Scientific Measurem

entChapter 3

1

Introduction • Measurements are key to any scientific

endeavor, including chemistry.

• All measurements have a numerical component and a unit component.

• The numerical component of a measurement must report the precision of the instrument.

• The SI system of unit is used in the sciences.

• Conversion factors, like density, allows us to convert from one unit to another. 2

Measurements and Their Uncertainty(Section 3.1)

• Using and Expressing Measurements

• Accuracy, Precision, and Error

• Significant Figures in Measurements

• Significant Figures in Calculations 3

I.) Using and Expressing Measurements

• Measurements are used to determine the magnitude of some quantity, like mass or volume.

• Measurements are a central part of all the sciences.

• Therefore, an important characteristic of a measurement is that it must be understood by anyone who looks at it.

4

What is a Measurement?What is a Measurement?

A quantity that has both a number and a unit.

Examples:1.5.54 mL2.3.00 x 108 m/s (speed of light in a vacuum)3.9.3 x 106 miles (distance to the sun)4.6.02 x 1023 mol-1 (Avogadro’s Number)

5

What are the parts of a What are the parts of a measurement?measurement?

A quantitative description has both a number and a unit.

We know what numbers are and how to represent them, but what about units?

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Scientific NotationScientific Notation

A number written as the product of two numbers: a coefficient and a 10 raised to a power.

This method of writing numbers is often used to express very large or very small values.

Examples:1.5.98 x 10 24 kg (mass of the Earth)2.9.11 x 10-28 g (mass of an electron)

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• Coefficient: This number is always greater than or equal to 1 and less than 10.

• Exponent: This value tells you how many times the coefficient must be multiplied or divided by 10 to equal the magnitude of the original number.

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How to Write in Scientific How to Write in Scientific NotationNotation

1. For large numbers: - start counting at the decimal point

- move towards the left

- stop right before the last digit

- the number of “spaces” moved is the exponent (expressed as a positive number)

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2. For small numbers:- Start counting at the decimal point

- Move towards the right.

- Stop when you pass the first non-zero digit.

- the number of “spaces” moved is the exponent (expressed as a negative number)

10

Let’s practice this.Write the following numbers in scientific notation:

1.) 6,300,000

2.) 0.0000008

3.) 0.000073611

Now let’s write the standard form for each of the following number in scientific notation.

1. 4 x 10-3

2. 5.4 x 106

3. 2.7 x 10-7

4. 8.9 x 10312

Adding and Subtracting Numbers in Scientific Notation

• Easiest way is to enter the numbers into you calculator. You must know how to use scientific notation on your calculator.

• If you don’t have a calculator:1. Make sure the exponents are the same.

2. Add/subtract the coefficients.

3. Keep the exponent the same.13

Multiplying and Dividing Numbers in Scientific Notation• Easiest way is to enter the numbers into

you calculator. You must know how to use scientific notation on your calculator.

• If you don’t have a calculator:1. For multiplying

a) Multiply coefficients

b) Add exponents

2. For dividinga) Divide the coefficients

b) Subtract the exponents 14

Let’s practice.Add, Subtract, Multiply, or Divide.

1.(3 x 104) x (2 x 102) =

2.(3.0 x 105) ÷ (6.0 x 102) =

3.(8.0 x 102) + (5.4 x 103) =

4.(3.42 x 10-5) – (2.5 x 10-6) =15

IncredulousUnwilling to admit or accept what is offered as true.

1 Thessalonians 5:21Test everything. Hold on to the good

16

II.) Accuracy, Precision, and Error

Error is introduced in how we carry out our experiment and how we choose to measure

what we observe.

Therefore, there is always error in experimentation. 17

Accuracy and Precision Accuracy and Precision ARE NOTARE NOT Synonymous Synonymous

in Sciencein Science

Accuracy A measure of how close a measurement comes to the actual value of whatever is measured (i.e. correctness)

Precision A measure of how close a series of measurements are to one Another (i.e. reproducibility).

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What do we strive for in science?

19

We must express the level of precision our

measurements have and always indicate the error

inherent in all measurements.

