Real-life applications - Density and Volume

Real-life applications - Density and Volume

8/20/13 6:43 PM

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Science Clarified ? Real-Life Chemistry Vol 3 - Physics Vol 1 ? Density and Volume

Density and Volume - Real-life applications

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M EASURING V OLUME

What about the volume of a solid

that is irregular in shape? Some

irregularly shaped objects, such as

a scooter, which consists

primarily of one round wheel and

a number of oblong shapes, can

be measured by separating them

into regular shapes. Calculus may

be employed with more complex

Photo by: thieury

problems to obtain the volume of

an irregular shape¡ªbut the most basic method is simply to immerse the object in water. This

procedure involves measuring the volume of the water before and after immersion, and

calculating the difference. Of course, the object being measured cannot be water-soluble; if it is,

its volume must be measured in a non-water-based liquid such as alcohol.

Measuring liquid volumes is easy, given the fact that liquids have no definite shape, and will



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simply take the shape of the container in which they are placed. Gases

are similar to liquids in the sense that they expand to fit their

container; however, measurement of gas volume is a more involved

process than that used to measure either liquids or solids, because

gases are highly responsive to changes in temperature and pressure.

If the temperature of water is raised from its freezing point to its

boiling point (32¡ã to 212¡ãF or 0 to 100¡ãC), its volume will increase by

only 2%. If its pressure is doubled from 1 atm (defined as normal air

pressure at sea level¡ª14.7 pounds-per-square-inch or 1.013 ¡Á 10 5 Pa)

to 2 atm, volume will decrease by only 0.01%.

Yet, if air were heated from 32¡ã to 212¡ãF, its volume would increase

by 37%; and if its pressure were doubled from 1 atm to 2, its volume

would decrease by 50%. Not only do gases respond dramatically to

changes in temperature and pressure, but also, gas molecules tend to

be non-attractive toward one another¡ªthat is, they do not tend to

stick together. Hence, the concept of "volume" involving gas is

essentially meaningless, unless its temperature and pressure are

known.

B UOYANCY : V OLUME AND D ENSITY

Consider again the description above, of an object with irregular

shape whose volume is measured by immersion in water. This is not

the only interesting use of water and solids when dealing with volume and density. Particularly

intriguing is the concept of buoyancy expressed in Archimedes's principle.

More than twenty-two centuries ago, the Greek mathematician, physicist, and inventor

Archimedes (c. 287-212 B.C. ) received orders from the king of his hometown¡ªSyracuse, a Greek

colony in Sicily¡ªto weigh the gold in the royal crown. According to legend, it was while bathing

that Archimedes discovered the principle that is today named after him. He was so excited,

legend maintains, that he jumped out of his bath and ran naked through the streets of Syracuse

shouting "Eureka!" (I have found it).

What Archimedes had discovered was, in short, the reason why ships float: because the buoyant,

or lifting, force of an object immersed in fluid is equal to the weight of the fluid displaced by the



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object.

HOW A STEEL SHIP FLOATS ON WATER.

Today most ships are made of steel, and therefore, it is even harder to understand why an aircraft

carrier weighing many thousands of tons can float. After all, steel has a weight density (the

preferred method for measuring density according to the British system of measures) of 480

pounds per cubic foot, and a density of 7,800 kilograms-per-cubic-meter. By contrast, sea water

has a weight density of 64 pounds per cubic foot, and a density of 1,030 kilograms-per-cubicmeter.

This difference in density should mean that the carrier would sink like a stone¡ªand indeed it

would, if all the steel in it were hammered flat. As it is, the hull of the carrier (or indeed of any

sea-worthy ship) is designed to displace or move a quantity of water whose weight is greater than

that of the vessel itself. The weight of the displaced water¡ªthat is, its mass multiplied by the

downward acceleration due to gravity¡ªis equal to the buoyant force that the ocean exerts on the

ship. If the ship weighs less than the water it displaces, it will float; but if it weighs more, it will

sink.

Put another way, when the ship is placed in the water, it displaces a certain quantity of water

whose weight can be expressed in terms of Vdg ¡ªvolume multiplied by density multiplied by the

downward acceleration due to gravity. The density of sea water is a known figure, as is g (32 ft or

9.8 m/sec 2 ); thus the only variable for the water displaced is its volume.

