1.6 Mathematical Treatment of Measurement Results

Question 1

Dimensional analysis (or the factor-label method)

Answer 1

Versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.

Question 2

The mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g

Answer 2

Given the conversion factor, the mass in ounces may be derived using an equation similar to the one used for converting length from inches to centimeters.
xoz=125 g×unit conversion factor

The unit conversion factor may be represented as:

1 oz28.349 gand28.349 g1 oz

The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.

x oz===125g×1 oz28.34912528.349) oz 4.41 oz (three significant figures)

Question 3

Convert a volume of 9.345 qt to liters.

Answer 3

8.844 L

Question 4

What is the density of common antifreeze in units of g/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.

Answer 4

Since density=mass volume
, we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A ×
unit conversion factor. The necessary conversion factors are given in Table 1.6: 1 lb. = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. Mass may be converted from pounds to grams as follows:
9.26lb×453.59 g1lb=4.20×103g

Volume may be converted from quarts to milliliters via two steps:

Step 1.
Convert quarts to liters.
4.00qt×1 L1.0567qt=3.78 L

Step 2.
Convert liters to milliliters.
3.78L×1000 mL1L=3.78×103mL

Then,

density=4.20×103g3.78×103mL=1.11 g/mL

Alternatively, the calculation could be set up in a way that uses three-unit conversion factors sequentially as follows:

9.26lb/4.00qt×453.59 g/1lb×1.0567qt/1L×1L/1000 mL=1.11 g/mL

Question 5

What is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?

Answer 5

2.957 × 10−2 L

Question 6

While being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.
(a) What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?

(b) If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?

Answer 6

(a) First convert distance from kilometers to miles:
1250km×0.62137 mi1km=777 mi

and then convert volume from liters to gallons:

213L×1.0567qt1L×1 gal4qt=56.3 gal

Finally,

(average) mileage=777 mi56.3 gal=13.8 miles/gallon=13.8 mpg

Alternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows:

1250km213L×0.62137 mi1km×1L1.0567qt×4qt1 gal=13.8 mpg

(b) Using the previously calculated volume in gallons, we find:

56.3gal×$3.801gal=$214

Question 7

A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).
(a) What (average) fuel economy, in miles per gallon, did the Prius get during this trip?

(b) If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?

Answer 7

(a) 51 mpg; (b) $62

Question 8

Fahrenheit

Answer 8

Unit of temperature; water freezes at 32 degrees Fahrenheit and boils at 212 degrees Fahrenheit on this scale.

Question 9

Normal body temperature has been commonly accepted as 37.0 °C (although it varies depending on time of day and method of measurement, as well as among individuals). What is this temperature on the kelvin scale and on the Fahrenheit scale?

Answer 9

K=°C+273.15=37.0+273.2=310.2 K

°F=95°C+32.0=(95×37.0)+32.0=66.6+32.0=98.6 °F

Question 10

Convert 80.92 °C to K and °F.

Answer 10

354.07 K, 177.7 °F

Question 11

Baking a ready-made pizza calls for an oven temperature of 450 °F. If you are in Europe, and your oven thermometer uses the Celsius scale, what is the setting? What is the kelvin temperature?

Answer 11

°C=59(°F−32) =59(450−32) =59×418=232 °C⟶set oven to 230 °C (two significant figures)

K=°C+273.15=59(°F−32) =59(450−32) +273.15=505.4K⟶5.1×102K

Question 12

Convert 50 °F to °C and K.

Answer 12

10 °C, 280 K

Question 13

Write conversion factors (as ratios) for the number of:
(a) yards in 1 meter

(b) liters in 1 liquid quart

(c) pounds in 1 kilogram

Answer 13

(a) 1.0936 yd/1 m
(b) 0.94635 L/ 1 qt
(c) 2.2046 lb./ 1 kg

Question 14

Write conversion factors (as ratios) for the number of:
(a) kilometers in 1 mile

(b) liters in 1 cubic foot

(c) grams in 1 ounce

Answer 14

The conversion factors as ratios for:
(a) kilometers in 1 mile: 1 mile is equivalent to 1.609 km.
(b) liters in 1 cubic foot: 1 cubic foot is equivalent to 28.317 L.
(c) grams in 1 ounce: 1 ounce is equivalent to 28.349g

Question 15

The label on a soft drink bottle gives the volume in two units: 2.0 L and 67.6 fl oz. Use this information to derive a conversion factor between the English and metric units. How many significant figures can you justify in your conversion factor?

