How Are Units of Measurement Related to One Another?

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How are Units of Measurement Related to One Another?

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind...

Lord Kelvin (1824-1907), developer of the absolute scale of temperature measurement

Engage: Is Your Locker Big Enough for Your Lunch and Your Galoshes?

A. Construct a list of ten units of measurement. Explain the numeric relationship among any three of the ten units you have listed.

Before Studying this Unit After Studying this Unit

High School Chemistry: An Inquiry Approach

1. Use the measuring instrument provided to you by your teacher to measure your locker (or other rectangular three-dimensional object, if assigned) in meters.

Table 1: Locker Measurements Measurement (in meters) Uncertainty in Measurement (in meters) Width

Height

Depth (optional)

Area of Locker Door or Volume of Locker

Show Your Work!

Pool class data as instructed by your teacher.

Table 2: Class Data Group 1 Group 2 Group 3 Group 4 Group 5 Group 6

Width

Height

Depth

Area of Locker Door or Volume of Locker

Unit 1 Measurement

2. Did all class members report their measurement data to the same number of decimal places? Did all class members report the same uncertainty in their measured quantities? Explain any differences.

3. Summarize your class discussion about uncertainty in measurement. What determines the number of digits that should be expressed in a measured quantity? How many digits should be expressed in reporting your locker’s (or other object’s) volume? Use the terminology from the class discussion where appropriate.

High School Chemistry: An Inquiry Approach

Explore 1: How do Scientists Express Uncertainty in Measurement?

4. Consider the illustration below.

Figure from: Cracolice, M.S., & Peters, E. I. (2007). Introductory Chemistry: An Active Learning Approach (4th Ed.). Belmont, CA: Brooks/Cole Cengage Learning.

Each part of the illustration shows a board being measured with meter sticks that have different graduation marks. Using a ± value to express uncertainty, state the length of the board in meters based on the accuracy of each meter stick.

Explain 1

5. Scientists sometimes use the ± convention to express the uncertainty in measured values. Another convention is called significant figures. The measured quantity is expressed as the number of digits known accurately plus one estimated or uncertain digit. Express each measurement using the significant figure convention.

Unit 1 Measurement

6. If you count the number of digits in a quantity expressed using the significant figures convention, you are counting the number of significant figures in that quantity. You begin with the first nonzero digit and end with the uncertain digit. How many significant figures are in each measurement? Explain.

7. Express the length of the board in each case in centimeters. How do the number of significant figures in each measured quantity compare when the quantity is expressed in meters? in centimeters? Why do the number of significant figures compare as they do?

8. What does the location of a decimal point have to do with significant figures? (Hint: Consider the measurements in meters in Item 5 with measurements of the same object in centimeters in Item 7.)

9. If you carefully count the number of paper clips in a box and determine that the box contains 14 clips, how many significant figures are in the counted quantity? Explain the role of the significant figure convention when applied to exact numbers.

10. The masses of two different items were measured on two different balances. Their masses were recorded and correctly expressed according to the significant figure convention: 43.1 g, 234.99 g. What is the total mass of the two objects? Explain how you decided upon the number of significant figures in the sum.

High School Chemistry: An Inquiry Approach

Explore 2: Are You a Fat Head?

11. Imagine living long ago in a medieval kingdom. The standard measurement units were often based on anatomical parts of the ruler of the time hence the origin of the foot.

You are the King or Queen of your kingdom and the new unit of measure will be based on the circumference of your royal cranium. Construct a standard length unit based on the circumference of your cranium by wrapping a string around the head, along the eyebrows and above the ears. Carefully cut the string so that it is matches the distance around your royal cranium. Name your unit after your royal highness, such as 1 Jennyhead or 1 Tommyhead.

Develop a method of improving the precision and accuracy of your measuring device. Note: you may not use a calibrated measuring device such as a ruler.

Using your personal measurement unit, determine the lengths of the objects provided by your teacher. For example, how many JennyHeads is a paperclip, textbook, or a desk? Record your measurements in the table below.

