CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING

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CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING

Problems: 1-64, 69-88, 91-120, 123-124

2.1 Measuring Global Temperatures

measurement: a number with attached units

When scientists collect data, it is important that they record the measurements as accurately as possible, and the measurements must reflect the accuracy and precision of the instruments used to collect that data.

Consider the following plot of global land-ocean temperatures based on measurements taken from meteorological stations and ship and satellite temperature (SST) measurements:

Source: Hansen, J., Mki. Sato, R. Ruedy, K. Lo, D.W. Lea, and M. Medina-Elizade, 2006: Global temperature change. Proc. Natl. Acad. Sci., 103, 14288-14293, doi:10.1073/pnas.0606291103.

(http://pubs.giss.nasa.gov/abstracts/2006/Hansen_etal_1.html)

The plot above shows annual mean (average) temperatures in black, 5-year mean temperatures in red, and the uncertainty as green bars.

Ex. 1: Based on this plot how have global land-ocean temperatures changed since the 1950s?

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2.3 SIGNIFICANT FIGURES (or SIG FIGS): Writing Numbers to Reflect Precision

To measure, one uses instruments = tools such as a ruler, balance, etc.

All instruments have one thing in common: UNCERTAINTY!

→ INSTRUMENTS CAN NEVER GIVE EXACT MEASUREMENTS!

When a measurement is recorded, all the given numbers are known with certainty (given the markings on the instrument), except the last number is estimated.

→ The digits are significant because removing them changes the measurement's uncertainty.

– Thus, when measurements are recorded,

– they are recorded to one more decimal place than the markings for analog instruments;

– they are recorded exactly as displayed on electronic (digital) instruments.

LENGTH

– generally reported in meters, centimeters, millimeters, kilometers, inches, feet, miles – Know the following English-English conversions: 1 foot ≡ 12 inches 1 yard ≡ 3 feet

Example: Using Rulers A, B, and C below, indicate the measurement to the line indicated for each ruler. Assume these are centimeter rulers, so show the each

measurement has units of cm. Circle the estimated digit for each measurement.

Ruler A Ruler B

0 1 2 3 4 5

Ruler C

4.1 4.2 4.3 4.4

A B C

Measurement

Increment of the smallest markings on ruler

# of sig figs

Thus, a measurement is always recorded with one more digit than the smallest

markings on the instrument used, and measurements with more sig figs are usually

more accurate.

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Guidelines for Sig Figs (if measurement is given):

Count the number of digits in a measurement from left to right:

1. When there is a decimal point:

– For measurements greater than 1, count all the digits (even zeros).

– 62.4 cm has 3 sig figs, 5.0 m has 2 sig figs, 186.100 g has 6 s.f.

– For measurements less than 1, start with the first nonzero digit and count all digits (even zeros) after it.

– 0.011 mL and 0.00022 kg each have 2 sig figs 2. When there is no decimal point:

– Count all non-zero digits and zeros between non-zero digits – e.g. 125 g has 3 sig figs, 1007 mL has 4 sig figs

– Placeholder zeros may or may not be significant – e.g. 1000 may have 1, 2, 3 or 4 sig figs

Example: Indicate the number of significant digits for the following:

a. 165.3 g ___ c. 90.40 m _ e. 0.19600 g ___

b. 105 cm ___ d. 100.00 L _ f. 0.0050 cm ___

2.5 THE BASIC UNITS OF MEASUREMENT

VOLUME: Amount of space occupied by a solid, gas, or liquid.

– generally in units of liters (L), milliliters (mL), or cubic centimeters (cm

³

) – Know the following:

1 L ≡ 1 dm

³

1 mL ≡ 1 cm

³

(These are both exact!)

Note: When the relationship between two units or items is exact, we use the “≡” to mean

“equals exactly” rather than the traditional “=” sign.

