = 1 m
pico
p
10-12
1012 pm = 1 m
femto
f
10-15
1015 fm = 1 m
Common Conversions. (more in textbook, memorize the bolded ones)
Length
Mass
2.54cm = 1 inch
453.6 g = 1 pound (lb)
1 km = 0.6214 mi
1 kg = 2.205 pounds (lbs)
Volume
1 ml = 1 cm3 = 1 cc
946 ml = 1 qt
Temperature °C = K -273.15
K = (°C) + 273.15
4.184 J = 1 calorie
Energy
°F = 1.8(°C) + 32
°C = [°F – 32]/(1.8 )
1.602 x 10-19 J = 1 eV
Ch 2:
Measurements and Problem Solving
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Dimensional Analysis:
• Aids problem solving using conversion factors/definitions
• A conversion factor or equivalence is multiplied as 1, changing the number and
units, without changing the “real value”.
• Units are multiplied, divided, and canceled like any other algebraic quantities.
• Using units as a guide to solving problems is called dimensional analysis.
Always write every number with its associated unit.
Always include units in your calculations, dividing them and multiplying
them as if they were algebraic quantities.
Do not let units disappear. Units must flow logically from beginning to end.
Multiple Step Conversions and Conversions taken to a Power (squared, cubed):
It is often necessary to use several conversion factors in one problem.
When squared, must square both the number and unit
When cubed, number and unit must be cubed
Problem Solving:
1) Identify the starting point (Write given quantity and units)
2) Identify the end point (Write down units to find)
3) Devise a way to get from the starting point to the end point using what is given
as well as what you already know or can look up (Write down appropriate
conversion factors or equations)
4) You can use a solution map to diagram steps and solve the problem
5) Round the answer to correct significant digits
6) Check/Estimate the answer, does it make sense?
Try this:
1) Convert 294.2 ml to ___quarts
2) The Earth’s surface area is 197 million square miles. What is the value in square kilometers?
Ch 2:
Measurements and Problem Solving
Density:
Density = Mass/Volume
For liquids/solids: units are generally g/ml or
equivalently g/cm3
For gases: density units are generally g/L
Calculate volume as l x w x h or by
displacement of a liquid
Density can be used as a conversion factor to
solve for mass or volume given the other.
Try this:
A titanium bicycle frame contains the same
amount of titanium as a titanium cube
measuring 6.8 cm on a side. Use the density
of titanium to calculate the mass in
kilograms of titanium in the frame. What
would be the mass of a similar frame composed of iron?
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Ch 2:
Measurements and Problem Solving
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Practice Problems:
1.
One lead (Pb) atom has a mass of 0.000 000 000 000 000 000 000 3441 g. Express
this in proper scientific notation.
2.
Round to 3 significant digits
a) 3.49621 g
3.
b) 8,973,641 ml
c) 1.67992 x 10-5 moles
Perform the calculations
a) (4.62)(75.234) ÷ (23.4 + 1.2) =
b) (43.2 – 12.8)2 x ( 4.29 x 104) =
4.
Convert 486 m to km
5.
Convert 8.22 x 102 nanoliters/min into units of quarts/year.
6.
A student finds that the weight of an empty beaker is 53.583 g. She places a solid
in the beaker to give a combined mass of 57.483 g. To how many significant
figures is the mass of the solid known?
7.
A 3.00 quart container weighs 302 grams when empty. When it is filled with
liquid, the container weighs 2.412 kilograms. What is the density of the liquid in
g/ml?
8.
Describe the difference between experimental data which is accurate and data
which is precise. Give examples.
9.
What is the volume of 50.0 grams of an object whose density is 1.326 g/ml?
10.
Monel metal is a corrosion-resistant copper-nickel alloy used in the electronics
industry. A particular alloy with a density of 8.70 g/cm3 and containing 0.024%
silicon by mass is used to make a rectangular plate that is 30.0 cm long, 17.0 cm
wide and 3.00 mm thick.
a) What is the volume of the alloy?
b) What is the mass of the alloy?
c) What is the mass of just the silicon in the sample?
11.
In 1999, NASA lost a $94 million Mars orbiter because two groups of engineers
failed to communicate to each other the units that they used in their calculations.
Consequently, the orbiter descended too far into the Martian atmosphere and
burned up.
Suppose that the Mars orbiter was to have established orbit at 155 km and that one
group of engineers specified this distance as 1.55 × 105 m and a second group of
engineers programmed the orbiter to go to 1.55 × 105 ft.
a) How low in km units was the probe programmed to orbit?
b) What was the difference in kilometers between the two altitudes?
Ch 2:
Measurements and Problem Solving
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Practice Problems: (ANSWERS)
1.
3.441 x 10-22 g
2.
a) 3.50 g
3.
a) 14.1
4.
0.486 km
5.
0.457 quarts/year.
6.
3.900 g; answer has 4 significant figures
7.
0.743 g/ml
8.
Precision (measurements agree with each other). Accuracy (measurement agrees
with true value). You may want to check the weight of a suitcase so you will not be
charged extra for too much baggage before going on an airline flight. You may
weigh your suitcase on the same bathroom scale several times and find the numbers
agree with each other (Precision). If the bathroom scale always reads 5 pounds
lighter that the true value you will not be accurate.
9.
37.7 ml
10.
a) 153 cm3 volume
b) 1330 g mass of the whole alloy
c) 0.32 g mass of just the silicon
11.
a) 4.72 x 104 km
b) 1.08 x 105 km
b) 8.97 x 106 ml
c) 1.68 x 10-5 moles
b) 3.96 x 107