Object Description Length Length (Personal Measuring (Centimeters) Unit)

12. Explain how you improved your measuring device and why it was improved as a result of your modifications.

Unit 1 Measurement

13. Things are going along well in your little kingdom until the evil French invader, Marquis de’ Centimeter arrives and forces your kingdom to adopt their standard units, called the metric system. The Marquis is ordering you to re-measure objects A-D using a meter stick and record the measurements, in centimeters, in the data table.

14. On the grid below, construct a plot of your unit of measurement with your improved device on the y-axis and length in centimeters on the x-axis. Use the grid on the next page to construct the opposite plot: length in centimeters on the y-axis and your unit of measurement on the x-axis.

High School Chemistry: An Inquiry Approach

Unit 1 Measurement

Explain 2

15. (a) Determine the slope of the line of best fit for the personal unit vs. metric length graph.

(b) Is (0,0) a valid point for these data? Why or why not?

(d) Determine the equation of the best-fit line for the metric length vs. personal unit graph.

16. What do the equations of the lines tell you?

17. What is the reciprocal of the slope of the line determined in the first graph? How does it compare with the slope of the line in the second graph? Why does this relationship exist?

18. Determine the length in centimeters of your personal unit of measure by using a meter stick calibrated in millimeters.

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19. Combine with another lab group to analyze your data and discuss the relationship between the two graphs, the slopes of the two best-fit lines, and the value found when your personal unit of measure was compared to meters. Summarize your findings.

Thinking About Your Thinking Proportional Reasoning

20. In the margin next to your answer to the previous question, there is a thinker icon and the words “Thinking About Your Thinking: Proportional Reasoning.” Why did your thinking about the answer to that question involve proportional reasoning? What is proportional reasoning?

21. Explain how the relationship between your personal unit of measure and the meter involves a proportionality.

22. Calculate the length of the following in your personal unit of measure and also in meters (the objects A–D that you measured now become a unti of measure.) You must document and support the reasoning or proportionalities you used to determine your specific values.

(a) 7.5 object A’s

(b) 0.75 object B’s

(d) 7.65 object D’s

Unit 1 Measurement

23. Individually graph the following data (see the corresponding table) and look for any trends or relationships between the variables and their corresponding graphs.

Set A Time Distance (s) (cm) 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16

Set B Time Distance (s) (m) 1 4 2 19 3 42 4 71 5 114 6 164 7 218

8 284

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Set C

Time Speed (h) (km/h)

1 5

2 2.4

3 1.7

4 1.3

5 1

6 0.9 7 0.7

8 0.6

Set D Month Temp

Number (°F)

1 36

2 43

3 41

4 55

5 60

6 69

7 82

8 91

9 71

10 58

11 43

12 39

Unit 1 Measurement

24. In which graph is y directly proportional to x? Explain.

25. Scientists use the symbol ∝ to indicate “is proportional to.” Use symbols to write the statement “y is proportional to x.”

What is a direct proportionality?

26. In which graph does y decrease as x increases?

27. In which set of data is y inversely proportional to x? Explain.

What is an inverse proportionality?

28. Which graph does not seem to picture a simple relationship?

High School Chemistry: An Inquiry Approach

Elaborate 1: Is Density a Matter of Size?

29. Liquid samples Procedure: A. Place a 25-mL graduated cylinder on a balance. The mass of the empty container is called the tare mass. Record this value below:

B. Using a pipette, add 5.0 mL of Liquid A to the cylinder, place it back on the balance, and measure the mass. Use the table below to record your mass and volume data, showing your work in the space provided. C. Repeat this process, using 5.0 mL increments, until you have added a total of 20.0 mL of liquid A into the cylinder. D. Dispose of liquid A according to your teacher’s instructions, clean the graduated cylinder, and then repeat steps A–C for liquid B. E. Repeat steps A–D for liquid C.

Mass (grams) Volume (mL) Liquid A Liquid B Liquid C

5.0

10.0

15.0

20.0

30. Solid samples: Use water displacement to determine the volume and respective masses for different size samples of the same substance. Show your work in the Volume of Solid column.

Mass of Solid Volume of Water Volume of Water + Volume of Solid (g) (mL) Solid (mL) (mL)

Unit 1 Measurement

Analysis: 31. From your data tables, graph and analyze the relationship between mass and volume for each liquid and solid given.