– also know the following equivalents in the English system

1 gallon ≡ 4 quarts 1 quart ≡ 2 pints 1 pint ≡ 2 cups

MASS: a measure of the amount of matter an object possesses – measured with a balance and NOT AFFECTED by gravity – usually reported in grams or kilograms

WEIGHT: a measure of the force of gravity

– usually reported in pounds (abbreviated lbs)

mass ≠ weight = mass × acceleration due to gravity

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Mass is not affected by gravity!

2.2 SCIENTIFIC NOTATION

Some numbers are very large or very small → difficult to express.

Avogadro’s number = 602,000,000,000,000,000,000,000

an electron’s mass = 0.000 000 000 000 000 000 000 000 000 91 kg

To handle such numbers, we use a system called scientific notation. Regardless of their magnitude, all numbers can be expressed in the form

N×10

ⁿ

where N =digit term= a number between 1 and 10, so there can only be one number to the left of the decimal point: #.####

n = an exponent = a positive or a negative integer (whole #).

To express a number in scientific notation:

– Count the number of places you must move the decimal point to get N between 1 and 10.

Moving decimal point to the right (if # < 1) → negative exponent.

Moving decimal point to the left (if # > 1) → positive exponent.

Example: Express the following numbers in scientific notation (to 3 sig figs):

555,000 → __________________

0.000888 → __________________

602,000,000,000,000,000,000,000 → ___________________________

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Some measurements may be rounded to a number of sig figs requiring scientific notation.

For example,

Express 100.0 g to 3 sig figs: _ → ____

Express 100.0 g to 2 sig figs: _ → ____

Express 100.0 g to 1 sig fig: _ → ____

UNBIASED ROUNDING (or ROUND-TO-EVEN METHOD) How do we eliminate nonsignificant digits?

• If first nonsignificant digit < 5, just drop the nonsignificant digits

• If first nonsignificant digit ≥ 5, raise the last sig digit by 1 and drop nonsignificant digits – e.g. 3.14501 ⎯ ⎯

^to³

⎯

^s.f.

⎯ → 3.15 (since the nonsig figs are 501 in 3.14501)

Express each of the following with the number of sig figs indicated:

a. 648.75

⎯ to 3 sig figs⎯ ⎯ ⎯ ⎯ ⎯ →

_______________________

b. 23.6500 ⎯ ⎯

^to³

⎯

^sig

⎯

^figs

⎯ ⎯ → _______________________

c. 64.55 ⎯ ⎯

^to³

⎯

^sig

⎯

^figs

⎯ ⎯ → _______________________

d. 0.00123456 ⎯ ⎯

^to³

⎯

^sig

⎯

^figs

⎯ ⎯ → _______________________

e. 1,234,567

⎯ to 5 sig figs⎯ ⎯ ⎯ ⎯ ⎯ →

_______________________

f. 1975 ⎯ ⎯

^to²

⎯

^sig

⎯

^figs

⎯ ⎯ → _______________________

When necessary express measurements in scientific notation to clarify the number of sig figs.

2.4 SIGNIFICANT FIGURES IN CALCULATIONS ADDING/SUBTRACTING MEASUREMENTS

When adding and subtracting measurements, your final value is limited by the measurement with the largest uncertainty—i.e. the measurement with the fewest decimal places.

MULTIPLYING/DIVIDING MEASUREMENTS

When multiplying or dividing measurements, the final value is limited by the measurement with

the least number of significant figures.

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6.02

x10

23 2.50131

x10

33

4.155

x10

9 Ex. 1: 7.4333 g + 8.25 g + 10.781 g = _________________________

Ex. 2: 13.50 cm × 7.95 cm × 4.00 cm = _________________________

Ex. 3: 9.75 mL − 7.35 mL = _________________________

Ex. 4:

cm 8.50 cm

10.25 cm

25.75 g 101.755

×

× = _________________________

MULTIPLYING/DIVIDING WITH EXPONENTIAL NUMBERS:

When multiplying or dividing measurements with exponents, use the digit term (N in “N ×10

ⁿ

”) to determine number of sig figs.

