Chapter 2 $ ₤ ¥ L m kg ml mm μg + - * / yx Chemistry 103 Chapter 2 $ ₤ ¥ L m kg ml mm μg + - * / yx

General Course structure Learning Tools Atoms ---> Compounds ---> Chemical Reactions

Outline Mathematics of Chemistry (Measurements) Units Significant Figures (Sig Figs) Calculations & Sig Figs Scientific Notation Dimensional Analysis Density

Importance of Units Job Offer: Annual Salary = 1,000,000.

Measurements Two components – Numerical component and Dimensional component

Everyday Measurements You make a measurement every time you Measure your height. Read your watch. Take your temperature. Weigh a cantaloupe.

Units and Measurements Scientists make many kinds of measurements The determination of the dimensions, capacity, quantity or extent of something Length, Mass, Volume, Density All measurements are made relative to a standard All measurements have uncertainty

Systems of Measurement English System Common measurements Pints, quarts, gallons, miles, etc. Metric System Units in the metric system consist of a base unit plus a prefix.

Measurement in Chemistry In chemistry we Measure quantities. Do experiments. Calculate results. Use numbers to report measurements. Compare results to standards. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Length Measurement Length Is measured using a meter stick. Has the unit of meter (m) in the metric (SI) system. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Inches and Centimeters The unit of an inch is equal to exactly 2.54 centimeters in the metric system. 1 in. = 2.54 cm Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Volume Measurement Volume Is the space occupied by a substance. Has the unit liter (L) in metric system. 1 L = 1.057 qt Uses the unit m3(cubic meter) in the SI system. Is measured using a graduated cylinder. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Mass Measurement The mass of an object Is the quantity of material it contains. Is measured on a balance. Has the unit gram(g) in the metric system. Has the unit kilogram(kg) in the SI system. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Temperature Measurement The temperature of a substance Indicates how hot or cold it is. Is measured on the Celsius (C) scale in the metric system. On this thermometer temperature is 18ºC or 64ºF. In the SI system uses the Kelvin (K) scale. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Units in the Metric System In the metric (SI) system, one unit is used for each type of measurement: Measurement Metric SI length meter (m) meter (m) volume liter (L) cubic meter (m3) mass gram (g) kilogram (kg) time second (s) second (s) temperature Celsius (C) Kelvin (K)

Metric Base Units

Learning Check For each of the following, indicate whether the unit describes A) length, B) mass, or C) volume.

Learning Check Identify the measurement with an SI unit. 1. John’s height is 2. The race was won in 3. The mass of a lemon is 4. The temperature is

Measured vs Exact numbers

Exact (Defined) and Inexact (Measured) Numbers Exact numbers Have no uncertainty associated with them They are known exactly because they are defined or counted Example: 12 inches = 1 foot Measured numbers Have some uncertainty associated with them Example: all measurements

Accuracy vs. Precision Accuracy How closely a measurement comes to the true, accepted value Precision How closely measurements of the same quantities come to each other

Significant Figures

Measured numbers convey Significant Figures Digits in any measurement are known with certainty, plus one digit that is uncertain. Measured numbers convey *Magnitude *Uncertainty *Units

The Calculator Problem 7.8 3.8 Calculator Answer: 2.05263…… Is this a realistic answer? Is it 2, 2.0, 2.1, 2.05, 2.06, 2.052, 2.053, 2.0526, etc.? Which is it? Answer must reflect uncertainty expressed in original measurements. Using Significant Figures. We will come back to this later.

