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Measurements
This chapter is crucial to the development of problem solving for the rest of the text. A short section covers the scientific method. Students review math and learn to do calculations while working on everyday exam- ples of problems in health and medicine using metric units. Students confront their fear of word problems by learning a system for analyzing and solving work problems. Scientific notations, accuracy and precision, and the use of a calculator are now included in the chapter. It is a chapter that is well worth taking some time on.
Study Goals
● (^) Describe the scientific method. ● (^) Learn the base units and abbreviations for the metric (SI) system. ● (^) Distinguish between measured numbers and exact numbers. ● (^) Determine the number of significant figures in a measurement. ● (^) Use prefixes to change base units to larger or smaller units. ● (^) Form conversion factors from units in an equality. ● (^) Use metric units, U.S. units, percent, and density as conversion factors. ● (^) In problem solving, convert the initial unit of a measurement to another unit. ● (^) Round off a calculator answer to report an answer with the correct number of significant figures. ● (^) Calculate temperature values in degrees Celsius and kelvins.
Chapter Outline
1.1 Scientific Method : Thinking Like A Scientist 1.2 Measurement and Scientific Notation Explore Your World: Units Listed on Labels 1.3 Measured Numbers and Significant Figures 1.4 Significant Figures in Calculations Career Focus: Phlebotomist 1.5 Prefixes and Equalities 1.6 Writing Conversion Factors Explore Your World: SI and Metric Equalities on Product Labels 1.7 Problem Solving Career Focus: Veterinary Technician 1.8 Density Explore Your World: Sink or Float? Health Note: Determination of Percentage of Body Fat 1.9 Temperature Career Focus: Surgical Technologist Health Note: Variation in Body Temperature
Chapter Summary and Demonstrations
1. Measurement and Significant Figures
Measurements in the U.S. and metric systems are compared. Consideration of real-life situations and health-related examples and at-home explore activities introduce or elaborate the discussion. The student is taught to recognize and count significant figures in measurements and calculations.
Demonstration : To demonstrate differences in measuring by different observers, I pass out several sheets of paper along with metric rulers and ask the students to measure the width and the length and to calculate the area. I list some of the responses, which are in centimeters, inches, or without units, some with two sig- nificant figures, and some with three. The last digit is often different. Since all of the measurements are done with the same size sheets of paper, we discuss measurements, differences in observing measuring tools, estimations, significant figures, units, and the importance of specifying units. Finally, we look at the calculation of area using their measurements and talk about the need to use significant figures when report- ing calculator answers. I also take some time here to discuss calculator operations.
2. Counting Numbers, Percent, and Conversion Factors
Equalities and conversion factors are carefully explained through many examples. Problem solving utilizes dimensional analysis to convert between U.S. and metric units. The emphasis on metric units, significant fig- ures, and numerical problem-solving exercises prepares the student for problem solving throughout the text.
Demonstration : To begin our study of numbers from measurement and exact numbers, I use the following kind of grid on a transparency but expand for the number of lecture teams. I bring in bags of small M&Ms and give one to each team. They are asked to predict the number of yellow, green, and brown M&Ms. You may want to add red, green, and tan. Each group could do this outside of class and bring in their results. Then they open the bag and determine the actual amounts. They calculate percent and grams per M&M (using the mass written on the package) and write their results on the transparency. We compare predictions and actual, discuss a counting number and a measured number (mass of the package contents and grams/M&M), and discuss what to do with significant figures. This is an exercise remembered by all of the students and helps to allay the fear of working with numbers in chemistry.
Chapter 1
M&Ms Statistics
M&Ms # yellow # brown pkg grams
Team predict/actual predict/actual predict/actual % yellow % brown mass M&M 1 2 3 4
3. Density
Density is defined and used as a conversion factor.
Demonstration : I introduce density by dropping an ice cube into a beaker containing water and into one containing isopropyl alcohol. This discrepant event (ice cube sinking in the alcohol) leads to questions about the densities of the substances and the liquids and the reason one substance sinks or floats in another. Discussion follows on what adjustments are made in density to make hot air balloons rise and divers descend or ascend.