Liquid Samples

Solid Samples

High School Chemistry: An Inquiry Approach

32. Determine the slope of each trend line.

In what units are the slopes expressed?

Write the mathematical equation for each proportionality.

Thinking About Your Thinking Discuss the significance of these proportionalities. Proportional Reasoning

How do the calculated density values for the three liquids and solid compare to slopes of the lines from the graphs?

33. Given accepted values for the density of each liquid and solid, identify each and calculate the absolute and percent error for each data set.

Unit 1 Measurement

Elaborate 2: How are Celsius Temperature Degrees Related to Fahrenheit Degrees?

34. Working with a partner, acquire a Fahrenheit and a Celsius thermometer and bind them together with tape or a rubber band.

Set up a ring stand and Bunsen burner. After recording the initial temperature of the water, begin to heat the water. Continue to heat the water for approximately 10 minutes recording temperature readings at 1-minute intervals, in both the Celsius and the Fahrenheit scales. (Do not allow the water to reach the boiling point.)

Temperature, °C Temperature, °F

35. Construct a graph from your data plotting the Celsius temperature as the independent variable and the Fahrenheit temperature as the dependent variable on Graph 1 and reverse the independent & dependent variables on Graph 2.

Graph 1

High School Chemistry: An Inquiry Approach

Graph 2

36. Construct a trend line for both graphs, and determine the slope of both trend lines.

37. How are the slopes of the two trend lines related to each other? Explain.

Unit 1 Measurement

38. How can you use the mathematical relationship from the previous question to solve °C–°F temperature questions?

39. Using information acquired in this lab, determine the corresponding change in Celsius temperature for a 78-degree change in Fahrenheit.

40. Using information acquired in this lab, determine the corresponding change in Fahrenheit temperature for a 123-degree change in Celsius temperature.

41. Using the data from your graphs write an equation for each trend line.

With this information, convert 77°C to its corresponding Fahrenheit temperature

Convert 44°F to its corresponding Celsius temperature.

High School Chemistry: An Inquiry Approach

Appendix 1: The Metric System Appendix 1 is from: Cracolice, M.S., & Peters, E. I. (2007). As you study this section, work to achieve these learning goals: Introductory Chemistry: An Active Learning Approach (4th Ed.). • Distinguish between mass and weight. Belmont, CA: Brooks/Cole • Identify the metric units of mass, length, and volume. Cengage Learning. • State and write with appropriate metric prefixes the relationship between any metric unit and its corresponding kilounit, deciunit, centiunit, and milliunit. • Using data from Table 1.1, state and write with appropriate metric prefixes the relationship between any metric unit and other larger and smaller metric units. • Given a mass, length, or volume expressed in metric units, kilounits, deciunits, centiunits, or milliunits, express that quantity in the other four units.

Mass and Weight Consider a tool carried to the moon by astronauts. Suppose that tool weighs 6 ounces on the earth. On the surface of the moon it will weigh about 1 ounce. Halfway between the earth and the moon, it will be essentially weightless. Released in mid-space, it will remain there, floating, until moved by an astronaut to some other location. Yet in all three locations it would be the same tool, having a constant quantity of matter. Mass is a measure of quantity of matter. Weight is a measure of the force of gravitational attraction. Weight is proportional to mass, but the ratio between them depends on where in the universe you happen to be. Fortunately, this proportionality is essentially constant over the surface of the earth. Therefore, when you weigh something— that is, measure the force of gravity on the object—you can express this weight in terms of mass. In effect, weighing an object is one way of measuring its mass. In the laboratory, mass is measured on a balance (Fig. 1.1). The SI unit of mass is the kilogram, kg. It is defined as the mass of a platinum-iridium cylinder that is stored in a vault in Sèvres, France. A kilogram weighs about 2.2 pounds, which is too large a unit for most small-scale work in the laboratory. Instead, the basic metric mass unit is used: the gram, g. One gram is 1/1000 kilogram, or 0.001 kg. Conversely, we can say that 1 kg is 1000 g. In the metric system, units that are larger than the basic unit are larger by multiples of 10, that is, 10 times larger, 100 times larger, 1000 times larger, and so on. Similarly, smaller units are 1/10 as large, 1/100 as large, and so forth. This is what makes the metric system so easy to work with. To convert from one unit to another, all you have to do is move the decimal point. Larger and smaller metric units are identified by metric symbols, or prefixes. The prefix • It is essential that upper- and for the unit 1000 times larger than the base unit is kilo-, and its symbol is k. When the kilo- lowercase metric prefixes and symbol, k, is combined with the unit symbol for grams, g, you have the symbol for kilogram, symbols are used kg. Similarly, milli-, symbol m, is the prefix for the unit that is 1/1000 as large as the unit. appropriately. For example, Thus, 1/1000 of a gram (0.001 g) is 1 milligram, mg. The unit 1/100 as large as the base unit the symbol for the metric is a centiunit. The symbol for centi- is c. It follows that 1 cg (centigram) is 0.01 g. prefix mega-, 1,000,000 times larger than the base unit, is M Table 1.1 lists many metric prefixes and their symbols. Entries for the kilo–, deci–, (uppercase), and the symbol centi–, and milli– units are shown in boldface. These should be memorized, and you should for milli-, 1,000 times smaller be able to apply them to any metric unit. We will have fewer occasions to use the prefixes than the base unit, is m and symbols for other units, but by referring to the table you should be able to work with (lowercase). them, too.

