We use **dimensional analysis **as a fail proof process to convert from one unit to another.

**Concept:** Understanding Unit Conversions

When it comes to conversions we belong in three different categories. We are going to say we are in length, volume, and mass. What we're going to say here, we're just going to take a look at some of the most basic and most common types of conversions that you're going to be expected to know.

If you look at length, we have conversions to go from kilometers to miles, between meters and yards, inches and centimeters, miles and feet.

For volume, we have gallons to liters. One liter is equal to one decimeter cubed. We're going to say that one milliliter is equal to one centimeter cubed. That one's especially important to remember. We have one liter equals that many quarts.

For mass, we're going to say one kilogram is equal to this many pounds. One pound is equal to 454 grams. And then finally, one ounce is equal to 28.35 grams.

Now, it may seem like a lot, but a majority of these calculations you'll be dealing with on a daily basis when it comes to chem one. Some of them will naturally become a part of your memory, so you will remember how to use them and when to use them through practice.

**Unit conversions** deal with three basic areas of measurement: l**ength, volume** and **mass**.

**Example:** Every Saturday morning Gregor has to travel from Main Campus to his parents’ home. If his car gets 58.5 km/L how many L will his car need to travel the 19.3 miles?

Set up the problem to begin with your given information and to end with your unknown information.

**Example:** A backyard swimming pool holds 315 cubic yards (yd** ^{3}**) of water. What is the mass of the water in pounds?

**Example:** An intravenous bag delivers medication to a patient at a rate of 2.75 drops a second. If a drop weighs 42 mg, how many grams of solution are delivered in 7.0 hours?

**Concept:** Examining Dimensional Analysis

We use dimensional analysis as a way of helping us figure out how to convert from one unit to another. We're going to say we design the problems to begin with your known, which is known values, and to end with the units of your unknown. That's the strategy we always take. What did they give me? What am I looking for? We're going to say be sure all of your units cancel out and that's the way we ensure we're using dimensional analysis correctly.

**Dimensional Analysis** is just simply a word problem dealing with unit conversions. It is design to begin with your given value and to end with the unknown value we need to find.

**Example:** Natty Light contains 4.2% alcohol. Steve from Kappa Epsilon Gamma, KEG for short, wants to get tanked tonight, and he is aiming to down at least 175 ml of alcohol in one night. If each can of Natty light contains 355 mL of beer, how many cans of Natty Light must Steve consume at minimum to reach his goal?

**Example:** A Volkswagen diesel engine consumes diesel at a rate of 25.83 L per hour. If the density of the diesel is 0.850 g/mL, what is the mass (in mg) of diesel needed to drive for a continuous 8.5 hours?

**Example:** An acetaminophen suspension for toddlers contains 95 mg/0.85 mL suspension. The recommended dose is 22 mg/kg body weight. How many liters of this suspension should be given to a toddler weighing 30.5 lbs?

**Concept:** Understanding Density

We're going to say that we use density to understand the relationship between mass and volume. Now usually when we're dealing with mass, mass will be in grams. And when we're dealing with volume, volume may be in milliliters or in liters.

Now we talked about there being three states of matter: solids, liquids, and gases. We're going to say that solids and liquids are more dense than gases. The density for solids and liquids will have the units of grams over milliliters.

Gases, on the other hand, are way less dense. We're going to say that gases, their density are in units of grams over liters. Later on in this semester when we pay more close attention to gases, we're going to see why that is. But for right now we're just going to when it comes to density it's going to be mass over volume. And if we're dealing with a solid or a liquid it's going to be grams over milliliters.

Density establishes a relationship between mass and volume.

**Example:** If the density of an unknown metal is 21.4 g/cm** ^{3}**, express its density in lb/ft

Density questions can include dimensional analysis so remember to use the same approach as you would normally.

**Example:** A piece of unknown metal weighs approximately 0.45 lbs. When a scientist places it in a glass beaker the water level increases from 1.85 L to 2.13 L. What is the density of the palladium metal in g/mL?