4. Temperature
Temperature is defined, and the scales used for measuring temperature are described.
Demonstration : To develop the idea of temperature units, I have students imagine that we have some unmarked thermometers that need temperature scales. We determine reference systems for a Fahrenheit scale and then for a metric scale and mark a line for the freezing points and another line for the boiling points. Then we talk about how many units of temperature must be placed between the freezing and boiling lines. Compare the size of the °F with a °C. Use the zero points and the degree conversion factor to set up
Lab 2 Conversion Factors in Calculations
From length measurements, areas are calculated and the volume of a solid is determined. Volume displace- ment of a matching object compares two methods of volume determination. Measurements in metric and U.S. systems are used to produce metric—U.S. conversion factors for length and volume. Students deter- mine the mass of objects on a balance and determine conversion factors using mass percent. Food products are used to derive a conversion factor for g/lb and lb/kg. The counting of significant figures in measured numbers and their effect on calculated answers is defined and studied. Students learn to round off calculator answers to the correct number of significant figures.
A. Rounding Off B. Significant Figures in Calculations C. Conversion Factors for Length D. Conversion Factors for Volume E. Conversion Factors for Mass F. Percent by Mass ( LM only )
Laboratory Skills to Demonstrate
Describe the units on a metric stick Use of a laboratory balance Give examples of measured and exact numbers Reading a graduated cylinder Techniques of measurement using a balance, meter stick, and graduated cylinders Review common operations on a calculator Give examples of counting significant figures and using them in calculations Review the formulas for calculating area and volume of solids Demonstrate rounding off numbers
Lab 3 Density and Specific Gravity
The mass and volume of a liquid and a solid are used to calculate density. Specific gravity for the liquids are determined using a hydrometer and compared to the value obtained earlier. The relationship between mass and volume of a liquid is graphed. The student is taught to prepare a data table, label axes with units of measurement, apply equal intervals, and plot the data points on a graph.
A. Density of a Solid B. Density of a Liquid C. Specific Gravity
Laboratory Skills to Demonstrate
Calculations of density and specific gravity Reading a hydrometer
Answers and Solutions to Text Problems
1.1 a. A hypothesis proposes a possible explanation for a natural phenomenon. b. An experiment is a procedure that tests the validity of a hypothesis. c. A theory is a hypothesis that has been validated many times by many scientists. d. An observation is a description or measurement of a natural phenomenon.
1.2 a. Hypothesis b. Observation c. Experiment d. Theory
Chapter 1
1.3 (1) A change in number of sales is an observation. (2) Changing the menu to improve sales is a hypothesis. (3) A taste test is an experiment. (4) The ratings of the taste test are observations. (5) Improvement in sales is an observation. (6) Better sales by changing the menu is a theory.
1.4 (1) Observation (2) Hypothesis (3) Experiment (4) Experiment (5) Observation (6) Hypothesis or theory
1.5 a. meter; length b. gram; mass c. liter; volume d. second; time e. degree Celsius; temperature
1.6 a. liter; volume b. meter, length c. kilogram; mass d. gram; mass e. kelvin; temperature
1.7 a. Move the decimal point left four decimal places to give m. b. Move the decimal point left two decimal places to give g. c. Move the decimal point right six decimal places to give cm. d. Move the decimal point right four decimal places to give s. e. Move the decimal point right three decimal places to give L. f. Move the decimal point left six decimal places to give kg.
1.8 a. g b. m c. g d. m e. s f.
1.9 a. The value , which is also , is greater than. b. The value , which is also is greater than. c. The value or 10 000 is greater than or 0.0001. d. The value or 0.068 is greater than 0.000 52.
1.10 a. b. c. d.
1.11 a. The standard number is 1.2 times the power of or 10 000, which gives 12 000. b. The standard number is 8.25 times the power of or 0.01, which gives 0.0825. c. The standard number is 4 times the power of or 1 000 000, which gives 4 000 000. d. The standard number is 5 times the power of or 0.001, which gives 0.005.
