Copyright©2004 by Houghton Mifflin Company. All rights reserved 1 Introductory Chemistry: A Foundation FIFTH EDITION by Steven S. Zumdahl University of Illinois

Copyright©2004 by Houghton Mifflin Company. All rights reserved 2 Measurements and Calculations Chapter 2

Copyright©2004 by Houghton Mifflin Company. All rights reserved 3 Measurement is important Correct I.V. solution concentration Correct lawn fertilizer application Correct amount of salt on food Correct amount of oil in engine Correct force acting on bridge girders

Copyright©2004 by Houghton Mifflin Company. All rights reserved 4 Automobile as an example Gallons of gas, quarts of oil Viscosity (thickness) of oil Antifreeze density (freezing temperature) Temperature Air pressure Voltage of battery Oxygen sensors on exhaust, intake

A gas pump measures the amount of gasoline delivered.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 6 What is measurement? Defined as a quantitative comparison of an unknown quantity to a known Standard. The measurement instrument must be standardized against the known standard Every measurement has a value (number) and a unit Standard kilogram of mass, officially known as the “International prototype of the kilogram” composed of platinum-iridium alloy, stored under glass in a vacuum since 1889

Copyright©2004 by Houghton Mifflin Company. All rights reserved 7 Standards of Measurement When we measure, we use a measuring tool to compare some dimension of an object to a standard. For example, at one time the standard for length was the king’s foot. What are some problems with this standard? Historical standard, platinum iridium meter bar The meter now is defined as the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. The speed of light is c = 299,792,458 m/s

Copyright©2004 by Houghton Mifflin Company. All rights reserved 8 Measurement In Action On 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s atmosphere 100 km lower than planned and was destroyed by heat. 1 lb = 1 N 1 lb = 4.45 N “This is going to be the cautionary tale that will be embedded into introduction to the metric system in elementary school, high school, and college science courses till the end of time.”

Copyright©2004 by Houghton Mifflin Company. All rights reserved Scientific Notation

Copyright©2004 by Houghton Mifflin Company. All rights reserved 10 Scientific Notation Technique Used to Express Very Large or Very Small Numbers Based on Powers of 10 To Compare Numbers Written in Scientific Notation –First Compare Exponents of 10 –Then Compare Numbers

Copyright©2004 by Houghton Mifflin Company. All rights reserved 11 Writing Numbers in Scientific Notation 1Locate the Decimal Point 2Move the decimal point to the right of the non-zero digit in the largest place –The new number is now between 1 and 10

Copyright©2004 by Houghton Mifflin Company. All rights reserved 12 Writing Numbers in Scientific Notation 3Multiply the new number by 10 n –where n is the number of places you moved the decimal point 4Determine the sign on the exponent n –If the decimal point was moved left, n is + –If the decimal point was moved right, n is – –If the decimal point was not moved, n is 0

Copyright©2004 by Houghton Mifflin Company. All rights reserved 13 Writing Numbers in Standard Form 1Determine the sign of n of 10 n –If n is + the decimal point will move to the right –If n is – the decimal point will move to the left 2Determine the value of the exponent of 10 –Tells the number of places to move the decimal point 3Move the decimal point and rewrite the number

Copyright©2004 by Houghton Mifflin Company. All rights reserved 14 When describing very small distances, such as the diameter of a swine flu virus, it is convenient to use scientific notation.

Copyright©2004 by Houghton Mifflin Company. All rights reserved Units of Measurement

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 17 The meter 1795 Royal jeweler made a bar of platinum 4 mm thick, 25 mm wide, and 1 m long. This was known as an “end measure” 1872 “line measure” on a platinum iridium alloy bar Now, the distance light travels in 1/299, 792, 458 th of a second

Copyright©2004 by Houghton Mifflin Company. All rights reserved 18 Length SI unit = meter (m) –About 3½ inches longer than a yard Commonly use centimeters (cm) for length

Copyright©2004 by Houghton Mifflin Company. All rights reserved 19 Length –1 m = 100 cm –1 cm = 0.01 m = 10 mm –1 inch = 2.54 cm (exactly)

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 21 Figure 2.1: Comparison of English and metric units for length on a ruler.

