Chapter 2 Measurement and Problem Solving

Chapter 2 Measurement and Problem Solving

Introductory Chemistry Fifth Edition Nivaldo J. Tro Chapter 2 Measurement and Problem Solving Dr. Sylvia Esjornson Southwestern Oklahoma State University Weatherford, OK 2015 Pearson Education, Inc. World of Chemistry Steven S. Zumdahl

Susan L. Zumdahl Donald J. DeCoste Chapter 5 Measurement and Calculations 2015 Pearson Education, Inc. Reporting the Measure of Global Temperatures The graph in this image displays average global temperatures (relative to the mean) over the past 100 years.

2015 Pearson Education, Inc. Uncertainty Indicated by Last Reported Digit The uncertainty is indicated by the last reported digit. Example: measuring global temperatures Average global temperatures have risen by 0.6 C in the last century. By reporting a temperature increase of 0.6 C, the scientists mean 0.6 +/ 0.1 C. The temperature rise could be as much as 0.7 C or as little as 0.5 C, but it is not 1.0 C. The degree of certainty in this particular measurement is critical, influencing political decisions that directly affect peoples lives.

2015 Pearson Education, Inc. Scientific Notation Has Two Parts A number written in scientific notation has two parts. A decimal part: a number that is between 1 and 10. An exponential part: 10 raised to an exponent, n. 2015 Pearson Education, Inc. Writing Very Large and Very Small Numbers A positive exponent means 1 multiplied by 10 n times. (large number) A negative exponent (n) means 1 divided by 10 n times. (small number)

2015 Pearson Education, Inc. To Convert a Number to Scientific Notation Find the decimal part. Find the exponent. Move the decimal point to obtain a number between 1 and 10. Multiply that number (the decimal part) by 10 raised to the power that reflects the movement of the decimal point. 2015 Pearson Education, Inc. To Convert a Number to Scientific Notation If the decimal point is moved to the left, the exponent is positive.

If the decimal point is moved to the right, the exponent is negative. 2015 Pearson Education, Inc. EXAMPLE 2.1 SCIENTIFIC NOTATION The 2013 U.S. population was estimated to be 315,000,000 people. Express this number in scientific notation. To obtain a number between 1 and 10, move the decimal point to the left eight decimal places; the exponent is 8. Because you move the decimal point to the left, the sign of the exponent is positive. SOLUTION 315,000,000 people = 3.15 108 people SKILLBUILDER 2.1 | Scientific Notation The total U.S national debt in 2013 was approximately $16,342,000,000,000. Express this number in scientific

notation. Answer: $1.6342 1013 FOR MORE PRACTICE Example 2.18; Problems 31, 32. 2015 Pearson Education, Inc. EXAMPLE 2.2 SCIENTIFIC NOTATION The radius of a carbon atom is approximately 0.000000000070 m. Express this number in scientific notation. To obtain a number between 1 and 10, move the decimal point to the right 11 decimal places; therefore, the exponent is 11. Because you moved the decimal point to the right, the sign of the exponent is negative. SKILLBUILDER 2.2 | Scientific Notation Express the number 0.000038 in scientific notation. Answer: 3.8 105

FOR MORE PRACTICE Problems 33, 34. 2015 Pearson Education, Inc. SOLUTION 0.000000000070 m = 7.0 1011 m Writing Numbers to Reflect Precision Penniescomeinwholenumbers,anda countofsevenpenniesmeansseven wholepennies. 2015 Pearson Education, Inc. Ourknowledgeoftheamountofgold

ina10-ggoldbardependsonhow preciselyitwasmeasured. Reporting Scientific Numbers The first four digits are certain; the last digit is estimated. The greater the precision of the measurement, the greater the number of significant figures. 2015 Pearson Education, Inc. Estimating Tenths of a Gram This balance has markings every 1 g. We estimate to the tenths place.

To estimate between markings, mentally divide the space into 10 equal spaces and estimate the last digit. This reading is 1.2 g. 2015 Pearson Education, Inc. Estimating Hundredths of a Gram This scale has markings every 0.1 g. We estimate to the hundredths place. The correct reading is 1.26 g.

2015 Pearson Education, Inc. EXAMPLE 2.3 REPORTING THE RIGHT NUMBER OF DIGITS The bathroom scale in Figure 2.3 has markings at every 1 lb. Report the reading to the correct number of digits. Figure 2.3 .3 Reading a bathroom scale SOLUTION Because the pointer is between the 147- and 148-lb markings, you mentally divide the space between the markings into 10 equal spaces and estimate the next digit. In this case, you should report the result as: 147.7 lb What if you estimated a little differently and wrote 147.6 lb? In general, one unit of difference in the last digit is acceptable because the last digit is estimated and different people might estimate it slightly differently.