Unit 1 Measurement

Table 1.1 Metric Prefixes* Large Units Small Units • Your instructor will probably Metric Metric Metric Metric tell you which prefixes from this table to memorize. If not, Prefix Symbol Multiple Prefix Symbol Multiple memorize the four prefixes in 12 0 tera- T 10 Unit (gram, meter, liter) 1 = 10 boldface. You need to recall giga- G 109 deci- d 0.1 = 10–1 and use the memorized mega- M 1,000,000 = 106 centi- c 0.01 = 10–2 prefixes, and you need to be kilo- k 1,000 = 103 milli- m 0.001 = 10–3 able to use the other prefixes, hecto- h 100 = 102 micro- µ 0.000001 = 10–6 given their value. deca- da 10 = 101 nano- n 10–9 Unit (gram, meter, liter) 1 = 100 pico- p 10–12 *The most important prefixes are printed in boldface.

Length The SI unit of length is the meter;∗ its abbreviation is m. The meter has a very precise but awesome definition: the distance light travels in a vacuum in 1/299,792,468 second. Modern technology requires such a precise definition. The meter is 39.37 inches long—about 3 inches longer than a yard. The common longer length unit, the kilometer (km) (1000 meters), is about 0.6 mile. Both the centimeter and the millimeter are used for small distances. A centimeter (cm) is about the width of a fingernail; a millimeter (mm) is roughly the thickness of a dime. Small metric and U.S. length units are compared in Figure 1.2.

Volume The SI volume unit is the cubic meter, m3. This is a derived unit because it consists of three base units, all meters, multiplied by each other. A cubic meter is too large a volume—larger than a cube whose sides are 3 feet long—to use in the laboratory. A more practical unit is the cubic centimeter, cm3. It is the volume of a cube with an edge of 1 cm (Fig. 1.3). A teaspoon holds about 5 cm3.

Figure 1.1 Three examples of laboratory balances. (a) A triple-beam balance measures mass with an error of ±0.01 g. It is usually used when high accuracy is not required. (b) A top-loading balance measures mass with an error of ±0.001 g. It has sufficient accuracy for most introductory chemistry applications. (c) An analytical balance measures mass with an error of ±0.0001 g. It is usually used in more advanced courses and in scientific laboratories. Figure from: Cracolice, M.S., & Peters, E. I. (2007). Introductory Chemistry: An Active Learning Approach (4th Ed.). Belmont,

CA: Brooks/Cole Cengage Learning.

∗ Outside the United States the length unit is spelled metre, and the liter, the volume unit that we will discuss shortly, is spelled litre. These spellings and their corresponding pronunciations come from France, where the metric system originated. In this book, we use the U.S. spellings, which match their pronunciations in the English language.

