chapter 2: measurements and calculations a decimal point is present: ... 2.5 significant figures in...

CHEM139: Chapter 2 & 10 page 1 of 19

CHAPTER 2: Measurements and Calculations MEASUREMENT UNITS measurement: a number with attached units 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! 2.4 - 2.5 SIGNIFICANT FIGURES (also called “Sig Figs” or “Significant Digits”) When a measurement is recorded, all the numbers known with certainty are given along with the last number, which is estimated. All the digits are significant because removing any of the digits changes the measurement's uncertainty. 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 Increment of the smallest

markings on ruler

# d.p. needed

Measurement

# 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.

Ruler A

Ruler B

0 1 2 3 4 5

0 1 2 3 4 5

Ruler C

4.1 4.2 4.3 4.4


Guidelines for Determining Number of Sig Figs (if the measurement is given): Count the number of digits in a measurement from left to right: 1. When a decimal point is present: – For measurements ≥1, count all the digits (even zeros). – 60.2 cm has 3 sig figs, 5.0 m has 2 sig figs, 150.00 g 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, 0.0050 g, 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 – 125 g has 3 sig figs and 107 mL has 3 sig figs – Placeholder zeros may or may not be significant – 1000 g may have 1, 2, 3 or 4 sig figs Example: How many significant digits do the following numbers have?

a. 105 _____ b. 90.40 _____ c. 100.00 _____ d. 0.0050 _____

2.1 LARGE AND SMALL NUMBERS (or 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 → ___________________________ Also, in some cases the number of sig figs in a measurement may be unclear: For example, Ordinary form Scientific Notation

Express 100.0 g to 3 sig figs: ___________ → ______________

Express 100.0 g to 2 sig figs: ___________ → ______________

Express 100.0 g to 1 sig fig: ___________ → ______________

Thus, some measurements—usually those expressing large amounts—must be expressed in scientific notation to accurately convey the number of sig figs.

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. The textbook explains unbiased rounding, and below we use “normal” rounding (biased toward even numbers). 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" # " " " " " _______________

Example: Express each of the following with the number of sig figs indicated: a. 376.276

!

to 3 sig figs" # " " " " " _______________________ b. 500.072

!

to 4 sig figs" # " " " " " _______________________ c. 0.00654321

!

to 3 sig figs" # " " " " " _______________________ d. 1,234,567

!

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

72.58643 g

last significant digit

first nonsignificant digit


e. 2,975

!

to 2 sig figs" # " " " " " _______________________ Be sure to express measurements in scientific notation when necessary to make it clear how many sig figs there are in the measurement. 2.5 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 number with the fewest decimal places.

Ex 1: 106.61 + 0.25 + 0.195 = 107.055 → 107.055 ⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to ______________

Ex 2: 725.50 – 103 = 622.50 → 622.50 ⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to ______________ 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: 106.61 × 0.25 × 0.195 = 5.1972375 → 5.1972375 ⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to ____________ Ex 2: 106.61 × 91.5 = 9754.815 → 9754.815 ⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to _____________ SOLVE: Ex. 1: 7.4333 g + 8.25 g + 10.781 g = _________________________

Ex. 2: 13.50 cm × 7.95 cm × 4.00 cm = _________________________ # s.f:

# dp: Ex. 3: 9.75 mL − 7.35 mL = _________________________

Ex. 4: cm 8.50 cm 10.25 cm 25.75

g 101.755 ××

= _________________________

# s.f: # dp:


6.02 x1023

2.50131 x1015

4.155 x10-9

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)(4.155×10-9) = 2.50131×1015 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 → 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 (Be sure to push the Negate button, not the Subtract mathematical operation.) → 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. Most of the calculations we do in chemistry involve very large and very small numbers with exponential terms. Ex. 2: (3.75×1015) (8.6×104) = 3.225×1020 ⎯⎯⎯⎯⎯⎯⎯ →⎯ figs sig of #correct to ___________________

Ex. 3: 1.90!1015

2.500!108 = 760000 ⎯⎯⎯⎯⎯⎯ →⎯ 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. 2.6 USING UNITS IN CALCULATIONS Unit equation: Simple statement of two equivalent values


Conversion factor = unit factor = equivalents: − Ratio of two equivalent quantities

Unit equation: 1 dollar = 10 dimes Unit factor: dollar 1dimes 10 or

dimes 10dollar 1

Unit factors are exact if we can count the number of units equal to another or if both quantities are in the same system of measurement—i.e., both in the metric system (e.g. cm and meters) or in the English system (inches and feet).