1.12 a. 0.000 036 b. 87 500 c. 0.03 d. 212 000
1.13 Measured numbers are obtained using some kind of measuring tool. Exact numbers are numbers obtained by counting or from a definition in the metric or the U.S. measuring system. a. measured b. exact c. exact d. measured
1.14 a. exact b. measured c. measured d. measured
1.15 Measured numbers are obtained using some kind of measuring tool. Exact numbers are numbers obtained by counting or from a definition in the metric or the U.S. measuring system. a. The value 6 oz of meat is obtained by measurement, whereas 3 hamburgers is a counted/exact number.
10 -^3
10 -^2
5.5 * 10 -^9 3.4 * 10 2 5 * 10 -^84 * 10 -^10
6.8 * 10 -^2
1 * 10 4 1 * 10 -^4
3.2 * 10 -^2 320 * 10 -^4 4.5 * 10 -^4
2.4 * 10 -^2 1.5 * 10 3 m^3
1.8 * 10 8 6 * 10 -^5 7.5 * 10 5 1.5 * 10 -^1
7.85 * 10 -^3
1.4 * 10 -^4
5 * 10 -^6
Measurements
c.
d.
1.30 a.
b.
c.
d.
1.31 a. 45.48 cm 8.057 cm 53.54 cm b. 23.45 g 104.1 g 0.025 g 127.6 g c. 145.675 mL 24.2 mL 121.5 mL d. 1.08 L 0.585 L 0.50 L
1.32 a. 5.08 g 25.1 g 30.2 g b. 85.66 cm 104.10 cm 0.025 cm 189.79 cm c. 24.568 mL 14.25 mL 10.32 mL d. 0.2654 L 0.2585 L 0.0069 L
1.33 The km/hr markings indicate how many kilometers (how much distance) will be traversed in 1 hour’s time if the speed is held constant. The mph markings indicate the same distance traversed but measured in miles during the 1 hour of travel.
You are not exceeding the 55-mph speed limit if your speedometer reads 80 kph.
1.35 Because the prefix kilo means one thousand times, a kilo gram is equal to 1000 grams.
1.36 Because the prefix centi means one hundredth, a centi meter is one hundredth of a meter.
1.37 a. mg b. dL c. km d. kg e. L
1.38 a. centimeter b. kilogram c. deciliter d. gigameter e. microgram
1.39 a. 0.01 b. 1000 c. 0.001 d. 0.1 e. 1 000 000
1.40 a. 1.0 decigram b. 1 microgram c. 1 kilogram d. 1 centigram e. 1 milligram
1.41 a. 100 cm b. 1000 m c. 0.001 m d. 1000 mL
1.42 a. 1 kg 1000 g b. 1 mL 0.001 L c. 1 g 0.001 kg d. 1 g 1000 mg
1.43 a. kilogram b. milliliter c. km d. kL
1.44 a. mg b. millimeter c. m d. dL
1.45 Because a conversion factor is unchanged when inverted; and
100 cm 1 m
1 m 100 cm
80 km hr
1000 m km
39.4 in. 1 mi.
1 ft 12 in.
1 mi. 5280 ft
= 50 mph
Measurements
1.46 Verify that the units cancel when the conversion factors are applied.
1.47 1 kg 1000 g
1.48 1 m 100 cm
1.49 a. 1 yd 3 ft
b. 1 mi 5280 ft
c. 1 min 60 s
d. 1 gal 27 mi.
e. 100 g sterling 93 g silver
1.50 a. 1 gal 4 qt
b. 1 lb $1.
c. 1 week 7 days
d. $1 4 quarters
e. 100 g alloy 58 g gold
1.51 Learning the relationships between the metric prefixes will help you write the following equalities and their resulting conversion factors.
a. 1 m 100 cm and
b. 1 g 1000 mg and
c. 1 L 1000 mL and
d. 1 dL 100 mL and
1.52 a. 1 in. 2.54 cm
b. 1 kg 2.20 lb and
c. 1 lb 454 g and
d. 1 qt 946 mL and
946 mL 1 qt
1 qt 946 mL
454 g 1 lb
1 lb 454 g
2.20 lb 1 kg
1 kg 2.20 lb
1 in. 2.54 cm
and
2.54 cm 1 in.