However, if you wrote 147.2 lb, you would clearly be wrong. 2015 Pearson Education, Inc. EXAMPLE 2.3 REPORTING THE RIGHT NUMBER OF DIGITS Continued SKILLBUILDER 2.3 | Reporting the Right Number of Digits You use a thermometer to measure the temperature of a backyard hot tub, and you obtain the reading shown in Figure 2.4. Record the temperature reading to the correct number of digits. Answer: 103.4 F FOR MORE PRACTICE Example 2.19; Problems 41, 42. 2015 Pearson Education, Inc.

Figure 2.3 .4 Reading a thermometer Significant Figures in a Correctly Reported Measurement 1. All nonzero digits are significant. 2. Interior zeros (zeros between two numbers) are significant. 3. Leading zeros (zeros to the left of the first nonzero number) are NEVER significant. They serve only to locate the decimal point. 4. Trailing zeros are significant only when a decimal point is in the number. 5. All numbers multiplied by 10 in scientific notation are considered significant. 2015 Pearson Education, Inc.

EXAMPLE 2.4 DETERMINING THE NUMBER OF SIGNIFICANT FIGURES IN A NUMBER How many significant figures are in each number? (a) (b) (c) (d) (e) (f) (g) 0.0035 1.080 2371 2.97 105

1 dozen = 12 100.00 100,000 SOLUTION (a) 0.0035 two significant figures The 3 and the 5 are significant (rule 1). The leading zeros only mark the decimal place and are not significant (rule 5). (b) 1.080 four significant figures The interior zero is significant (rule 2), and the trailing zero is significant (rule 3). The 1 and the 8 are also

significant (rule 1). (c) 2371 four significant figures All digits are significant (rule 1). All digits in the decimal part are significant (rule 1). 2015 Pearson Education, Inc. (d) 2.97 105 three significant figures EXAMPLE 2.4 DETERMINING THE NUMBER OF SIGNIFICANT FIGURES IN A NUMBER Continued Defined numbers are exact and therefore have an

unlimited number of significant figures. (e) 1 dozen = 12 The 1 is significant (rule 1), and the trailing zeros before the decimal point are significant (rule 4). The trailing zeros after the decimal point are also significant (rule 3). (f) 100.00 five significant figures This number is ambiguous. Write as 1 105 to indicate one significant figure or as 1.00000 105 to indicate six significant figures.

(g) 100,000 ambiguous 2015 Pearson Education, Inc. unlimited significant figures Identifying Exact Numbers Exact numbers have an unlimited number of significant figures. Exact counting of discrete objects Integral numbers that are part of an equation Defined quantities Some conversion factors are defined quantities, while others are not.

1 in. = 2.54 cm exact 2015 Pearson Education, Inc. Counting Significant Figures How many significant figures are in each number? 0.0035 1.080 2371 2.9105 1dozen=12 100.00 2015 Pearson Education, Inc.

EXAMPLE 2.4 DETERMINING THE NUMBER OF SIGNIFICANT FIGURES IN A NUMBER Continued SKILLBUILDER 2.4 | Determining the Number of Significant Figures in a Number How many significant figures are in each number? (a) (b) (c) (d) (e) (f) 58.31 0.00250 2.7 103

1 cm = 0.01 m 0.500 2100 Answers: (a) four significant figures (b) three significant figures (c) two significant figures (d) unlimited significant figures (e) three significant figures (f) ambiguous FOR MORE PRACTICE Example 2.20; Problems 43, 44, 45, 46, 47, 48. 2015 Pearson Education, Inc. Significant Figures in Calculations

Rules for Rounding: When numbers are used in a calculation, the result is rounded to reflect the significant figures of the data. For calculations involving multiple steps, round only the final answerdo not round off between steps. This practice prevents small rounding errors from affecting the final answer. 2015 Pearson Education, Inc. Significant Figures in Calculations

Rules for Rounding: Use only the last (or leftmost) digit being dropped to decide in which direction to roundignore all digits to the right of it. Round down if the last digit dropped is 4 or less; round up if the last digit dropped is 5 or more. 2015 Pearson Education, Inc. Significant Figures in Calculations

Multiplication and Division Rule: The result of multiplication or division carries the same number of significant figures as the factor with the fewest significant figures. 2015 Pearson Education, Inc. Significant Figures in Calculations Multiplication and Division Rule: The intermediate result (in blue) is rounded to two significant figures to reflect the least precisely known factor (0.10), which has two significant figures. 2015 Pearson Education, Inc.

