World of
CHEMISTRY
Zumdahl • Zumdahl • DeCoste
Chapter 5
Measurements and
Calculations
Goals of Chapter 5:
Measurement & Calculations
• Express numbers in scientific notation
• Learn English, metric, & SI system of measurement
• Use metric system to measure length, volume, and
mass
• Significant digits
• Dimensional Analysis
• Temperature Scales
• Density/Specific Gravity
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Qualitative vs. Quantitative
• Qualitative: appearance.
– the substance was a white powder
• Quantitative: Measurement
– The substance had a mass of 5.6 grams
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Why are measurements important?
• Pay for gasoline by the gallon
• Construction: must have accurate measurements
• Welding: measurements must be accurate
• Automobiles: Most items on cars have a measurement
• Landscaping: Measurements important for spatial
relationships
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Measurements: 2 Parts
• Number + Unit
– Number alone is meaningless
• If I tell you a 2 x 4 is 5 long – what does this mean: 5
feet, 5 inches, 5 meters?
– Unit alone is meaningless
• If I tell you the 2 x 4 is feet long – what does this mean:
2 ft, 4 ft, 8 ft?
• ALWAYS INCLUDE BOTH NUMBER AND UNIT!
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Scientific Notation
• Used to express very large or very small
numbers
• Express using number between 1 & 10
multiplied by a power of 10
– 10 is to positive power for large numbers
– 10 is to a negative number for small numbers
(decimals)
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Large Numbers
Decimal point moved to left
Power of 10 = # places moved = + number
125 = 1.25 x 100 = 1.25 x 102
Decimal point moved 2 places left
1700 = 1.7 x 1000 = 1.7 x 103
Decimal point moved 3 places left
93,000,000 = 9.3 x 107
Decimal point moved 7 places left
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Small Numbers (Decimals < 1.0)
Decimal point moved to the right
Power of 10 = # places moved = negative number
0.010 = 1.0 x 0.010 = 1.0 x 10-2
Decimal point moved right 2 places
0.000167 = 1.67 x 10-4
Decimal point moved right 4 places
0.089 = 8.9 x 10-2
Decimal point moved right 2 places
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Units
• Tell us what scale or standard is being used to
represent measurement
• Scientists need common units to represent quantities
like mass, length, time, and temperature
• If everyone had own set of units – chaos would result
• US uses English system, most of world (& scientists)
use metric system, also SI system
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BASIC UNITS
Physical Quantity
English System
Metric System
Mass
Pound or slug
Gram or kilogram
Length
inch, foot, yard
Centimeter or meter
Volume
Quart or gallon
Liter, milliliter, or cm3
Hours, minutes, seconds
Seconds
Degrees Fahrenheit
°C or Kelvin
Time
Temperature
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Metric Prefixes
Kilok
hectoh
dekadk
Move decimal point left
UNIT
deci-
centi-
milli-
d
c
m
Move decimal point right
Power of 10 between each increment
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Examples
1000 meters = 1 km (decimal moved 3 places left)
1 meter = 100 cm (decimal moved 2 places right)
10 mm = 1 cm (decimal moved 1 place left)
1 Liter = 1000 mL (decimal moved 3 places to right)
600 grams = 0.6 kg (decimal moved 3 places left)
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Table 5.2
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Table 5.3
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Figure 5.1: Comparison of English and
metric units.
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Figure 5.2:
Cube representations.
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Figure 5.3:
A 100 mL
graduated cylinder.
1 mL = 1 cm3
1 milliliter = 1 cubic centimeter
100 mL = 100 cm3
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• Dimensional Analysis
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Uncertainty of Measurement
• When using instrument to measure (such
as a ruler or graduated cylinder), we
visualize divisions between markings and
estimate
• When making measurement, record all
certain numbers and first uncertain
number
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Figure 5.5: Measuring a pin.
Reading is between
2.8 cm & 2.9 cm
These divisions
were visualized
2.85 cm is
measurement
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“5” is uncertain
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Significant Figures
Ex.