Answer 15

2.0 L/67.6 fl oz = 0.030 L/ 1 fl oz

Question 16

The label on a box of cereal gives the mass of cereal in two units: 978 grams and 34.5 oz. Use this information to find a conversion factor between the English and metric units. How many significant figures can you justify in your conversion factor?

Answer 16

28.3 g/oz with 3 significant figures. This aligns well with the standard conversion factor of 1 ounce being approximately 28.35 grams

Question 17

Soccer is played with a round ball having a circumference between 27 and 28 in. and a weight between 14 and 16 oz. What are these specifications in units of centimeters and grams?

Answer 17

68-71 cm; 400-450 g

Question 18

A woman’s basketball has a circumference between 28.5 and 29.0 inches and a maximum weight of 20 ounces (two significant figures). What are these specifications in units of centimeters and grams?

Answer 18

Therefore, the maximum weight is 570 grams.
Circumference: 72.4 cm to 73.7 cm
Maximum weight: 570 grams

Question 19

How many milliliters of a soft drink are contained in a 12.0-oz can?

Answer 19

355 mL

Question 20

A barrel of oil is exactly 42 gal. How many liters of oil are in a barrel?

Answer 20

159 L

Question 21

The diameter of a red blood cell is about 3 ×10^−4 in. What is its diameter in centimeters?

Answer 21

8 x 10^-4 cm

Question 22

The distance between the centers of the two oxygen atoms in an oxygen molecule is 1.21 ×
10^−8 cm. What is this distance in inches?

Answer 22

Approximately 4.76 × 10^− 10 inches.

Question 23

Is a 197-lb weightlifter light enough to compete in a class limited to those weighing 90 kg or less?

Answer 23

Yes, weight = 89.4 kg

Question 24

A very good 197-lb weight lifter lifted 192 kg in a move called the clean and jerk. What was the mass of the weight lifted in pounds?

Answer 24

To convert the mass of the weight lifted from kilograms to pounds, you can use the conversion factor: 1 kilogram is approximately 2.20462 pounds.
So, for 192 kilograms:
192kg×2.20462kg/lb.​≈423.29lb
Therefore, the mass of the weight lifted in the clean and jerk was approximately 423.29 pounds.

Question 25

Many medical laboratory tests are run using 5.0 μL blood serum. What is this volume in milliliters?

Answer 25

5.0 x 10^-3 mL

Question 26

If an aspirin tablet contains 325 mg aspirin, how many grams of aspirin does it contain?

Answer 26

To convert milligrams to grams, you can use the conversion factor: 1 gram (g) = 1000 milligrams (mg). So, for 325 milligrams: 325mg×1000mg/1g​=0.325g Therefore, an aspirin tablet containing 325 mg of aspirin contains 0.325 grams of aspirin

Question 27

Use scientific (exponential) notation to express the following quantities in terms of the SI base units in Table 1.2: (a) 0.13 g (b) 232 Gg (c) 5.23 pm (d) 86.3 mg (e) 37.6 cm (f) 54 μm (g) 1 Ts (h) 27 ps (i) 0.15 mK

Answer 27

(a) 1.3 x 10^-4 kg (b) 2.32 x 10^8 kg (c) 5.23 x 10-12 m (d) 8.63 x 10^-5 kg (e) 3.76 x 10^-1 m (f) 5.4 x 10^-5 m (g) 1 x 10^12 s (h) 2.7 x 10^-11 s (i) 1.5 x 10^-4 K

Question 28

Complete the following conversions between SI units. (a) 612 g = ________ mg (b) 8.160 m = ________ cm (c) 3779 μg = ________ g (d) 781 mL = ________ L (e) 4.18 kg = ________ g (f) 27.8 m = ________ km (g) 0.13 mL = ________ L (h) 1738 km = ________ m (i) 1.9 Gg = ________ g

