1 Chapter 2 Measurements
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Chapter 2
Measurements
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CHAPTER OUTLINE
Scientific Notation Error in Measurements Significant Figures Rounding Off Numbers SI Units Conversion of Factors Conversion of Units Volume & Density
What is a Measurement? quantitative
observation comparison to an
agreed upon standard
every measurement has a number and a unit
A Measurement the unit tells you what standard you
are comparing your object to the number tells you
1. what multiple of the standard the object measures
2. the uncertainty in the measurement
Scientists have measured the average global temperature rise over the past century to be 0.6°C
°C tells you that the temperature is being compared to the Celsius temperature scale
0.6 tells you that1. the average temperature rise is
0.6 times the standard unit2. the uncertainty in the
measurement is such that we know the measurement is between 0.5 and 0.7°C
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SCIENTIFICNOTATION
Scientific Notation is a convenient way to express very large or very small quantities.
Its general form is
A x 10n
coefficient
1 A < 10
n = exponent
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SCIENTIFICNOTATION
To convert from decimal to scientific notation: 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.
7 5 0 0 0 0 0 0
7.5 x 10 7
Scientific Notation: Writing Large and Small Numbers
A positive exponent means 1 multiplied by 10 n times. A negative exponent (–n) means 1 divided by 10 n
times.
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SCIENTIFICNOTATION
For numbers smaller than 1, the decimal moves to the left and the power becomes negative.
0 0 0 6 4 2
6.42 x 103
10
1. Write 6419 in scientific notation.
64196419.641.9x10164.19x1026.419 x 103
decimal after first nonzero
digitpower of 10
Examples:
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2. Write 0.000654 in scientific notation.
0.0006540.00654 x 10-10.0654 x 10-20.654 x 10-3 6.54 x 10-4
decimal after first nonzero
digit
power of 10
Examples:
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CALCULATIONS WITHSCIENTIFIC NOTATION
To perform multiplication or division with scientific notation:
1. Change numbers to exponential form.
2. Multiply or divide coefficients.
3. Add exponents if multiplying, or subtract exponents if dividing.
4. If needed, reconstruct answer in standard exponential form.
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Multiply 30,000 by 600,000
Example 1:
Convert to exponential form
(3 x 104) (6 x 105) =
Multiply coefficients
18 x 10
Add exponents
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Reconstruct answer
1.8 x 1010
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Divided 30,000 by 0.006
Example 2:
Convert to exponential form
(3 x 104)
(6 x 10-3)
Divide coefficients
= 0.5 x 10
Subtract exponents
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Reconstruct answer
5 x 106
4 – (-3)
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Follow-up Problems:
(5.5x103)(3.1x105) = 17.05x108 = 1.7x109
(9.7x1014)(4.3x1020) = 41.71x106 = 4.2x105
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2
2.6x10=
5.8x10 0.4483x104 = 4.5x103
1.7x10 5
8.2x10 8= 0.2073x103 = 2.1x102
(3.7x106)(4.0x108) = 14.8x102 = 1.5x103
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Follow-up Problems:
(8.75x1014)(3.6x108) = 31.5x1022 = 3.2x1023
1.48x10 28
13=
7.25x10 0.2041x1041 = 2.04x1042
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Precision is the reproducibility of a measurement compared to other similar measurements.
Precision describes how close measurements are to one another.
Precision is affected by random errors.
ACCURACY & PRECISION
Avg mass = 3.12± 0.01 g
This measurement has high precision because the deviation of multiple trials is small.
Is this measurement
precise?
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Accuracy is the closeness of a measurement to an accepted value (external standard).
Accuracy describes how true a measurement is. Accuracy is affected by systematic errors.
ACCURACY & PRECISION
Avg mass = 3.12± 0.01 g
Accuracy cannot be determined without knowledge of the accepted value.
Is this measurement
accurate?
True mass = 3.03 g
This measurement has low accuracy because the deviation from true value is large.
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ACCURACY & PRECISION
Good precision
Good accuracy
Good precision
Poor accuracy
Poor precision
Good accuracy
Poor precision
Poor accuracy
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Two types of error can affect measurements: Systematic errors: those errors that are controllable, and cause
measurements to be either higher or lower than the actual value. Random errors: those errors that are uncontrollable, and cause
measurements to be both higher and lower than the average value.
ACCURACY & PRECISION
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ERROR INMEASUREMENTS
Two kinds of numbers are used in science:
Counted or defined:
exact numbers; have no uncertainty
Measured:
are subject to error; have uncertainty
Every measurement has uncertainty because of instrument limitations and human error.
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ERROR IN MEASUREMENTS
What is this measurement?
8.65certain uncertain
What is this measurement?
8.6certain uncertain
The last digit in any measurement is the estimated one.
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RECORDING MEASUREMENTSTO THE PROPER NO OF DIGITS
What is the correct valuefor each measurement?
a) 28ml (1 certain, 1 uncertain)
b) 28.2ml (2 certain, 1 uncertain)
c) 28.31ml (3 certain, 1 uncertain)
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SIGNIFICANTFIGURES RULES
Significant figures are the certain and uncertain digits in a measurement.