Rules for Significant Figures It’s ALL about the ZEROs

Rules for Sig Figs All non-zero numbers in a measurement are significant. 4573 4573 has 4 sig figs

Rules for Sig Figs All zeros between sig figs are significant. 23007 23007 has 5 sig figs

Rules for Sig Figs In a number less than 1, zeros used to fix the position of the decimal are not significant. 0.00021 0.00021 has 2 sig figs

Rules for Sig Figs When a number has a decimal point, zeros to the right of the last nonzero digit are significant 0.0002100 0.0002100 has 4 sig figs

Rules for Sig Figs _ 820000 meters - 3 sig figs 820000 When a number without a decimal point explicitly shown ends in one or more zeros, we consider these zeros not to be significant. If some of the zeros are significant, bar notation is used. _ 820000 meters - 3 sig figs 820000

Practice Identifying Sig Figs

Significant Figures How many assuming all numbers are measured?

Measured Numbers A measuring tool Is used to determine a quantity such as height or the mass of an object. Provides numbers for a measurement called measured numbers. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

Reading a Meter Stick . l2. . . . l . . . . l3 . . . . l . . . . l4. . cm The markings on the meter stick at the end of the orange line are read as The first digit 2 plus the second digit 2.7 The last digit is obtained by estimating. The end of the line might be estimated between 2.7–2.8 as about half-way (0.5) which gives a reported length of 2.75 cm

Known + Estimated Digits In the length reported as 2.75 cm, The digits 2 and 7 are certain (known) The final digit 5 was estimated (uncertain) All three digits (2.75) are significant including the estimated digit

Learning Check . l8. . . . l . . . . l9. . . . l . . . . l10. . cm What is the length of the red line? 1) 9.0 cm 2) 9.03 cm 3) 9.04 cm

Solution . l8. . . . l . . . . l9. . . . l . . . . l10. . cm The length of the red line could be reported as 2) 9.03 cm or 3) 9.04 cm The estimated digit may be slightly different. Both readings are acceptable.

Zero as a Measured Number . l3. . . . l . . . . l4. . . . l . . . . l5. . cm For this measurement, the first and second known digits are 4.5. Because the line ends on a mark, the estimated digit in the hundredths place is 0. This measurement is reported as 4.50 cm.

Significant Figures in Measured Numbers Obtained from a measurement include all of the known digits plus the estimated digit. Reported in a measurement depend on the measuring tool.

Rounding off Numbers The number of significant figures in measurements affects any calculations done with these measurements Your calculated answer can only be as certain as the numbers used in the calculation

Calculator: Friend or Foe? Sometimes, the calculator will show more (or fewer) significant digits than it should If the first digit to be deleted is 4 or less, simply drop it and all the following digits If the first digit to be deleted is 5 or greater, that digit and all that follow are dropped and the last retained digit is increased by one

Sig Fig Rounding Example: Round the following measured number to 4 sig figs: 82.56702

Adding Significant Zeros Sometimes a calculated answer requires more significant digits. Then one or more zeros are added. Calculated Answer Zeros Added to Give 3 Significant Figures 4 1.5 0.2 12

Practice Rounding Numbers

Significant Figures Round each to 3 sig figs

Multiplication and Division When multiplying or dividing, use The same number of significant figures in your final answer as the measurement with the fewest significant figures. Rounding rules to obtain the correct number of significant figures. Example: 110.5 x 0.048 = 5.304 = 5.3 (rounded) 4 SF 2 SF calculator 2 SF

Addition and Subtraction When adding or subtracting, use The same number of decimal places in your final answer as the measurement with the fewest decimal places. Use rounding rules to adjust the number of digits in the answer. 25.2 one decimal place + 1.34 two decimal places 26.54 calculated answer 26.5 answer with one decimal place

Math operations with Sig Figs

Report Answer with Correct Number of Sig Figs A) 124.54 x 2.2 = 273.98800 B) 3420. + 2400. + 1095 = 6915.0000 C) 3420 + 2400 + 1095 = 6915.0000 D) 98.5564 = 2.1575394 45.68

When Math Operations Are Mixed If you have both addition/subtraction and multiplication/division in a formula, carry out the operations in parenthesis first, and round according to the rules for that type of operation. -complete the calculation by rounding according to the rules for the final type of operation.