Ex. 1: (6.02×10

²³

)(4.155×10

⁹

) = 2.50131×10

³³

How do you calculate this using your scientific calculator?

Step 1. Enter “6.02×10

²³

” by pressing:

6.02 then EE or EXP (which corresponds to “×10”) then 23

→ Your calculator should look similar to:

Step 2. Multiply by pressing: ×

Step 3. Enter “4.155× 10

⁹

” by pressing:

4.155 then EE or EXP (which corresponds to “×10”) then 9

→ Your calculator should look similar to:

Step 4. Get the answer by pressing: =

→ Your calculator should now read

The answer with the correct # of sig figs = ___________________

Be sure you can do exponential calculations with your calculator. Many calculations we do in chemistry involve numbers in scientific notation.

Ex. 2: (3.25×10

¹²

) (8.6×10

⁴

) = 2.795 ×10

¹⁷

⎯ ⎯

^to^correct

⎯ ⎯

^#^of

⎯

^sig

⎯

^figs

⎯ → _________________

Ex. 3:

¹⁵₄ 10 8.605

10 3.75

×

= 4.357931435×10

¹⁰

⎯ ⎯

^to^correct

⎯ ⎯

^#^of

⎯

^sig

⎯

^figs

⎯ → __________________

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SIGNIFICANT DIGITS AND EXACT NUMBERS

Although measurements can never be exact, we can count an exact number of items. For example, we can count exactly how many students are present in a classroom, how many M&Ms are in a bowl, how many apples in a barrel. We say that exact numbers of objects have an infinite number of significant figures.

2.6 CONVERTING FROM ONE UNIT TO ANOTHER (or DIMENSIONAL ANALYSIS) UNIT EQUATIONS AND UNIT FACTORS

Unit equation: Simple statement of two equivalent values Conversion factor = unit factor = equivalents:

- Ratio of two equivalent quantities

Unit equation Unit factor 1 dollar ≡ 10 dimes 1 dollar

10 dimes or 10 dimes 1 dollar

Unit factors are exact if we can count the number of units equal to another.

For example, the following unit factors and unit equations are exact:

7 days

1 week 24 hours

1 day 1 gallon

4 quarts 100 cm

1 m and 1 yard ≡ 3 feet Exact equivalents have an infinite number of sig figs

→ never limit the number of sig figs in calculations!

Other equivalents are inexact or approximate because they are measurements or approximate relationships, such as

1.61 km

1 mile 55 miles

1 hour 454 g lb Approximate equivalents do limit the sig figs for the final answer.

2.7 SOLVING MULTSTEP CONVERSION PROBLEMS (or DIMENSIONAL ANALYSIS PROBLEM SOLVING) 1. Write the units for the answer.

2. Determine what information to start with.

3. Arrange all unit factors (showing them as fractions with units), so all of the units cancel except those needed for the final answer.

4. Check for the correct units and the correct number of sig figs in the final answer.

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Example 1: If a marathon is 26.2 miles, then a marathon is how many yards?

(1 mile≡5280 feet, 1 yard≡3 feet)

Example 2: You and a friend decide to drive to Portland, which is about 175 miles from Seattle. If you average 99 kilometers per hour with no stops, how many hours does it take to get there? (1 mile = 1.609 km)

Example 3: The speed of light is about 2.998×10

⁸

meters per second.

Express this speed in miles per hour. (1 mile=1.609 km, 1000 m≡1 km)

2.5 Basic Units of Measurement

International System or SI Units (from French "le Système International d’Unités") – standard units for scientific measurement

Metric system: A decimal system of measurement with a basic unit for each type of measurement

quantity basic unit (symbol) quantity SI unit (symbol)

length meter (m) length meter (m)

mass gram (g) mass kilogram (kg)

volume liter (L) time second (s)

time second (s) temperature Kelvin (K)

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Metric Prefixes

− Multiples or fractions of a basic unit are expressed as a prefix

→ Each prefix = power of 10

→ The prefix increases or decreases the base unit by a power of 10.