High School Chemistry: An Inquiry Approach

Figure 1.3 One cubic centimeter. This full-scale illustration of a cube 1 cm on each side will help you visualize the volume of 1 cm3. One milliliter and one cubic centimeter are the same volume, 1 mL = 1 cm3. The milliliter and the cubic centimeter are the most common volume units used in chemistry. Figure from: Cracolice, M.S., & Peters, E. I. (2007). Introductory Figure 1.2 Length measurements: inches, centimeters, and millimeters. This illustration is very close to Chemistry: An Active full scale. One inch is equal to 2.54 centimeters (numbered lines on metric scale) or 25.4 millimeters Learning Approach (4th Ed.). (unnumbered lines). Figure from: Cracolice, M.S., & Peters, E. I. (2007). Introductory Chemistry: An Belmont, CA: Brooks/Cole Active Learning Approach (4th Ed.). Belmont, CA: Brooks/Cole Cengage Learning. Cengage Learning.

Liquids and gases are not easily weighed, so we usually measure them in terms of the volumes they occupy. The common unit for expressing their volumes is the liter, L, which is defined as exactly 1000 cubic centimeters. Thus, there are 1000 cm3/L. This volume is equal to 1.06 U.S. quarts. Smaller volumes are given in milliliters, mL. Notice that there are 1000 mL in 1 liter (there are always 1000 milliunits in a unit), and 1 liter is 1000 cm3. This makes 1 mL and 1 cm3 exactly the same volume: Learn It Now! 3 This simple relationship is 1 mL = 0.001 L = 1 cm often missed. There is 1 mL in 1 cm3, not 1000. Figure 1.4 shows some laboratory devices for measuring volume.

Unit Conversions within the Metric System Conversions from one metric unit to another are applications of dimensional analysis. You should be able to make these conversions among the unit, kilounit, deciunit, centiunit, and milliunit. In this sense, unit (u) may be gram (g), meter (m), or liter (L). These relationships are summarized here as PER expressions and their resulting conversion factors:

1000 units per kilounit 1000 units/kilounit 1000 u/ku 10 deciunits per unit 10 deciunits/unit 10 du/u 100 centiunits per unit 100 centiunits/unit 100 cu/u 1000 milliunits per unit 1000 milliunits/unit 1000 mu/u

Your instructor may add other units to those you are required to know from memory. If you are given the relationship between any two metric units other than the four you should memorize, you should be able to make conversions with the given relationship.

Unit 1 Measurement

Example 1.1

How many meters are in 5948 centimeters? You can solve this problem by using the dimensional analysis procedure given at the end of the previous section. Identify the GIVEN quantity and units, the WANTED units, and the PER/PATH that you will use.

GIVEN: 5948 cm WANTED: m PER:

PATH: cm m Figure 1.4 Volumetric glassware. The beaker is only Now you have all the information before you. Set up the problem and calculate the answer. for estimating volumes. The tall graduated cylinder is used to measure volume more accurately. The flask with the tall neck (volumetric flask) and the pipet are used to obtain samples of fixed but precisely measured volumes. The buret is used to dispense variable 5948 cm × = 59.48 m volumes with high precision. Figure from: Cracolice, M.S., & CHECK: More centimeters (smaller unit) than meters (larger unit). OK. Peters, E. I. (2007). Introductory Chemistry: An Active Learning Approach (4th Ed.). Belmont, CA: Brooks/Cole Cengage Learning. Conversions between metric units are an example of proportional reasoning. Let’s examine the 1000-units-per-kilounit relationship more closely. In any measured quantity, the number of units is directly proportional to the number of kilounits: (# of units) ∝ (# of kilounits). Using the form (# of units) = m × (# of kilounits) and solving for the proportionality constant, m, gives Thinking About Your Thinking: Proportional Reasoning m =

which, by definition, is (1000 units PER kilounit).

The closest U.S. unit to the centimeter is the inch, and the closest U.S. unit to the meter is the yard. You can compare unit conversions in the metric system with similar conversions in the U.S. system by calculating the number of yards in 5948 inches. Again, it is a dimensional analysis problem: • In order to avoid confusion with the word in, the symbol “in.” for inches 5948 in. × = 165.222 . . . yd includes a period. This is the only unit symbol in this book that has a period. Which calculation is easier?