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 23 Volume Measure of the amount of three- dimensional space occupied by a substance SI unit = cubic meter (m 3 ) Commonly measure solid volume in cubic centimeters (cm 3 ) –1 m 3 = 10 6 cm 3 –1 cm 3 = m 3 = m 3

Copyright©2004 by Houghton Mifflin Company. All rights reserved 24 Volume Commonly measure liquid or gas volume in milliliters (mL) –1 L is slightly larger than 1 quart –1 L = 1 dm 3 = 1000 mL = 10 3 mL –1 mL = L = L –1 mL = 1 cm 3

Figure 2.2: The largest drawing represents a cube that has sides 1 m in length and a volume of 1 m 3.

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Figure 2.3: A 100-mL graduated cylinder.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 28 Mass Measure of the amount of matter present in an object SI unit = kilogram (kg)

Copyright©2004 by Houghton Mifflin Company. All rights reserved 29 Mass Commonly measure mass in grams (g) or milligrams (mg) –1 kg = 2.20 pounds –1 kg = 1000 g = 10 3 g – 1 g = 1000 mg = 10 3 mg –1 g = kg = kg

Figure 2.4: An electronic analytical balance used in chemistry labs.

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 32 Related Units in the Metric System All units in the metric system are related to the fundamental unit by a power of 10 The power of 10 is indicated by a prefix The prefixes are always the same, regardless of the fundamental unit

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 35 Uncertainty in Measured Numbers A measurement always has some amount of uncertainty Uncertainty comes from limitations of the techniques used for comparison To understand how reliable a measurement is, we need to understand the limitations of the measurement

Copyright©2004 by Houghton Mifflin Company. All rights reserved 36 Exact Numbers Exact Numbers are numbers known with certainty Unlimited number of significant figures They are either –counting numbers number of sides on a square –or defined 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm 1 kg = 1000 g, 1 LB = 16 oz 1000 mL = 1 L; 1 gal = 4 qts. 1 minute = 60 seconds

Copyright©2004 by Houghton Mifflin Company. All rights reserved 37 Figure 2.5: Measuring a pin.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 38 Precision Refers to the reproducibility of a measurement. Measurements with more “decimal places” are more precise. Devices with finer markings are more precise.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 39 Estimation in measurement When we measure, the quantity rarely falls exactly on the calibration marks of the scale we are using. Because of this we are estimating the last digit of the measurement. For instance, we could measure “A” above as about 2.3 cm. We are certain of the digit “2”, but the “.3” part is a guess - an estimate. What is your estimate for B and C? 1 cm2 cm3 cm4 cm5 cm ABC

Copyright©2004 by Houghton Mifflin Company. All rights reserved 40 Higher Precision A measuring device with more marks on the scale is more precise. I.e., we are estimating less, and get a more accurate reading. Here we are estimating the hundredths place instead of the tenths. Here, we can measure A as 1.25 cm. Only the last digit is uncertain. Usually we assume the last digit is accurate ± 1. How would you read B, C, and D? 1 cm2 cm3 cm ABCD

Copyright©2004 by Houghton Mifflin Company. All rights reserved 41 Accuracy Refers to how close a measurement comes to the true or accepted value. This depends on both the measuring device and the skill of the person using the measuring device. This can be determined by comparing the measured value to the known or accepted value.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 42 Accurate or Precise? Precise! (but not accurate) What is the temperature at which water boils? Measurements: 95.0°C, 95.1°C, 95.3°C True value: 100°C

Copyright©2004 by Houghton Mifflin Company. All rights reserved 43 Accurate or Precise? Accurate! (it’s hard to be accurate without being precise) What is the temperature at which water freezes? Measurements: 1.0°C, 1.2°C, -5.0°C True value: 0.0°C

