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Measurements and

Calculations

Chapter 2

A gas pump measures the amount ofgasoline delivered.

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When describing very small distances, such as the diameter of a swine flu virus, it is convenient to use scientific notation.

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Scientific Notation

Technique Used to Express Very Large or Very Small Numbers

Based on Powers of 10To Compare Numbers Written in

Scientific Notation First Compare Exponents of 10 Then Compare Numbers

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Writing Numbers in Scientific Notation1 Locate the Decimal Point2 Move the decimal point to the right of the

non-zero digit in the largest place The new number is now between 1 and 10

3 Multiply the new number by 10n where n is the number of places you moved the

decimal point4 Determine the sign on the exponent n

If the decimal point was moved left, n is + If the decimal point was moved right, n is – If the decimal point was not moved, n is 0

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Examples of Scientific Notation

Write the number 25,000 in scientific notation 2.5 x 104

Write the number 0.0036 in scientific notation

3.6 x 10-3

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Writing Numbers in Standard Form

1 Determine the sign of n of 10n

If n is + the decimal point will move to the right If n is – the decimal point will move to the left

2 Determine the value of the exponent of 10 Tells the number of places to move the decimal

point3 Move the decimal point and rewrite the

number

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Converting to Decimal (Standard) Notation

Convert 3.2 x 10-4 to decimal notation0.00032

Convert 6.25 x 103

6,250

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Measurement

Quantitative ObservationComparison Based on an Accepted Scale

e.g. Meter Stick, Graduated Cylinder,BalanceHas 2 Parts – the Number and the Unit

Number Tells Comparison Unit Tells Scale

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Artist's conception of the lost Mars Climate Orbiter.

Lost because JPL in CA assumed that Lockheed Martin here in Denver used metric units.

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Related Units in the Metric System

All units in the metric system are related to the fundamental unit by a power of 10

The power of 10 is indicated by a prefixThe prefixes are always the same,

regardless of the fundamental unit

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Be sure you can immediately identify what physical quantity isbeing measured when you see a unit or it’s abbreviation. (This table can be found on page 18 of your text book)

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Memorize this table (page 19 of your textbook) and be able to useit functionally, that is know what millimeter or kilogram means.

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Length SI unit = meter (m)

About 39.37 in or about 3½ inches longer than a yard1 meter = one ten-millionth the distance from the North Pole to

the Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light =distance covered by light in a vacuum in a 1/299,792,458 of a second (approximately 3.3 nanoseconds).

Commonly use centimeters (cm) in a lab 1 m = 100 cm 1 cm = 0.01 m 1cm = 10 mm 1 inch = 2.54 cm (exactly)

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Figure 2.1: Comparison of English and metric units for length on a ruler.

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Volume Measure of the amount of three-dimensional space

occupied by a substance SI unit = cubic meter (m3) Commonly measure solid volume in cubic centimeters

(cm3) 1 m3 = 106 cm3 1 cm3 = 10-6 m3 = 0.000001 m3

Commonly measure liquid or gas volume in milliliters (mL) 1 L is slightly larger than 1 quart 1 L = 10 dL = 1000 mL = 103 mL 1 mL = 0.001 L = 10-3 L 1 mL = 1 cm3 REALLY IMPORTANT!

Figure 2.2:The largest drawing represents a cube that has sides 1 m in length (10dm) and a volume of 1 m3 (10 x 10 x 10 dm3). The smaller cube has sides 1dm in length (10cm) and a volume of 1 dm3 (10 x 10 x 10cm3)1000dm3 = 1m3

1000cm3 = 1dm3 = 1L1cm3= 1mL1000mL = 1L

Figure 2.3:A 100-mL graduated cylinder.Used for approximate volume measurements

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Very important relationship, memorize and be sure you don’t invert this relationship, remember 1 mL = 20 drops

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Mass

Measure of the amount of matter present in an object SI unit = kilogram (kg) Commonly measure mass in grams (g) or milligrams

(mg) 1 kg = 2.2046 pounds, 1 lbs. = 453.59 g 1 kg = 1000 g = 1x103 g, 1 g = 1000 mg = 1x103 mg 1 g = 0.001 kg = 1x10-3 kg, 1 mg = 0.001 g = 1x10-3 g

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Figure 2.4:An electronic analytical balance used in chemistry labs.Determines mass to within 0.0001g (0.1mg)

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Some other Examples

A doorknob is usually installed 1 meter above the floor

A paperclip (depending on the size) is 0.5 to 1 gram

20 drops from an eyedropper =1 mL, 5mL = 1 teaspoon

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Uncertainty in Measured Numbers

A measurement always has some amount of uncertainty

Uncertainty comes from limitations of the techniques used for comparison

To understand how reliable a measurement is, we need to understand the limitations of the measurement

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Reporting Measurements

To indicate the uncertainty of a single measurement scientists use a system called significant figures (digits)

The last digit written in a measurement is the number that is considered to be uncertain (estimated)

Unless stated otherwise, the uncertainty in the last digit is ±1

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Figure 2.5: Measuring a pin.