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Reporting ErrorReporting Error• Error is inherent in all measurements.

• Accurate values are difficult to attain and require multiple measurements.

• Accepted Value: The “correct” value based on reliable references.

• Experimental Value: The value measured in the lab

21

Error and Percent ErrorError and Percent Error

• Error

Error = experimental value – accepted value

• Percent error

Percent error = error /accepted value x 100

22

Determining PrecisionDetermining Precision

• Precision is determined by the instruments used to make a measurement.

• Significant figures Significant figures are used to report precision in a measurement.

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III.) Significant Figures in Measurements• The significant figures in a measurement

include all of the digits that are known and a last digit that is estimated.

• Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

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Significant Figures Rule #1Significant Figures Rule #1

Every nonzero digit in a reported measurement is assumed to be significant.

Examples:1.) 24.7 meters2.) 0.743 meter3.) 714 meters

3 significant figures

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Significant Figures Rule #2Significant Figures Rule #2

Zeros appearing between nonzero digits are significant.

Examples:1.) 7003 meters2.) 40.79 meters3.) 1.503 meters

4 significant figures

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Significant Figures Rule #3Significant Figures Rule #3

Leftmost zeros appearing in front of nonzero digits are not significant. They are placeholders. By writing the measurements in scientific notation, you can eliminate such placeholding zeros.

Examples:1.) 0.0071 meter2.) 0.42 meter3.) 0.000000099 meter

2 significant figures

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Significant Figures Rule #4Significant Figures Rule #4

Zeros at the end of a number and to the right of a decimal point are always significant.

Examples:1.) 43.00 meters2.) 1.010 meters3.) 9.000 meters

4 significant figures

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Significant Figures Rule #5Significant Figures Rule #5

Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not significant if they serve as a placeholder.

Examples:1.) 300 meters2.) 7000 meters

1 significant figure

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Significant Figures Rule #6Significant Figures Rule #6

There are two situations in which numbers have an unlimited number of significant figures:

1.) Counted quantities

2.) Defined quantities

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Let’s practice.Indicate the number of significant figures in each of the following measurements.

1.456 mL.

2.70.4 m.

3.444,000 g.

4.0.00406 mg.

5.0.90 L.

6.56 eggs in a basket

7.12 eggs in 1 dozen31

IV.) Significant Figures in Calculations

A calculated answer cannot be more precise than the least precise measurement from which it was calculated.

What does this mean?

32

Find the measurement with the least number of significant figures and this will tell you how many significant figures you can have in your answer.You will be required to round.

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Rounding RulesRounding Rules1. Decide how many significant figures the

answer should have.

2. Round to that many digits counting from the left.

3. If the digit immediately to the right of the last significant digit is less than 5, the value of the last significant figure stays the same.

4. If the digit immediately to the right of the last significant digit is 5 or greater, the value of the last significant figure is increased by one.

5. Drop all other digits. 34

Let’s practice.Round each of measurement to three significant figures. Write your answers in scientific notation.

1. 87.073 meters

2. 4.3621 x 108 meters

3. 0.01552 meter

4. 9009 meter

5. 1.7777 x 10-3 meter

6. 629.55 meters35

Significant Figures in Addition Significant Figures in Addition and Subtraction Problemsand Subtraction Problems

The answer to an addition or a subtraction problem should be rounded to the same number of decimal places decimal places (not digits) as the measurement with the least number of decimal places.

36

Sample ProblemSample Problem

Calculate the sum of the three measurements. Give the answer to the correct number of significant figures.

12.52 meters + 349.0 meters + 8.24 meters

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Let’s practice.

Find the total mass of three diamonds that have masses of 14.2 g., 8.73 g., and 0.912 g.

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Significant Figures in Multiplication Significant Figures in Multiplication and Division Problemsand Division Problems

The product or quotient must have the same number of significant figures as the measurement with the least number of significant figures.

Note: In these problems the place of the decimal point has nothing to do with the rounding process

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Sample ProblemSample ProblemCalculate the product or quotient of the three measurements. Give the answer to the correct number of significant figures.