Prefix Symbol Multiple/Fraction

kilo k 1000

deci d 0.1 ≡

10 1

centi c 0.01 ≡

100 1

milli m 0.001 ≡

1000 1

micro μ (Greek “mu”) 0.000 001 ≡

1,000,000 1

KNOW the metric units above!

Metric Conversion Factors

Ex. 1 Complete the following unit equations:

a. 1 dollar ≡ cents → ¹ ^m ≡ cm b. 1 dollar ≡ dimes → 1 m ≡ dm

Note: To help remember the number of centimeters or decimeters in a meter, just think of the number of cents or dimes in a dollar!

Ex. 2 Complete the following unit equations:

a. 1 kg ≡ g c. 1 L ≡ mL e. 1 m ≡ mm

b. 1 g ≡ cg d. 1 s ≡ ds f. 1 g ≡ μ g Note: Although scientists use μ g to abbreviate microgram, hospitals avoid using the

Greek letter μ in handwritten orders since it might be mistaken for an m for milli

— i.e., an order for 200 μ g might be mistaken to be 200 mg which would lead to an overdose that’s 1000 times more concentrated.

Instead, hospitals use the abbreviation mcg to indicate micrograms.

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Writing Unit Factors

Example: Complete the following unit equations then write two unit factors for each equation:

a. 1 km ≡ m b. 1 g ≡ _ mg

Metric-Metric Conversions: Solve the following using dimensional analysis.

Ex. 1 Convert 175 ms into units of seconds.

Ex. 2 Convert 0.120 kilograms into milligrams.

Ex. 3 Convert 3.00×10

⁸

m/s into kilometers per hour.

Ex. 4 Convert 3.50×10

⁷

cm to units of kilometers.

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Metric-English Conversions

English system: Our general system of measurement.

Scientific measurements are exclusively metric. However, most Americans are more familiar with inches, pounds, quarts, and other English units.

→ Conversions between the two systems are often necessary.

These conversions will be given to you on quizzes and exams.

Quantity English unit Metric unit English–Metric conversion length 1 inch (in) 1 cm 1 in. ≡ 2.54 cm (exact)

mass 1 pound (lb) 1 g 1 lb = 453.6 g (approximate) volume 1 quart (qt) 1 mL 1 qt = 946 mL (approximate)

Ex. 1 What is the mass in kilograms of a person weighing 155 lbs?

Ex. 2 A 2.0-L bottle can hold how many cups of liquid? (1 qt. ≡ 2 pints, 1 pint ≡ 2 cups)

Ex. 3 A light-year (about 5.88×10

¹²

miles) is the distance light travels in one year.

Calculate the speed of light in meters per second. (1 mile=1.609 km)

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Temperature:

– A measure of the average energy of a single particle in a system.

The instrument for measuring temperature is a thermometer.

Temperature is generally measured with these units:

References

Fahrenheit scale (°F) English system

Celsius scale (°C) Metric system

freezing point for water 32°F 0°C

boiling point for water 212°F 100°C

Nice summer day in Seattle 77°F 25°C

Conversion between Fahrenheit and Celsius scales:

°C = (°F - 32) 1.8 °F = (°C ×1.8) + 32

Kelvin Temperature Scale

– There is a third scale for measuring temperature: the Kelvin scale.

– The unit for temperature in the Kelvin scale is Kelvin (K, NOT °K!).

– The Kelvin scale assigns a value of zero kelvins (0 K) to the lowest possible temperature, which we call absolute zero and corresponds to –273.15°C.

– The term absolute zero is used to indicate the theoretical lowest temperature.

Conversion between °C and K: K = ˚C + 273 ˚C = K – 273

Ex. 1 Liquid nitrogen is so cold, it can be used

to make a banana hammer. If liquid nitrogen’s

temperature is 77 K, calculate the equivalent

temperature in ˚C and in ˚F?