Significant Figures in Calculations Multiplication and Division Rule: The intermediate result (in blue) is rounded to three significant figures to reflect the least precisely known factor (6.10), which has three significant figures. 2015 Pearson Education, Inc. EXAMPLE 2.5 SIGNIFICANT FIGURES IN MULTIPLICATION AND DIVISION Perform each calculation to the correct number of significant figures. (a) 1.01 0.12 53.51 96 (b) 56.55 0.920 34.2585 Round the intermediate result (in blue) to two

significant figures to reflect the two significant figures in the least precisely known quantities (0.12 and 96). SOLUTION Round the intermediate result (in blue) to three significant figures to reflect the three significant figures in the least precisely known quantity (0.920). (b) 56.55 0.920 34.2585 = 1.51863 = 1.52 (a) 1.01 0.12 53.51 96 = 0.067556 = 0.068 SKILLBUILDER 2.5 | Significant Figures in Multiplication and Division Perform each calculation to the correct number of significant figures. (a) 1.10 0.512 1.301 0.005 3.4

(b)4.562 3.99870 89.5 Answers: (a) 0.001 or 1 103 (b)0.204 FOR MORE PRACTICE Examples 2.21, 2.22; Problems 57, 58, 59, 60. 2015 Pearson Education, Inc. Significant Figures in Calculations Addition and Subtraction Rule: In addition or subtraction calculations, the result carries the same number of decimal places as the quantity carrying the fewest decimal places.

2015 Pearson Education, Inc. Significant Figures in Calculations Addition and Subtraction Rule: We round the intermediate answer (in blue) to two decimal places because the quantity with the fewest decimal places (5.74) has two decimal places. 2015 Pearson Education, Inc. Significant Figures in Calculations Addition and Subtraction Rule: We round the intermediate answer (in blue) to one decimal place because the quantity with the fewest decimal places (4.8) has one decimal place.

2015 Pearson Education, Inc. EXAMPLE 2.6 SIGNIFICANT FIGURES IN ADDITION AND SUBTRACTION Perform the calculations to the correct number of significant figures. (a) (b) Round the intermediate answer (in blue) to one decimal place to reflect the quantity with the fewest decimal places (125.1). Notice that 125.1 is not the quantity with the fewest significant figuresit has four while the other quantities only have threebut because it has the fewest decimal places, it determines the number of decimal places in the answer. Round the intermediate answer (in blue) to two decimal

places to reflect the quantity with the fewest decimal places (5.98). 2015 Pearson Education, Inc. SOLUTION (a) (b) EXAMPLE 2.6 SIGNIFICANT FIGURES IN ADDITION AND SUBTRACTION Continued SKILLBUILDER 2.6 | Significant Figures in Addition and Subtraction Perform the calculations to the correct number of significant figures. (a)

(b) Answers: (a) 7.6 (b) 131.11 FOR MORE PRACTICE Example 2.23; Problems 61, 62, 63, 64. 2015 Pearson Education, Inc. Both Multiplication/Division and Addition/Subtraction In calculations involving both multiplication/division and addition/subtraction, do the steps in parentheses first; determine the correct number of significant figures

in the intermediate answer without rounding; then do the remaining steps. 2015 Pearson Education, Inc. Both Multiplication/Division and Addition/Subtraction 3.489 (5.67 2.3), do the step in parentheses first. 5.67 2.3 = 3.37 In the calculation Use the subtraction rule to determine that the intermediate answer has only one significant decimal place. To avoid small errors, it is best not to round at this point; instead, underline the least significant figure as a reminder. 3.489 3.37 = 11.758 = 12

Use the multiplication rule to determine that the intermediate answer (11.758) rounds to two significant figures (12) because it is limited by the two significant figures in 3.37. 2015 Pearson Education, Inc. EXAMPLE 2.7 SIGNIFICANT FIGURES IN CALCULATIONS INVOLVING BOTH MULTIPLICATION/DIVISION AND ADDITION/SUBTRACTION Perform the calculations to the correct number of significant figures. (a) 6.78 5.903 (5.489 5.01) (b) 19.667 (5.4 0.916) Do the step in parentheses first. Use the subtraction rule to mark 0.479 to two decimal places because 5.01, the number in the parentheses with the least number of decimal places, has two. Then perform the multiplication and round the answer

to two significant figures because the number with the least number of significant figures has two. Do the step in parentheses first. The number with the least number of significant figures within the parentheses (5.4) has two, so mark the answer to two significant figures. Then perform the subtraction and round the answer to one decimal place because the number with the least number of decimal places has one. 2015 Pearson Education, Inc. SOLUTION (a) 6.78 5.903 (5.489 5.01) = 6.78 5.903 (0.479) = 6.78 5.903 0.479 6.78 5.903 0.479

= 19.1707 = 19 (b) 19.667 (5.4 0.916) = 19.667 (4.9464) = 19.667 4.9464 19.667 4.9464 = 14.7206 = 14.7 EXAMPLE 2.7 SIGNIFICANT FIGURES IN CALCULATIONS INVOLVING BOTH MULTIPLICATION/DIVISION AND ADDITION/SUBTRACTION Continued SKILLBUILDER 2.7 | Significant Figures in Calculations Involving Both Multiplication/Division and Addition/Subtraction Perform each calculation to the correct number of significant figures.

