﻿ Density - Measurements and Calculations

# Measurements and CalculationsDensity

Objective

· To define density and its units.

When you were in elementary school, you may have been embarrassed by your answer to the question “Which is heavier, a pound of lead or a pound of feathers?” If you said lead, you were undoubtedly thinking about density, not mass. Density can be defined as the amount of matter present in a given volume of substance. That is, density is mass per unit volume, the ratio of the mass of an object to its volume:

It takes a much bigger volume to make a pound of feathers than to make a pound of lead. This is because lead has a much greater mass per unit volume—a greater density.

The density of a liquid can be determined easily by weighing a known volume of the substance, as illustrated in Example 2.13.

Interactive Example 2.13. Calculating Density

Suppose a student finds that mL of a certain liquid weighs g. What is the density of this liquid?

Solution

We can calculate the density of this liquid simply by applying the definition

This result could also be expressed as because .

The volume of a solid object is often determined indirectly by submerging it in water and measuring the volume of water displaced. In fact, this is the most accurate method for measuring a person’s percent body fat. The person is submerged momentarily in a tank of water, and the increase in volume is measured (Fig. 2.10). It is possible to calculate the body density by using the person’s weight (mass) and the volume of the person’s body determined by submersion. Fat, muscle, and bone have different densities (fat is less dense than muscle tissue, for example), so the fraction of the person’s body that is fat can be calculated. The more muscle and the less fat a person has, the higher his or her body density. For example, a muscular person weighing lb has a smaller body volume (and thus a higher density) than a fat person weighing lb.

Figure 2.10. Example 2.14. Determining Density

At a local pawn shop a student finds a medallion that the shop owner insists is pure platinum. However, the student suspects that the medallion may actually be silver and thus much less valuable. The student buys the medallion only after the shop owner agrees to refund the price if the medallion is returned within two days. The student, a chemistry major, then takes the medallion to her lab and measures its density as follows. She first weighs the medallion and finds its mass to be g. She then places some water in a graduated cylinder and reads the volume as mL. Next she drops the medallion into the cylinder and reads the new volume as mL. Is the medallion platinum or silver ?

Solution

The densities of platinum and silver differ so much that the measured density of the medallion will show which metal is present. Because by definition

to calculate the density of the medallion, we need its mass and its volume. The mass of the medallion is g. The volume of the medallion can be obtained by taking the difference between the volume readings of the water in the graduated cylinder before and after the medallion was added.

The volume appeared to increase by mL when the medallion was added, so mL represents the volume of the medallion. Now we can use the measured mass and volume of the medallion to determine its density:

or

The medallion is really platinum.

Self-Check: Exercise 2.9

· A student wants to identify the main component in a commercial liquid cleaner. He finds that mL of the cleaner weighs g. Of the following possibilities, which is the main component of the cleaner?

 Substance Density, chloroform diethyl ether isopropyl alcohol toluene

See Problems 2.89 and 2.90.

Interactive Example 2.15. Using Density in Calculations

Mercury has a density of g/mL. What volume of mercury must be taken to obtain g of the metal?

Solution

and then rearrange this equation to isolate the required quantity. In this case we want to find the volume. Remember that we maintain an equality when we do the same thing to both sides. For example, if we multiply both sides of the density definition by volume,

volume cancels on the right, leaving

We want the volume, so we now divide both sides by density,

to give

Now we can solve the problem by substituting the given numbers:

We must take mL of mercury to obtain an amount that has a mass of g.

The densities of various common substances are given in Table 2.8. Besides being a tool for the identification of substances, density has many other uses. For example, the liquid in your car’s lead storage battery (a solution of sulfuric acid) changes density because the sulfuric acid is consumed as the battery discharges. In a fully charged battery, the density of the solution is about . When the density falls below , the battery has to be recharged. Density measurement is also used to determine the amount of antifreeze, and thus the level of protection against freezing, in the cooling system of a car. Water and antifreeze have different densities, so the measured density of the mixture tells us how much of each is present. The device used to test the density of the solution—a hydrometer—is shown in Fig. 2.11.

Table 2.8. Densities of Various Common Substances at

 Substance Physical State Density oxygen gas * hydrogen gas * ethanol liquid benzene liquid water liquid magnesium solid salt (sodium chloride) solid aluminum solid iron solid copper solid silver solid lead solid mercury liquid gold solid

Figure 2.11. 