For the buoyant force on the ship, g will of course be the same, and the value of V will be the

same as for the water. In order for the ship to float, then, its density must be much less than that

of the water it has displaced. This can be achieved by designing the ship in order to maximize

displacement. The steel is spread over as large an area as possible, and the curved hull, when

seen in cross section, contains a relatively large area of open space. Obviously, the density of this

space is much less than that of water; thus, the average density of the ship is greatly reduced,

which enables it to float.

C OMPARING D ENSITIES

As noted several times, the densities of numerous materials are known quantities, and can be

easily compared. Some examples of density, all expressed in terms of kilograms per cubic meter,

are:



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Hydrogen: 0.09 kg/m 3

Air: 1.3 kg/m 3

Oak: 720 kg/m 3

Ethyl alcohol: 790 kg/m 3

Ice: 920 kg/m 3

Pure water: 1,000 kg/m 3

Concrete: 2,300 kg/m 3

Iron and steel: 7,800 kg/m 3

Lead: 11,000 kg/m 3

Gold: 19,000 kg/m 3

Note that pure water (as opposed to sea water, which is 3% denser) has a density of 1,000

kilograms per cubic meter, or 1 gram per cubic centimeter. This value is approximate; however,

at a temperature of 39.2¡ãF (4¡ãC) and under normal atmospheric pressure, it is exact, and so,

water is a useful standard for measuring the specific gravity of other substances.

SPECIFIC GRAVITY AND THE DENSITIES OF PLANETS.

Specific gravity is the ratio between the densities of two objects or substances, and it is expressed

as a number without units of measure. Due to the value of 1 g/cm 3 for water, it is easy to

determine the specific gravity of a given substance, which will have the same number value as its

density. For example, the specific gravity of concrete, which has a density of 2.3 g/cm 3 , is 2.3.

The specific gravities of gases are usually determined in comparison to the specific gravity of dry

air.

Most rocks near the surface of Earth have a specific gravity of somewhere between 2 and 3, while

the specific gravity of the planet itself is about 5. How do scientists know that the density of Earth

is around 5 g/cm 3 ? The computation is fairly simple, given the fact that the mass and volume of

the planet are known. And given the fact that most of what lies close to Earth's surface¡ªsea

water, soil, rocks¡ªhas a specific gravity well below 5, it is clear that Earth's interior must contain

high-density materials, such as nickel or iron. In the same way, calculations regarding the density

of other objects in the Solar System provide a clue as to their interior composition.

ALL THAT GLITTERS.



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Closer to home, a comparison of density makes it possible to determine whether a piece of

jewelry alleged to be solid gold is really genuine. To determine the answer, one must drop it in a

beaker of water with graduated units of measure clearly marked. (Here, figures are given in cubic

centimeters, since these are easiest to use in this context.)

Suppose the item has a mass of 10 grams. The density of gold is 19 g/cm 3 , and since V = m / d =

10/19, the volume of water displaced by the gold should be 0.53 cm 3 . Suppose that instead, the

item displaced 0.91 cm 3 of water. Clearly, it is not gold, but what is it?

Given the figures for mass and volume, its density would be equal to m / V = 10/0.91 = 11 g/cm 3

¡ªwhich happens to be the density of lead. If on the other hand the amount of water displaced

were somewhere between the values for pure gold and pure lead, one could calculate what

portion of the item was gold and which lead. It is possible, of course, that it could contain some

other metal, but given the high specific gravity of lead, and the fact that its density is relatively

close to that of gold, lead is a favorite gold substitute among jewelry counterfeiters.

WHERE TO LEARN MORE

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Chahrour, Janet. Flash! Bang! Pop! Fizz!: Exciting Science for Curious Minds. Illustrated by Ann

Humphrey Williams. Hauppauge, N.Y.: Barron's, 2000.

"Density and Specific Gravity" (Web site). (March 27,

2001).

"Density, Volume, and Cola" (Web site).

(March 27, 2001).

"The Mass Volume Density Challenge" (Web site). ................
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