CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING


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1 CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING Problems: 164, 6988, , Measuring Global Temperatures measurement: a number with attached units When scientists collect data, it is important that they record the measurements as accurately as possible, and the measurements must reflect the accuracy and precision of the instruments used to collect that data. Consider the following plot of global landocean temperatures based on measurements taken from meteorological stations and ship and satellite temperature (SST) measurements: Source: Hansen, J., Mki. Sato, R. Ruedy, K. Lo, D.W. Lea, and M. MedinaElizade, 2006: Global temperature change. Proc. Natl. Acad. Sci., 103, , doi: /pnas ( The plot above shows annual mean (average) temperatures in black, 5year mean temperatures in red, and the uncertainty as green bars. Ex. 1: Based on this plot how have global landocean temperatures changed since the 1950s? CHEM 121: Chapter 2 v0912 page 1 of 17
2 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 EnglishEnglish 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 Ruler B Ruler C A 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 v0912 page 2 of 17
3 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) cm has 3 sig figs, 5.0 m has 2 sig figs, g has 6 s.f. For measurements less than 1, start with the first nonzero digit and count all digits (even zeros) after it ml and kg each have 2 sig figs 2. When there is no decimal point: Count all nonzero digits and zeros between nonzero 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 may have 1, 2, 3 or 4 sig figs Example: Indicate the number of significant digits for the following: a g c m e g b. 105 cm d L f 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 (cm 3 ) Know the following: 1 L 1 dm 3 1 ml 1 cm 3 (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 CHEM 121: Chapter 2 v0912 page 3 of 17
4 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 = 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 10 n 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, ,000,000,000,000,000,000,000 CHEM 121: Chapter 2 v0912 page 4 of 17
5 Some measurements may be rounded to a number of sig figs requiring scientific notation. For example, Express g to 3 sig figs: Express g to 2 sig figs: Express g to 1 sig fig: UNBIASED ROUNDING (or ROUNDTOEVEN METHOD) 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 e.g to 3 s.f (since the nonsig figs are 501 in ) Express each of the following with the number of sig figs indicated: a to 3 sig figs to 3 sig figs b to 3 sig figs c to 3 sig figs d e. 1,234,567 to 5 sig figs to 2 sig figs f When necessary express measurements in scientific notation to clarify the number of sig figs. 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 v0912 page 5 of 17
6 Ex. 1: Ex. 2: Ex. 3: g g g = cm 7.95 cm 4.00 cm = 9.75 ml 7.35 ml = Ex. 4: g cm cm 8.50 cm = MULTIPLYING/DIVIDING WITH EXPONENTIAL NUMBERS: When multiplying or dividing measurements with exponents, use the digit term (N in N 10 n ) to determine number of sig figs. Ex. 1: ( )( ) = How do you calculate this using your scientific calculator? Step 1. Enter 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 by pressing: then EE or EXP (which corresponds to 10 ) then x10 23 Your calculator should look similar to: Step 4. Get the answer by pressing: = Your calculator should now read x x10 9 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. Ex. 2: ( ) ( ) = to correct # of sig figs Ex. 3: = to correct # of sig figs CHEM 121: Chapter 2 v0912 page 6 of 17
7 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 CONVERTING FROM ONE UNIT TO ANOTHER (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 1 dollar 10 dimes 1 dollar 10 dimes Unit factor 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 1 m 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 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. CHEM 121: Chapter 2 v0912 page 7 of 17
8 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 = km) Example 3: The speed of light is about meters per second. Express this speed in miles per hour. (1 mile=1.609 km, 1000 m 1 km) 2.5 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) CHEM 121: Chapter 2 v0912 page 8 of 17
9 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. Metric Conversion Factors Prefix Symbol Multiple/Fraction kilo k 1000 deci d centi c milli m micro μ (Greek mu ) ,000,000 Ex. 1 Complete the following unit equations: KNOW the metric units above! 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 μg 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. CHEM 121: Chapter 2 v0912 page 9 of 17
10 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 MetricMetric Conversions: Solve the following using dimensional analysis. Ex. 1 Convert 175 ms into units of seconds. Ex. 2 Convert kilograms into milligrams. Ex. 3 Convert m/s into kilometers per hour. Ex. 4 Convert cm to units of kilometers. CHEM 121: Chapter 2 v0912 page 10 of 17
11 MetricEnglish 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 cm (exact) mass 1 pound (lb) 1 g 1 lb = 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 2.0L bottle can hold how many cups of liquid? (1 qt. 2 pints, 1 pint 2 cups) Ex. 3 A lightyear (about 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 v0912 page 11 of 17
12 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 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 C. The term absolute zero is used to indicate the theoretical lowest temperature. Conversion between C and K: K = C 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 v0912 page 12 of 17
13 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 cm 3 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 v0912 page 13 of 17
14 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 mass m density = or d = volume V generally in units of g/cm 3 or g/ml For water, 1.00 g of water occupies a volume of 1.00 cm 3 : m 1.00 g d = = 3 V 1.00 cm = 1.00 g/cm 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/cm 3, a cork's density must be less than 1.00 g/cm 3, and a penny's density must be greater. 3 Ex.: Consider the figure at the right and the following solids and liquids and their densities: ice (d=0.917 g/cm 3 ) honey (d=1.50 g/cm 3 ) iron cube (7.87 g/cm 3 ) hexane (d=0.65 g/cm 3 ) rubber cube (d=1.19 g/cm 3 ) Identify L 1, L 2, S 1, and S 2 by filling in the blanks below: L 1 = and L 2 = S 1 =, S 2 =, and S 3 = CHEM 121: Chapter 2 v0912 page 14 of 17
15 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 g/cm g cm 3 Two unit factors would be: or cm g Ex. 1 Give 2 unit factors for each of the following: a. density of lead = 11.3 g/cm 3 b. density of chloroform = 1.48 g/ml Ex. 2 Aluminum has a density of 2.70 g/cm 3. What is the volume (in ml) of a piece of aluminum with a mass of kg? Ex. 3 Ethanol is used in alcoholic beverages and has a density of g/ml. What is the mass of ethanol that has a volume of 1.50 L? Ex. 4 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. CHEM 121: Chapter 2 v0912 page 15 of 17
16 CALCULATING 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 = one part total sample 100% 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 v0912 page 16 of 17
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 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 g penny cast in Ex. 4: Calculate the mass of pennies cast in the 1970s that contains 1.00 lbs. of copper. (1 lb. = g) CHEM 121: Chapter 2 v0912 page 17 of 17
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