Answer 28

(a) 612 g = 612,000 mg To convert grams to milligrams, multiply by 1000: 612g×1000mg/g​=612,000mg (b) 8.160 m = 816 cm To convert meters to centimeters, multiply by 100: 8.160m×100cm/m​=816cm (c) 3779 μg = 0.003779 g To convert micrograms to grams, divide by 1,000,000: 3779μg×1g/1,000,000μg​=0.003779g (d) 781 mL = 0.781 L To convert milliliters to liters, divide by 1000: 781mL×1L/1000mL=0.781L (e)4.18 kg = 4180 g To convert kilograms to grams, multiply by 1000: 4.18kg×1000g/kg=4180g (f) 27.8 m = 0.0278 km To convert meters to kilometers, divide by 1000: 27.8m×1km/1000mL=0.0278km (g) 0.13 mL = 0.00013 L To convert milliliters to liters, divide by 1000: 0.13mL×1L/1000mL=0.00013L (h) 1738 km = 1,738,000 m To convert kilometers to meters, multiply by 1000: 1738km×1000 m/km​=1,738,000m (i) 1.9 Gg = 1,900,000,000 g To convert gigagrams to grams, multiply by 1,000,000,000: 1.9Gg×1,000,000,000 g/Gg=1,900,000,000g

Question 29

Gasoline is sold by the liter in many countries. How many liters are required to fill a 12.0-gal gas tank?

Answer 29

45.4 L

Question 30

Milk is sold by the liter in many countries. What is the volume of exactly 1/2 gal of milk in liters?

Answer 30

To convert 1/2 gallon to liters, you can use the conversion factor: 1 gallon (US) is approximately 3.78541 liters. So, for 1/2 gallon: ​1/2 gal×3.78541L/gal ​≈ 1.89271L Therefore, the volume of exactly 1/2 gallon of milk is approximately 1.89271 liters

Question 31

A long ton is defined as exactly 2240 lb. What is this mass in kilograms?

Answer 31

1.0160 x 10^3 kg

Question 32

Make the conversion indicated in each of the following: (a) the men’s world record long jump, 29 ft 4¼ in., to meters (b) the greatest depth of the ocean, about 6.5 mi, to kilometers (c) the area of the state of Oregon, 96,981 mi2, to square kilometers (d) the volume of 1 gill (exactly 4 oz) to milliliters (e) the estimated volume of the oceans, 330,000,000 mi3, to cubic kilometers. (f) the mass of a 3525-lb car to kilograms (g) the mass of a 2.3-oz egg to grams

Answer 32

(a) Men’s world record long jump, 29 ft 4¼ in., to meters First, convert feet and inches to inches: 29ft×12 in/ft + 4.25in = 348.25in Then, convert inches to meters (1 inch = 0.0254 meters): 348.25in × 0.0254 m/in ​≈8.85m (b) Greatest depth of the ocean, about 6.5 mi, to kilometers Convert miles to kilometers (1 mile = 1.60934 kilometers): 6.5mi × 1.60934 km/mi ≈ 10.46km (c) Area of the state of Oregon, 96,981 mi², to square kilometers Convert square miles to square kilometers (1 square mile = 2.58999 square kilometers): 96,981mi^2×2.58999 km^2/mi^2 ≈ 251,419 km^2 (d) Volume of 1 gill (exactly 4 oz) to milliliters Convert ounces to milliliters (1 ounce = 29.5735 milliliters): 4 oz × 29.5735 mL/oz ​≈ 118.29mL (e) Estimated volume of the oceans, 330,000,000 mi³, to cubic kilometers Convert cubic miles to cubic kilometers (1 cubic mile = 4.16818 cubic kilometers): 330,000,000mi^3 × 4.16818 km^3/mi^3 ​≈ 1,375,499,400 km^3 (f) Mass of a 3525-lb car to kilograms Convert pounds to kilograms (1 pound = 0.453592 kilograms): 3525 lb. × 0.453592 kg/lb. ≈ 1,598.34kg (g) Mass of a 2.3-oz egg to grams Convert ounces to grams (1 ounce = 28.3495 grams): 2.3 oz × 28.3495 g/oz ​≈ 65.20g

Question 33

Make the conversion indicated in each of the following: (a) the length of a soccer field, 120 m (three significant figures), to feet (b) the height of Mt. Kilimanjaro, at 19,565 ft, the highest mountain in Africa, to kilometers (c) the area of an 8.5- × 11-inch sheet of paper in cm2 (d) the displacement volume of an automobile engine, 161 in.3, to liters (e) the estimated mass of the atmosphere, 5.6 × 1015 tons, to kilograms (f) the mass of a bushel of rye, 32.0 lb, to kilograms (g) the mass of a 5.00-grain aspirin tablet to milligrams (1 grain = 0.00229 oz)