In a **water displacement **question the volume of the water increases with the added volume of the unknown substance.

**Example:** The U.S. Environmental Protection Agency sets the maximum safe level of lead in blood at 24 μg per dL of blood. The average person has 60 mL of blood per kilogram of body weight. For a 63.7-kg (140.459 lb) person, what is the total maximum safe content of lead in blood?

A sample has a density of 1.05 g/mL. What is the volume in mL (written in scientific notation) that 2.9 kg of the sample would occupy? You must use dimensional analysis

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You must use Dimensional Analysis and answer the question.

A gaseous sample of methane, CH_{4}, has a density of 0.66 g/L. How many hydrogen atoms are in 2.41 L of the sample?

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How many inches per minute (in./min) are in 4.65 x 10^{7} Ångstrom per millisecond (Å/ms)?

A. 1.10 x 10^{6} in./min

B. 1.10 x 10^{4} in./min

C. 11.0 in./min

D. 1.10 x 10^{-2} in./min

E. None of these choices are correct.

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Precious metals are sometimes weighed in old-timey units like “troy ounces”, where 1 troy ounce = 31.1035 grams. A platinum coin is made that weighs 0.5003 troy ounces. The coin has a radius of 13.5 mm and a thickness of 1.28 mm. (Useful information: volume of a cylinder = (π*r*^{2})*h* where *r* is radius, and *h* is height or thickness)

a) What is its density, in g/cm^{3}?

b) How many atoms of platinum are in the coin?

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A car averages 35 miles per gallon. How many liters of gasoline will be needed for a trip of 450 kilometers? Some Conversion factors which may be helpful are: 1 qt = 946 mL; 4qts = 1 gallon; 1 mile = 1.609 km;

a. 8 L

b. 25 L

c. 30 L

d. 34 L

e. 40 L

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The density of lead is 11.4 g/cm^{3} at 25^{o}C. The volume occupied by 25.0 g of lead is

a. 2.19 cm^{3}

b. 1.24 cm^{3}

c. 0.456 cm^{3}

d. 285 cm^{3}

e. 6.05 cm^{3}

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An atom of an element weights 9.12 x 10^{-23} g. What is the atomic mass of this element in atomic mass units?

a. 45.0

b. 54.9

c. 55.8

d. 58.7

e. 63.5

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A sample mercury has a volume of 0.50 mL and a mass of 6.765 g

a) What is the density of the material in mg/mL?

b) How many atoms are in this 0.50 mL sample of mercury?

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Dimensional Analysis

How many milligrams in 125.0 gram ____________________

The boiling point of H_{2} is -253°C, what is the boiling point in Kelvin. ________________

Oil has a density of 0.916 g/ml. What is the mass of 225 mL of the oil? ________________

What is the formula and charge of the phosphate ion? ___________________

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An aluminum sphere contains 8.55 x 10 ^{22 }aluminum atoms. What is the sphere's radius in centimeters? The density of aluminum is 2.70 g/ cm^{3}. Recall that the volume of a sphere is given by the formula: 4/3 π r^{3}^{ }

A. 1.4187 cm ^{3}

B. 0.697 cm

C. 0.339 cm

D. 0.822 cm

E. 7.8583 x10^{-1 }cm

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Your infant child weighs 13lbs and has a fever. You want to give the baby a dose of Tylenol that contains 160mg acetaminophen/5 mL, but administering too much could cause damage to the child's liver. Your doctor says you can give your child 14 mg acetaminophen/kg. What volume (mL) of Tylenol should you give your baby? Note: 1kg =2.2 lbs

A. 2.6 mL

B. 13 mL

C. 0.013 mL

D. 1.0 mL

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A cube of aluminum metal contains 3.35 x 10^{23} atoms. The density of aluminum is 2.70 g/cm^{3}. What is the length of the edge of the cube? Recall, volume = length ^{3} (for a cube).

A) 1.77 cm

B) 5.56 cm

C) 2.70 cm

D) 15.0 cm

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In the year 1900, the total area of the mountain covered by glaciers was 1.60 km ^{2}. What is this surface area with units cm^{2}?

