1 chem 10: chapter two measurement & problem solving

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CHEM 10: CHAPTER TWO

MEASUREMENT & PROBLEM SOLVING

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Scientific Notation

Very large and very small numbers are often encountered in science.

Large: 602210000000000000000000

And small: 0.00000000000000000000625

Very large and very small numbers like these are awkward and difficult to work with.

A method for representing these numbers in a simpler form is called scientific notation.

Large: 6.022 x 1023 And small: 6.25 x 10-21

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Scientific Notation

To write a number as a power of 10

- 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.

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Write 6419 in scientific notation.

64196419.641.9x10164.19x1026.419 x 103

decimal after first nonzero

digitpower of 10

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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

digitpower of 10

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Scientific Notation

Adding and subtracting in exponential or scientific notation:

- must be in same power of ten

Try adding: 3.47x102 + 5.93267x105

0.00347x105 + 5.93267x105 = 5.93614x105

Scientific Notation

Multiplying and dividing: 10a * 10b = 10a+b 10a/10b = 10a-b

Your calculator will do all this for you, if you enter the numbers correctly!

Group practice:A. 3.47 x 102 * 1.20 x 10-3 =B. 0.0012 + 1.3 x 10-2 =C. 3.47 x 102 / 1.20 x 10-3 =A. 4.16 x 10-1

B. 1.4 x 10-2

C. 2.89 x 105

Measurements

Experiments are performed.

Numerical values or data are obtained from these measurements.

The values are recorded to the most significant digits provided by the measuring device.

The units (labels) are recorded with the values.

Form of a Measurement

70.0 kilograms = 154 pounds

numerical value

unit

Significant Figures

The number of digits that are known plus one estimated digit are considered significant in a measured quantity

estimated5.16143

known

estimated6.06320

Significant Figures

The number of digits that are known plus one estimated digit are considered significant in a measured quantity

known

Temperature is estimated to be 21.2oC. The last 2 is uncertain.

The temperature 21.2oC is expressed to 3 significant figures.

12 inches = 1 foot100 centimeters = 1 meter

Exact numbers have an infinite number of significant figures.

Exact numbers occur in simple counting operations

Exact Numbers

• Defined numbers are exact.

12345

461

All nonzero numbers are significant.

Significant Figures

461

3 Significant Figures

All nonzero numbers are significant.

Significant Figures

401


A zero is significant when it is between nonzero digits.

Significant Figures



600.39

Significant Figures


30.9


Significant Figures

A zero is significant at the end of a number that includes a decimal point.


000.55

Significant Figures

A zero is significant at the end of a number that includes a decimal point.


0391.2

Significant Figures

A zero is not significant when it is before the first nonzero digit.

1 Significant Figure

600.0

Significant Figures

A zero is not significant when it is before the first nonzero digit.


907.0

Significant Figures

A zero is not significant when it is at the end of a number without a decimal point.

1 Significant Figure

00005

Significant Figures

A zero is not significant when it is at the end of a number without a decimal point.


01786

Significant Figures

Rounding Off Numbers

Often when calculations are performed extra digits are present in the results.

It is necessary to drop these extra digits so as to express the answer to the correct number of significant figures.

When digits are dropped the value of the last digit retained is determined by a process known as rounding off numbers.

80.873

Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed.

4 or less


1.875377

Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed.

4 or less


5 or greater

5.459672

Rule 2. When the first digit after those you want to retain is 5 or greater, that digit and all others to its right are dropped. The last digit retained is increased by 1.

drop these figuresincrease by 1

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CALCULATIONS AND SIGNIFICANT FIGURES

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

Learn how to determine number of sig figs in answers after performing calculations, including multiply/divide and add/subtract.

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.

CALCULATIONS AND SIGNIFICANT FIGURES

The results of an addition or a subtraction must be expressed to the same precision as the least precise measurement.

- The result must be rounded to the same number of decimal places as the value with the fewest decimal places.

(190.6)(2.3) = 438.38

438.38

Answer given by calculator.

2.3 has two significant figures.

190.6 has four significant figures.

The answer should have two significant figures because 2.3 is the number with the fewest significant figures.

Drop these three digits.

Round off this digit to four.

The correct answer is 440 or 4.4 x 102

Add 125.17, 129 and 52.2

125.17129.

52.2306.37


Least precise number.

Round off to the nearest unit.

306.37

Correct answer.

1.039 - 1.020Calculate

1.039

1.039 - 1.020 = 0.018286814

1.039


1.039 - 1.020 = 0.019

0.019 = 0.018286814

1.039

The answer should have two significant figures because 0.019 is the number with the fewest significant figures.