High School Chemistry: An Inquiry Approach

The most common error made in metric–metric conversions is moving the decimal the wrong way. The best protection against that mistake is to set up the problem by dimensional analysis, including all units. Always check your result with the larger/smaller rule: If your givens and wanteds have a large number of small units and a small number of large units, and if you’ve moved the decimal the right number of places, the answer should be correct.

Example 1.2

How many millimeters are in 2.35 centimeters?

If you are sufficiently familiar with the metric system, you can solve this problem in one step. However, at this point we recommend that you convert from the given unit to the base unit, and then from the base unit to the wanted unit. PLAN how you will solve the problem, set it up, and calculate the answer. Be sure to check the answer.

GIVEN: 2.35 cm WANTED: mm PER:

PATH: cm m mm

2.35 cm × × = 23.5 mm

CHECK: More mm (smaller unit) than cm (larger unit). OK.

• When you are first learning The two conversion factors in the preceding example can be combined: 1000 mm/m conversions among metric units, and 100 cm/m show that both 1000 mm and 100 cm equal 1 m. Therefore, 1000 mm = we recommend following the 100 cm. This equality can be written as a PER expression, 1000 mm/100 cm, which reduces path GIVEN unit → base unit → to 10 mm/cm. This relationship is well known to those accustomed to working with metric WANTED unit whenever you have units. Thus, Example 1.2 can be solved by either of the following one-step setups: to convert between units that are not the base unit. After you have worked a large number of 2.35 cm × = 23.5 mm or 2.35 cm × = 23.5 mm conversion problems, however, you may find the suggestion in this paragraph to be useful.

Unit 1 Measurement

Example 1.3

A fruit drink is sold in bottles that contain 2216 mL. Express the volume in cubic centimeters and in liters.

2216 mL = 2216 cm3. Recall that 1 mL and 1 cm3 are the same volume.

GIVEN: 2216 mL WANTED: L PER:

PATH: mL L

2216 mL × = 2.216 L

CHECK: More mL (smaller unit) than L (larger unit). OK.

Appendix 2: Significant Figures

The significant figure rule for addition and subtraction can be stated as follows:

The sum or difference must correspond with the most uncertain decimal place. To do this, round off the answer to the same number of decimal places as the factor with the fewest decimal places.

The significant figure rule for multiplication and division is as follows:

Round off the answer to the same number of significant figures as the smallest number of significant figures in any factor.

When a calculation contains both addition/subtraction and multiplication/division, you must apply each individual rule for significant figures separately. If two numbers are to be added and their sum is to be divided by another number, such as in , first perform the addition to the correct number of significant figures. Then perform the division, applying the multiplication/division significant-figure rule. The correct answer is 3.03, which you should now verify by performing the calculation for yourself.

Zeros are sometimes confusing when it comes to counting significant figures, so let’s examine them more closely. Suppose you weigh an object on a centigram balance with a precision of ±0.01 gram, and the mass is 50.30 grams. This mass is correctly recorded as 50.30 grams, not 50.3 grams. The tail-end zero must be shown because it is the uncertain digit. Beginning with the 5 and counting to the zero—the uncertain digit, the last digit shown—the measurement has four significant figures.

High School Chemistry: An Inquiry Approach

Now suppose you wish to express the same measurement—a measurement with four significant figures—in kilograms. 50.30 grams is the same mass as 0.05030 kilogram. Counting significant figures begins at the first nonzero digit, the 5. Specifically, note that counting significant figures does not begin at the decimal point. If the same measurement, still a four-significant-figure number, is expressed in milligrams, the value is 50,300 milligrams. Here a predicament arises. Knowing the history of the measurement, you know that the zero after the 3 is the uncertain digit. It is significant. The second zero after the 3 is necessary to express the value in standard form, but it is not significant. This violates the convention that the last digit shown is the uncertain digit. Tail- end zeros in a value with no decimal point are ambiguous when it comes to counting significant figures. This predicament is resolved simply by writing the value in exponential notation. The coefficient is written with the proper number of significant figures, with the uncertain digit being the last digit shown. Therefore, the best way to express the mass in milligrams is to write 5.030 × 104 milligrams. If the measurement was justified to three significant figures, you would write 5.03 × 104 milligrams, and if it should have five significant figures, write 5.0300 × 104 milligrams.