When Math Operations Are Mixed _____5.681g_____ = (52.15ml - 32.4ml) carry out the operations in parenthesis first, and round according to the rules for that type of operation.

When Math Operations Are Mixed _____5.681g_____ = 5.681g (52.15ml - 32.4ml) 19.8ml carry out the operations in parenthesis first, and round according to the rules for that type of operation.

Mixed Operations and Significant Figures What is the result (to the correct number of significant figures) of the following calculations? Assume all numbers are measured. (23 - 21) x (24.4 - 23.1) (298 - 270) x (322)

Back To The Calculator Problem 7.8 3.8 Calculator Answer: 2.05263…… Is this a realistic answer? Is it 2, 2.0, 2.1, 2.05, 2.06, 2.052, 2.053, 2.0526, etc.? Which is it? Answer must reflect uncertainty expressed in original measurements.

Scientific Notation Scientific notation Is used to write very large or very small numbers The width of a human hair, 0.000 008 m is written as: 8 x 10-6 m A large number such as 2 500 000 s is written as: 2.5 x 106 s Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Scientific Notation A number in scientific notation contains a coefficient (1 or greater, less than 10) and a power of 10. 150 0.000735 coefficient power of ten coefficient power of ten 1.5 x 102 7.35 x 10-4 To write a number in scientific notation, the decimal point is moved after the first non zero digit. The spaces moved are shown as a power of ten. 52 000 = 5.2 x 104 0.00378 = 3.78 x 10-3 4 spaces left 3 spaces right

Some Powers of Ten

Comparing Numbers in Standard and Scientific Notation Standard Format Scientific Notation Diameter of Earth 12 800 000 m Mass of a human 68 kg Length of a pox virus 0.000 03 cm

Comparing Numbers in Standard and Scientific Notation Standard Format Scientific Notation Diameter of Earth 12 800 000 m 1.28 x 107 m (3 sig figs) Mass of a human 68 kg 6.8 x 101 kg (2 sig figs) Length of a pox virus 0.000 03 cm 3 x 10-5 cm (1 sig fig) NOTE: The Coefficient identifies or indicates the number of significant figures in the measurement.

Defining Conversion Factors Dimensional Analysis Defining Conversion Factors

Conversion Factors Conversion factors A ratio that specifies how one unit of measurement is related to another Creating conversion factors from equalities 12 in.= 1 ft 1 L = 1000 mL

Dimensional Analysis How many seconds are in 2 minutes? ? seconds = 2 minutes 60 seconds = 1 minute ? seconds = 2 minutes x 60 seconds = 1 minute 120 seconds (exactly)

Dimensional Analysis If we assume there are exactly 365 days in a year, how many seconds are in one year? ? seconds = 1 year

Dimensional Analysis A problem solving method in which the units (associated with numbers) are used as a guide in setting up the calculations. Conversion Factor

Exact vs Measured Relationships Metric to Metric – exact English to English – exact Metric to English – typically measured (must consider sig figs)

English to Metric Conversion Factors

Dimensional Analysis What is 165 lb in kg? STEP 1 Given: 165 lb Need: kg STEP 2 Plan STEP 3 Equalities/Factors 1 kg = 2.205 lb 2.205 lb and 1 kg 1 kg 2.205 lb STEP 4 Set Up Problem ? kg = 165 lb

Learning Check If a ski pole is 3.0 feet in length, how long is the ski pole in mm? (1000mm = 1m, 12 inches=1ft, 1m=39.37inches)

Learning Check If a ski pole is 3.0 feet in length, how long is the ski pole in mm? (1000mm = 1m, 12 inches=1ft, 1m=39.37inches) 3.0 feet mm? Plan

Learning Check If a bucket contains 4.65L of water. How many gallons of water is this? (1 gallon = 4qts, 1L = 1.057qt)

Dimensional Analysis If Jules Vern expressed the title of his famous book, “Twenty Thousand Leagues Under the Sea” in feet, what would the title be? (1mile = 5280ft, 1 League = 3.450miles)

Converting from squared units to squared units or cubed units to cubed units Warning: This type of conversions give many students difficulties!!!!! The one thing you have to remember: What does it mean to say that a unit is squared or cubed? m2 = m x m; s3 = s x s x s When there are squared or cubed units, you have multiple units to cancel out!

