Density and Percent Composition: Their Use in Problem Solving

Throughout this blog, we will encounter new concepts about the structure and behavior of matter. One means of firming up our understanding of these new concepts is to work problems that relate concepts that we already know to those we are trying to understand. In this section, we will introduce two quantities frequently required in problem solving: density and percent composition.

Density

Here is an old riddle: What weighs more, a ton of bricks or a ton of cotton? If you answer that they weigh the same, you demonstrate a clear understanding of the meaning of weight and, indirectly, of the quantity of matter to which weight is proportional, that is, mass. Anyone who answers that the bricks weigh more than the cotton has confused the concepts of weight and density. Matter in a brick is more concentrated than in cotton that is, the matter in a brick is confined to a smaller volume. Bricks are denser than cotton. Density is the ratio of mass to volume.

Mass and volume are both extensive properties. An extensive property is dependent on the quantity of matter observed. However, if we divide the mass of a substance by its volume, we obtain density, an intensive property. An intensive property is independent of the amount of matter observed. Thus, the density of pure water at 25 °C has a unique value, whether the sample fills a small beaker (small mass/small volume) or a swimming pool (large mass/large volume). Intensive properties are especially useful in chemical studies because they can often be used to identify substances.

The SI base units of mass and volume are kilograms and cubic meters, respectively, but chemists generally express mass in grams and volume in cubic centimeters or milliliters. Thus, the most commonly encountered density unit is grams per cubic centimeter or the identical unit grams per milliliter (g/mL).

The mass of 1.000 L of water at 4 degrees Celsius is 1.000 kg. The density of water at 4 degrees Celsius 1000 g/1000 mL, or 1.000 g/mL. At 20 °C, the density of water is 0.9982 g/mL, whereas mass remains constant. One reason that climate change is a concern is because as the average temperature of seawater increases, the seawater will become less dense, its volume will increase, and sea level will rise even if no continental ice melts. Like temperature, the state of matter affects the density of a substance. In general, solids are denser than liquids and both are denser than gases, but there are notable overlaps in densities between solids and liquids. Following are the ranges of values generally observed for densities; this information should prove useful in solving problems.

> Solid densities: from about 0.2 g/cm3 to 20 g/cm3

> Liquid densities: from about 0.5 g/mL to 3-4 g/mL

> Gas densities mostly in the range of a few grams per liter

In general, densities of liquids are known more precisely than those of solids (which may have imperfections in their microscopic structures). Also, densities of elements and compounds are known more precisely than densities of materials with variable compositions (such as wood or rubber). There are several important consequences of the different densities of solids and liquids. A solid that is insoluble and floats on a liquid is less dense than the liquid, and it displaces a mass of liquid equal to its own mass. An insoluble solid that sinks to the bottom of a liquid is denser than the liquid and displaces a volume of liquid equal to its own volume. Liquids that are immiscible in each other separate into distinct layers, with the densest liquid at the bottom and the least dense liquid at the top. 

Density in Conversion Pathways

If we measure the mass of an object and its volume, simple division gives us its density. Conversely, if we know the density of an object, we can use density as a conversion factor to determine the object s mass or volume. For example, a cube of osmium 1.000 cm on edge weighs 22.59 g. The density of osmium (the densest of the elements) is 22.59 g/cm³. What would be the mass of a cube of osmium that is 1.25 in. on edge (1 in. = 2.54 cm)? To solve this problem, we begin by relating the volume of a cube to its length, that is, V = l³ and then we can map out the conversion pathway:

At 25 °C the density of mercury, the only metal that is liquid at this temperature, is 13.5g/mL.  Suppose we want to know the volume, in mL, of 1.000 kg of mercury   at 25 °C. We proceed by (1) identifying the known information: 1.000 kg of mercury and d = 13.5g/mL (at 25 °C); (2) noting what we are trying to determine a volume in milliliters (which we designate mL mercury); and (3) looking for the relevant conversion factors. Outlining the conversion pathway will help us find these conversion factors:

We need the factor 1000 g/kg to convert from kilograms to grams. Density provides the factor to convert from mass to volume. But in this instance, we need to use density in the inverted form. That is,

Examples below further illustrate that numerical calculations involving density are generally of two types: determining density from mass and volume measurements and using density as a conversion factor to relate mass and volume.

Percent Composition as a Conversion Factor

Above we described composition as an identifying characteristic of a sample of matter. A common way of referring to composition is through percentages. Percent (per centum) is the Latin for per (meaning for each) and centum (meaning 100). Thus, percent is the number of parts of a constituent in 100 parts of the whole. To say that a seawater sample contains 3.5% sodium chloride by mass means that there are 3.5 g of sodium chloride in every 100 g of the seawater. We make the statement in terms of grams because we are talking about percent by mass. We can express this percent by writing the following ratios:

In the follow example we will use one of these ratios as a conversion factor.