100 mL 1 dL
1 dL 100 mL
1000 mL 1 L
1 L
1000 mL
1000 mg 1 g
1 g 1000 mg
100 cm 1 m
1 m 100 cm
58 g gold 100 g alloy
100 g alloy 58 g gold
4 quarters
4 quarters $
1 week 7 day
7 day 1 week
1 lb $ 1.
1 lb
1 gal 4 qt
4 qt 1 gal
93 g silver 100 g sterling
and
100 g sterling 93 g silver
1 gal 27 mi
and
27 mi 1 gal
1 min 60 s
and
60 s 1 min
1 mi 5280 ft
and
5280 ft 1 mi
1 yd 3 m
and
3 ft 1 yd
Chapter 1
d.
1.59 a. Plan: ft in. cm m
b. Plan: ft in. cm m m^2
c. Plan: m km hr min s
1.60 a. 91.4 m b. 41 m c. 2.7 s
1.61 a. Plan: L qt gal
b. Plan: g mg tablet
c. Plan: lb g kg mg ampicillin
1.62 a.
b.
c.
1.63 Each of the following require a percent factor from the problem information. a. Plan: g crust g oxygen (percent equality: 100.0 g crust 46.7 g oxygen)
b. Plan: g crust g magnesium (percent equality: 100.0 g crust 2.1 g magnesium)
c. Plan: oz lb g g nitrogen (percent equality: 100.0 g fertilizer 15 g nitrogen)
10.0 oz fertilizer *
1 lb 16 oz
454 g 1 lb
15 g nitrogen 100.0 g fertilizer
= 43 g nitrogen
1.25 g crust *
2.1 g magnesium 100.0 g crust
= 0.026 g magnesium
325 g crust *
46.7 g oxygen 100.0 g crust
= 152 g oxygen
325 mg *
1 g 1000 mg
1 mL 0.50 g
= 0.65 mL
425 mg *
1 kg body wt 5.00 mg
2.20 lb 1 kg
= 187 lb
1 day *
24 hr 1 day
1.0 g tetracycline 6 hr
1000 mg 1 g
1 tablet 500 mg
= 8 tablets
34 lb body weight *
454 g 1 lb
1 kg 1000 g
115 mg ampicillin 1 kg body weight
= 1.8 * 10 3 mg
0.024 g *
1000 mg 1 g
1 tablet 8 mg
= 3. tablets
250 L *
1.06 qt 1 L
1 gal 4 qt
= 66 gal
23.8 m *
1 km 1000 m
1 hr 185 km
60 min 1 hr
60 s 1 min
= 0.463 s
Area = 23.8 m * 8.23 m = 196 m^2
27.0 ft *
12 in. 1 ft
2.54 cm 1 in.
1 m 100 cm
= 8.23 m (width)
78.0 ft *
12 in. 1 ft
2.54 cm 1 in.
1 m 100 cm
= 23.8 m (length)
18.5 gal - 12.2 gal = 6.3 gal
46.0 L *
1.06 qt 1 L
1 gal 4 qt
= 12.2 gal
Chapter 1
d. Plan: kg pecans kg choc. bars lb (percent equality: 100.0 kg bars 22.0 kg pecans)
1.65 a. 0.045 kg b. 2530 g c. 29 g fiber/cake d. 4.0 oz
1.65 Because the density of aluminum is 2.70 g/cm 3 , silver is 10.5 g/cm 3 , and lead is 11.3 g/cm 3 , we can identify the unknown metal by calculating its density as follows:
The metal is lead.
1.66 The volume of a cube, 2.0 cm on each edge, is calculated as follows:
(Both cubes have the same volume, but their masses differ.) A cube will displace its volume when submerged in water, so the final volume reading in each graduated cylinder is:
1.67 Density is the mass of a substance divided by its volume. The densities of solids and liquids are usu- ally stated in g/ml or g/cm^3.
a.
b.
c. volume of gem: 34.5 mL total 20.0 mL water 14.5 mL
density of gem:
d.
