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Inside Collection (Textbook):

Textbook by: Erik Christensen. E-mail the author

Density

Module by: OpenStax College. E-mail the author

Summary:

• Define density.
• Calculate the mass of a reservoir from its density.
• Compare and contrast the densities of various substances.

Which weighs more, a ton of feathers or a ton of bricks? This old riddle plays with the distinction between mass and density. A ton is a ton, of course; but bricks have much greater density than feathers, and so we are tempted to think of them as heavier. (See Figure 1.)

Density, as you will see, is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid. Density is the mass per unit volume of a substance or object. In equation form, density is defined as

ρ=mV,ρ=mV, size 12{ρ= { {m} over {V} } } {}
(1)

where the Greek letter ρρ size 12{ρ} {} (rho) is the symbol for density, mm size 12{m} {} is the mass, and VV size 12{V} {} is the volume occupied by the substance.

Density:

Density is mass per unit volume.

ρ=mV,ρ=mV, size 12{ρ= { {m} over {V} } } {}
(2)

where ρρ size 12{ρ} {} is the symbol for density, mm size 12{m} {} is the mass, and VV size 12{V} {} is the volume occupied by the substance.

In the riddle regarding the feathers and bricks, the masses are the same, but the volume occupied by the feathers is much greater, since their density is much lower. The SI unit of density is kg/m3kg/m3 size 12{"kg/m" rSup { size 8{3} } } {}, representative values are given in Table 1. The metric system was originally devised so that water would have a density of 1g/cm31g/cm3 size 12{1"g/cm" rSup { size 8{3} } } {}, equivalent to 103kg/m3103kg/m3 size 12{"10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } } {}. Thus the basic mass unit, the kilogram, was first devised to be the mass of 1000 mL of water, which has a volume of 1000 cm3.

Table 1: Densities of Various Substances
Substance ρ ( 10 3 kg/m 3 or g/mL ) ρ ( 10 3 kg/m 3 or g/mL ) size 12{ρ $$"10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } "or""g/mL"$$ } {} Substance ρ ( 10 3 kg/m 3 or g/mL ) ρ ( 10 3 kg/m 3 or g/mL ) size 12{ρ $$"10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } "or""g/mL"$$ } {} Substance ρ ( 10 3 kg/m 3 or g/mL ) ρ ( 10 3 kg/m 3 or g/mL ) size 12{ρ $$"10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } "or""g/mL"$$ } {}
Solids   Liquids   Gases
Aluminum 2.7 Water (4ºC) 1.000 Air 1 . 29 × 10 3 1 . 29 × 10 3 size 12{1 "." "29" times "10" rSup { size 8{ - 3} } } {}
Brass 8.44 Blood 1.05 Carbon dioxide 1 . 98 × 10 3 1 . 98 × 10 3 size 12{1 "." "98" times "10" rSup { size 8{ - 3} } } {}
Copper (average) 8.8 Sea water 1.025 Carbon monoxide 1 . 25 × 10 3 1 . 25 × 10 3 size 12{1 "." "25" times "10" rSup { size 8{ - 3} } } {}
Gold 19.32 Mercury 13.6 Hydrogen 0 . 090 × 10 3 0 . 090 × 10 3 size 12{0 "." "090" times "10" rSup { size 8{ - 3} } } {}
Iron or steel 7.8 Ethyl alcohol 0.79 Helium 0 . 18 × 10 3 0 . 18 × 10 3 size 12{0 "." "18" times "10" rSup { size 8{ - 3} } } {}
Lead 11.3 Petrol 0.68 Methane 0 . 72 × 10 3 0 . 72 × 10 3 size 12{0 "." "72" times "10" rSup { size 8{ - 3} } } {}
Polystyrene 0.10 Glycerin 1.26 Nitrogen 1 . 25 × 10 3 1 . 25 × 10 3 size 12{1 "." "25" times "10" rSup { size 8{ - 3} } } {}
Tungsten 19.30 Olive oil 0.92 Nitrous oxide 1 . 98 × 10 3 1 . 98 × 10 3 size 12{1 "." "98" times "10" rSup { size 8{ - 3} } } {}
Uranium 18.70     Oxygen 1 . 43 × 10 3 1 . 43 × 10 3 size 12{1 "." "43" times "10" rSup { size 8{ - 3} } } {}
Concrete 2.30–3.0     Steam 100º C100º C size 12{ left ("100""°C" right )} {} 0 . 60 × 10 3 0 . 60 × 10 3 size 12{0 "." "60" times "10" rSup { size 8{ - 3} } } {}
Cork 0.24
Glass, common (average) 2.6
Granite 2.7
Earth’s crust 3.3
Wood 0.3–0.9
Ice (0°C) 0.917
Bone 1.7–2.0

As you can see by examining Table 1, the density of an object may help identify its composition. The density of gold, for example, is about 2.5 times the density of iron, which is about 2.5 times the density of aluminum. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space.

Take-Home Experiment Sugar and Salt:

A pile of sugar and a pile of salt look pretty similar, but which weighs more? If the volumes of both piles are the same, any difference in mass is due to their different densities (including the air space between crystals). Which do you think has the greater density? What values did you find? What method did you use to determine these values?