Copyright©2004 by Houghton Mifflin Company. All rights reserved 44 Accurate or Precise? Not Accurate & Not Precise (don’t quit your day job) What is the atmospheric pressure at sea level? Measurements: atm, 0.25 atm, atm True value: 1.00 atm

Copyright©2004 by Houghton Mifflin Company. All rights reserved 45 Accurate or Precise? Accurate & Precise (it’s time to go pro) What is the mass of one Liter of water? Measurements: kg, kg, kg True value: kg

Copyright©2004 by Houghton Mifflin Company. All rights reserved 46 Reporting Measurements To indicate the uncertainty of a single measurement scientists use a system called significant figures The last digit written in a measurement is the number that is considered to be uncertain Unless stated otherwise, the uncertainty in the last digit is ±1

Copyright©2004 by Houghton Mifflin Company. All rights reserved 47 Rules for Counting Significant Figures Nonzero integers are always significant Zeros –Leading zeros never count as significant figures –Captive zeros are always significant –Trailing zeros are significant if the number has a decimal point Exact numbers have an unlimited number of significant figures

Copyright©2004 by Houghton Mifflin Company. All rights reserved 48 Counting Significant Figures RULE 1. All non-zero digits in a measured number are significant. Only a zero could indicate that rounding occurred. Number of Significant Figures cm ft lb ___ m m ___

Copyright©2004 by Houghton Mifflin Company. All rights reserved 49 Leading Zeros RULE 2. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures mm oz lb____ mL mL ____

Copyright©2004 by Houghton Mifflin Company. All rights reserved 50 Sandwiched Zeros RULE 3. Zeros between nonzero numbers are significant. (They can not be rounded unless they are on an end of a number.) Number of Significant Figures 50.8 mm min lb ____

Copyright©2004 by Houghton Mifflin Company. All rights reserved 51 Trailing Zeros RULE 4. Trailing zeros in numbers without decimals are NOT significant. They are only serving as place holders. Number of Significant Figures 25,000 in. 2 25,000 in yr yr3 48,600 gal____ 48,600 gal____

Copyright©2004 by Houghton Mifflin Company. All rights reserved 52 Examples EXAMPLES# OF SIG. DIG.COMMENT 4533 All non-zero digits are always significant Zeros between two significant digits are significant Additional zeros to the right of decimal and a significant digit are significant Placeholders are not significant Trailing zeros in numbers with no decimal point are not significant (= placeholder)

Copyright©2004 by Houghton Mifflin Company. All rights reserved 53 Learning Check A. Which answers contain 3 significant figures? 1) ) ) 4760 B. All the zeros are significant in 1) ) ) x 10 3 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x ) 535 2) 535,000 3) 5.35 x 10 5

Copyright©2004 by Houghton Mifflin Company. All rights reserved 54 Learning Check In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and ) 22.0 and ) and 40 3) and 150,000

Copyright©2004 by Houghton Mifflin Company. All rights reserved 55 State the number of significant figures in each of the following: A m B L C g D m E. 2,080,000 bees Learning Check

Copyright©2004 by Houghton Mifflin Company. All rights reserved 56 Practice How many significant digits in the following? Number# Significant Digits

Copyright©2004 by Houghton Mifflin Company. All rights reserved 57 The Problem Area of a rectangle = length x width We measure: Length = cm Width = cm Punch this into a calculator and we find the area as: cm x cm = cm 2 But there is a problem here! This answer makes it seem like our measurements were more accurate than they really were. By expressing the answer this way we imply that we estimated the thousandths position, when in fact we were less precise than that!

