113. DensityLearning Objectives
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.)
where the Greek letter $\rho $ (rho) is the symbol for density, $m$ is the mass, and $V$ is the volume occupied by the substance. Density:Density is mass per unit volume. $$\rho =\frac{m}{V},$$where $\rho $ 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 ${\text{kg/m}}^{3}$, representative values are given in Table 1. The metric system was originally devised so that water would have a density of $1\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$, equivalent to ${\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$. 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 cm^{3}. 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. TakeHome 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 VolumeA reservoir has a surface area of $\text{50}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{km}}^{2}$ 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 $V$ of the reservoir from its dimensions, and find the density of water $\rho $ in Table 1. Then the mass $m$ can be found from the definition of density $$\rho =\frac{m}{V}.$$Solution Solving equation $\rho =m/V$ for $m$ gives $m=\rho V$. The volume $V$ of the reservoir is its surface area $A$ times its average depth $h$: $$\begin{array}{lll}V& =& \text{Ah}=\left(\text{50.0}\phantom{\rule{0.25em}{0ex}}{\text{km}}^{2}\right)\left(\text{40.0}\phantom{\rule{0.25em}{0ex}}\text{m}\right)\\ & =& \left[\left(\text{50.0 k}{\text{m}}^{2}\right){\left(\frac{{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m}}{1\phantom{\rule{0.25em}{0ex}}\text{km}}\right)}^{2}\right]\left(\text{40.0 m}\right)=2\text{.}\text{00}\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\end{array}$$The density of water $\rho $ from Table 1 is $1\text{.}\text{000}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$. Substituting $V$ and $\rho $ into the expression for mass gives $$\begin{array}{lll}m& =& \left(1\text{.}\text{00}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left(2\text{.}\text{00}\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\right)\\ & =& 2.00\times {\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\text{kg.}\end{array}$$Discussion A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is $\text{mg}=1\text{.}\text{96}\times {\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{N}$, where $g$ is the acceleration due to the Earth’s gravity (about $9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$). 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
Conceptual QuestionsExercise 1Approximately how does the density of air vary with altitude? Exercise 2Give 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 3Figure 3 shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer. Figure 3 Problems & ExercisesExercise 1Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold? Show/Hide Solution Solution$1\text{.}\text{610}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ Exercise 2Mercury 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. Show/Hide Solution 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 4A 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 240g rock that displaces $\text{89}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ 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.) Show/Hide Solution Solution$2\text{.}\text{70}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$ Exercise 5Suppose 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 0.500m wide by 0.900m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car. Show/Hide Solution Solution(a) 0.163 m (b) Equivalent to 19.4 gallons, which is reasonable Exercise 7A 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 8A 2.50kg 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? Show/Hide Solution Solution$7\text{.}9\times {\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0ex}{0ex}}{\text{kg/m}}^{3}$ Exercise 9What is the density of 18.0karat 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. Show/Hide Solution Solution$\text{15}\text{.}6\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$ Exercise 10There 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 ${\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$. The nucleus of an atom has a radius about ${\text{10}}^{5}$ 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 $7\times {\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m}$)? Show/Hide Solution Solution(a) ${\text{10}}^{\text{18}}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ (b) $2\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{m}$
