# Density and Volume - Real-life applications

Photo by: thieury

### 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 problems to obtain the volume of an irregular shape—but 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 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° to 212°F or 0 to 100°C), its volume will increase by only 2%. If its pressure is doubled from 1 atm (defined as normal air pressure at sea level—14.7 pounds-per-square-inch or 1.013 × 10 5 Pa) to 2 atm, volume will decrease by only 0.01%.

Yet, if air were heated from 32° to 212°F, 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—that 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—Syracuse, a Greek colony in Sicily—to 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 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-cubic-meter.

This difference in density should mean that the carrier would sink like a stone—and 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—that is, its mass multiplied by the downward acceleration due to gravity—is 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 —volume 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:

• 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
• 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°F (4°C) 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—sea water, soil, rocks—has 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.

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 —which 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.

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