A) 1.60 x 10^{10} cm^{2}

B) 1.60 x 10^{5} cm^{2}

C) 1.60 x 10^{-10} cm^{2}

D) 1.60 x 10^{-5} cm^{2}

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The mountain is 19, 341 feet tall. Convert this height to the * SI unit *for length. To get you started, 1km = 3280.84 ft.

A) 5.8951 km

B) 6.3455 x 10^{7} km

C) 5895.1 m

D) 6.3455 x 10^{8} m

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Because of the high heat and low humidity in the summer in Death Valley, California, a visitor requires about one quart of water for every two miles traveled on foot. If the density of water is 0.990 g/mL at 45°C, how many kilograms of water are required for a person to walk 30 kilometers in Death Valley?

A) 8.7 kg

B) 70 kg

C) 7.0 × 10 ^{2} kg

D) 3.5 × 10 ^{2} kg

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A mass of mercury occupies 0.750 L. What volume would an equal mass of ethanol occupy? The density of mercury is 13.546 g/mL and the density of ethanol is 0.789 g/mL.

A) 0.0437 L

B) 22.9 L

C) 12.9 L

D) 0.0777 L

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The estimated mass of the planet Jupiter is 1.90 × 10 ^{27} kg and the density is believed to be 1.34 g/cm^{3}. If Jupiter were a perfect sphere, what would be its diameter?

A) 6.96 × 10^{7} m

B) 1.39 × 10^{7} m

C) 1.39 × 10^{8} m

D) 6.96 × 10^{6} m

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A gold ingot weighs 5.50 lbs. If the density of gold is 19.31 g/cm ^{3}, and the length and width of the ingot are 12.0 cm and 3.00 cm respectively, what is the height of the ingot?

A) 10.2 cm

B) 1.34 × 10^{3} cm

C) 3.59 cm

D) 6.50 × 10^{-3} cm

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If the walls in a room are 955 square feet in area, and a gallon of paint covers 15 square yards, how many gallons of paint are needed for the room? (3 ft = 1 yd)

A) 7.0 gallons

B) 2.3 gallons

C) 21 gallons

D) 47 gallons

E) 24 gallons

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Osmium is the densest known element, with density 22.59 g/cm^{3}. The tip of a fountain pen contains 152 mg of Osmium. What is the volume of Osmium in the pen in units of microliters, μL?

a. 0.152 μL

b. 6.73 μL

c. 0.149 μL

d. 149 μL

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The atmospheric pressure is 700 mm Hg. What is the pressure in inches of Hg?

A) 16.0 in Hg

B) 0.921 in Hg

C) 13.5 in Hg

D) 27.6 in Hg

E) 32.5 in Hg

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An ibuprofen (C_{13}H_{18}O_{2}) suspension for infants contains 100 mg/5.0 mL suspension. The recommended dose is 10 mg/kg body weight. How many mL of this suspension should be given to an infant weighing __18 lb__??

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The air pressure in a volleyball is 75 psi. What is this pressure in torr ( 1 psi = 14.7 atm, 1 atm = 101,325 Pa = 760 torr)?

a. 75,000 torr

b. 393,300 torr

c. 76,500 torr

d. 837,900 torr

e. 562 torr

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One electron has a charge of 1.602 x 10 ^{-19 }C (Coulomb) and its mass is 9.11 x 10^{-28}g. If a particlar mass of electron exerts a charge of 3.84 X 10 ^{-2}, what would be the mass of the electrons?