SUMMARY Counting Significant Figures

1. The significant figures convention applies only to measured quantities. 2. Begin counting significant figures at the first nonzero digit. 3. Stop counting significant figures at the uncertain digit. 4. The uncertain digit is the last digit expressed. 5. Use exponential notation when necessary to make the uncertain digit the last digit expressed.

Homework Questions

What Mathematical Concepts Should be Reviewed to Understand Measurement?

1. The distance from the Earth to Saturn is 12000000000 kilometers.

a) Use this measurement to explain why scientists prefer to use exponential notation to express large numbers.

b) Write the distance in exponential notation.

2. Solve the following without using a calculator. Express your answers in decimal form.

a) b) 10–2 × 103 c) 10–2 ÷ 10–4

3. Solve the following without using a calculator. Express your answers in decimal form.

a) 3.1 × 103 + 2 × 102 b) 3.9887 × 104 – 3.456 × 103

Unit 1 Measurement

4. Solve the following without using a calculator. Express your answers in decimal form.

a) 5–2 b)

5. Use a calculator to complete the following operations.

a) 2.88722 × 103 + 6.1420 × 102 b) 5.731 × 105 – 5.99 × 104

6. Use a calculator to complete the following operations.

a) (2 × 103)(99 × 10–5)(9.949 × 102) b)

How do Scientists Express Uncertainty in Measurement?

7. You weigh a beaker on an electronic balance. It displays 45.55 grams. You re-zero the balance and weigh the beaker again, and the balance now displays 45.56 grams. Intrigued, you re-zero and reweigh the beaker two more times, yielding readings of 45.54 grams and 45.55 grams.

a) What digits (ones, tenths, etc.) in the mass of the beaker are known with certainty?

b) What digit in the mass of the beaker is uncertain?

c) Precision is a measure of uncertainty. Using a ± value, express the precision of the balance in grams.

d) What is the significant figure convention, as it is used in chemistry?

e) Express the mass of the beaker, using the significant figure convention.

f) Accuracy is a measure of the closeness of a measured value to the true value. What do you know about the accuracy of the mass of the beaker? Explain. How can you learn more about the accuracy of the mass of the beaker?

8. The significant figure convention does not apply to counting numbers. Explain why.

9. You know that there are 12 inches in 1 foot. Is is correct to say that there are 12.0 inches per foot? 12.00 inches per foot? 12.000 inches per foot? How many significant figures are justified in the “12” part of 12 inches in 1 foot?

Questions 10 and 11: To how many significant figures is each quantity expressed?

10. (a) 75.9 g sugar, (b) 89.583 mL weed killer, (c) 0.366 in. diameter glass fiber, (d) 48,000 cm wire, (d) 0.80 ft spaghetti, (e) 0.625 kg silver, (f) 9.6941 × 106 cm thread, (g) 8.010 × 10–3 L acid

11. (a) 4.5609 g salt, (b) 0.10 in. diameter wire, (c) 12.3 × 10–3 kg fat, (d) 5310 cm3 copper, (e) 0.0231 ft licorice, (f) 6.1240 × 106 L salt brine, (g) 328 mL ginger ale, (h) 1200.0 mg dye

High School Chemistry: An Inquiry Approach

Questions 12 and 13: Round off each quantity to three significant figures.

12. (a) 6.398 × 10–3 km rope, (b) 0.0178 g silver nitrate, (c) 79,000 m cable, (d) 42,150 tons fertilizer, (e) $649.85

13. (a) 52.20 mL helium, (b) 17.963 g nitrogen, (c) 78.45 mg MSG, (d) 23,642,000 µm wavelength, (e) 0.0041962 kg lead

14. A moving-van crew picks up the following items: a couch that weighs 147 pounds, a chair that weighs 67.7 pounds, a piano at 3.6 × 102 pounds, and several boxes having a total weight of 135.43 pounds. Calculate and express in the correct number of significant figures the total weight of the load.

15. A solution is prepared by dissolving 2.86 grams of sodium chloride, 3.9 grams of ammonium sulfate, and 0.896 grams of potassium iodide in 246 grams of water. Calculate the total mass of the solution and express the sum in the proper number of significant figures.