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

cm 100m 1

foot 1in. 12

hours 24day 1

year1days 365.25 and 1 yard ≡ 3 feet

Exact equivalents have an infinite number of sig figs → never limit number of sig figs!

Note: When the relationship between two units or items is exact, the “≡” (meaning “equals exactly”) is used instead of the basic “=” sign.

Equivalents based on measurements or relating measurements from two different systems are inexact or approximate because they contain uncertainty, such as

s

m 103.00 hour

mi 65 mile 1

km 1.61 8×

Approximate equivalents do limit the sig figs for the final answer. 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. 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) 2.2 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

Note: Although scientists use µg 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 µg 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. Writing Unit Factors: Complete the following unit equations then write two unit factors for each equation: a. 1 km ≡ __________ m b. 1 g ≡ ___________ mg


METRIC-METRIC CONVERSIONS Ex. 1 Convert 25.0 dm into micrometers, µm (also called “microns”). 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.


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. → A method of conversion between the two systems is 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 5.88×1012 miles) is the distance light travels in one year.

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


DETERMINING VOLUME

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?


2.8 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:

25.0 g zinc cube (d=7.14 g/cm3) 5.0 g ice cube (d=0.917 g/cm3) 50.0 g gold cube (d=19.3 g/cm3) 10.0 g aluminum cube (d=2.70 g/cm3) 35.0 g lead cube (d=11.4 g/cm3) 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= ___________________


2.7 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

Ex. 1. The book Fahrenheit 451 is based on 451˚F, the temperature at which paper ignites

and burns. What is this temperature in degrees Celsius?

Ex. 2. The sea water temperature around Maui in June is 77°F while the average temperature in Paris in June is 22°C. Which is warmer?

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 kelvin (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 On a hot summer day in Reykjavik, Iceland the temperature is 16˚C. What is the equivalent Kelvin temperature? Ex. 2 In the movie Terminator 2, a tanker crashes and pours out liquid nitrogen. This freezes

the T-1000 because liquid nitrogen has a temperature of 77K. What is the equivalent temperature in degrees Celsius?

CHAPTER 10: ENERGY Energy: the capacity to do work or produce heat potential energy (PE): energy due to position or its composition (chemical bonds) – A 10-lb bowling ball has higher PE when it is 10 feet

off the ground compared to 10 inches off the ground → Greater damage on your foot after falling 10 feet

compared to falling only 10 inches – In terms of chemical bonds, the stronger the bond, → more energy is required to break the bond, → the higher the potential energy of the bond

kinetic energy (KE): energy associated with an object’s motion – e.g. a car moving at 75 mph has much greater KE than the same car moving at 15 mph → Greater damage if the car crashes at 75 mph than at 15 mph Energy changes accompany physical and chemical changes due to changes in potential and kinetic energy.

Kinetic Energy and Physical States

Solids have the lowest KE of the three physical states – Highest attraction between particles → particles are fixed


Liquids have slightly higher KE than solids – Particles are still attracted to each other but can move past one another → particles are less restricted Gases have greatest KE compared to solids and liquids – Attractive forces completely overcome, so particles fly freely within container → particles are completely unrestricted

Temperature: Random Molecular and Atomic Motion Heat: Energy that is transferred from a body at a higher temperature to one at a lower

temperature → heat always transfers from the hotter to the cooler object! – "heat flow" means heat transfer

Heat Transfer and Temperature – One becomes hotter by gaining heat. – One becomes colder by losing heat—i.e., when you “feel cold”, you are actually losing heat! Ex. 1: Fill in the blanks to indicate how heat is transferred:

a. You burn your hand on a hot frying pan. ________________ loses heat, and ______________ gains the heat. b. Your tongue feels cold when you eat ice cream. ________________ loses heat, and ______________ gains the heat.