Significant figures rules are used to determine which digits are significant and which are not.
1. All non-zero digits are significant.
2. All sandwiched zeros are significant.
3. Leading zeros (before or after a decimal) are NOT significant.
4. Trailing zeros (after a decimal) are significant.
0 . 0 0 4 0 0 4 5 0 0
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Examples:
Determine the number of significant figures in each of the following measurements.
461 cm 3 sig figs
1025 g 4 sig figs
0.705 mL 3 sig figs
93.500 g 5 sig figs
0.006 m 1 sig fig
5500 km 2 sig figs
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ROUNDING OFFNUMBERS
If rounded digit is less than 5, the digit is dropped.
51.234 Round to 3 sig figs
Less than 51.875377
Round to 4 sig figs
Less than 5
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ROUNDING OFFNUMBERS
If rounded digit is equal to or more than 5, the digit is increased by 1.
51.369 Round to 3 sig figs
More than 5
4
5.4505Round to 4 sig figs
Equal to 5
1
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SIGNIFICANTFIGURES & CALCULATIONS
The results of a calculation cannot be more precise than the least precise measurement.
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.
For addition and subtraction, the answer must have the same number of decimal places as there are in the measurement with the fewest decimal places.
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(9.2)(6.80)(0.3744) = 23.4225
Calculator answer
3 sig figs 4 sig figs
The answer should have two significant figures because 9.2 is the number with the fewest significant figures.
The correct answer is 23
2 sig figs
MULTIPLICATION& DIVISION
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Add 83.5 and 23.28
83.523.28
106.78Calculator
answer
Least precise number
106.8Correct answer
ADDITION &SUBTRACTION
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Example 1:
5.008 + 16.2 + 13.48 = 34.688
Least precise number
Round to
34.7
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Example 2:
3.15 x 1.53=
0.786.1788
2 sig figs
3 sig figs
Round to
6.2
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SI UNITS
Measurements are made by scientists to determine size, length and other properties of matter.
For measurements to be useful, a measurement standard must be used.
A standard is an exact quantity that people agree to use for comparison.
SI is the standard system of measurement used worldwide by scientists.
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SI (METRIC)BASE UNITS
Quantity Measured Metric Units Symbol English
Units
Length Meter m yd
Mass Kilogram kg lb
Time Seconds s s
Temperature Kelvin K F
Amount of substance Mole mol mol
Basic Units of Measurement
The kilogram is a measure of mass, which is different from weight.
The mass of an object is a measure of the quantity of matter within it.
The weight of an object is a measure of the gravitational pull on that matter.
Consequently, weight depends on gravity while mass does not.
Derived Units
A derived unit is formed from other units. Many units of volume, a measure of
space, are derived units. Any unit of length, when cubed (raised to
the third power), becomes a unit of volume.
Cubic meters (m3), cubic centimeters (cm3), and cubic millimeters (mm3) are all units of volume.
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DERIVED UNITS
Quantity Measured Units Symbol
Volume Liter L
Density grams/cc g/cm3
In addition to the base units, several derived units are commonly used in SI system.
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SI PREFIXES
The SI system of units is easy to use because it is based on multiples of ten.
Common prefixes are used with the base units to indicate the multiple of ten that the unit represents.
SI Prefixes
Prefixes Symbol Multiplying factor
mega- M 1,000,000
kilo- k 1000
centi- c 0.01
milli- m 0.001
micro- 0.000,001
106
103
10-2
10-3
10-6
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SI UNITS & PREFIXES
SI system used a common set of prefixes for use with the base units.
Base Unit
10 10 1010
kilo
103
10 10 10
mega
106
10 10
decicenti
milli
103
101010
micro
106
Smaller units Larger units
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SI CONVERSION FACTORS
Base Unit
10 10 1010
kilo
103
10 10 10
mega
106
10 10
decicenti
milli
103
101010
micro
106
1 m = 103 mm or 1 mm = 103 m
1 mm = 103 m or 1 m = 103 mm
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SI PREFIXES
How many mm are in a cm? 10How many cm are in a km? 10x10x10x10x10100000 or 105
Prefix Multipliers
Choose the prefix multiplier that is most convenient for a particular measurement.
Pick a unit similar in size to (or smaller than) the quantity you are measuring.
A short chemical bond is about 1.2 × 10–10
m. Which prefix multiplier should you use? The most convenient one is probably the
picometer. Chemical bonds measure about 120 pm.
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CONVERSIONFACTORS
Many problems in chemistry and related fields require a change of units.
Any unit can be converted into another by use of the appropriate conversion factor.