A. 0.220 pg

B. 2.63 x 10^{44} g

C. 2.19 x 10^{-10} ng

D. 2.60 x 10^{-48} g

E. 218 pg

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The painting American Gothic, by Grant Wood, is a rectanglar shaped canvas with a length of 29 inches and width of 24 inches. What is the area of the painting in cm? Recal 1 inch = 2.54 cm

A. 1.3 X 10^{2} cm^{2}

B. 4.5 X 10^{3} cm^{2}

C. 2.7 X 10^{2} cm^{2}

D. 1.1 X 10^{2} cm^{2}

E. 1.8 X 10^{3} cm^{2}

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The density of cobalt (II) oxide is 6.45 g/cm^{3}. What is the volume occupied by a 5.57 kg sample of cobalt (II) oxide?

a. 3.40 x 10^{3} cm^{3}

b. 817 cm^{3}

c. 122 cm^{3}

d. 34.0 cm^{3}

e. 864 cm^{3}

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The dimensions of an ameoba are approximately 87 um. Express this length in Mm.

a. 8.7 x 10^{-11} Mm

b. 8.7 x 10^{19} Mm

c. 8.7 x 10^{7} Mm

d. 8.7 x 10^{-5} Mm

e. 8.7 x 10^{-14} Mm

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A planet is travelling around the sun at 68.2 miles/hour. What will be the speed of the planet in m/s?

a. 30.5 m/s

b. 3.05 x 10^{-5} m/s

c. 11.8 m/s

d. 3.95 x 10^{8} m/s

e. 1.52 x 10^{8} m/s

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A student in their chemistry lab is asked to measure out the area of the classroom and report the answer in m^{2}. Using a measuring tape, the student found that the room is 18.8 by 15.5 ft in area, What should the student report for the area in m^{2}? ( 1 in. = 2.54 cm )

a. 27.1 m^{2}

b. 291 m^{2}

c. 88.8 m^{2}

d. 5.20 m^{2}

e. 17.1 m^{2}

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A sheet of copper is 18.0 in x 7.00 in and a mass of 0.0318 g. Given that the density of copper is 8.96 g/cm^{3}. What is the thickness of the copper sheet?

a. 282 nm

b. 4.4 x 10^{-6} nm

c. 0.044 nm

d. 43.7 nm

e. 3.51 x 10^{3} nm

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The human nose is so sensitive to vanillin (used to flavor vanilla ice cream) that it can detect it at a concentration of 200 picograms/L of air. The current market price for 50.0 grams of vanillin is $148. Determine the cost to supply enough vanillin so that the aroma could be detected in a large aircraft hanger with a volume of 6.40 x 10^{9} ft^{3}.

a. $ 107

b. $ 124

c. $ 1.07 x 10^{8}

d. $ 0.115

e. $ 35.9

Watch Solution

The average person breathes 14 times per minute. Each breath is about 350 mL. The exhaled air is about 5.3% carbon dioxide. In all about 5,300 moles (44 g in a mole) of carbon dioxide are exhaled per person per year. If the human population of the earth is 7.2 x 10** ^{9}**, how many kilograms of carbon dioxide are released into the atmosphere by all people in a year?

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What mass of silver is required for an object that is to be coated in a uniform 0.32 mm thick layer of silver that has surface area of 162 cm ^{2}? The density of silver is 10.49 g/cm ^{3}.

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A cylindrical tube has a length that is 15.2 cm and is filled with bromine liquid. If the mass of the bromine is 112.3 g and has a density of 25.3 g/mL, then what is the inner diameter of the glass tube? ( V = π • r^{2} • h )

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The recommended adult dose of a drug used to treat asthma is 10.0 mg/kg of body mass. Calculate the dose in mg for a 200-lb person . (1 kg = 2.205 lb)

A. 0.110 mg

B. 907 mg

C. 4.41 mg

D. 2.00 x 10^{3} mg

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A gold cylinder has a length of 6.25 cm and a radius of 1.25 cm. The density of gold is 19.3 g/cm^{3}. V_{cylinder} = π r ^{2 }l

How many gold atoms are contained in the gold cylinder?

A. 1.94 x 10 ^{-19} atoms

B. 1.81 x 10 ^{24} atoms

C. 1.92 x 10 ^{21} atoms

D. 9.85 x 10 ^{23} atoms

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If an object has a density of 8.65 g/cm^{3}, what is its density in units of kg/m^{3}?