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Determining Volume

– Volume is determined in three principal ways:

1. The volume of any liquid can be measured directly using calibrated glassware in the laboratory (e.g. graduated cylinder, pipets, burets, etc.)

2. The volume of a solid with a regular shape (rectangular, cylindrical, uniformly spherical or cubic, etc.) can be determined by calculation.

3. Volume of solid with an irregular shape can be found indirectly by the amount of liquid it displaces. This technique is called volume by displacement.

VOLUME BY CALCULATION

The volume of a rectangular solid can be calculated as follows:

volume = length × width × thickness

Ex. 1 What is the volume of a gold bar that is 5.25 cm long, 3.50 cm wide, and 2.75 cm thick?

Ex. 2 A rectangular bar of gold with a volume of 35.5 cm

³

is 7.50 cm long and 3.50 cm wide.

How thick is the bar?

VOLUME BY DISPLACEMENT

a. Fill a graduated cylinder halfway with water, and record the initial volume.

b. Carefully place the object into the graduated cylinder so as not to splash or lose any water.

c. Record the final volume.

d. Volume of object = final volume – initial volume

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Example: What is the volume of the piece of green jade in the figure below?

2.9 DENSITY: The amount of mass in a unit volume of matter

V d m volume

density = mass or = generally in units of g/cm

³

or g/mL

For water, 1.00 g of water occupies a volume of 1.00 cm

³

:

₃ _1.00_g/cm³ cm

1.00 g 1.00 V

d= m = =

Density also expresses the concentration of mass – i.e., the more concentrated the mass in an object

→ the heavier the object → the higher its density Sink or Float

Some objects float on water (e.g. a cork), but others sink (e.g. a penny). Thus, objects with a higher density than a liquid will sink in the liquid, but those with a lower density than the liquid will float.

Since water's density is about 1.00 g/cm

³

, a cork's density must be less than 1.00 g/cm

³

, and a penny's density must be greater.

Ex.: Consider the figure at the right and the following solids and liquids and their densities:

ice (d=0.917 g/cm

³

) honey (d=1.50 g/cm

³

) iron cube (7.87 g/cm

³

) hexane (d=0.65 g/cm

³

) rubber cube (d=1.19 g/cm

³

)

Identify L

1

, L

2

, S

1

, and S

2

by filling in the blanks below:

L

1

= _______________ and L

2

= _______________

S

1

= _______________, S

2

= _______________, and S

3

= _______________

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Applying Density as a Unit Factor

Given the density for any matter, you can always write two unit factors. For example, the density of ice is 0.917 g/cm

³

.

Two unit factors would be:

0.917g cm3 or cm3 0.917g

Ex. 1 Give 2 unit factors for each of the following:

a. density of lead = 11.3 g/cm

³

b. density of chloroform = 1.48 g/mL

Ex. 2 Aluminum has a density of 2.70 g/cm

³

. What is the volume (in mL) of a piece of aluminum with a mass of 0.125 kg?

Ex. 3 Ethanol is used in alcoholic beverages and has a density of 0.789 g/mL. What is the mass of ethanol that has a volume of 1.50 L?

Ex. 4 A chunk of silver metal weighing 168 g is placed in a graduated cylinder with 21.0 mL of

water. The volume of water now reads 37.0 mL. Calculate the density of silver.

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CALCULATING PERCENTAGES

Percent: Ratio of parts per 100 parts → 10% is 100

10 , 25% is 100

25 , etc.

To calculate percent, divide one quantity by the total of all quantities in sample:

Percentage =

sample total

part

one ×100%

Ex. 1 In a chemistry class with 25 women and 20 men, what percentage of the class is female? What percentage is male? (Express your answers to 3 sig figs.)

Writing out Percentage as Unit Factors

Ex. 1: Water is 88.8% oxygen by mass. Write two unit factors using this info.

Ex. 2: Pennies cast between 1963 and 1982 are a mixture of 95.0% copper and 5.0% zinc by

mass. Write four unit factors using this information.

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CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING