2.E: Measurements, Unit Conversions, and Density (Exercises)

Scientific Notation and Decimal Form

PROBLEM 2-1 Express each number in scientific notation.

a. 67,000,000,000	b. 1,689	c. 12.6	d. 1,492
e. 102,000,000	f. 101,325	g. 0.000006567	h. −0.0004004
i. 0.000000000000123	j. 0.000355	k. 0.314159	l. −0.051204

PROBLEM 2-2 Change the number in scientific notation to standard form.

a. 5.27 × 10^4	b. 1.0008 × 10^6	c. 6.98×10^8	d. 1.005×10^2
e. 6.22 × 10^−2	f. 9.9 × 10^−9	g. 9.98 × 10^−5	h. 5.109 × 10^−8

PROBLEM 2-3 Why is it easier to use scientific notation to express very large or very small numbers?

Measurements

PROBLEM 2-4 Identify the number and the unit in each quantity.

1. One dozen eggs
2. 2.54 centimeters
3. a box of pencils
4. 88 meters per second

PROBLEM 2-5 Give the abbreviation for each unit and define the abbreviation in terms of the base unit.

a. kiloliter	b. microsecond	c. decimeter	d. nanogram
e. kilometer	f. milligram	g. nanosecond	h. centiliter

Temperature

PROBLEM 2-6 Convert each temperature to °C.

1. The temperature of the surface of the sun (5800. K)
2. The boiling point of gold (3080. K)
3. The boiling point of liquid nitrogen (77.36 K)

PROBLEM 2-7 Convert each temperature to K.

1. 98.6 ^oC
2. 25.0 ^oC
3. 0.0 ^oC
4. -40.0 ^oC

PROBLEM 2-8

1. Convert the temperature of scalding water, 54 °C, into degrees Fahrenheit and kelvin.
2. Convert the temperature of the coldest area in a freezer, −10 °F, to degrees Celsius and kelvin.
3. Convert the temperature of dry ice, −77 °C, into degrees Fahrenheit and kelvin.
4. Convert the boiling temperature of liquid ammonia, −28.1 °F, into degrees Celsius and kelvin.
5. 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?
6. 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?

Uncertainty

PROBLEM 2-9 Indicate whether each of the following can be determined exactly or must be measured with some degree of uncertainty:

1. the number of seconds in an hour
2. the number of pages in this book
3. the number of grams in your weight
4. the number of grams in 3 kilograms
5. the volume of water you drink in one day
6. the distance from San Francisco to Kansas City

Significant Figures

PROBLEM 2-11 How many significant figures does each number have?

a- 6,798,000	b- 6,000,798	c- 6,000,798.00	d- 0.0006798	e- 2.1828
f- 0.005505	g- 55,050	h- 5	i- 500

PROBLEM 2-11 The following quantities were reported on the labels of commercial products. Determine the number of significant figures in each.

1. 0.0055 g active ingredients
2. 12 tablets
3. 3% hydrogen peroxide
4. 5.5 ounces
5. 473 mL
6. 1.75% bismuth
7. 0.001% phosphoric acid
8. 99.80% inert ingredients

PROBLEM 2-12 Round off each of the following numbers to two significant figures:

(a) 0.436; (b) 9.000; (c) 27.2; (d) 135; (e) 1.497 \(\times\) 10⁻³; (f) 0.445

PROBLEM 2-13 Solve and write the answer with the appropriate number of significant figures.

a- 23.096 × 90.300 =	b- 125 × 9.000 =	c- 1,027 + 610.0 + 363.06 =	d- 217 ÷ 903 =
e- 0.00666 x 321 =	f- 255.0 − 99 =	g- 13.77 + 908.226 + 515 =

PROBLEM 2-14 Explain why the concept of significant figures is important in scientific measurements.

PROBLEM 2-15 State the rules for determining the significant figures in a measurement.

PROBLEM 2-16 When do you round a number up, and when do you not round a number up?

Dimensional Analysis

Set up all your calculations using dimensional analysis. If you were training using a different method, you can use it while fully showing it.

PROBLEM 2-17 The average volume of blood in an adult male is 4.7 L. What is this volume in milliliters? [1L =1000 mL]

PROBLEM 2-18 A hummingbird can flap its wings once in 18 ms. How many seconds are in 18 ms? [1 s = 1000 ms]

PROBLEM 2-19 Complete the following conversions using dimensional analysis.

a. 5.67 mm = ________ m	b. 5.67 nm = ________ m	c. 0.3 cm = ________ m	d. 1.5 km = ________ mm
e. 24 mcg = ________ mg	f. 15 dg = ________ cg	g. 0.712 mg = ________ kg	h. 41 kg = ________ g
i. 612 g = ________ mg	j. 8.160 m = ________ cm	k. 3779 μg = ________ g	l. 781 mL = ________ L
m. 4.18 kg = ________ cg	n. 27.8 m = ________ km	o. 0.13 mL = ________ L	p. 1738 km = ________ m
q. 582 s = ________ minutes	r. 181.2 g = ________ kg

Density

For the following problems, assume that all measurements are done at a room temperature of 20 °C.

PROBLEM 2-20 Calculate the density of a 30.2 mL sample of ethyl alcohol with a mass of 23.7102 g.

PROBLEM 2-21 If you have a 2.130 mL sample of acetic acid with mass 0.002234 kg, what is the density in kg/L?

PROBLEM 2-22 A mercury thermometer used to measure a patient’s temperature contains 0.750 g of mercury. What is the volume of this mass of mercury? The density of Hg is 13.6 g/mL.

PROBLEM 2-23 You have a sample of aluminum that has a volume of 7.88 mL. What is the mass of Al if the density is 2.70 g/mL?

PROBLEM 2-24 Octane is the main component of the gas that you use in your car. If you fill a plastic container with 250.0 mL of octane, how much heavier is the container? Consider the mass of the container 0 g. Density of Octane = 0.702 g/cm³?

PROBLEM 2-25 Calculate these masses.

1. What is the mass of 4.00 cm³ of sodium, density = 0.97 g/cm?
2. What is the mass of 125 mL gaseous chlorine, density = 3.16 g/L?

PROBLEM 2-26 Calculate these volumes.

1. What is the volume of 25 g iodine, density = 4.93 g/cm³?
2. What is the volume of 3.28 g gaseous hydrogen, density = 0.089 g/L?
3. What is the volume of 11.3 g graphite, density = 2.25 g/cm³?
4. What is the volume of 39.657 g bromine, density = 2.928 g/cm³?

PROBLEM 2-27 A large piece of jewelry has a mass of 132.6 g. A graduated cylinder initially contains 48.6 mL water. When the jewelry is submerged in the graduated cylinder, the total volume increases to 61.2 mL.

1. Determine the density of this piece of jewelry.
2. Assuming that the jewelry is made from only one substance, what substance is it likely to be? Explain.

PROBLEM 2-28 Visit this PhET density simulation and select the Same Volume Blocks.

1. What are the mass, volume, and density of the yellow block?
2. What are the mass, volume, and density of the red block?
3. List the block colors in order from smallest to largest mass.
4. List the block colors in order from lowest to highest density.
5. How are mass and density related for blocks of the same volume?

PROBLEM 2-29 Visit this PhET density simulation and select Custom Blocks and then My Block.

1. Enter mass and volume values for the block such that the mass in kg is less than the volume in L. What does the block do? Why? Is this always the case when mass < volume?
2. Enter mass and volume values for the block such that the mass in kg is more than the volume in L. What does the block do? Why? Is this always the case when mass > volume?
3. How would (a) and (b) be different if the liquid in the tank were ethanol instead of water?
4. How would (a) and (b) be different if the liquid in the tank were mercury instead of water?

PROBLEM 2-30 Visit this PhET density simulation and select Mystery Blocks.

1. Pick one of the Mystery Blocks and determine its mass, volume, density, and its likely identity.
2. Pick a different Mystery Block and determine its mass, volume, density, and its likely identity.
3. Order the Mystery Blocks from least dense to most dense. Explain.

Dosage Calculations I

Set up all your calculations using dimensional analysis. If you were training using a different method, you can use it while fully showing it.

PROBLEM 2-31 Epinephrine is used to treat peanut allergies. A 5-year-old patient needs immediate treatment of 0.12 mg of epinephrine. The solution is available in a concentration of 0.50 mg/ml. Calculate the volume (mL) required for the patient.

PROBLEM 2-32 A physician ordered 100 mg of Demerol. Demerol is available as 50 mg per tablet. How many tablets should the nurse administer?

PROBLEM 2-33 The physician ordered 20 mg of Valium. Valium is available as 10 mg per tablet. How many tablets should the nurse administer?

PROBLEM 2-34 The recommended daily allowance (RDA) for magnesium for 19-30 year old+ men is 400 mg. Magnesium supplements come in 200 mg capsules. How many capsules need to be taken to meet RDA?

PROBLEM 2-35 The safe dosage of an IV antibiotic for children weighing more than 2.0 kg is 60. mg per kilogram of body weight. How many mg should be administered to a child weighing 16 kg?

PROBLEM 2-36 The tranquilizer valium is sold in 2.0 mL syringes that contain 50.0 mg of the drug per 1.0 mL of liquid. If a physician prescribes 25 mg of this drug, how many mL should be administered?

PROBLEM 2-37 An antibiotic is sold in 3.0 mL ampoules that contain 60.0 mg of the drug. How many mL of the antibiotic should be withdrawn from the ampoule if 45 mg are to be administered to a patient?

Dosage Calculations II

Set up your calculations using dimensional analysis. If you were training using a different method, you can use it while fully showing it.

PROBLEM 2-38 Neomycin is an antibiotic. A patient needs 2.5 grams of Neomycin sulfate. 500 mg tablets are available. How many tablets will you give to the patient?

PROBLEM 2-39 The doctor’s order is 1.2 g of folic acid. Folic Acid is available as 800 mg per tablet. How many tablets should be taken?

PROBLEM 2-40 The doctor’s order is 1 g of Calcium. What is on hand is Calcium, as 500 mg per tablet. How many tablets should be taken?

PROBLEM 2-41 Demerol is ordered 1.5mg/kg for a patient who is 220 lbs. Demerol is available as 50 mg per tablet. How many tablets should the nurse administer?

PROBLEM 2-42 A doctor prescribes amoxicillin 30 mg/kg to a child weighing 73.5 lbs. Amoxicillin is available as 500 mg tablets. How many tablets should the nurse administer?

PROBLEM 2-43 Vitamin C tablets can come in 500 mg tablets. How many of these tablets are needed to obtain 10 g of vitamin C?

PROBLEM 2-44 A 175 lb patient is to undergo surgery and will be given an intravenous anesthetic. The safe dosage of anesthetic is up to 12 mg/kg of body weight. Determine the mg of the dose of anesthetic that should be used. (2.205 lbs = 1 kg)

PROBLEM 2-45 A drug dose of 1.5 mg/kg is ordered for a child weighing 70.4 lbs. How many mg of the drug should be administered? If the drug is available as 60 mg/2 mL. How many mL must the nurse administer?

PROBLEM 2-46 The antiviral drug Stavudine is used for the treatment of AIDS. The daily recommended dosage of Stavudine is 1.0 mg/kg of body weight. How many grams of the drug should be administered to a 150 lb patient?

PROBLEM 2-47 Tetracycline is an antibiotic used to mark polar bears. If 25 mg/kg is an effective dose, how much tetracycline is needed in grams to treat a 100 kg polar bear?

PROBLEM 2-48 Ivermectine is used to treat dogs that have intestinal parasites. The effective dosage for this drug is 10.5 mcg/kg of body weight. How much ivermectin should be given to a 9.0 kg dog?

PROBLEM 2-49 Chloroquine is used to treat malaria. Studies have shown that an effective dose for children is 3.5 mg/kg of body weight every 6 hours. If a child weighs 12 kg, how many milligrams of this drug should be given in a 24-hour period?

Selected Answers

PROBLEM 2-8 (b) −23 °C, 250 K. (d) −33.4 °C, 239.8 K. (f) 113 °F

PROBLEM 2-9 (a) exact; (b) exact; (c) uncertain; (d) exact; (e) uncertain; (f) uncertain

PROBLEM 2-12 (a) 0.44; (b) 9.0; (c) 27; (d) 140; (e) 1.5 \(\times\) 10⁻³; (f) 0.44

PROBLEM 2-24 176.0 g

PROBLEM 2-26 (a) 5.1 mL; (b) 37 L

PROBLEM 2-29 (a) 8.00 kg, 5.00 L, 1.60 kg/L; (b) 2.00 kg, 5.00 L, 0.400 kg/L; (c) red < green < blue < yellow; (d) If the volumes are the same, then the density is directly proportional to the mass.

PROBLEM 2-30 (a) and (b) Answer is one of the following:

A/yellow: mass = 65.14 kg, volume = 3.38 L, density = 19.3 kg/L, likely identity = gold.

B/blue: mass = 0.64 kg, volume = 1.00 L, density = 0.64 kg/L, likely identity = apple.

C/green: mass = 4.08 kg, volume = 5.83 L, density = 0.700 kg/L, likely identity = gasoline.

D/red: mass = 3.10 kg, volume = 3.38 L, density = 0.920 kg/L, likely identity = ice; and

E/purple: mass = 3.53 kg, volume = 1.00 L, density = 3.53 kg/L, likely identity = diamond.

(c) B/blue/apple (0.64 kg/L) < C/green/gasoline (0.700 kg/L) < D/red/ice (0.920 kg/L) < E/purple/diamond (3.53 kg/L) < A/yellow/gold (19.3 kg/L)