• Includes all numbers recorded in a
measurement
– For pin, length = 2.85 cm: 3 significant figures
– All certain numbers plus first uncertain
• Assume to be accurate to ± 1 in last #
– Pin length is 2.85 ± 0.01 cm
– Pin is somewhere between 2.84 & 2.86 cm
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Rules for Counting Significant Figures
• Significant
– Captive zeros (between nonzero digits)
• Example: 1.008 (4 s.f. - all numbers are
significant)
– Trailing zeros (right end of number)
• Significant if bar is placed over zero or over zero
to right
• Example: 200 (only #2 is significant = 1 s.f.)
2Ō0 (2 s.f. - 2 & Ō both significant)
20Ō (3 s.f. - 2 and both zeros are
significant)
– Following decimal to right of nonzero digit =
significant
• 5.00 = 3 s.f.
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Non-significant zeros
– Leading zeros, precede non-zero digits
• Example: 0.0025 = 2 s.f. - zeros are nonsignificant
• Both: some numbers are sig. some are not
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Rules for counting significant figures
(continued)
• All nonzero integers are significant
– Example: 1457 has 4 s.f.
• Exact numbers have unlimited number of s.f.
– Determined by counting
• 8 apples, 21 students
• Not obtained from measuring devices
• From definitions
• 1 inch is exactly 2.54 cm
• Will not limit numbers in calculations
• Use same rules for scientific notation (10x not
s.f.)
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To give answer with correct number
of significant figures – round off
• Look at number to right of last s.f.
– If number is <5 round down
– If number is ≥5 round up
• Do not round off until end of calculations
Rules for s.f. in calculations
• Multiplication & Division
– Answer should have same number of s.f. as
measurement with smallest number of s.f.
– Example: 4.56 x 1.4 = 6.384 → 6.4
(3 s.f.) (2 s.f.)*
(2 s.f.)*
• Addition & Subtraction
– Limited by smallest number of decimal places
– Example:
12.11 (2 decimal places)
18.0
(1 decimal place)*
+ 1.013 (3 decimal places)
31.123 → 31.1 (1 decimal place)*
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Figure 5.6: The three major temperature scales.
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Temperature Scales
Temperture Scale
Fahrenheit
Celsius
Kelvin
Abbreviation
°F
°C
K
Boiling point H2 0 Freezing Point H 2O
212°F
32°F
100°C
0°C
373 K
273 K
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Converting between Kelvin & Celsius
To convert from Kelvin to Celsius:
T°C = TK – 273
Liquid Nitrogen boils at 77K, what is this in Celsius?
T°C = 77 – 273 = -196 °C
To convert from Celsius to Kelvin:
TK = T°C + 273
The bp of water on top of Mt. Everest is 70 °C. Convert to K.
TK = 70 + 273 = 343 K
Converting 70
degrees
Celsius to Kelvin
units.
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Fahrenheit/Celsius Conversions
To convert from Celsius to Fahrenheit:
T°F = 1.80(T°C) + 32
If the temperature is 28°C, what is this in °F?
T°F = 1.80(28) + 32 = 50.4 + 32 = 82°F (2 s.f.)
To convert from Fahrenheit to Celsius:
T°C = (T°F – 32)/1.80
If you have a temperature of 101°F, what is this in °C?
T°C = (101 – 32)/1.80 = 69/1.8 = 38°C
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Figure 5.8: Comparison of the Celsius and Fahrenheit scales.
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Density: amount of matter in a
given volume of substance
Density = mass / volume
•Determine mass using a balance
•Determine volume by calculations,
cylinder, or water displacement
•Units are in g/cm3, g/mL, kg/L, lb/gal
graduated
Determining volume by water
displacement
• Place water in graduated cylinder & record
level
• Add object
• Record volume after addition of object
• Volume is difference between second
volume and first volume
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Figure 5.9: Tank of water.
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Figure 5.9: Person submerged in the tank.
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