Examples Convert 127.4 cm3 to m3. (100cm = 1m) Convert .572 miles2 to km2. (1km = .621miles)

Displacement volume for a stock engine in a 1984 Corvette is specified at 350 in3. What is the displacement in L?

Percent Factor in a Problem If the thickness of the skin fold at the waist indicates an 11% body fat, how much fat is in a person with a mass of 86 kg? percent factor 86 kg mass x 11 kg fat 100 kg mass = 9.5 kg fat Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Even MORE Practice with Conversion Factors A lean hamburger is 22% fat by weight. How many grams of fat are in 0.25 lb of the hamburger? (1lb = 453.6g)

Density Density = Mass/Volume A ratio of the mass of an object divided by its volume Density = Mass/Volume Typical units = g/mL (NOTE: 1mL=1cm3) We have an unknown metal with a mass of 59.24 g and a volume of 6.64 mL. What is its density?

Density Density = Mass/Volume A ratio of the mass of an object divided by its volume Density = Mass/Volume Typical units = g/mL (NOTE: 1mL=1cm3) We have an unknown metal with a mass of 59.24 g and a volume of 6.64 mL. What is its density? Density = 59.24g = 8.92g/mL 6.64mL

Densities of Common Substances Is Density a Physical or a Chemical Property?

Measuring Density in Lab

Learning Check What is the density (g/cm3) of 48.0 g of a metal if the level of water in a graduated cylinder rises from 25.0 mL to 33.0 mL after the metal is added? A) 0.17 g/cm3 B) 6.0 g/cm3 C) 380 g/cm3 25.0 mL 33.0 mL object

Sink or Float Ice floats in water because the density of ice is less than the density of water. Aluminum sinks because its density is greater than the density of water. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Learning Check Which diagram correctly represents the liquid layers in the cylinder? Karo (K) syrup (1.4 g/mL), vegetable (V) oil (0.91 g/mL,) water (W) (1.0 g/mL) A B C W K V V W K V W K

Learning Check Osmium is a very dense metal. What is its density in g/cm3 if 50.0 g of osmium has a volume of 2.22 cm3? a) 2.25 g/cm3 b) 22.5 g/cm3 c) 111 g/cm3

Density as a Conversion Factor Density can be written as an equality. For a substance with a density of 3.8 g/mL, the equality is: 3.8 g = 1 mL From this equality, two conversion factors can be written for density. Conversion 3.8 g and 1 mL factors 1 mL 3.8 g

Density Example You have been given 150.g of ethyl alcohol which has a density of 0.785g/mL. Will this quantity fit into a 150mL beaker?

DENSITY PRACTICE

Learning Check The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane? A) 0.614 kg B) 614 kg C) 1.25 kg

Temperature Temperature Is a measure of how hot or cold an object is compared to another object Indicates that heat flows from the object with a higher temperature to the object with a lower temperature Is measured using a thermometer Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Temperature Scales

Solving a Temperature Problem A person with hypothermia has a body temperature of 34.8°C. What is that temperature in °F? TF = 1.8 TC + 32 TF = 1.8 (34.8°C) + 32° exact tenths exact = 62.6 + 32° = 94.6°F tenths Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Converting between Temperature Scales ***Conversions between Celsius and Kelvin (Temperature in K) = (temperature in oC) + 273 (temperature in oC) = (temperature in K) – 273 Conversions between Celsius and Fahrenheit oF = 9/5 (oC) + 32 or 1.8 (oC) + 32 oC = 5/9(oF – 32) or 1/1.8 (oF – 32) 9/5 = 1.8/1 or 5/9 = 1/1.8