Download chapter 2: measurement and problem solving

CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING
Problems: 1-64, 69-86, 95-103, 112-118, 120, 122-124, 126
2.1 Measuring Global Temperatures
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: Between about 1950 and 1980, what was the general range in variation in the average
global surface temperature?
Ex. 2: Between about 1980 and 2010, what was the general range in variation in the average
global surface temperature?
CHEM 121: Chapter 2 v0916 page 1 of 17 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
0
1
2
3
4
5
0
1
2
3
4
5
Ruler B
Ruler C
4.1
A
4.2
4.3
4.4
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.
CHEM 121: Chapter 2 v0916 page 2 of 17 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
MASS:
1 quart ≡ 2 pints
1 pint ≡ 2 cups
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
CHEM 121: Chapter 2 v0916 page 3 of 17 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):
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 → _____________________
CHEM 121: Chapter 2 v0916 page 4 of 17 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
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
to 3 s.f.
– e.g. 3.14501 ⎯⎯
⎯⎯→ 3.15 (since the nonsig figs are 501 in 3.14501)
Express each of the following with the number of sig figs indicated:
to 3 sig figs
a. 648.75
"""""
"→
b. 23.6500
⎯⎯ ⎯ ⎯ ⎯
⎯→
c. 64.55
⎯⎯ ⎯ ⎯ ⎯
⎯→
d. 0.00123456
⎯⎯ ⎯ ⎯ ⎯
⎯→
e. 1,234,567
"""""
"→
f. 1975
⎯⎯ ⎯ ⎯ ⎯
⎯→
to 3 sig figs
to 3 sig figs
to 3 sig figs
to 5 sig figs
to 2 sig figs
_______________________
_______________________
_______________________
_______________________
_______________________
_______________________
Express measurements in scientific notation when the number of sig figs is unclear!
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.
CHEM 121: Chapter 2 v0916 page 5 of 17 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:
150.175 g
= _________________________
7.55 cm × !1.85 cm × !1.20 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: (6.02×1023)(4.155×109) = 2.50131×1033
How do you calculate this using your scientific calculator?
Step 1.
Enter “6.02×1023” by pressing:
6.02 then
EE or EXP (which corresponds to “×10”) then 23
6.02
→ Your calculator should look similar to:
Step 2.
Multiply by pressing: ×
Step 3.
Enter “4.155× 109” by pressing:
4.155 then
23
EE or EXP (which corresponds to “×10”) then 9
4.155
→ Your calculator should look similar to:
Step 4.
x10
x10
9
Get the answer by pressing: =
→ Your calculator should now read
2.50131
x10
33
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.
to correct # of sig figs
Ex. 2: (3.25×1012) (8.6×104) = 2.795 ×1017 ⎯⎯
⎯⎯⎯⎯⎯→ _________________
Ex. 3:
3.75 × 1015
8.605 × 10
CHEM 121: Chapter 2 v0916 4
to correct # of sig figs
= 4.357931435×1010 ⎯⎯
⎯⎯⎯⎯⎯→ __________________
page 6 of 17 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 PROBLEM SOLVING AND UNIT CONVERSION (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
1m
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 MULTISTEP UNIT 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.
CHEM 121: Chapter 2 v0916 page 7 of 17 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 2.998×108 meters per second.
Express this speed in miles per hour. (1 mile=1.609 km, 1000 m≡1 km)
2.5 THE BASIC UNITS OF MEASUREMENT
Show the “Powers of Ten” video.
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)
CHEM 121: Chapter 2 v0916 page 8 of 17 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 ≡
centi
c
0.01 ≡
milli
m
0.001 ≡
micro
µ (Greek “mu”)
but
mc (in medicine)
1
10
1
100
1
0.000 001 ≡
1000
1
1,000,000
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 ≡ ________ mcg
Note: Although scientists use µ g to abbreviate microgram, hospitals use mcg instead of
the µ g 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.
Note that I will use and accept either mcg or µ g to indicate micrograms unless I
specifically ask for the abbreviation used in medicine or hospitals.
CHEM 121: Chapter 2 v0916 page 9 of 17 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 3.00×108 m/s into kilometers per hour.
Ex. 4 Convert 3.50×107 cm to units of kilometers.
CHEM 121: Chapter 2 v0916 page 10 of 17 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)
1g
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 5.88×1012 miles) is the distance light travels in one year.
Calculate the speed of light in meters per second. (1 mile=1.609 km)
CHEM 121: Chapter 2 v0916 page 11 of 17 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:
Fahrenheit scale (°F)
English system
References
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?
CHEM 121: Chapter 2 v0916 page 12 of 17 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 cm3 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
CHEM 121: Chapter 2 v0916 page 13 of 17 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
density =
mass
volume
or
d=
m
V
generally in units of g/cm3 or g/mL
For water, 1.00 g of water occupies a volume of 1.00 cm3: d =
m
V
=
1.00 g
3
1.00 cm
= 1.00 g/cm3
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/cm3, a cork's density must be
less than 1.00 g/cm3, 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/cm3)
iron cube (7.87 g/cm3)
rubber cube (d=1.19 g/cm3)
honey (d=1.50 g/cm3)
hexane (d=0.65 g/cm3)
Identify L1, L2, S1, and S2 by filling in the blanks below:
L1= _______________ and L2= _______________
S1= _______________, S2= _______________, and S3= _______________
CHEM 121: Chapter 2 v0916 page 14 of 17 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.
0.917g
cm3
Two unit factors would be:
or
0.917g
cm3
Ex. 1 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
Ex. 2 Ethanol is used in alcoholic beverages and has a density of 0.789 g/mL. What is the
mass (in kg) of ethanol that has a volume of 1.50 L?
Ex. 3 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.
Ex. 4 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?
CHEM 121: Chapter 2 v0916 page 15 of 17 CALCULATING PERCENTAGES
Percent: Ratio of parts per 100 parts
→ 10% is 10 , 25% is 25 , etc.
100
100
To calculate percent, divide one quantity by the total of all quantities in sample:
Percentage =
one part
×100%
total sample
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.
CHEM 121: Chapter 2 v0916 page 16 of 17 Percentage Practice Problems
Ex. 1 An antacid sample was analyzed and found to be 10.0% aspirin by mass. What mass of
aspirin is present in a 3.50 g tablet of antacid?
Ex. 2 Water is 88.8% oxygen and 11.2% hydrogen by mass. How many grams of hydrogen
are present in 250.0 g (about a cup) of water?
Ex. 3: Pennies cast between 1963 and 1982 are a mixture of 95.0% copper and 5.0% zinc.
Calculate the mass of copper present in a 2.505 g penny cast in 1968.
Ex. 4: Calculate the mass of pennies cast in the 1970s that contains 1.00 lb. of copper.
(1 lb. = 453.6 g)
CHEM 121: Chapter 2 v0916 page 17 of 17