1.68 a. 1220 g/3500 mL 0.35 g/mL b. 155 g/125 mL 1.24 g/mL c. 5.025 g/5.00 mL 1.01 g/mL d. 140 g/10 000 mL 0.014 g/mL
1.69 a.
b.
c. 225 mL *
7.8 g 1 mL
1 lb 454 g
16 oz 1 lb
= 62 oz
6.5 mL *
13.6 g 1 mL
= 88 g
1.5 kg alcohol *
1000 g 1 kg alcohol
1 mL 0.79 g
1 L
1000 mL
= 1.9 L
density of syrup =
67.23 g 47.3 mL
= 1.42 g/mL
mass of syrup = 182.48 g - 115.25 g = 67.23 g
0.100 pint *
1 qt 2 pints
1 L
1.06 qt
1000 mL 1 L
= 47.3 mL
45.0 g 14.5 mL
= 3.10 g/mL
0.250 lb
- mL
454 g 1 lb
0.873 g mL
24.0 g 20.0 mL
= 1.20 g/mL
Density =
mass (grams) Volume (mL)
40.0 mL water + 8.0 mL metal = 48.0 mL total volume
(2.0 cm)^3 * 1 mL/1 cm^3 = 8.0 mL
217 g metal 19.2 cm^3 metal
= 11.3 g/cm 3
5.0 kg pecans *
100 kg choc. bars 22.0 kg pecans
2.20 lb 1 kg
= 50. lb chocolate bars
Measurements
f.
1.76 a. b.
c.
d. e. f.
1.77 a.
b.
No, there is no need to phone the doctor. The child’s temperature is less than 40.0°C.
1.78 a.
b.
1.79 Yes. Sherlock’s investigation includes observations (gathering data), formulating a hypothesis, test- ing the hypothesis, and modifying it until one of the hypotheses is validated.
1.80 Sherlock meant that you should not propose a theory until you have data from experiments and observations.
1.81 a. The number of legs is a counted number; it is exact. b. The height is measured with a ruler or tape measure; it is a measured number. c. The number of chairs is a counted number; it is exact. d. The area is measured with a ruler or tape measure; it is a measured number.
1.82 a. Length is 3.7 cm; the 7 is the estimated digit. b. Length is 2.50 cm; the 0 is the estimated digit. c. Length is 4.10 cm; the 0 is the estimated digit.
1.83 a. length 6.96 cm; width 4.75 cm b. length 69.6 mm; width 47.5 mm c. There are three significant figures in the length measurement. d. There are three significant figures in the width measurement. e. 33.3 cm 2 f. Since there are three significant figures in the width and length measurements, there are three sig- nificant figures in the area.
1.84 a. This is cube C, since C has sunk to the bottom. b. This is cube D, since D is floating about one-third out of the water. c. This is cube A, since A is floating about one-half out of the water. d. This is cube B, since B is floating just at the surface of the water.
1.85 The volume of the object is 23.1 mL 18.5 mL 4.6 mL
The mass is 8.24 g and the density is
8.24 g 4.6 mL
= 1.8 g/mL
1.8 (20.6°C) + 32 = 37.1 + 32 = 69.1°F (32 is exact)
(145°F - 32)
= 62.8°C (1.8 is exact)
(103°F - 32)
= 39°C
(106°F - 32)
= 41°C
875 K - 273 = 602°C; 1.8 (602°C) + 32 = 1080 + 32 = 1110°F
545 K - 273 = 295°C
224 K - 273 = -49°C
- 25°F - 32
= -32°C
1.8 (155°C) + 32 = 279 + 32 = 311°F
1.8 (25°C) + 32 = 45 + 32 = 77°F
(72°F - 32)
= 22°C; 22°C + 273 = 295 K
Measurements
1.86 A would be gold; it has the highest density and the smallest volume. B would be silver; its density is intermediate and the volume is intermediate. C would be aluminum; it has the lowest density and the largest volume.