(a) 3.897 (782.3 451.88) (b)(4.58 1.239) 0.578 Answers: (a) 1288 (b) 3.12 FOR MORE PRACTICE Example 2.24; Problems 65, 66, 67, 68. 2015 Pearson Education, Inc. The Basic Units of Measurement The unit system for science measurements, based on the metric system, is called the International System of Units (Systme International dUnits) or SI units.

2015 Pearson Education, Inc. Basic Units of Measurement: Length The standard of length The definition of a meter, established by international agreement in 1983, is the distance that light travels in vacuum in 1/299,792,458 s. (The speed of light is 299,792,458 m/s.) 2015 Pearson Education, Inc.

Basic Units of Measurement: Mass The standard of mass The kilogram is defined as the mass of a block of metal kept at the International Bureau of Weights and Measures at Svres, France. A duplicate is kept at the National Institute of Standards and Technology near Washington, D.C. 2015 Pearson Education, Inc. Basic Units of Measurement: Time

The standard of time The second is defined, using an atomic clock, as the duration of 9,192,631,770 periods of the radiation emitted from a certain transition in a cesium-133 atom. 2015 Pearson Education, Inc. Weight vs. Mass The kilogram is a measure of mass, which is different from weight. The mass of an object is a measure of the

quantity of matter within it. The weight of an object is a measure of the gravitational pull on that matter. Consequently, weight depends on gravity while mass does not. 2015 Pearson Education, Inc. SI Prefix Multipliers 2015 Pearson Education, Inc. Choosing Prefix Multipliers Choose the prefix multiplier that is most convenient for a particular measurement. Pick a unit similar in size to (or smaller than) the

quantity you are measuring. A short chemical bond is about 1.2 1010 m. Which prefix multiplier should you use? pico = 1012; nano = 109 The most convenient one is probably the picometer. Chemical bonds measure about 120 pm. 2015 Pearson Education, Inc. Volume as a Derived Unit A derived unit is formed from other units. Many units of volume, a measure of space, are derived units. Any unit of length, when cubed (raised to the third power), becomes a unit of volume. Cubic meters (m3), cubic centimeters (cm3), and

cubic millimeters (mm3) are all units of volume. 2015 Pearson Education, Inc. Problem-Solving and Unit Conversions Getting to an equation to solve from a problem statement requires critical thinking. No simple formula applies to every problem, yet you can learn problem-solving strategies and begin to develop some chemical intuition. Unit conversion type: Many of the problems can be thought of as unit conversion problems, in which you are given one or more quantities and asked to convert them into different units. Specific equation type: Other problems require the use of specific equations to

get to the information you are trying to find. 2015 Pearson Education, Inc. Using Dimensional Analysis to Convert Between Units Units are multiplied, divided, and canceled like any other algebraic quantities. Using units as a guide to solving problems is called dimensional analysis. Always write every number with its associated unit. Always include units in your calculations, dividing them and multiplying them as if they were algebraic quantities. Do not let units appear or disappear in calculations. Units must flow logically from beginning to end. 2015 Pearson Education, Inc.

For most conversion problems, we are given a quantity in some units and asked to convert the quantity to another unit. These calculations take the form: 2015 Pearson Education, Inc. Converting Between Units Conversion factors are constructed from any two quantities known to be equivalent. We construct the conversion factor by dividing both sides of the equality by 1 in. and canceling the units. The quantity is equal to 1 and can be used to convert between inches and centimeters. 2015 Pearson Education, Inc.

Converting Between Units In solving problems, always check if the final units are correct, and consider whether or not the magnitude of the answer makes sense. Conversion factors can be inverted because they are equal to 1 and the inverse of 1 is 1. 2015 Pearson Education, Inc. EXAMPLE 2.8 UNIT CONVERSION Convert 7.8 km to miles. PROBLEM-SOLVING PROCEDURE SORT Begin by sorting the information in the problem into given and find.

GIVEN: 7.8 km STRATEGIZE Draw a solution map for the problem. Begin with the given quantity and symbolize each step with an arrow. Below the arrow, write the conversion factor for that step. The solution map ends at the find quantity. (In these examples, the relationships used in the conversions are below the solution map.) SOLUTION MAP FIND: mi RELATIONSHIPS USED 1 km = 0.6214 mi

(This conversion factor is from Table 2.3.) 2015 Pearson Education, Inc. EXAMPLE 2.8 UNIT CONVERSION Continued SOLVE Follow the solution map to solve the problem. Begin with the given quantity and its units. Multiply by the appropriate conversion factor, canceling units to arrive at the find quantity. SOLUTION Round the answer to the correct number of significant figures. (If possible, obtain conversion factors to

enough significant figures so that they do not limit the number of significant figures in the answer.) Round the answer to two significant figures, because the quantity given has two significant figures. CHECK Check your answer. Are the units correct? Does the answer make sense? The units, mi, are correct. The magnitude of the answer is reasonable. A mile is longer than a kilometer, so the value in miles should be smaller than the value in kilometers. SKILLBUILDER 2.8 | Unit Conversion

