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Chapter 1B

Measurement

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CHAPTER OUTLINE���

§  SI Units §  Scientific Notation §  Error in Measurements §  Significant Figures §  Rounding Off Numbers §  Conversion of Factors §  Conversion of Units §  Volume & Density

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SI UNITS���

q  Measurements are made by scientists to determine size, length and other properties of matter.

q  For measurements to be useful, a measurement standard must be used.

q  A standard is an exact quantity that people agree to use for comparison.

q  SI is the standard system of measurement used worldwide by scientists.

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SI BASE UNITS���

Quantity Measured Units Symbol Length Meter m Mass Kilogram kg Time Seconds s Temperature Kelvin K Amount of substance Mole mol Electric current Ampere A Intensity of light Candela cd

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DERIVED UNITS���

Quantity Measured Units Symbol Volume Liter L Density grams/cc g/cm3

q  In addition to the base units, several derived units are commonly used in SI system.

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SCIENCTIFIC���NOTATION���

q  Scientific Notation is a convenient way to express very large or very small quantities.

q  Its general form is

A x 10n

coefficient

1 A < 10

n = integer

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SCIENTIFIC���NOTATION���

To convert from decimal to scientific notation: q  Move the decimal point in the original number so

that it is located after the first nonzero digit. q  Follow the new number by a multiplication sign

and 10 with an exponent (power). q  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

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SCIENTIFIC���NOTATION���

q  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 10 -3

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1. Write 6419 in scientific notation.

6419 6419. 641.9x101 64.19x102 6.419 x 103

decimal after first nonzero

digit power of 10

Examples:���

10

2. Write 0.000654 in scientific notation.

0.000654 0.00654 x 10-1 0.0654 x 10-2 0.654 x 10-3 6.54 x 10-4

decimal after first nonzero

digit

power of 10

Examples:���

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CALCULATIONS WITH���SCIENTIFIC NOTATION���

q  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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ERROR IN���MEASUREMENTS���

q  Two kinds of numbers are used in science:

Counted or defined:

exact numbers; have no uncertainty

Measured:

are subject to error; have uncertainty

q  Every measurement has uncertainty because of instrument limitations and human error.

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ERROR IN ���MEASUREMENTS���

What is this measurement?

8.65 certain uncertain

What is this measurement?

8.6 certain uncertain

q  The last digit in any measurement is the estimated one.

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SIGNIFICANT���FIGURES RULES���

q  Significant figures are the certain and uncertain digits in a measurement.

q  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 OFF���NUMBERS���

q  If rounded digit is less than 5, the digit is dropped.

51.234 Round to 3 sig figs

Less than 5 1.875377 Round to 4 sig figs

Less than 5

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ROUNDING OFF���NUMBERS���

q  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

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5.4505 Round to 4 sig figs

Equal to 5

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SIGNIFICANT���FIGURES & CALCULATIONS���

q  The results of a calculation cannot be more precise than the least precise measurement.

q  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.

q  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 83.5 23.28

106.78 Calculator

answer

Least precise number

106.8 Correct 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 =0.78

6.1788

2 sig figs

3 sig figs

Round to

6.2

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SI PREFIXES���

q  The SI system of units is easy to use because it is based on multiples of ten.

q  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- µ 0.000,001

106

103 10-2 10-3 10-6

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SI PREFIXES���

How many mm are in a cm? 10 How many cm are in a km? 10x10x10x10x10 100000 or 105

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CONVERSION���FACTORS���

q  Many problems in chemistry and related fields require a change of units.

q  Any unit can be converted into another by use of the appropriate conversion factor.

q  Any equality in units can be written in the form of a fraction called a conversion factor. For example:

1 m = 100 cm Equality

Conversion Factors 1 m 100 cm

100 cm 1 m or

Metric-Metric Factor

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CONVERSION���FACTORS���

1 kg = 2.20 lb Equality

Conversion Factors 1 kg

2.20 lb 2.20 lb

1 kg or

Metric-English Factor

Percent quantity:

q  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 fat

or

Percentage Factor

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CONVERSION���OF UNITS���

beginning unit x = final unitbeginning unit

final unit

Conversion factor

q  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: English-Metric

factor lb kg

Step 3: 1 kg 2.20 lb or

2.20 lb 1 kg

Step 4: 1 kg164 lb x = 74.5 kg

2.20 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

factor cm mm

Step 3: 1 cm 10 mm or

10 mm 1 cm

Step 4: 10 mm2.5 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

factor ft in

Step 3: 1 ft 12 in and

1 in 2.54 cm

Step 4: 12 in2.0 ft x x1 ft

cm English-Metric factor

2.54 cm =1 in

60.96 cm 61 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?

Step 1: Given: 1.75 lb bronze Need: g of copper

Step 2: English-Metric factor

lb brz

g brz

g Cu

Percentage factor

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Example 4:���

Step 3: 1 lb

454 g and 80.0 g Cu 100 g brz

Step 4: = 635.6 g = 636 g 1.75 lb brz x 454 g 1 lb x

80.0 g Cu 100 g brz

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VOLUME���

q  Volume is the amount of space an object occupies.

q  Common units are cm3 or liter (L) and milliliter (mL).

1 L = 1000 mL 1 mL = 1 cm3

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DENSITY���

q  Density is mass per unit volume of a material. q  Common units are g/cm3 (solids) or g/mL (liquids).

Which has greatest density?

Density is directly proportional to the mass of an object.

Density is indirectly proportional 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 cm3. What is the density of copper?

m = 44.65 g V = 5.0 cm3 d = ???

d = m V =

44.65 g 5.0 mL

= 8.93 g/cm3 = 8.9 g/cm3

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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 cm3 d = ???

d = m V =

294 g 28.0 mL

= 10.5 g/cm3

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Example 3:���If the density of gold is 19.3 g/cm3 , how many grams does a 5.00 cm3 nugget weigh?

Step 1: Given: 5.00 cm3 Need: g

Step 2: density cm3 g

Step 3: 19.3 g 1 cm3 or 1 cm3

19.3 g

Step 4: 33

19.3 g5.00 cm x = 96.5 g1 cm

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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 gt Need: g

Step 2: English-metric Factor

qt mL

Step 3: and 1.04 g 1 mL

Step 4: 1000 mL0.50 qt x 1.06 qt

g density

1L 1.06 qt

1000 mL

1.04 gx 1 mL

= 490.57 g = 490 g

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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

=g mL

60.0 g 4.41 1 13.6

inverse of density

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DENSITY & FLOATING���

q  Objects float in liquids when their density is lower relative to the density of the liquid.

q  The density column shown was prepared by layering liquids of various densities.

q  See demo Honey

Syrup

Soap

Water

Vegetable oil Isopropyl alcohol less dense

more dense

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IS UNIT CONVERSION���IMPORTANT?���

o  In 1999 Mars Climate orbiter was lost in space because engineers failed to make a simple conversion from English units to metric, an embarrassing lapse that sent the $125 million craft fatally close to the Martian surface.

q  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