A) 8.65 × 10^{-3} kg/m^{3}

B) 8.65 × 10^{-7} kg/m^{3}

C) 8.65 × 10^{3} kg/m^{3}

D) 8.65 × 10^{1} kg/m^{3}

E) 8.65 × 10^{-1} kg/m^{3}

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Molecular hydrogen is used as the “liquid fuel” (liquefied hydrogen) for rockets. How much water is generated from 1000 gallons of liquefied hydrogen? (1 gallon = 3.80 L. The density of liquefied hydrogen at 20 K is 71 g/L.)

a. 4860 kg

b. 2430 kg

c. 1220 kg

d. 135 kg

e. 17 kg

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A normal breath takes in about 1.0 L of air. Assuming that air has an average molar mass of 28.8 g/mol, and that its density is 0.97 g/L, how many molecules of air do you take in with each breath?

A. 1.7 x 10^{25}

B. 1.8 x 10^{25}

C. 2.0 x 10^{22}

D. 2.2 x 10^{22}

E. 5.8 x 10^{23}

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A pascal is a newton per square meter and a newton is a kilogram-meter per second squared, while a joule is a newton-meter. What is a pascal-meter cubed?

A. newton

B. atmosphere

C. joule

D. Watt

E. volt

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The diameter of a hydrogen atom is 212 pm. Find the length in km of a row of 6.02 x 10^{23} hydrogen atoms. 1 pm = 10 ^{-12} m.

A. 2.41 x 10^{19} km

B. 1.28 x 10^{11} km

C. 3.60 x 10^{12} km

D. 5.59 x 10^{23} km

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What is the mass (in grams) of 4 L of gasoline if the density of gasoline is 0.79 g/ml?

a. 2.2e3

b. 3.2e3

c. 4.3e3

d. 5.4e3

e. 6.5e3

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The density of gold is 19.31 g/cm ^{3}. A rectangular bar of gold has a mass of 66.75 g and measures 3.04 cm width and 25.5 cm in height. What is the length of the gold bar?

a) 0.0446 cm b) 0.0550 cm c) 0.0189 cm d) 0.0592 cm

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An aluminum sphere contains 8.66 x 10^{22} aluminum atoms. What is the sphere’s radius in centimeters? The density of aluminum is 2.70 g/cm^{3} . Recall that the volume of a sphere is given by the formula: 4/3 π r^{3}

- 1.418 cm
^{3} - 0.699 cm
- 0.339 cm
- 0.822 cm
- 7.8586 x 10
cm^{-1}

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Osmium is the densest known element, with density 22.59 g/cm ^{3} . The tip of a fountain pen contains 152 mg osmium. What is the volume of osmium in the pen in units of microliters, µL?

A. 0.152 µL

B. 6.73 µL

C. 0.149 µL

D. 149 µL

Watch Solution

In ancient times, the closest thing to a formal unit area was the yoke (Hebrew semed) which referred to the amount of land that a pair of yoked oxen could plough in a single day. The standard estimate of this was usually taken as equal to 6480 square cubits. How many acres is this? (1 cubit = 18 units, 12 inches = 1 foot, 1 acre = 43,560 square feet)

a) 7.9

b) 0.25

c) 0.33

d) 10

e) none of the given answers is close.

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The man in the wilderness said to me, How many buckets could empty the sea? Who-who’s that crying in the cold night air? Who-who? Who-who? Who-who goes there? How many buckets could empty the sea? Three times ten to the power of twenty Is the number of gallons in all the oceans. According to the best of the mathematicians. Now if each bucket is of good parameters Say five hundred cubic centimeters How many moles of these do we need To completely empty the sea. And where will we put all that water? Never mind that, my little daughter. Just tell me how many moles please Of buckets to buy, to empty the seas. (1 gal=3.78 L)

a) 4 x 10^{-3}

b) 1 x 10^{45}

c) 2 x 10^{15}

d) 2 x 10^{21}

e) none of the given answers is even within an order of magnitude

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