2 80.018 6814

Two significant figures.

Drop these 6 digits.

0.018286814

Correct answer.

The Metric System

The metric or International System (SI, Systeme International) is a decimal system of units.

It is built around standard units.

It uses prefixes representing powers of 10 to express quantities that are larger or smaller than the standard units.

International System’s Standard Units of Measurement

Quantity Name of Unit Abbreviation

Length meter m

Mass kilogram kg Temperature Kelvin K

Time second s

Amount of substance mole mol

Prefixes and Numerical Values for SI Units Power of 10

Prefix Symbol Numerical Value Equivalent

exa E 1,000,000,000,000,000,000 1018

peta P 1,000,000,000,000,000 1015

tera T 1,000,000,000,000 1012

giga G 1,000,000,000 109

mega M 1,000,000 106

kilo k 1,000 103

hecto h 100 102

deca da 10 101

— — 1 100

Prefixes and Numerical Values for SI Units

deci d 0.1 10-1

centi c 0.01 10-2

milli m 0.001 10-3

micro 0.000001 10-6

nano n 0.000000001 10-9

pico p 0.000000000001 10-12

femto f 0.00000000000001 10-15

atto a 0.000000000000000001 10-18

Power of 10Prefix Symbol Numerical Value Equivalent

GROUP RACE FOR ANSWERS IN METRIC

a. one millionth of a scope = ____scope

b. 0.01 mental = ____mental

c. 1,000,000 phones = ___phones

d. 2000 mockingbird =___bird

e. 1/1000 tary =___tary

MEMORIZE THESE ENGLISH/METRIC CONVERSIONS: (table 2.3 plus others)

1 lb = 453.59 grams 1 inch = 2.54 cm

(exactly)

1 mile = 1.609 km 1.0567 qt = 1 L

1000 mL = 1 L

1 mL = 1 cm3

1 cal = 4.184 Joule 1 atm = 760.00 torroF = 1.8oC + 32 K = oC + 273.15

Dimensional Analysis

Dimensional analysis converts one unit to another by using conversion factors.

unit1 x conversion factor = unit2

Basic Steps

1. Read the problem carefully. Determine what is to be solved for and write it down.

2. Tabulate the data given in the problem.Label all factors and measurements with the proper units.


Basic steps – continued:3. Determine which principles are involved

and which unit relationships are needed to solve the problem.

You may need to refer to tables for needed data.

4. Set up the problem in a neat, organized and logical fashion.

Make sure unwanted units cancel. Use sample problems in the text as guides for setting up the problem.


Basic steps continued:

5. Proceed with the necessary mathematical operations.

Make certain that your answer contains the proper number of significant figures.

6. Check the answer to make sure it is reasonable.

Metric Units of Length Exponential

Unit Abbreviation Metric Equivalent Equivalent

kilometer km 1,000 m 103 m

meter m 1 m 100 m

decimeter dm 0.1 m 10-1 m

centimeter cm 0.01 m 10-2 m

millimeter mm 0.001 m 10-3 m

micrometer m 0.000001 m 10-6 m

nanometer nm 0.000000001 m 10-9 m

Use the conversion factor with millimeters in the numerator and meters in the denominator.

1000 mmx

1 m2.5 m = 2500 mm

32.5 x 10 mm

How many millimeters are there in 2.5 meters?

Convert 3.7 x 103 cm to micrometers.

33.7 x 10 cm1 m

x 100 cm

610 μmx

1 m7 = 3.7 x 10 μm

Centimeters can be converted to micrometers by writing down conversion factors in succession.

cm m meters

Centimeters can be converted to micrometers by two stepwise conversions.

cm m meters

33.7 x 10 cm1 m

x 100 cm

1 = 3.7 x 10 m

610 μmx

1 m7 = 3.7 x 10 μm13.7 x 10 m

Convert 3.7 x 103 cm to micrometers.

16.0 in2.54 cm

x 1 in

= 40.6 cm

2.54 cm1 in

Use this conversion factor

Convert 16.0 inches to centimeters.

Metric Units of mass Exponential

Unit Abbreviation Gram Equivalent Equivalent

kilogram kg 1,000 g 103 g

gram g 1 g 100 g

decigram dg 0.1 g 10-1 g

centigram cg 0.01 g 10-2 g

milligram mg 0.001 g 10-3 g

microgram g 0.000001 g 10-6 g

Convert 45 decigrams to grams.

45 dg1 g

x 10 dg

= 4.5 g

1 g = 10 dg

An atom of hydrogen weighs 1.674 x 10-24 g. How many ounces does the atom weigh?

