CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING€¦ · CHEM 121: Chapter 2 page 7 of 15 Your calculator should look similar to: 6.02 x10 23 2.50131 x10 33 4.155 x10 9 Step 1. Enter “6.02×1023”

CHEM 121: Chapter 2 page 1 of 15

CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING

Suggested Problems: 1-8, 11-12, 14-22, 27-62, 67-122 2.1 Measuring Global Temperatures measurement: a number with attached units When scientists collect data, it is important that they record the measurements are accurately as possible, and they also report the measurements taken to reflect the accuracy and precision of the instruments they 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. This plot indicates that global land-ocean temperatures have increased by about 0.6°C since the 1960s.


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 Consider the length of the sample indicated with each of the following rulers:

Example: Record the measurement in centimeters using each of the rulers above.

Ruler Measurement/quantity # of sig figs

Top

Middle

Bottom


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 (cm3) – Know the following:

1 L ≡ 1 dm3 1 mL ≡ 1 cm3 (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


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×10n 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 → ___________________________


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: ___________ → ______________

ROUNDING OFF NONSIGNIFICANT DIGITS It is safer to NEVER round or truncate, but to indicate the last significant digit by underlining it and keeping one extra digit. You must be able to round answers if necessary using the normal method, but only to present final results. How do we eliminate nonsignificant digits? • If first nonsignificant digit < 5, just drop ALL nonsignificant digits • If first nonsignificant digit ≥ 5, raise the last sig digit by 1 then drop ALL nonsignificant digits

For example, express 72.58643 with 3 sig figs: 72.58643

!

to 3 sig figs" # " " " " " _______________

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

a. 648.75

!

to 3 sig figs" # " " " " " _______________________

b. 23.6500 ⎯⎯⎯⎯⎯ →⎯ figs sig 3 to _______________________

c. 64.55 ⎯⎯⎯⎯⎯ →⎯ figs sig 3 to _______________________ d. 0.00123456 ⎯⎯⎯⎯⎯ →⎯ figs sig 3 to _______________________ e. 1,234,567

!

to 5 sig figs" # " " " " " _______________________

f. 1975 ⎯⎯⎯⎯⎯ →⎯ figs sig 2 to _______________________ When necessary express measurements in scientific notation to clarify the number of sig figs.

72.58643 g

last significant digit

first nonsignificant digit


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.

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

7.4333 8.25 10.781

# decimal. places # sig. figures

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

13.50 7.95 4.00 # decimal. places

# sig. figures Ex. 3: 9.75 mL − 7.35 mL = _________________________

9.75 7.35 # decimal. places

# sig. figures

Ex. 4: cm 8.50 cm 10.25 cm 25.75

g 101.755 ××

= _________________________

101.755 25.75 10.25 8.50

# decimal. places # sig. figures

MULTIPLYING/DIVIDING WITH EXPONENTIAL NUMBERS:

When multiplying or dividing measurements with exponents, use the digit term (N in “N ×10n”) to determine number of sig figs. Ex. 1: (6.02×1023)(4.155×109) = 2.50131×1033 How do you calculate this using your scientific calculator?


6.02 x1023

2.50131 x1033

4.155 x109

Step 1. Enter “6.02×1023” 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× 109” 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: (3.25×1012) (8.6×104) = 2.795 ×1017 ⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to _________________

Ex. 3: 4

15

108.605103.75×

× = 4.357931435×1010 ⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to __________________

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 dollar10 dimes

or 10 dimes1 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 days1 week

24 hours1 day

1 gallon4 quarts

100 cm1 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 km1 mile

55 miles1 hour

454 glb

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. Ex. 1: The Hope Diamond is a large, blue diamond that weighs 0.020 lb. How many carats is

the Hope Diamond? (1 lb. = 453.6 g and 1 carat = 0.200 g) Ex. 2 The distance from the Earth to the Sun is about 93 million miles. If light travels at a

speed of 2.998×108 m/s, how many minutes does it take for light from the Sun to reach the Earth? (1 mile = 1.609 km and 1 km≡1000 m)


