CHEM 139: Zumdahl Chapter 2 page 1 of 1 CHAPTER 2: MEASUREMENT AND CALCULATIONS Active Learning Problems: 1-4, 6, 8-21 End-of-Chapter Problems: 1-97, 99-101, 103-107, 109-110, 112-119, 121-137, 141-153 measurement: a number with attached units When scientists collect data, they record the measurements as accurately as possible, and they 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: “This graph illustrates the change in global surface temperature relative to 1951-1980 average temperatures. The 10 warmest years in the 134-year record all have occurred since 1998, with 2010 and 2005 ranking as the warmest years on record (Source: NASA/GISS ). This research is broadly consistent with similar constructions prepared by the Climatic Research Unit and the National Atmospheric and Oceanic Administration .” Source: http://climate.nasa.gov/vital-signs/global-temperature/ Ex. 1: What was the general range for the annual mean global surface temperature between 1950 and 1980? ______________ between 1980 and 2010? ______________ Ex. 2: What does this data indicate about any trend for the mean global surface temperature between 1950 and 2010?
measurement: a number with attached units When scientists collect data, they record the measurements as accurately as possible, and they 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:
“This graph illustrates the change in global surface temperature relative to 1951-1980 average temperatures. The 10 warmest years in the 134-year record all have occurred since 1998, with 2010 and 2005 ranking as the warmest years on record (Source: NASA/GISS). This research is broadly consistent with similar constructions prepared by the Climatic Research Unit and the National Atmospheric and Oceanic Administration.”
Ex. 1: What was the general range for the annual mean global surface temperature between 1950 and 1980? ______________ between 1980 and 2010? ______________ Ex. 2: What does this data indicate about any trend for the mean global surface temperature
between 1950 and 2010?
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2.5 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.
A B C
Measurement
Increment of the smallest markings on ruler
# of sig figs
Thus, for analog instruments (e.g. a ruler) 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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2.4 UNCERTAINTY IN MEASUREMENT 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, 3.0 cm has 2 sig figs, 184.00 m has 5 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 _____
2.3 MEASUREMENTS OF LENGTH, VOLUME, AND MASS 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
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Mass is not affected by gravity!
2.1 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):
Height of Mt. Rainier: 14,400 ft. → __________________
Diameter of a human hair: 0.0110 cm → __________________
Avogadro’s Number: 602,000,000,000,000,000,000,000 → _____________________ Some measurements may be rounded to a number of sig figs requiring scientific notation.
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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 ≥ 6, raise the last sig digit by 1 and drop nonsignificant digits
• If first nonsignificant digit =5 and
– nonzero digits follow 5, raise the last sig digit by 1 and drop nonsignificant digits – e.g. 3.14501 3.15 (since nonsig figs are “501” in 3.14501)
– no digits or only zeros follow the 5, leave it alone or raise the last sig digit to get an even number and drop nonsignificant zeros
– e.g. 3.145 or 3.145000 3.14 (to get last sig fig to be an even number) – e.g. 3.175 or 3.175000 3.18 (to get last sig fig to be an even number)
Express each of the following with the number of sig figs indicated:
a. 648.75 _______________________
b. 23.6500 _______________________
c. 64.55 _______________________
d. 0.00123456 _______________________
e. 1,234,567 _______________________
f. 1975 _______________________ Express measurements in scientific notation whenever the number of sig figs is unclear!
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6.022 x1023
2.7099 x1033
4.50 x109
SIGNIFICANT FIGURES IN CALCULATIONS
ADDING/SUBTRACTING MEASUREMENTS
When adding or subtracting measurements, the final value is limited by the measurement with the largest uncertainty—i.e. the measurement with the fewest number of 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 = _________________________ Ex. 2: 13.50 cm × 7.95 cm × 4.00 cm = _________________________ Ex. 3: 9.75 mL − 7.35 mL = _________________________
Ex. 4: 101.755 g10.75 cm ! !2.25 cm ! !1.50 cm
= _________________________
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: (6.022×1023)(4.50×109) = 2.7099×1033 How do you calculate this using your scientific calculator?
Step 1. Enter “6.02×1023” by pressing:
6.022 then EE or EXP (which corresponds to “×10”) then 23 → Your calculator should look similar to:
Step 2. Multiply by pressing: ×
Step 3. Enter “4.50× 109” by pressing:
4.50 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 = ___________________
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Be sure you can do exponential calculations with your calculator. Many calculations we do in chemistry involve numbers in scientific notation.
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.
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
Note: The “≡” is used to mean “exactly equal to” in place of the usual equal sign, “=”. 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:
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
Approximate equivalents do limit the sig figs for the final answer.
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2.6 PROBLEMS SOLVING AND DIMENSIONAL ANALYSIS 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 Star of India Sapphire is the largest and most famous star sapphire in the world. It
weighs 0.2484 lb. How many carats is the Star of India Sapphire? (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)
2.2 UNITS Show “Powers of Ten” video: https://www.youtube.com/watch?v=0fKBhvDjuy0 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
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
centi c 0.01 ≡ ≡ 10-2
milli m 0.001 ≡ ≡ 10-3
micro µ (Greek “mu”) 10–6
nano n 10–9
pico p 10–12
femto f 10–15
Metric Conversion Factors
Example 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, b, and c.