Convert 56.0 cm to inches. Answer: 22.0 in. FOR MORE PRACTICE Example 2.25; Problems 73, 74, 75, 76. 2015 Pearson Education, Inc. The Solution Map A solution map is a visual outline that shows the strategic route required to solve a problem. For unit conversion, the solution map focuses on units and how to convert from one unit to another. 2015 Pearson Education, Inc. Diagram Conversions Using a Solution Map

The solution map for converting from inches to centimeters is as follows: The solution map for converting from centimeters to inches is as follows: 2015 Pearson Education, Inc. General Problem-Solving Strategy Identify the starting point (the given information). Identify the end point (what you must find). Devise a way to get from the starting point to the end point using what is given as well as what you already know or can look up. You can use a solution map to diagram the steps

required to get from the starting point to the end point. In graphic form, we can represent this progression as Given Solution Map Find 2015 Pearson Education, Inc. General Problem-Solving Strategy Sort. Begin by sorting the information in the problem. Strategize. Create a solution mapthe series of steps that will get you from the given information to the information you are trying to find. Solve. Carry out mathematical operations (paying attention to the rules for significant figures in calculations) and cancel units as needed. Check. Does this answer make physical sense? Are the units correct?

2015 Pearson Education, Inc. EXAMPLE 2.9 UNIT CONVERSION Convert 0.825 m to millimeters. PROBLEM-SOLVING PROCEDURE SORT Begin by sorting the information in the problem into given and find. GIVEN: 0.825 m STRATEGIZE Draw a solution map for the problem. Begin with the given quantity and symbolize each step with an arrow. Below the arrow, write the conversion factor for that

step. The solution map ends at the find quantity. (In these examples, the relationships used in the conversions are below the solution map.) SOLUTION MAP FIND: mm RELATIONSHIPS USED 1 mm = 103 m (This conversion factor is from Table 2.2.) 2015 Pearson Education, Inc. EXAMPLE 2.9 UNIT CONVERSION Continued SOLVE

Follow the solution map to solve the problem. Begin with the given quantity and its units. Multiply by the appropriate conversion factor, canceling units to arrive at the find quantity. SOLUTION Round the answer to the correct number of significant figures. (If possible, obtain conversion factors to enough significant figures so that they do not limit the number of significant figures in the answer.) Leave the answer with three significant figures, because the quantity given has three significant figures and the conversion factor is a definition and therefore does not limit the number of significant figures in the answer.

CHECK Check your answer. Are the units correct? Does the answer make sense? The units, mm, are correct and the magnitude is reasonable. A millimeter is shorter than a meter, so the value in millimeters should be larger than the value in meters. SKILLBUILDER 2.9 | Unit Conversion Convert 5678 m to kilometers. Answer: 5.678 km FOR MORE PRACTICE Problems 69, 70, 71, 72. 2015 Pearson Education, Inc. Solving Multistep Unit Conversion Problems

Each step in the solution map should have a conversion factor with the units of the previous step in the denominator and the units of the following step in the numerator. SOLUTION MAP 2015 Pearson Education, Inc. Follow the Solution Map to Solve the Problem SOLUTION 2015 Pearson Education, Inc. EXAMPLE 2.10 SOLVING MULTISTEP UNIT CONVERSION

PROBLEMS A recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups of cream should you use? (4 cups = 1 quart) SORT Begin by sorting the information in the problem into given and find. GIVEN: 0.75 L STRATEGIZE Draw a solution map for the problem. Begin with the given quantity and symbolize each step with an arrow. Below the arrow, write the conversion factor for that step. The solution map ends at the find quantity. SOLUTION MAP

FIND: cups RELATIONSHIPS USED 1.057 qt = 1 L (from Table 2.3) 4 cups = 1 qt (given in problem statement) 2015 Pearson Education, Inc. EXAMPLE 2.10 SOLVING MULTISTEP UNIT CONVERSION PROBLEMS Continued SOLVE Follow the solution map to solve the problem. Begin with 0.75 L and multiply by the appropriate conversion factor, canceling units to arrive at qt. Then, use the

second conversion factor to arrive at cups. SOLUTION Round the answer to the correct number of significant figures. In this case, you round the answer to two significant figures, because the quantity given has two significant figures. CHECK Check your answer. Are the units correct? Does the answer make physical sense? The answer has the right units (cups) and seems reasonable. A cup is smaller than a liter, so the value in cups should be larger than the value in liters.

