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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

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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