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a) 2.85 cm b) 2.850 cm

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Rules for Counting Significant Figures

Nonzero integers are always significant Zeros

Leading zeros never count as significant figures (Example of leading zeros 0.012, so two sig. figs)

Captive zeros are always significant(Example of captive zero 102, so three sig. figs.)

Trailing zeros are significant if the number has a decimal point (Example of trailing zeros significant, 25.0, this number has three sig.figs.)(Example of trailing zero not significant, 250, this number has two sig. figs)

Remember Never, Always, If

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Exact NumbersExact Numbers are numbers known with

certainty Unlimited number of significant figuresThey are either

counting numbersFor example, number of sides on a square

or defined (they are not measurements)100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm1 kg = 1000 g, 1 LB = 16 oz1000 mL = 1 L; 1 gal = 4 qts.1 minute = 60 seconds

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Adding/Subtracting Numbers with Significant Figures

Result is limited by the number with the smallest number of significant decimal places (or said another way, the largest estimation)

Find last significant figure in each measurement Find which one is “left-most” Round answer to the same decimal place

450 mL + 27.5 mL = 480 mLprecise to 10’s place precise to 0.1’s place precise to 10’s place

Why isn’t it 477.5?

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Multiplication/Division with Significant Figures Result has the same number of significant

figures as the measurement with the smallest number of significant figures

Count the number of significant figures in each measurement

Round the result so it has the same number of significant figures as the measurement with the smallest number of significant figures

4.5 cm x 0.200 cm = 0.90 cm2

2 sig figs 3 sig figs 2 sig figs

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Calculations with Significant Figures

Calculators/computers do not know about significant figures!!! You have to learn to be the boss!

Exact numbers do not affect the number of significant figures in an answer

Answers to calculations must be rounded to the proper number of significant figures round at the very end of the calculation

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Rules for Rounding Off

If the digit to be removed• is less than 5, the preceding digit stays the same• is equal to or greater than 5, the preceding digit

is increased by 1 In a series of calculations, carry the extra

digits to the final result (keep track of where rounding should occur by placing a line above the last digit that should be retained in each step) and then round off at the end

Don’t forget to add place-holding zeros if necessary to keep value the same!!

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Examples of Rounding

Round the following number to 3 significant figures93,101,000 two ways to do this9.31 x 107 or

Round the following number to 2 significant figures0.000295 two ways to do this0.00030 or 3.0 x 10-4

93,100,000

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Problem Solving and Dimensional Analysis

Many problems in chemistry involve using equivalence statements to convert one unit of measurement to another

Conversion factors are relationships between two units May be exact or measured Both parts of the conversion factor should have the

same number of significant figures Conversion factors generated from equivalence

statements e.g. 1 inch = 2.54 cm can give or

in1cm54.2

cm54.2in1

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Arrange conversion factors so starting unit cancels Arrange conversion factor so starting unit

is on the bottom of the conversion factorMay “string” conversion factors

Problem Solving and Dimensional Analysis

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Converting One Unit to Another

Find the relationship(s) between the starting and goal units. Write an equivalence statement for each relationship.

Write a conversion factor for each equivalence statement.

Arrange the conversion factor(s) to cancel starting unit and result in goal unit.

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Converting One Unit to Another

Check that the units cancel properly Multiply and Divide the numbers to

give the answer with the proper unit. Check your significant figures Check that your answer makes sense!

39More equivalents can be found on page A8 and the inside back cover of your textbook.

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Temperature Scales

Fahrenheit Scale, °F Water’s freezing point = 32°F, boiling point = 212°F

Celsius Scale, °C Temperature unit larger than the Fahrenheit Water’s freezing point = 0°C, boiling point = 100°C

Kelvin Scale, K Temperature unit same size as Celsius Water’s freezing point = 273 K, boiling point = 373 K

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Figure 2.7: The three major temperature scales.

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Figure 2.6: Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

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Figure 2.9: Comparison of the Celsius and Fahrenheit scales.

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Figure 2.8: Converting 70. 8C to the Kelvin scale.

T (K) = T (°C) + 273.15 70.8 + 273.15 =343.95K343.95K rounds to 344.0K

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Density

For equal volumes, the denser object has a larger mass For equal masses, the denser object has a smaller volume Solids = density measured in g/cm3

1 cm3 = 1 mL Liquids =density measured in g/mL Gases = density is measured in g/L Volume of a solid can be determined by water displacement Density : solids > liquids >>> gases In a heterogeneous mixture, the denser object sinks

VolumeMass

Density

Density is an intensive property of matter Density represents the mass per unit volume

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Figure 2.10: (a) Tank of water. (b) Person submerged in the tank, raising the level of the water.

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Spherical droplets of mercury, a very dense liquid.

Liquid gallium expands within a carbon nanotube as the temperature increases (left to right a to b) and then shrinks as it is cooled (in c).

Source: Glenn Izett/U.S. Geological Survey

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Using Density in Calculations

DensityMass

Volume

Volume Density Mass