1 lbx

454 g-241.674 x 10 g -27 3.69 x 10 lb

16 ozx

1 lb-26 5.90 x 10 oz-273.69 x 10 lb

1 lb = 454 g

16 oz = 1 lb

Grams can be converted to ounces by a series of stepwise conversions.

An atom of hydrogen weighs 1.674 x 10-24 g. How many ounces does the atom weigh?

-241.674 x 10 g1 lb

x454 g

16 ozx

1 lb-26 5.90 x 10 oz

Grams can be converted to ounces using a linear expression by writing down conversion factors in succession.

Derived Units: Area & Volume

Area: Measure of the amount of two-dimensional space occupied. It is a derived unit from the two dimensions of area: length x width.

Volume: Measure of the amount of three-dimensional space occupied. It is a derived unit from the three dimensions of volume: length x width x height.SI unit = cubic meter (m3)

Usually measure liquid or gas volume in milliliters (mL)1 L is slightly larger than 1 quart (1 L = 1.0567 qt)1 L = 1000 mL = 103 mL 1 mL = 0.001 L = 10-3 L1 mL = 1 cm3

Convert 4.61 x 102 microliters to milliliters.

Microliters can be converted to milliliters by a series of stepwise conversions.

L L mL

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1 Lx

10 μL24.61x10 μL -4 4.61x10 L

-1 = 4.61 x 10 mL-44.61x10 L1000 mL

x1 L

Microliters can be converted to milliliters using a linear expression by writing down conversion factors in succession.

L L mL

24.61x10 μL 6

1 Lx

10 μL1000 mL

x1 L

-1= 4.61 x 10 mL

Convert 4.61 x 102 microliters to milliliters.

Heat

A form of energy that is associated with the motion of small particles of matter.

Heat refers to the quantity of this energy associated with the system.

The system is the entity that is being heated or cooled.

Temperature

A measure of the intensity of heat.

It does not depend on the size of the system.

Heat always flows from a region of higher temperature to a region of lower temperature.

Temperature Measurement

The SI unit of temperature is the Kelvin.

There are three temperature scales: Kelvin, Celsius and Fahrenheit.

In the laboratory temperature is commonly measured with a thermometer.

180 Farenheit Degrees

= 100 Celcius degrees

180 =1.8

100

It is not uncommon for temperatures in the Canadian plains to reach –60.oF and below

during the winter. What is this temperature in oC and K?

oo F - 32C =

1.8

o o60. - 32C = = -51 C

1.8

It is not uncommon for temperatures in the Canadian planes to reach –60.oF and below

during the winter. What is this temperature in oC and K?

oK = C + 273.15

oK = -51 C + 273.15 = 222 K

Density is the ratio of the mass of a substance to the volume occupied by that substance.

massd =

volume

Density varies with temperature

o

2

4 CH O

1.0000 g gd = = 1.0000

1.0000 mL mL

o

2

80 CH O

1.0000 g gd = = 0.97182

1.0290 mL mL

Density examples

Try to rank: air, lead, feathers, water, gasoline

Air < feathers < gasoline < water < lead

1.28 g/L <0.5 g/mL <0.7 g/mL< 1.0 g/mL < 11.3 g/mL

gas (solid) liquid liquid solid

The density of ether is 0.714 g/mL. What is the mass of 25.0 milliliters of ether?

Method 1 (a) Solve the density equation for mass.

massd =

volume

(b) Substitute the data and calculate.

mass = density x volume

0.714 g25.0 mL x = 17.9 g

mL

The density of ether is 0.714 g/mL. What is the mass of 25.0 milliliters of ether?

Method 2 Dimensional Analysis. Use density as a conversion factor. Convert:

0.714 g25.0 ml x = 17.9 g

mL

mL → g

gmL x = g

mLThe conversion of units is

The density of oxygen at 0oC is 1.429 g/L. What is the volume of 32.00 grams of oxygen at this temperature?

Method 1

(a) Solve the density equation for volume.

massd =

volume

(b) Substitute the data and calculate.

massvolume =

density

2

2

32.00 g Ovolume = = 22.40 L

1.429 g O /L

The density of oxygen at 0oC is 1.429 g/L. What is the volume of 32.00 grams of oxygen at this temperature?

Method 2 Dimensional Analysis. Use density as a conversion factor. Convert:

2 22

1 L32.00 g O x = 22.40 L O

1.429 g O

g → L

Lg x = L

gThe conversion of units is

SPECIFIC GRAVITY

• An older concept still used in many places.

• Specific gravity of a liquid or solid is always its density referenced to water at the same temperature.

• Specific gravity of a gas is always referenced to air at the same temperature and pressure.

1 chem 10: chapter two measurement & problem solving

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