SKILLBUILDER 2.10 | Solving Multistep Unit Conversion Problems A recipe calls for 1.2 cups of oil. How many liters of oil is this? Answer: 0.28 L FOR MORE PRACTICE Problems 85, 86. 2015 Pearson Education, Inc. EXAMPLE 2.11 SOLVING MULTISTEP UNIT CONVERSION PROBLEMS One lap of a running track measures 255 m. To run 10.0 km, how many laps should you run? SORT Begin by sorting the information in the problem into given and find. You are given a distance in km and asked to find the distance in laps. You are also given the quantity 255 m per lap, which is a conversion

factor between m and laps. GIVEN: 10.0 km 255 m = 1 lap STRATEGIZE Build the solution map beginning with km and ending at laps. Focus on the units. SOLUTION MAP FIND: number of laps RELATIONSHIPS USED 1 km = 103 m (from Table 2.2) 1 lap = 255 m

(given in problem) 2015 Pearson Education, Inc. EXAMPLE 2.11 SOLVING MULTISTEP UNIT CONVERSION PROBLEMS Continued SOLVE Follow the solution map to solve the problem. Begin with 10.0 km and multiply by the appropriate conversion factor, canceling units to arrive at m. Then, use the second conversion factor to arrive at laps. Round the intermediate answer (in blue) to three significant figures, because it is limited by the three significant figures in the given quantity, 10.0 km.

SOLUTION CHECK Check your answer. Are the units correct? Does the answer make physical sense? The units of the answer are correct, and the value of the answer makes sense. If a lap is 255 m, there are about 4 laps to each km (1000 m), so it seems reasonable that you would have to run about 40 laps to cover 10 km. 2015 Pearson Education, Inc. EXAMPLE 2.11 SOLVING MULTISTEP UNIT CONVERSION PROBLEMS Continued

SKILLBUILDER 2.11 | Solving Multistep Unit Conversion Problems A running track measures 1056 ft per lap. To run 15.0 km, how many laps should you run? (1 mi = 5280 ft) Answer: 46.6 laps SKILLBUILDER PLUS An island is 5.72 nautical mi from the coast. How far away is the island in meters? (1 nautical mi = 1.151 mi) Answer: 1.06 104 m FOR MORE PRACTICE Problems 83, 84. 2015 Pearson Education, Inc. Converting Units Raised to a Power When converting quantities with units raised to a power, the conversion factor must also be raised to that power.

2015 Pearson Education, Inc. Conversion with Units Raised to a Power We cube both sides to obtain the proper conversion factor. We can do the same thing in fractional form. 2015 Pearson Education, Inc. EXAMPLE 2.12 CONVERTING QUANTITIES INVOLVING UNITS RAISED TO A POWER A circle has an area of 2659 cm2. What is its area in square meters? SORT You are given an area in square centimeters and asked to convert the area to square meters.

GIVEN:2659 cm2 STRATEGIZE Build a solution map beginning with cm2 and ending with m2. Remember that you must square the conversion factor. SOLUTIONMAP FIND:m2 RELATIONSHIPSUSED 1 cm = 0.01 m (from Table 2.2) 2015 Pearson Education, Inc.

EXAMPLE 2.12 CONVERTING QUANTITIES INVOLVING UNITS RAISED TO A POWER Continued SOLVE Follow the solution map to solve the problem. Square the conversion factor (both the units and the number) as you carry out the calculation. SOLUTION Round the answer to four significant figures to reflect the four significant figures in the given quantity. The conversion factor is exact and therefore does not limit the number of significant figures. CHECK Check your answer. Are the units correct? Does the answer make physical sense?

2015 Pearson Education, Inc. The units of the answer are correct, and the magnitude makes physical sense. A square meter is much larger than a square centimeter, so the value in square meters should be much smaller than the value in square centimeters. EXAMPLE 2.12 CONVERTING QUANTITIES INVOLVING UNITS RAISED TO A POWER Continued SKILLBUILDER 2.12 | Converting Quantities Involving Units Raised to a Power An automobile engine has a displacement (a measure of the size of the engine) of 289.7 in. 3 What is its displacement in cubic centimeters? Answer: 4747 cm3

FOR MORE PRACTICE Example 2.26; Problems 87, 88, 89, 90, 91, 92. 2015 Pearson Education, Inc. EXAMPLE 2.13 SOLVING MULTISTEP CONVERSION PROBLEMS INVOLVING UNITS RAISED TO A POWER The average annual per person crude oil consumption in the United States is 15,615 dm . What is this value 3 in cubic inches? SORT You are given a volume in cubic decimeters and asked to convert it to cubic inches.