Answer 33

(a) 394 ft (b) 5.9634 km (c) 6.0 x 10^2 (d) 2.64 L (e) 5.1 x 10^18 kg (f) 14.5 kg (g) 324 mg

Question 34

Many chemistry conferences have held a 50-Trillion Angstrom Run (two significant figures). How long is this run-in kilometers and in miles? (1 Å = 1 × 10−10 m)

Answer 34

5 kilometers or 3.11 miles

Question 35

A chemist’s 50-Trillion Angstrom Run (see Exercise 1.78) would be an archeologist’s 10,900 cubit run. How long is one cubit in meters and in feet? (1 Å = 1 × 10−8 cm)

Answer 35

0.46 m; 1.5 ft/cubit

Question 36

The gas tank of a certain luxury automobile holds 22.3 gallons according to the owner’s manual. If the density of gasoline is 0.8206 g/mL, determine the mass in kilograms and pounds of the fuel in a full tank.

Answer 36

1. Convert gallons to milliliters: 1 gallon = 3,785.41 milliliters So, 22.3 gallons = (22.3 \times 3,785.41) milliliters ≈ 84,416.643 milliliters 2. Calculate the mass in grams: Density of gasoline = 0.8206 g/mL Mass (in grams) = Volume (in mL) × Density (g/mL) Mass = 84,416

Question 37

As an instructor is preparing for an experiment, he requires 225 g phosphoric acid. The only container readily available is a 150-mL Erlenmeyer flask. Is it large enough to contain the acid, whose density is 1.83 g/mL?

Answer 37

Yes, the acid's volume is 123 mL

Question 38

To prepare for a laboratory period, a student lab assistant needs 125 g of a compound. A bottle containing 1/4 lb. is available. Did the student have enough of the compound?

Answer 38

1. Convert 125 grams to pounds: 1 gram ≈ 0.00220462 pounds So, 125 grams ≈ (125 \times 0.00220462) pounds ≈ 0.2756 pounds 2. Convert 1/4 pound to grams: 1 pound = 453.592 grams So, 1/4 pound = (1/4 times 453.592) grams ≈ 113.4 grams Since 125 grams (0.2756 pounds) is more than 113.4 grams (1/4 pound), the student does not have enough of the compound.

Question 39

A chemistry student is 159 cm tall and weighs 45.8 kg. What is her height in inches and weight in pounds?

Answer 39

62.6 in (about 5 ft 3 in.) and 101 lb.

Question 40

In a recent Grand Prix, the winner completed the race with an average speed of 229.8 km/h. What was his speed in miles per hour, meters per second, and feet per second?

Answer 40

Therefore, the winner’s average speed was approximately 142.8 miles per hour, 63.83 meters per second, and 209.4 feet per second.

Question 41

Solve these problems about lumber dimensions. (a) To describe to a European how houses are constructed in the US, the dimensions of “two-by-four” lumber must be converted into metric units. The thickness × width × length dimensions are 1.50 in. × 3.50 in. × 8.00 ft in the US. What are the dimensions in cm × cm × m? (b) This lumber can be used as vertical studs, which are typically placed 16.0 in. apart. What is that distance in centimeters?

Answer 41

(a) 3.81 cm x 8.89 cm x 2.44 m (b) 40.6 cm

Question 42

The mercury content of a stream was believed to be above the minimum considered safe—1 part per billion (ppb) by weight. An analysis indicated that the concentration was 0.68 parts per billion. What quantity of mercury in grams was present in 15.0 L of the water, the density of which is 0.998 g/ml? (1 ppb Hg=1 ng Hg/1 g water)

Answer 42

Therefore, the quantity of mercury present in 15.0 liters of water is approximately 0.01018 grams.

Question 43

Calculate the density of aluminum if 27.6 cm3 has a mass of 74.6 g.

Answer 43

2.70 g/cm^3

Question 44

Osmium is one of the densest elements known. What is its density if 2.72 g has a volume of 0.121 cm^3?