Any equality in units can be written in the form of a fraction called a conversion factor. For example:
1 m = 100 cmEquality
Conversion Factors 1 m100 cm
100 cm1 mor
Metric-Metric Factor
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CONVERSIONFACTORS
1 kg = 2.20 lbEquality
Conversion Factors1 kg
2.20 lb2.20 lb
1 kgor
Metric-English Factor
Percent quantity:
Sometimes a conversion factor is given as a percentage. For example:
18% body fat by mass
Conversion Factors
18 kg body fat
100 kg body mass
100 kg body mass
18 kg body fator
Percentage Factor
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CONVERSIONOF UNITS
beginning unit x = final unit
beginning unit
final unit
Conversion factor
Problems involving conversion of units and other chemistry problems can be solved using the following step-wise method:
1. Determine the intial unit given and the final unit needed.2. Plan a sequence of steps to convert the initial unit to the final unit.
3. Write the conversion factor for each units change in your plan.
4. Set up the problem by arranging cancelling units in the numerator and denominator of the steps involved.
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Example 1:
Convert 164 lb to kg (1 kg = 2.20 lb)
Step 1: Given: 164 lb Need: kg
Step 2:Metric-English
factorlb kg
Step 3: 1 kg2.20 lb or
2.20 lb1 kg
Step 4:1 kg
164 lb x = 74.5 kg2.20 lb
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Example 2:The thickness of a book is 2.5 cm. What is this measurement in mm?
Step 1: Given: 2.5 cm Need: mm
Step 2:Metric-Metric
factorcm mm
Step 3: 1 cm10 mm or
10 mm1 cm
Step 4:10 mm
2.5 cm x = 25 mm1 cm
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Example 3:
How many centimeters are in 2.0 ft? (1 in=2.54 cm)
Step 1: Given: 2.0 ft Need: cm
Step 2:English-English
factorft in
Step 3: 1 ft12 in and
1 in2.54 cm
Step 4:12 in
2.0 ft x x1 ft
cmMetric-English factor
2.54 cm =
1 in60.96 cm61 cm
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Example 4:
Bronze is 80.0% by mass copper and 20.0% by mass tin. A sculptor is preparing to case a figure that requires 1.75 lb of bronze. How many grams of copper are needed for the brass figure (1lb = 454g)?
Step 1: Given: 1.75 lb bronze Need: g of copper
Step 2:English-Metric
factorlb
brzg
brzg
CuPercentage
factor
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Example 4:
Step 3:1 lb
454 g and80.0 g Cu100 g brz
Step 4: = 635.6 g= 636 g1.75 lb brz x454 g1 lb x
80.0 g Cu100 g brz
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VOLUME
Volume is the amount of space an object occupies.
Common units are cm3 or liter (L) and milliliter (mL).
1 L = 1000 mL 1 mL = 1 cm3
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VOLUME
Volume of various regular shapes can be calculated as follows:
Cube V = s x s x s Rect. V = l x w x h
Cylinder V = π x r2 x h Sphere V =4/3 πr3
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DENSITY
Density is mass per unit volume of a material. Common units are g/cm3 (solids) or g/mL (liquids).
Which has greatest density?
Density is directly related to the mass of an object.
Density is indirectly related to the volume of an object.
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Example 1:
A copper sample has a mass of 44.65 g and a volume of 5.0 mL. What is the density of copper?
m = 44.65 g
v = 5.0 mL
d = ???
md =
v
44.65 g =
5.0 mL= 8.93 g/mL= 8.9 g/mL
Round to 2 sig figs
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Example 2:
A silver bar with a volume of 28.0 cm3 has a mass of 294 g. What is the density of this bar?
m = 294 g
v = 28.0 mL
d = ???
md =
v
294 g =
28.0 mL= 10.5 g/mL
3 sig figs
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Example 3:
Step 1: Given: 5.00 cm3 Need: g
Step 2: densitycm3 g
Step 3:
If the density of gold is 19.3 g/cm3, how many grams does a 5.00 cm3 nugget weigh?
5.00 cm33
19.3 gx
1 cm = 96.5 g
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Example 4:
If the density of milk is 1.04 g/mL, what is the mass of 0.50 qt of milk? (1L = 1.06 qt)
Step 1: Given: 0.5 qt Need: g
Step 2: English-Metric factor
qt L
Step 3:1 L
1.06 qtand 103 mL
1 L
Step 4:1 L
0.50 qt x 1.06 qt
Metric-Metric factor
310 mLx
1 L= 490.57 g= 490 g
mL gdensity
and 1.04 g1 mL
1.04 gx
1 mL
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Example 5:
What volume of mercury has a mass of 60.0 g if its density is 13.6 g/mL?
x mL
=gmL
60.0 g 4.41113.6
inverse of density
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IS UNIT CONVERSIONIMPORTANT?
In 1999 Mars Climate orbiter was lost in space because engineers failed to make a simple conversion from English units to metric units, an embarrassing lapse that sent the $125 million craft fatally close to the Martian surface.
Further investigation showed that engineers at Lockheed Martin, which built the aircraft, calculated navigational measurements in English units. When NASA’s JPL engineers received the data, they assumed the information was in metric units, causing the confusion.
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THE END