Copyright©2004 by Houghton Mifflin Company. All rights reserved 58 Calculations with Significant Figures Calculators/computers do not know about significant figures!!! Exact numbers do not affect the number of significant figures in an answer Answers to calculations must be rounded to the proper number of significant figures –round at the end of the calculation

Copyright©2004 by Houghton Mifflin Company. All rights reserved 59 Multiplication/Division with Significant Figures Result has the same number of significant figures as the measurement with the smallest number of significant figures Count the number of significant figures in each measurement

Copyright©2004 by Houghton Mifflin Company. All rights reserved 60 Adding/Subtracting Numbers with Significant Figures Result is limited by the number with the smallest number of significant decimal places Find last significant figure in each measurement

Copyright©2004 by Houghton Mifflin Company. All rights reserved 61 Adding/Subtracting Numbers with Significant Figures Find which one is “left-most” Round answer to the same decimal place 450 mL mL = 480 mL precise to 10’s place precise to 0.1’s place precise to 10’s place

Copyright©2004 by Houghton Mifflin Company. All rights reserved 62 Rules for Rounding Off If the digit to be removed is less than 5, the preceding digit stays the same is equal to or greater than 5, the preceding digit is increased by 1 In a series of calculations, carry the extra digits to the final result and then round off Don’t forget to add place-holding zeros if necessary to keep value the same!!

A student performing a titration in the laboratory.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 64 Problem Solving and Dimensional Analysis Many problems in chemistry involve using equivalence statements to convert one unit of measurement to another Conversion factors are relationships between two units –May be exact or measured –Both parts of the conversion factor should have the same number of significant figures

Copyright©2004 by Houghton Mifflin Company. All rights reserved 65 Problem Solving and Dimensional Analysis Conversion factors generated from equivalence statements –e.g. 1 inch = 2.54 cm can give or

Copyright©2004 by Houghton Mifflin Company. All rights reserved 66 Arrange conversion factors so starting unit cancels –Arrange conversion factor so starting unit is on the bottom of the conversion factor May string conversion factors together Problem Solving and Dimensional Analysis

Copyright©2004 by Houghton Mifflin Company. All rights reserved 67 Converting One Unit to Another Find the relationship(s) between the starting and goal units. Write an equivalence statement for each relationship. Write a conversion factor for each equivalence statement. Arrange the conversion factor(s) to cancel starting unit and result in goal unit.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 68 Converting One Unit to Another Check that the units cancel properly Multiply and Divide the numbers to give the answer with the proper unit. Check your significant figures Check that your answer makes sense!

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 70 Temperature Scales Fahrenheit Scale, °F –Water’s freezing point = 32°F, boiling point = 212°F Celsius Scale, °C –Temperature unit larger than the Fahrenheit –Water’s freezing point = 0°C, boiling point = 100°C

Copyright©2004 by Houghton Mifflin Company. All rights reserved 71 Temperature Scales Kelvin Scale, K –Temperature unit same size as Celsius –Water’s freezing point = 273 K, boiling point = 373 K

Copyright©2004 by Houghton Mifflin Company. All rights reserved 72 Figure 2.7: The three major temperature scales.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 73 Figure 2.9: Comparison of the Celsius and Fahrenheit scales.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 74 Figure 2.6: Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 75 Figure 2.8: Converting 70. 8C to units measured on the Kelvin scale.

Liquid gallium expands within a carbon nanotube as the temperature increases (left to right). Source: Glenn Izett/U.S. Geological Survey

Copyright©2004 by Houghton Mifflin Company. All rights reserved 77 Density Density is a property of matter representing the mass per unit volume For equal volumes, denser object has larger mass For equal masses, denser object has small volume

Copyright©2004 by Houghton Mifflin Company. All rights reserved 78 Density Solids = g/cm 3 –1 cm 3 = 1 mL Liquids = g/mL Gases = g/L Volume of a solid can be determined by water displacement Density : solids > liquids >>> gases In a heterogeneous mixture, denser object sinks

Copyright©2004 by Houghton Mifflin Company. All rights reserved 79 Spherical droplets of mercury, a very dense liquid.

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Copyright©2004 by Houghton Mifflin Company. All rights reserved 81 Figure 2.10: (a) Tank of water. (b) Person submerged in the tank, raising the level of the water.

Figure 2.11: A hydrometer being used to determine the density of the antifreeze solution in a car’s radiator.

Copyright©2004 by Houghton Mifflin Company. All rights reserved 83 Using Density in Calculations