GIVEN:15,615 dm3 STRATEGIZE Build a solution map beginning with dm3 and ending with in.3 You must cube each of the conversion actors, because the quantities involve cubic units. SOLUTIONMAP 2015 Pearson Education, Inc. FIND:in.3 EXAMPLE 2.13 SOLVING MULTISTEP CONVERSION PROBLEMS INVOLVING UNITS RAISED TO A POWER Continued

RELATIONSHIPSUSED 1 dm = 0.1 m (from Table 2.2) 1 cm = 0.01 m (from Table 2.2) 2.54 cm = 1 in. (from Table 2.3) . 2015 Pearson Education, Inc. EXAMPLE 2.13 SOLVING MULTISTEP CONVERSION PROBLEMS INVOLVING UNITS RAISED TO A POWER Continued SOLVE Follow the solution map to solve the problem. Begin with the given value in dm3 and multiply by the string of conversion factors to arrive at in.3 Be sure to cube each conversion factor as you carry out the calculation.

SOLUTION Round the answer to five significant figures to reflect the five significant figures in the least precisely known quantity (15,615 dm3). The conversion factors are all exact and therefore do not limit the number of significant figures. CHECK Check your answer. Are the units correct? Does the answer make physical sense? The units of the answer are correct, and the magnitude makes sense. A cubic inch is smaller than a cubic decimeter, so the value in cubic inches should be larger than the value in cubic decimeters.

SKILLBUILDER 2.13 | Solving Multistep Problems Involving Units Raised to a Power How many cubic inches are there in 3.25 yd3? FOR MORE PRACTICE Problems 93, 94. 2015 Pearson Education, Inc. Answer: 1.52 105 in.3 Physical Property: Density Why do some people pay more than $3000 for a bicycle made of titanium? For a given volume of metal, titanium has less mass than steel. We describe this property

by saying that titanium (4.50 g/cm3) is less dense than iron (7.86 g/cm3). 2015 Pearson Education, Inc. Density The density of a substance is the ratio of its mass to its volume. 2015 Pearson Education, Inc. Calculating Density We calculate the density of a substance by dividing the mass of a given amount of the substance by its volume.

For example, a sample of liquid has a volume of 22.5 mL and a mass of 27.2 g. To find its density, we use the equation d = m/V. 2015 Pearson Education, Inc. EXAMPLE 2.14 CALCULATING DENSITY A jeweler offers to sell a ring to a woman and tells her that it is made of platinum. Noting that the ring feels a little light, the woman decides to perform a test to determine the rings density. She places the ring on a balance and finds that it has a mass of 5.84 g. She also finds that the ring displaces 0.556 cm3 of water. Is the ring made of platinum? The density of platinum is 21.4 g/cm3. (The displacement of water is a common way to measure the volume of irregularly shaped objects. To say that an object displaces 0.556 cm3 of water means that when the object is submerged in a container of water filled to the brim, 0.556 cm 3 overflows. Therefore, the volume of the object is 0.556 cm3.) SORT You are given the mass and volume of the ring and asked to find the density.

GIVEN: m = 5.84 g V = 0.556 cm3 FIND: density in g/cm3 STRATEGIZE If the ring is platinum, its density should match that of platinum. Build a solution map that represents how you get from the given quantities (mass and volume) to the find quantity (density). Unlike in conversion problems, where you write a conversion factor beneath the arrow, here you write the equation for density beneath the arrow. 2015 Pearson Education, Inc. SOLUTION MAP

RELATIONSHIPS USED EXAMPLE 2.14 CALCULATING DENSITY Continued SOLVE Follow the solution map. Substitute the given values into the density equation and calculate the density. SOLUTION Round the answer to three significant figures to reflect the three significant figures in the given quantities. The density of the ring is much too low to be platinum; therefore the ring is a fake.

CHECK Check your answer. Are the units correct? Does the answer make physical sense? The units of the answer are correct, and the magnitude seems like it could be an actual density. As you can see from Table 2.4, the densities of liquids and solids range from below 1 g/cm3 to just over 20 g/cm3. 2015 Pearson Education, Inc. EXAMPLE 2.14 CALCULATING DENSITY Continued

SKILLBUILDER 2.14 | Calculating Density The woman takes the ring back to the jewelry shop, where she is met with endless apologies. The jeweler had accidentally made the ring out of silver rather than platinum. The jeweler gives her a new ring that she promises is platinum. This time when the customer checks the density, she finds the mass of the ring to be 9.67 g and its volume to be 0.452 cm3. Is this ring genuine? Answer: Yes, the density is 21.4 g/cm3 and matches that of platinum. FOR MORE PRACTICE Example 2.27; Problems 95, 96, 97, 98, 99, 100. 2015 Pearson Education, Inc. A Solution Map Involving the Equation for Density In a problem involving an equation, the solution map shows how the equation takes you from the given quantities to the find quantity.