16. A buret contains 22.93 milliliters of sodium hydroxide solution. A student drains some solution from the buret into the sink and pours the solution remaining in the buret into a graduated cylinder, which reads 19.4 mL. How many milliliters of solution were drained from the buret?

17. An empty beaker has a mass of 94.33 grams. After some chemical has been added, the mass on a more precise balance is 101.209 grams. What is the mass of the chemical in the beaker?

18. The bleem is the Martian unit for the amount of a substance. The mass of one bleem of pure table sugar is 342.3 grams. How many grams of sugar are in exactly bleem? What is the mass of 0.764 bleem?

19. Exactly one liter of a solution contains 31.4 grams of a certain chemical. How many grams are in exactly 2 liters? How about 7.37 liters? Express the results in the proper number of significant figures.

20. An empty beaker with a mass of 42.3 g is filled with a liquid, and the resulting mass of the liquid and the beaker when measured on a more precise balance is 62.87 g. The volume of this liquid is 19 mL. What is the density of the liquid?

21. Use the definition density ≡ to calculate the density of a liquid with a volume of 50.6 mL if that liquid is placed in an empty beaker with a mass of 32.344 g and the mass of the liquid plus the beaker on a less precise balance is 84.64 g.

How is Proportionality Used to Convert Among Units of Measurement?

22. How many inches are in 4.3 feet?

23. Convert 682 mL to dL.

Unit 1 Measurement

24. George weighs 185 pounds. What is his mass in kilograms?

25. How many liters of gasoline will fill a 25-gallon tank?

26. A speed limit is posted at 55 mph. What is the speed limit in meters per minute?

27. Convert 75°F to degrees Celsius.

Is Density a Matter of Size?

28. What is the mass of 50.0 mL of isopropyl alcohol, which has a density of 0.80 g/mL?

29. A rectangular solid has the dimensions 25 cm × 0.10 m × 50.0 m. Its mass is 7.50 × 102 kg. Will the object float in water?

30. A lab procedure requires 40.0 mL of concentrated sulfuric acid, which has a density of 1.85 g/mL. Explain how to measure the required volume without using a graduated cylinder or any other form of volumetric glassware.

31. While hiking along a creek, you discover a rock that resembles pure gold. You bring back to your chemistry classroom and measure that it weighs 539 grams and has a volume of 27.9 cubic centimeters. Is the rock gold (Hint: You need additional data to answer this question)? If so, what is its value at the current market price (additional data is needed to answer this question, as well)? If not, what other substance might it be?

32. The densest pure element is osmium, which has a density of 22.6 g/cm3. What is the volume (in cubic centimeters) of 1.0 ounce of osmium?

Additional Questions

33. In the United States, nutrition labels are required on all packaged foods. The table below lists the Calorie count for some food items, along with the energy expressed in kilojoules, which is the unit used in the rest of the world. Construct a plot of the data and write an equation that will allow a person to convert between Calories and kilojoules.

Food Item Energy (Calories) Energy (kilojoules) 20 oz cola 240 1003 One molasses cookie 120 502 One slice chocolate cake 485 2027 One apricot (raw) 60 251 One piece hard candy 12 50

High School Chemistry: An Inquiry Approach

34. The graph below shows the mass–volume relationship of two liquids.

mass, g

volume, mL

One of the lines represents water and the other represents carbon tetrachloride; a liquid once used in numerous applications, such as refrigeration, aerosol propellants, dry cleaning, and in fire extinguishers. However, it is now known that exposure to carbon tetrachloride can cause nervous system, liver, and kidney damage, so it now has limited use. Carbon tetrachloride has a density of 1.6 g/mL. Which line represents water and which line represents carbon tetrachloride? Explain.

35. (a) A student measures the volume of a liquid sample as 20.0 mL. She determines that the mass of the sample is 19.989 g. Which of the two measurements has the greatest uncertainty? Explain. Which has the greatest number of significant figures? Explain how uncertainty and the number of significant figures are related in general.

(b) The student uses the data from part (a) to calculate the density of the sample. Which measurement limits the certainty in value of the sample’s density? Explain. How many significant figures should be used to express the density? Explain.