Example 1: Calculating the Mass of a Reservoir From Its Volume

A reservoir has a surface area of 50.0km250.0km2 size 12{"50" "." 0"km" rSup { size 8{2} } } {} and an average depth of 40.0 m. What mass of water is held behind the dam? (See Figure 2 for a view of a large reservoir—the Three Gorges Dam site on the Yangtze River in central China.)

Strategy

We can calculate the volume VV size 12{V} {} of the reservoir from its dimensions, and find the density of water ρρ size 12{ρ} {} in Table 1. Then the mass mm size 12{m} {} can be found from the definition of density

ρ=mV.ρ=mV. size 12{ρ= { {m} over {V} } } {}
(3)

Solution

Solving equation ρ=m/Vρ=m/V size 12{ρ= {m} slash {V} } {} for mm size 12{m} {} gives m = ρ V m = ρ V size 12{ρ= {m} slash {V} } {}.

The volume VV size 12{V} {} of the reservoir is its surface area AA size 12{A} {} times its average depth hh size 12{h} {}:

V = Ah = 50.0 km 2 40.0 m = 50.0 k m 2 10 3 m 1 km 2 40.0 m = 2 . 00 × 10 9 m 3 V = Ah = 50.0 km 2 40.0 m = 50.0 k m 2 10 3 m 1 km 2 40.0 m = 2 . 00 × 10 9 m 3
(4)

The density of water ρρ size 12{ρ} {} from Table 1 is 1.000×103kg/m31.000×103kg/m3 size 12{1 "." "000" times "10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } } {}. Substituting VV size 12{V} {} and ρρ size 12{ρ} {} into the expression for mass gives

m = 1 . 00 × 10 3 kg/m 3 2 . 00 × 10 9 m 3 = 2.00 × 10 12 kg. m = 1 . 00 × 10 3 kg/m 3 2 . 00 × 10 9 m 3 = 2.00 × 10 12 kg. alignl { stack { size 12{m= left (1 "." "00" times "10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } right ) left (2 "." "00" times "10" rSup { size 8{9} } m rSup { size 8{3} } right )} {} # =2 "." "00" times "10" rSup { size 8{"12"} } "kg" "." {} } } {}
(5)

Discussion

A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is mg=1.96×1013Nmg=1.96×1013N size 12{ ital "mg"=1 "." "96" times "10" rSup { size 8{"13"} } N} {}, where gg size 12{g} {} is the acceleration due to the Earth’s gravity (about 9.80m/s29.80m/s2 size 12{9 "." "80""m/s" rSup { size 8{2} } } {}). It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.

Section Summary

• Density is the mass per unit volume of a substance or object. In equation form, density is defined as
ρ=mV.ρ=mV. size 12{ρ= { {m} over {V} } } {}
(6)
• The SI unit of density is kg/m3kg/m3 size 12{"kg/m" rSup { size 8{3} } } {}.

Conceptual Questions

Exercise 1

Approximately how does the density of air vary with altitude?

Exercise 2

Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?

Exercise 3

Figure 3 shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.

Problems & Exercises

Exercise 1

Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?

Solution

1 . 610 cm 3 1 . 610 cm 3 size 12{1 "." "610""cm" rSup { size 8{3} } } {}

Exercise 2

Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in liters of this much mercury?

Exercise 3

(a) What is the mass of a deep breath of air having a volume of 2.00 L? (b) Discuss the effect taking such a breath has on your body’s volume and density.

Solution

(a) 2.58 g

(b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.

Exercise 4

A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces 89.0cm389.0cm3 size 12{"89" "." 0"cm" rSup { size 8{3} } } {} of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)

Solution

2 . 70 g/cm 3 2 . 70 g/cm 3 size 12{2 "." "70""g/cm" rSup { size 8{3} } } {}

Exercise 5

Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.

Exercise 6

(a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.

Solution

(a) 0.163 m

(b) Equivalent to 19.4 gallons, which is reasonable

Exercise 7

A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?

Exercise 8

A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?

Solution

7 . 9 × 10 2 kg/m 3 7 . 9 × 10 2 kg/m 3 size 12{7 "." 9 times "10" rSup { size 8{2} } "kg/m" rSup { size 8{3} } } {}

Exercise 9

What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.

Solution

15 . 6 g/cm 3 15 . 6 g/cm 3 size 12{"15" "." 6"g/cm" rSup { size 8{3} } } {}

Exercise 10

There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately 103kg/m3103kg/m3 size 12{"10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } } {}. The nucleus of an atom has a radius about 105105 size 12{"10" rSup { size 8{ - 5} } } {} that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is 7×108m7×108m size 12{7 times "10" rSup { size 8{8} } m} {})?

Solution

(a) 1018kg/m31018kg/m3 size 12{"10" rSup { size 8{"18"} } "kg/m" rSup { size 8{3} } } {}

(b) 2×104m2×104m size 12{2 times "10" rSup { size 8{4} } m} {}

Glossary

density:
the mass per unit volume of a substance or object

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