1.8.1 Density

Physics Department of Douglas College; OpenStax; and Joey Wu

Summary

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

In our exploration of physical properties, we’ve discovered various ways to describe and measure matter. Now, let’s dive into one of the most fascinating and important physical properties: density. You might remember that physical properties are characteristics we can observe or measure without changing the substance itself. Density is a perfect example of this. It tells us how much mass is packed into a given volume, providing crucial information about materials without altering their chemical makeup.

Have you ever wondered why a massive ship made of steel can float on water, while a small pebble sinks to the bottom of a pond? The answer lies in their densities. By understanding density, we can explain a wide range of phenomena in our everyday world, from why ice floats in your drink to how hot air balloons rise in the sky. In this chapter, we’ll explore how density relates to other physical properties we’ve studied, and discover its vital role in science and our daily lives.

A pile of feathers measuring a ton and a ton of bricks are placed on either side of a plank that is balanced on a small support.
Figure 1. A ton of feathers and a ton of bricks have the same mass, but the feathers make a much bigger pile because they have a much lower density.

Density is like a secret code that tells us a lot about the substances around us. It’s crucial in determining whether an object sinks or floats in a fluid. For instance, density explains why oil floats on water when you’re making salad dressing, or why helium balloons rise to the ceiling at a birthday party. Imagine you’re at a lake, tossing in different objects. A wooden stick floats, while a pebble sinks. Why? It’s all about their densities compared to the water!

So what exactly is density? Simply put, density is how much “stuff” (mass) is packed into a certain amount of space (volume). Scientists define it as the mass per unit volume of a substance or object. It’s like comparing how heavy your backpack feels when it’s stuffed full versus when it’s nearly empty. The fuller backpack has more mass in the same space, so it has a higher density.

DENSITY

Density is mass per unit volume.

In equation form, density is defined as                ρ = mass / Volume

where the Greek letter ρ is the symbol for density, m is the mass, and 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 / m3   representative values are given in Table 1. The metric system was originally devised so that water would have a density of 1 gram / cubic centimeter = 1 g/cm 3   equivalent to  1000 kg / m3  .  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.

Substance ρ (103 kg/m3 or g/mL) Substance ρ (103 kg/m3 or g/mL) Substance ρ (103 kg/m3 or g/mL)
Solids Liquids Gases
Aluminum 2.7 Water (4ºC) 1.000 Air 1.29 x 10-3  
Brass 8.44 Blood 1.05 Carbon dioxide 1.29 x 10-3  
Copper (average) 8.8 Sea water 1.025 Carbon monoxide 1.25 x 10-3  
19.32 Mercury 13.6 Hydrogen 0.090 x 10-3  
Iron or steel 7.8 Ethyl alcohol 0.79 Helium 0.18 x 10-3  
Lead 11.3 Petrol 0.68 Methane 0.72 x 10-3  
Polystyrene 0.10 Glycerin 1.26 Nitrogen 1.25 x 10-3  
Tungsten 19.30 Olive oil 0.92 Nitrous oxide 1.98 x 10-3  
Uranium 18.70 Oxygen 1.43 x 10-3  
Concrete 2.30–3.0 Steam 100º C  0.60 x 10-3  
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
Table 1. Densities of Various Substances
A pile of feathers measuring a ton and a ton of bricks are placed on either side of a plank that is balanced on a small support.
Figure 1. A ton of feathers and a ton of bricks have the same mass, but the feathers make a much bigger pile because they have a much lower density.

As you can see by examining the tables of density, 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 50.0 km 2  and an average depth of 40.0 m. What mass of water is held behind the dam? (See the figure below, or Google search it, 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 V  of the reservoir from its dimensions, and find the density of water ρ from the table.  Remember that by definition 1 gram = 1 millilitre or 1 kg = 1 litre or 1000 kg = 1 m3 of water.    Then the mass can be found from the definition of density

ρ = mass / Volume   gives  mass =  volume  ρ   

Solution

Solving equation    

The volume of the reservoir V  is its surface area A  times its average depth h.

Volume =  50 km2  x 40 m =  50 x (1000)2 m2  x 40 m  =  2.00 x 109 m3

 

\The density of water ρ water = 1000 kg/m3     gives  mass =  volume  ρ 

mass = (  2.00 x 109 m3)   x ( 1000 kg/m3)   = 2.00 x 1012  kg

Discussion

A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is  1.96 x 1013 N , where g is the acceleration due to the Earth’s gravity (about 9.80 m/s2 . 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.

As we’ve explored in this chapter, density is a fundamental physical property that helps us understand and explain many phenomena in our world. From determining whether objects float or sink to explaining why different layers form in our planet’s structure, density plays a crucial role in science and everyday life. As future educators, your understanding of density will enable you to demystify many natural occurrences for your students, fostering their curiosity about the physical world. Remember, density is more than just a formula – it’s a key to unlocking a deeper understanding of the matter around us. By grasping this concept, you’re well-equipped to inspire the next generation of scientific thinkers.
Photograph of the Three Gorges Dam in central China.
Figure 2. Three Gorges Dam in central China. When completed in 2008, this became the world’s largest hydroelectric plant, generating power equivalent to that generated by 22 average-sized nuclear power plants. The concrete dam is 181 m high and 2.3 km across. The reservoir made by this dam is 660 km long. Over 1 million people were displaced by the creation of the reservoir. (credit: Le Grand Portage)

Section Summary

  • Density is the mass per unit volume of a substance or object. In equation form, density is defined as ρ = mass / Volume 
  • The SI unit of density is kg/m3 

Conceptual Questions

1: Approximately how does the density of air vary with altitude?

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?

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

A glass filled to the brim with water and ice cubes.
Figure 3.

Problems & Exercises

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

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

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.

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  89.0 cm3 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.)

Glossary

density
the mass per unit volume of a substance or object

Solutions

Problems & Exercises

1: 1.610 cm 3

3: (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.

4:  2.70  g/cm3 

 

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1.8.1 Density Copyright © 2022 by Physics Department of Douglas College; OpenStax; and Joey Wu is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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