# Density and Volume

# DENSITY AND VOLUME

## CONCEPT

Density and volume are simple topics, yet in order to work within any of the hard sciences, it is essential to understand these two types of measurement, as well as the fundamental quantity involved in conversions between them—mass. Measuring density makes it possible to distinguish between real gold and fake gold, and may also give an astronomer an important clue regarding the internal composition of a planet.

## HOW IT WORKS

There are four fundamental standards by which most qualities in the physical world can be measured: length, mass, time, and electric current. The volume of a cube, for instance, is a unit of length cubed: the length is multiplied by the width and multiplied by the height. Width and height, however, are not distinct standards of measurement: they are simply versions of length, distinguished by their orientation. Whereas length is typically understood as a distance along an *x* -axis in one-dimensional space, width adds a second dimension, and height a third.

Of particular concern within this essay are length and mass, since volume is measured in terms of length, and density in terms of the ratio between mass and volume. Elsewhere in this book, the distinction between mass and weight has been presented numerous times from the standpoint of a person whose mass and weight are measured on Earth, and again on the Moon. Mass, of course, does not change, whereas weight does, due to the difference in gravitational force exerted by Earth as compared with that of its satellite, the Moon. But consider instead the role of the fundamental quality, mass, in determining this significantly less fundamental property of weight.

According to the second law of motion, weight is a force equal to mass multiplied by acceleration. Acceleration, in turn, is equal to change in velocity divided by change in time. Velocity, in turn, is equal to distance (a form of length) divided by time. If one were to express weight in terms of *l, t,* and *m,* with these representing, respectively, the fundamental properties of length, time, and mass, it would be expressed as

—clearly, a much more complicated formula than that of mass!

### Mass

So what is mass? Again, the second law of motion, derived by Sir Isaac Newton (1642-1727), is the key: mass is the ratio of force to acceleration. This topic, too, is discussed in numerous places throughout this book; what is actually of interest here is a less precise identification of mass, also made by Newton.

Before formulating his laws of motion, Newton had used a working definition of mass as the quantity of matter an object possesses. This is not of much value for making calculations or measurements, unlike the definition in the second law. Nonetheless, it serves as a useful reminder of matter's role in the formula for density.

Matter can be defined as a physical substance not only having mass, but occupying space. It is composed of atoms (or in the case of subatomic particles, it is part of an atom), and is convertible with energy. The form or state of matter itself is not important: on Earth it is primarily observed as a solid, liquid, or gas, but it can also be found (particularly in other parts of the universe) in a fourth state, plasma.

Yet there are considerable differences among types of matter—among various elements and states of matter. This is apparent if one imagines three gallon jugs, one containing water, the second containing helium, and the third containing iron filings. The volume of each is the same, but obviously, the mass is quite different.

The reason, of course, is that at a molecular level, there is a difference in mass between the compound H_{2}O and the elements helium and iron. In the case of helium, the second-lightest of all elements after hydrogen, it would take a great deal of helium for its mass to equal that of iron. In fact, it would take more than 43,000 gallons of helium to equal the mass of the iron in one gallon jug!

### Density

Rather than comparing differences in molecular mass among the three substances, it is easier to analyze them in terms of density, or mass divided by volume. It so happens that the three items represent the three states of matter on Earth: liquid (water), solid (iron), and gas (helium). For the most part, solids tend to be denser than liquids, and liquids denser than gasses.

One of the interesting things about density, as distinguished from mass and volume, is that it has nothing to do with the amount of material. A kilogram of iron differs from 10 kilograms of iron both in mass and volume, but the density of both samples is the same. Indeed, as discussed below, the known densities of various materials make it possible to determine whether a sample of that material is genuine.

### Volume

Mass, because of its fundamental nature, is sometimes hard to comprehend, and density requires an explanation in terms of mass and volume. Volume, on the other hand, appears to be quite straightforward—and it is, when one is describing a solid of regular shape. In other situations, however, volume is more complicated.

As noted earlier, the volume of a cube can be obtained simply by multiplying length by width by height. There are other means for measuring the volume of other straight-sided objects, such as a pyramid. That formula applies, indeed, for any polyhedron (a three-dimensional closedsolid bounded by a set number of plane figures) that constitutes a modified cube in which thelengths of the three dimensions are unequal—that is, an oblong shape.

For a cylinder or sphere, volume measurements can be obtained by applying formulaeinvolving radius (*r* ) and the constant π, roughly equal to 3.14. The formula for volume of a cylinderis *V* = π*r* ^{2}*h,* where *h* is the height. A sphere's volume can be obtained by the formula (4/3) π*r* ^{3}. Even the volume of a cone can be easily calculated: it is one-third that of a cylinder of equal base and height.

## REAL-LIFE APPLICATIONS

### Measuring Volume

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.

### Buoyancy: Volume and Density

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.

### Comparing Densities

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} - 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°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.

### 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). <http://www.tpub.com/fluid/ch1e.htm> (March 27, 2001).

*"Density, Volume, and Cola"* (Web site). <http://student.biology.arizona.edu/sciconn/density/density_coke.html> (March 27, 2001).

*"The Mass Volume Density Challenge"* (Web site). <http://science-math-technology.com/mass_volume_density.html> (March 27, 2001).

*"Metric Density and Specific Gravity"* (Web site). <http://www.essex1.com/people/speer/density.html> (March 27, 2001).

*"Mineral Properties: Specific Gravity" The Mineral and Gemstone Kingdom* (Web site). <http://www.minerals.net/resource/property/sg.htm> (March 27, 2001).

Robson, Pam. *Clocks, Scales and Measurements.* New York: Gloucester Press, 1993.

*"Volume, Mass, and Density"* (Web site). <http://www.nyu.edu/pages/mathmol/modules/water/density_intro.html> (March 27, 2001).

Willis, Shirley. *Tell Me How Ships Float.* Illustrated by the author. New York: Franklin Watts, 1999.

## KEY TERMS

### ARCHIMEDES'S PRINCIPLE:

A rule of physics which holds that the buoyant force of an object immersed in fluid is equal to the weight of the fluid displaced by the object. It is named after the Greekmathematician, physicist, and inventor Archimedes (c. 287-212 b.c.), who first identified it.

### BUOYANCY:

The tendency of an objectimmersed in a fluid to float. This can be explained by Archimedes's principle.

### DENSITY:

The ratio of mass to volume—in other words, the amount of matter within a given area.

### MASS:

According to the second law of motion, mass is the ratio of force to acceleration. Mass may likewise be defined, though much less precisely, as the amount of matter an object contains. Mass is also the product of volume multiplied by density.

### MATTER:

Physical substance that occupies space, has mass, is composed of atoms (or in the case of subatomic particles, is part of an atom), and is convertible intoenergy.

### SPECIFIC GRAVITY:

The density of an object or substance relative to the density of water; or more generally, the ratio between the densities of two objects or substances.

### VOLUME:

The amount of three-dimensional space an object occupies. Volume is usually expressed in cubic units of length.

### WEIGHT DENSITY:

The proper term for density within the British system of weights and measures. The pound is a unit of weight rather than of mass, and thus British units of density are usually rendered in terms of weight density—that is, pounds-per-cubic-foot. By contrast, the metric or international units measure mass density (referred to simply as "density"), which is rendered in terms of kilograms-per-cubic-meter, or grams-per-cubic-centimeter.

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