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Metric-Metric Conversions: Solve the following using dimensional analysis. Ex. 1 The Space Needle is 184.41 m tall. a. Convert this to units of kilometers. b. Convert this to units of centimeters.
Ex. 2 Human hair can vary from about 17-181 µm (micrometers, also called “microns”) in diameter. What is this range in millimeters?
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?
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Ex. 2 A 2.0-L bottle can hold how many cups of liquid? (1 qt. ≡ 2 pints, 1 pint ≡ 2 cups) Ex. 3: The speed of sound in dry air is about 343 meters per second. Express this speed in miles
per hour. (1 mile=1.609 km) Ex. 4: In Europe fuel efficiency is given in liters of gasoline required to drive 100 kilometers. If a
car averages 6.72 L per 100 km, what is its fuel efficiency in miles per gallon?
2.7 TEMPERATURE CONVERSIONS: AN APPROACH TO PROBLEM SOLVING
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
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Conversion between Fahrenheit and Celsius scales:
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 + 273.15 ˚C = K – 273.15
Ex. 1 Liquid nitrogen boils at 77 K, so anything placed in a container with liquid nitrogen can be
cooled to extremely low temperatures. For example, a rose can be cooled in liquid nitrogen until the water in it crystallizes, so the rose can shatter just like glass.
What is the equivalent temperature in ˚C and in ˚F?
Liquid Nitrogen Video: https://www.youtube.com/watch?v=-gvxOBfHiE4&t=1m52s and https://www.youtube.com/watch?v=-gvxOBfHiE4&t=6m17s DETERMINING VOLUME – Volume is determined in three principal ways:
1. The volume of any liquid can be measured directly using calibrated glassware in the
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 = πr3
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.
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VOLUME BY DISPLACEMENT
a. Fill a graduated cylinder halfway with water, and record the initial volume.
b. Carefully place the object in 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?
2.8 DENSITY: The amount of mass in a unit volume of matter The Density Concept: The amount of mass in a unit volume of matter
generally in units of g/cm3 or g/mL
For water, 1.00 g of water occupies a volume of 1.00 cm3:
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:
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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 The density of pure silver is 10.5 g/cm3. A silvery piece of metal weighing 55.195 g is
placed in a graduated cylinder containing 3.50 mL of water. The water level rises until the volume of water is 9.25 mL. a. Calculate the density of the metal.
b. Is the metal pure silver? Yes No 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?
https://www.youtube.com/watch?v=aADExWV1bsM&t=1m22s Ex. 3 The average density of the Earth is 5,515 kg/m3. What is its density in grams per cubic
centimeter?
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Ex. 4: Given 25.0 g of each of the following, rank the objects in terms of increasing volume:
a. lead (Pb) cube (d=11.4 g/cm3) c. ice cube (d=0.917 g/cm3) b. gold (Au) cube (d=19.3 g/cm3) d. aluminum (Al) cube (d=2.70 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= ___________________ PERCENTAGES
Percent: Ratio of parts per 100 parts → 10% is , 25% is , etc.
To calculate percent, divide one quantity by the total of all quantities in sample:
Percentage = ×100%
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Example: Carbon dioxide consists of carbon and oxygen atoms. A 0.500 g sample of carbon dioxide was analyzed and found to contain 0.136 g of carbon. Calculate the percentage by mass of carbon and oxygen in carbon dioxide.
Writing out Percentage as Unit Factors Example: 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. Percentage Practice Problems
Ex. 1 Steel is an alloy of iron mixed with elements like carbon and chromium. If high carbon steel is 1.35% carbon by mass, what mass (in kg) of steel contains 25.0 g of carbon?
Ex. 2 What mass (in kg) of copper is present in 100.0 lbs. of pennies cast in the 1970’s?
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Ex. 4 Methane is the primary component (about 70-95%) of natural gas, which burns to produce carbon dioxide. In Washington state burning natural gas produces 10.7% of the state’s electricity,1 and Washington State Ferries (WSF) are considering conversion of their diesel fuel to liquid natural gas (LNG). If a sample of methane undergoes complete combustion, it should produce 2.74 g of carbon dioxide per gram of methane burned.
If the typical WSF ferry burns 28,000 gallons of LNG per week, calculate number of tons of
CO2 emitted each week by a WSF ferry fueled by LNG based on its combustion of CH4. (Use dLNG=0.45 g/cm3, the LNG is 95% CH4, 1 gal.≡4 qt., 1 qt.=946 mL, 1 lb.=453.6 g, and 1 ton≡2000 lb.)
“Gases that trap heat in the atmosphere are called greenhouse gases.”2 Consider the image below indicating the primary greenhouse gases and the main sources of these gases. Note that methane not only burns to produce carbon dioxide, but it is itself a greenhouse gas that traps heat more effectively than carbon dioxide.