Answer 44

Density = Volume/Mass​ Given: Mass = 2.72 g Volume = 0.121 cm³ So, Density = 2.72 g/0.121 cm^3 ≈22.48g/cm^3 Therefore, the density of osmium is approximately 22.48 g/cm³

Question 45

Calculate these masses. (a) What is the mass of 6.00 cm3 of mercury, density = 13.5939 g/cm^3? (b) What is the mass of 25.0 mL octane, density = 0.702 g/cm^3?

Answer 45

(a) 81.6 g (b) 17.6 g

Question 46

Calculate these masses. (a) What is the mass of 4.00 cm3 of sodium, density = 0.97 g/cm3 ? (b) What is the mass of 125 mL gaseous chlorine, density = 3.16 g/L?

Answer 46

(a) Mass of 4.00 cm³ of sodium: Given: Volume = 4.00 cm³ Density of sodium = 0.97 g/cm³ Using the formula for mass: Mass = Density × Volume So, Mass = 0.97 g/cm^3 ​×4.00cm^3 = 3.88g Therefore, the mass of 4.00 cm³ of sodium is 3.88 grams.

Question 47

Calculate these volumes. (a) What is the volume of 25 g iodine, density = 4.93 g/cm3? (b) What is the volume of 3.28 g gaseous hydrogen, density = 0.089 g/L?

Answer 47

(a) 5.1 mL (b) 37 L

Question 48

Calculate these volumes. (a) What is the volume of 11.3 g graphite, density = 2.25 g/cm3? (b) What is the volume of 39.657 g bromine, density = 2.928 g/cm3?

Answer 48

(a) Volume of 11.3 g of graphite: Given: Mass = 11.3 g Density = 2.25 g/cm³ Using the formula for volume: Volume = Mass/Density So, Volume = 11.3 g/ 2.25 g/cm^3​ ≈ 5.02cm^3 Therefore, the volume of 11.3 g of graphite is 5.02 cm³

Question 49

Convert the boiling temperature of gold, 2966 °C, into degrees Fahrenheit and kelvin.

Answer 49

5371 °F; 3239 K

Question 50

Convert the temperature of scalding water, 54 °C, into degrees Fahrenheit and kelvin.

Answer 50

1. Convert Celsius to Fahrenheit: The formula to convert Celsius to Fahrenheit is: °F = °C × 9/5 + 32 So, 54°C × 9/5 + 32=129.2°F 2. Convert Celsius to Kelvin: The formula to convert Celsius to Kelvin is: K = °C + 273.15 So, 54°C + 273.15 = 327.15K Therefore, the temperature of scalding water at 54 °C is 129.2 °F and 327.15 K

Question 51

Convert the temperature of the coldest area in a freezer, −10 °F, to degrees Celsius and kelvin.

Answer 51

-23°C, 250 K

Question 52

Convert the temperature of dry ice, −77 °C, into degrees Fahrenheit and kelvin.

Answer 52

1. Convert Celsius to Fahrenheit: The formula to convert Celsius to Fahrenheit is: °F = °C × 9/5 ​+ 32 So, −77°C × 9/5 ​+ 32 = −106.6°F 2. Convert Celsius to Kelvin: The formula to convert Celsius to Kelvin is: K = °C + 273.15 So, −77°C + 273.15 = 196.15K Therefore, the temperature of dry ice at −77 °C is −106.6 °F and 196.15 K

Question 53

Convert the boiling temperature of liquid ammonia, −28.1 °F, into degrees Celsius and kelvin.

Answer 53

-33.4°C, 239.8 K

Question 54

The label on a pressurized can of spray disinfectant warns against heating the can above 130 °F. What are the corresponding temperatures on the Celsius and kelvin temperature scales?

Answer 54

1. Convert Fahrenheit to Celsius: The formula to convert Fahrenheit to Celsius is: °C = (°F−32) × 5/9 So, 130°F = (130−32) × 5/9 ​≈ 54.44°C 2. Convert Fahrenheit to Kelvin: The formula to convert Fahrenheit to Kelvin is: K = (°F + 459.67) × 5/9 So, 130°F = (130+459.67) × 5/9 ​≈ 327.59K Therefore, the corresponding temperatures are approximately 54.44 °C and 327.59 K

Question 55

The weather in Europe was unusually warm during the summer of 1995. The TV news reported temperatures as high as 45 °C. What was the temperature on the Fahrenheit scale?

Answer 55

113°F