Ex. 2: A small chunk of gold is heated in beaker #1, which contains boiling water. The gold chunk is then transferred to beaker #2, which contains room-temperature water.

Before: Au in Beaker #1, with boiling water After: Au into Beaker #2 Beaker #2, water at room temp. What happen to the gold temperature? a. The temperature of the water in beaker #2 _____. ↑ ↓ stays the same

b. Fill in the blanks: _____________ loses heat, and ____________ gains the heat.


Energy and Chemical and Physical Change endothermic change: a physical or chemical change that requires energy or heat to occur

– boiling water requires energy: H2O(l) + heat energy → H2O(g)

– electrolysis of water requires energy: 2 H2O(l) + electrical energy → 2 H2(g) + O2(g)

exothermic change: a physical or chemical change that releases energy or heat

– water condensing releases energy: H2O(g) → H2O(l) + heat energy

– hydrogen burning releases energy: 2 H2(g) + O2(g) → 2 H2O(g) + heat energy For physical changes, consider whether the reactants or products have more kinetic energy. – If the reactants have greater kinetic energy than the products → exothermic process. – If the products have greater kinetic energy than the reactants → endothermic process.

system: that part of the universe being studied surroundings: the rest of the universe outside the system For chemical changes, observe if the surroundings (including you) feel hotter or colder after the reaction has occurred.


– If the surroundings are hotter, the reaction released heat → exothermic reaction. – If the surroundings are colder, the reaction absorbed heat → endothermic reaction. Ex. 1: Circle all of the following changes that are exothermic:

freezing vaporizing sublimation melting deposition Ex. 2: A student adds ammonium chloride (NH4Cl) salt to a test tube containing water and

notices that the test tube feels colder as the ammonium chloride dissolves. Is this process exothermic or endothermic? Explain.

Ex. 3: A student mixes two solutions, hydrochloric acid and sodium hydroxide, and notices the

beaker with the substances feels hotter as they mix. Is this reaction exothermic or endothermic? Explain.

Units of Energy calorie (cal): unit of energy used most often in the US – amount of energy required to raise the temperature of 1 g of water by 1˚C

1 cal ≡ 4.184 J (Note: This is EXACT!) – But a nutritional calorie (abbreviated Cal) is actually 1000 cal: 1 Cal = 1 kcal = 4.184 kJ joule (J): SI unit of energy

– To recognize the size of a joule, note that 1 watt = 1 sJ

→ So a 100-watt light bulb uses 100 J every second.

– Heat is also often reported in kilojoules (kJ), where 1 kJ = 1000 J Temperature Changes: Heat Capacity heat capacity: amount of heat necessary to raise the temperature of a given amount of any

substance by 1°C; in units of J/°C specific heat capacity: amount of heat necessary to raise the temperature of 1 gram of any (or specific heat) substance by 1°C; has units of J/g°C


– Water has a relatively high specific heat (4.184 J/g·°C) compared to the specific heats of rocks and other solids (1.3 J/g°·C for dry Earth, 0.9 J/g°·C for concrete, 0.46 J/g°·C for iron)

– Because water covers most of the Earth, water can absorb a lot more energy before its temperature starts to rise. → Water helps to regulate temperatures on Earth within a comfortable range for humans. → Why coastal regions have less extreme temperatures compared to desert regions

The heat transferred (in the absence of a phase change) is proportional to the change in temperature: Ex 1: Gallium is a solid metal at room temperature, but melts at 29.9 °C. If you hold gallium in your hand, it melts from body heat. How much heat must 2.5 g of gallium absorb from your hand to raise its temperature from 25.0 °C to 29.9 °C? The heat capacity of gallium is 0.372 J/g°C. Ex 2: A 328-g sample of water absorbs 5.78 kJ of heat. Find the temperature change for the water. If the water is initially at 25.0 °C, what is the final temperature?

chapter 2: measurements and calculations a decimal point is present: ... 2.5 significant figures in...

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