1.87 A hypothesis, which is a possible explanation for an observation, can be tested with experiments.
1.88 Experimentation is used to test and verify a hypothesis.
1.89 b. Another hypothesis needs to be written when experimental results do not support the previous hypothesis. c. More experiments are needed for a new hypothesis.
1.90 b.
1.91 a. Determination of a melting point with a thermometer is an observation. b. Describing a reason for the extinction of dinosaurs is a hypothesis or theory. c. Measuring the speed of a race is an observation.
1.92 a. observation b. observation c. hypothesis or theory
1.93 This problem requires several conversion factors. Let’s take a look first at a possible unit plan. When you write out the unit plan, be sure you know a conversion factor you can use for each step. Plan: ft in. cm m min
1.94 a. 22 kg salmon 5.5 kg crab 3.48 kg oysters 31 kg seafood
b.
1.95 Plan: lb g onions
Because the number of onion is a counting number, the value for onions, 15.8, is rounded to a whole number, 16.
1.97 a. Plan: oz crackers
b. Plan: crackers servings g lb oz
c. Plan: boxes oz servings mg g
50 boxes *
8.0 oz 1 box
1 serving 0.50 oz
140 mg sodium 1 serving
1 g 1000 mg
= 110 g sodium
10 crackers *
1 serving 6 crackers
4 g fat 1 serving
1 lb 454 g
16 oz 1 lb
= 0.2 oz fat
8.0 oz *
6 crackers 0.50 oz
= 96 crackers
1 lb $1.
1 kg 2.20 lb
= 4 * 10 2 kg
4.0 lb onions *
454 g 1 lb
1 onion 115 g onion
= 16 onions
31 kg seafood (total) *
2.20 lb 1 kg
= 68 lb
7500 ft *
12 in. 1 ft
2.54 cm 1 in.
1 m 100 cm
1 min 55.0 m
= 42 min
Chapter 1
1.109 a. Plan: kg mass kg fat lb (percent equality: 100.0 kg mass 3.0 kg fat)
lb fat
b. Plan: L fat mL g lb
lb fat
1.110 5.77 kg
1.111 Plan: cm 3 g g silver lb oz (percent equality: 100 g sterling 92.5 g silver)
oz of pure silver
1.112 Plan: kg lb body mass lb water
lb water
1.114 115.7°F, 319.5 K
1.115 a. (1) observation b. (2) hypothesis c. (3) experiment d. (2) hypothesis
1.116 a. (1) observation b. (2) hypothesis c. (1) observation d. (1) observation
1.117 You should record the mass as 34.075 g. Since your balance will weigh to the nearest 0.001 g, the mass values should be reported to 0.001 g.
1.118 Any measurement contains some estimation. The three different students were estimating the final digit, and they came up with different estimates. The student who reported 5.8 cm should have given the value of 5.80.
1.120 Silver:
Gold:
New water level: 75.5 ml 4.76 mL 2.59 mL 82.9 mL
1.121 Volume:
Radius:
h =
V
pr^2
0.644 cm^3 3.14 (3.81 cm)^2
= 0.0141 cm *
10 mm 1 cm
= 0.141 mm
3.00 in * 12 *
2.54 cm 1 in.
= 3.81 cm
1.50 g *
1 cm^3 2.33 g
= 0.644 cm 3
50.0 g *
1 mL 19.3 g
= 2.59 mL
50.0 g *
1 mL 10.5 g
= 4.76 mL
3.0 hr *
55 mi 1 hr
1 km 0.621 mi
1 L
11 km
1.06 qt 1 L
1 gal 4 qt
= 6.4 gal
K = -15°C + 273 = -247 K
TC =
(TF - 32°)
= -26°C
65 kg body mass *
2.20 lb 1 kg
55 lb water 100 lb body mass
27.0 cm 3 *
10.3 g
1 cm^3
92.5 g silver 100 g
1 lb 454 g
16 oz 1 lb
3.0 L fat *
1000 mL 1 L
0.94 g fat 1 mL fat
1 lb 454 g
45 kg body weight *
3.0 kg fat 100.0 kg body mass
2.20 lb 1 kg
Chapter 1