1. 7.55 meters x 0.34 meter =

2. 2.4526 meter ÷ 8.4 =

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Let’s practice.Calculate the volume of a warehouse that has inside dimensions of 22.4 meters by 11.3 meters by 5.2 meters (Volume = length x height x width).

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The International System of Units

(Section 3.2)• Measuring

with SI Units

• Units and Quantities

42

I.) Measuring with SI Units

What is SI?What is SI?

• It is “Le Systeme International d’Unites” (or The International System of Units)–It is a modified version of the metric

system.

–Adopted internationally in 196043

Why SI?Why SI?• It is simple and is widely used in the

sciences.

• All metric units are based on multiples of 10.

• Conversions between units are quite easy.

• There are 7 base SI units, 5 of which are commonly used in chemistry. 44

The 7 Base SI UnitsThe 7 Base SI Units

Quantity SI Base Unit Symbol

Length Meter m

Mass Kilogram kg

Temperature Kelvin K

Time Second s

Amount of Substances Mole mol

Luminous Intensity Candela cd

Electric Current Ampere A

45

Commonly Used PrefixesCommonly Used PrefixesPrefix Meaning Factor

mega (M) 1 million times larger than the unit it precedes

106

kilo (k) 1000 times larger than the unit it precedes 103

deci (d) 10 times smaller than the unit it precedes 10-1

centi (c) 100 times smaller than the unit it precedes 10-2

milli (m) 1000 times smaller than the unit it precedes 10-3

micro (µ) 1 million times smaller than the unit it precedes

10-6

nano (n) 1000 million times smaller than the unit it precedes

10-9

pico (p) 1 trillion times smaller than the unit it precedes

10-12 46

II.) Units and Quantities

• Different quantities require different units of measurements. Length = meter (m) Volume = liter (L) Mass = kilogram (kg) Temperature = Celsius (C) or Kelvin

(K) Energy = joule (J) 47

Length (cm, m, km)The SI basic unit of length is the meter (m).

(Adding the prefixes to the basic unit of length allows us to add scale to it.)

A meter is about the height of the doorknob to the floor.

48

kilometer ~ 5 city blocks

decimeter~ diameter of an orange

millimeter ~ thickness of a dime

micrometer ~ diameter of bacterial cell

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Volume (L, mL, cm3, µL)The SI unit for volume is the cubic meter (m3).

A cubic meter (m3) is about thevolume of an automatic dishwasher.

More often the non-SI unit of liter (L) is used for volume. 50

Liter ~ quart of milk

Cubic centimeter (cm3) ~cube of sugar

microliter ~ a crystal of table salt

1L = 1000 cm3 = 1000 mL51

Mass (kg, g, mg, µg) The SI basic unit of mass is the kilogram.

A kilogram is about the same as a small textbook

Mass: The measure of the amount of matter an object contains.

Weight: A force that measures the pull on a given mass by gravity.

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Gram ~ mass of dollar

Milligram~ 10 grains of NaCl

Microgram~ 1 particle Of baking soda(NaHCO3)

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Temperature (oC, K)

A measure of how hot or cold an object is.

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• “Hotness” or “coldness” depends on the direction that heat flows.

• Almost all objects expand with increasing temperature and contracts with decreasing temperature. (What is one common and very important exception?)

• There are two SI units for temperature being used: the degree Celsius and the kelvin.

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The Degree Celsius Scale

• Named after the Swedish astronomer Anders Celsius.

• Uses the freezing and boiling points of water as two key reference points.

• The distance between these two points are divided into 100 equal intervals or degrees Celsius (oC).

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The Kelvin Scale (aka: the absolute scale)

• The scale is named after the Scottish physicist and mathematician Lord Kelvin

• Absolute zero (0 K)

• There are no degrees or negative signs used.

• Water freezes at 273.15 K. and boils at 373.15 K.