2015 Pearson Education, Inc. Density as a Conversion Factor We can use the density of a substance as a conversion factor between the mass of the substance and its volume. For a liquid substance with a density of 1.32 g/cm3, what volume should be measured to deliver a mass of 68.4 g? 2015 Pearson Education, Inc. Density as a Conversion Factor Solution Map

Solution Measure 51.8 mL to obtain 68.4 g of the liquid. 2015 Pearson Education, Inc. Densities of Some Common Substances Table 2.4 provides a list of the densities of some common substances. These data are needed when solving homework problems. 2015 Pearson Education, Inc. Example: Comparing Densities

A titanium bicycle frame contains the same amount of titanium as a titanium cube measuring 6.8 cm on a side. Use the density of titanium to calculate the mass in kilograms of titanium in the frame. What would be the mass of a similar frame composed of iron? 2015 Pearson Education, Inc. EXAMPLE 2.15 DENSITY AS A CONVERSION FACTOR The gasoline in an automobile gas tank has a mass of 60.0 kg and a density of 0.752 g/cm 3. What is its volume in cm3? SORT You are given the mass in kilograms and asked to find the volume in cubic centimeters. Density is the conversion factor between mass and volume.

GIVEN: 60.0 kg Density = 0.752 g/cm3 STRATEGIZE Build the solution map starting with kg and ending with cm3. Use the density (inverted) to convert from g to cm3. SOLUTION MAP FIND: volume in cm3 RELATIONSHIPS USED 0.752 g/cm3 (given in problem) 1000 g = 1 kg (from Table 2.2)

2015 Pearson Education, Inc. EXAMPLE 2.15 DENSITY AS A CONVERSION FACTOR Continued SOLVE Follow the solution map to solve the problem. Round the answer to three significant figures to reflect the three significant figures in the given quantities. CHECK Check your answer. Are the units correct? Does the answer make physical sense? SOLUTION The units of the answer are those of volume, so they are correct. The magnitude seems reasonable because the density is somewhat less than 1 g/cm3; therefore the volume of 60.0 kg should be somewhat more than 60.0 103 cm3. 2015 Pearson Education, Inc.

EXAMPLE 2.15 DENSITY AS A CONVERSION FACTOR Continued SKILLBUILDER 2.15 | Density as a Conversion Factor A drop of acetone (nail polish remover) has a mass of 35 mg and a density of 0.788 g/cm 3. What is its volume in cubic centimeters? Answer: 4.4 102 cm3 SKILLBUILDER PLUS A steel cylinder has a volume of 246 cm3 and a density of 7.93 g/cm 3. What is its mass in kilograms? Answer: 1.95 kg FOR MORE PRACTICE Example 2.28; Problems 101, 102. 2015 Pearson Education, Inc.

Chapter 2 in Review Uncertainty: Scientists report measured quantities so that the number of digits reflects the certainty in the measurement. Write measured quantities so that every digit is certain except the last, which is estimated. 2015 Pearson Education, Inc. Chapter 2 in Review Units: Measured quantities usually have units associated with them. The SI units:

length: meter, mass: kilogram, time: second Prefix multipliers such as kilo- or milli- are often used in combination with these basic units. The SI units of volume are units of length raised to the third power; liters or milliliters are often used as well. 2015 Pearson Education, Inc. Chapter 2 in Review Density: The density of a substance is its mass divided by its volume, d = m/V, and is usually reported in units of grams per cubic centimeter or grams per milliliter. Density is a fundamental property of all substances and generally differs from one substance to another.

2015 Pearson Education, Inc. Chemical Skills Learning Objectives 1. LO: Express very large and very small numbers using scientific notation. 2. LO: Report measured quantities to the right number of digits. 3. LO: Determine which digits in a number are significant. 4. LO: Round numbers to the correct number of significant figures. 2015 Pearson Education, Inc. Chemical Skills Learning Objectives

5. LO: Determine the correct number of significant figures in the results of multiplication and division calculations. 6. LO: Determine the correct number of significant figures in the results of addition and subtraction calculations. 7. LO: Determine the correct number of significant figures in the results of calculations involving both addition/subtraction and multiplication/division. 2015 Pearson Education, Inc. Chemical Skills Learning Objectives 8. LO: Convert between units.

9. LO: Convert units raised to a power. 10. LO: Calculate the density of a substance. 11. LO: Use density as a conversion factor. 2015 Pearson Education, Inc. Highlight Problem Involving Units In 1999, NASA lost a $94 million orbiter because two groups of engineers failed to communicate to each other the units that they used in their calculations. Consequently, the orbiter descended too far into

the Martian atmosphere and burned up. 2015 Pearson Education, Inc. Highlight Problem Involving Units Suppose that the Mars orbiter was to have established orbit at 155 km and that one group of engineers specified this distance as 1.55 105 m. Suppose further that a second group of engineers programmed the orbiter to go to 1.55 105 ft. What was the difference in kilometers between the two altitudes? How low did the probe go? 2015 Pearson Education, Inc.

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