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) 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

tera T 1,000,000,000,000 ≡ 1012

giga G 1,000,000,000 ≡ 109

mega M 1,000,000 ≡ 106

kilo k 1000 ≡ 103

deci d 0.1 ≡ 10

1 ≡ 10-1

centi c 0.01 ≡ 100

1 ≡ 10-2

milli m 0.001 ≡ 1000

1 ≡ 10-3

micro µ (Greek “mu”) 10–6

nano n 10–9

pico p 10–12

femto f 10–15


Metric Conversion Factors

Ex. 1 Complete the following unit equations:

a. 1 kg ≡ ________ g d. 1 L ≡ ________ mL g. 1 s ≡ _______ fs

b. 1 m ≡ ________ nm e. 1 g ≡ ________ µg h. 1 m ≡ _______ pm

c. 1 cm ≡ ________ m f. 1 megaton ≡ ________ tons

Writing Unit Factors: Write two unit factors using the unit equations for examples a, and c. Metric-Metric Conversions: Solve the following using dimensional analysis. Ex. 1 Convert 175 kilograms into milligrams.

Ex. 2 Convert 25.0 dm into micrometers, µm (also called “microns”). 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 175 lbs? Ex. 2 What is the volume in cups for a 2.0-L bottle? Ex. 3: The speed of light is about 2.998×108 meters per second. Express this speed in

miles per hour. (1 mile=1.609 km) Ex. 4: If a car averages 6.72 L per 100 kilometers, what is its fuel efficiency in miles per

gallon? (1 mile=1.609 km)


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. – e.g. volume of rectangular solid = length × width × thickness

volume of a sphere = 34πr3

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 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

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

Vwater = 50.0 mL Vwater + Vjade = 60.5 mL - Vwater = 50.0 mL Vwater = 10. 5 mL jade


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

Vmd

volumemassdensity = or = generally in units of g/cm3 or g/mL

For water, 1.00 g of water occupies a volume of 1.00 cm3: 33 g/cm 1.00

cm 1.00g 1.00

Vmd ===

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/cm3.

Two unit factors would be: 0.917g

cm or cm

0.917g 3

3

Example: Give 2 unit factors for each of the following: a. density of lead = 11.3 g/cm3 b. density of chloroform = 1.48 g/mL Solve the following problems: Ex. 1 A piece of silver metal weighing 194.295 g is placed in a graduated cylinder containing

42.0 mL of water. The volume of water now reads 60.5 mL. Calculate the density of silver.

Ex. 2 In the opening sequence of “Raiders of the Lost Ark,” Indiana Jones steals a gold

statue by replacing it with a bag of sand. If the statue has a volume of about 1.5 L and gold has a density of 19.3 g/cm3, how much does the statue weigh in pounds?


Ex. 3 The average density of the Earth is 5,515 kg/m3. What is its density in grams per cubic

centimeter? Ex. 4: Rank the following objects in terms of increasing volume:

a. 25.0 g zinc cube (d=7.14 g/cm3) d. 5.0 g ice cube (d=0.917 g/cm3) b. 50.0 g gold cube (d=19.3 g/cm3) e. 10.0 g aluminum cube (d=2.70 g/cm3) c. 35.0 g lead cube (d=11.4 g/cm3) f. 15.0 g copper cube (d=8.96 g/cm3)

__________ < __________ < __________ < __________ < __________ < __________ smallest volume largest volume


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

Note how some objects float on water (e.g. a cork), but others sink (e.g. a penny). That's because objects that have 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/cm3, cork's density must be less than 1.00 g/cm3, and a penny's density must be greater than 1.00 g/cm3. Ex. 1: Consider the figure at the right and the following solids and liquids and their densities:

ice (d=0.917 g/cm3) honey (d=1.50 g/cm3) iron cube (7.87 g/cm3) hexane (d=0.65 g/cm3) rubber cube (d=1.19 g/cm3) Identify L1, L2, S1, and S2 by filling in the blanks below: L1= ___________________ and L2= ___________________

S1= ___________________ S2= ___________________ and S3= ___________________ Ex. 2 A 1.35 g cube made of African teak wood has an edge length of 1.114 cm. Will the cube

of teak wood sink or float in a sample of ethanol which has a density of 0.789 g/mL?

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CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING€¦ · CHEM 121: Chapter 2 page 7 of 15 Your calculator should look similar to: 6.02 x10 23 2.50131 x10 33 4.155 x10 9 Step 1. Enter “6.02×1023”