57

Converting Between Temperature Converting Between Temperature ScalesScales

1. A one-to-one relationship exist between the Celsius and Kelvin scale.

2. Conversion between the two scales is easy and straight forward.

K = oC + 273 oC = K – 273

3. Converting between degree Celsius and degree Fahrenheit is not as straight forward.

tF = (9oF/5oC)tC + 32oF 58

Let’s practice.

Normal human body temperature is 37o C. What is that temperature in kelvins?

59

Energy (J, cal)

The capacity to do work or to produce heat.

• The SI basic unit of energy is the joule (J).

• One calorie (cal) is the quantity of heat that raises the temperature of 1 g. of pure water by 1oC.

• Covert between joule and calorie: 1 J = 0.2390 cal or 1 cal = 4.184 J

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Conversion Problems(Section 3.3)

• Conversion Factors

• Dimensional Analysis

• Converting Between Units

61

I.) Conversion Factors

• Because many quantities are expressed in many different ways, we need a ways to convert one expression to another (i.e. currencies)

• Conversion factor: A ratio of equivalent measurements.

62

Examining a Conversion FactorExamining a Conversion Factor

1 m

100 cm

Smaller number is with the larger unit

Larger number iswith the smaller unit.

In a conversion factor, the measurement in thenumerator is equivalent to the measurement in the denominator. 63

Conversion factors expresses equivalent amounts as a fraction.

1 m =10 dm =100 cm = 1000 mm 1m 10 dm

1m100 cm

1m1000mm

1 m.

1 dm

64

When two measurements are equivalent, a ratio of the two measurements will equal one, or unity.

1 m. 100 cm.

1 m. 1 m.= = 1

Conversion factors are useful when we need to change from one unit of measure to another

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Some Things to Keep in Mind Some Things to Keep in Mind When Using Conversion FactorsWhen Using Conversion Factors

1. When multiplying by a conversion factor the numerical value changes, but the actual size of the quantity measured remains the same.

2. Conversion factors are defined quantities and thus have an unlimited number of sig. figs.

66

II.) Dimensional Analysis

A method to analyze and solve problems using units, or dimensions, of a measurement.

67

DA is a technique used to change any unitfrom one to another.

Steps in Using dimensional Analysis1. Identify the starting measurement or

quantity.

2. Identify the end measurement or quantity.

3. Identify the conversion factors.

4. Use the conversion factors in sequence to cancel out starting unit and get the end unit.

5. Multiply/divide the numbers to get the final numerical value. 68

Let’s practice.How many seconds are in a work day that lasts

exactly eight hours?

69

Starting unit:

Ending unit:

Conversion factors:

III.)Converting Between Units

In chemistry we often need to express measurements in units different from the initial measurement. How do we do it?

By using dimensional analysis

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Let’s practice.Express 750 dg. in grams. 1 g. = 10 dg.

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Density(Section 3.4)

• Determining Density

• Density and Temperature

72

DensityDensity

The ratio of the mass of an object to its volume.

• Density is an intensive property that depends only on the composition of a substance, not on the size of the sample.

• We can use density to ID a substance.

• If we know the volume and mass of an object we can calculate its density.

73

Lead, Pbd=11.4 g/cm3 Lithium, Li

d = 0.53 g/cm3

Water, H2O d = 1.0 g/cm3

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Substances that have less density will float on substances that have greater density.

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I.) Determining Density

Since density is defined as the ratio between mass and volume for a substance, if we can measure the mass and volume of the substance we can calculate its density.

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Density =MassVolume

Let’s practice.

A bar of silver has a mass of 68.0 g. and a volume of 6.48 cm3. What is the density of silver.

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Density as a Conversion FactorDensity as a Conversion Factor

Density can be used as a conversion to convert mass to volume and vice versa.

78

Since density is defined as the ratio of mass to volume for a substance:

Let’s practice.

What is the volume of a pure silver coin that has a mass of 14 g. The density of silver (Ag) is 10.5 g/cm3.

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II.) Density and Temperature

• The volume of most substances increase with increasing temperature.

• The mass does not change with temperature.

80

What then happens to the density when temperature is increased?

The density decreases, with one important exception.

Scientific Measurem

entChapter 3

The End81