Density - Theory, Experimental, and Graphical Methods

What is Density?

Conceptually, density is best described as the compactness of a material. That is to say that dense materials contain a large amount of material in a small amount of space. Less dense materials have a smaller concentration of mass in a given volume, or a mass spread out over a large volume of space. Gases are not particularly dense because of the large amounts of empty space between particles. Even a very large mass of gas (e.g. the atmosphere) is spread out over such a large volume that its overall density is small.
Mathematically, density is expressed as a ratio of mass to volume. It is a characteristic quantity, that is the density value is unique for every substance.
Graphically, density is the slope of a graph of mass vs. volume. Slope is defined as a ratio of the change in y to the change in x and since the y-axis displays mass and the x-axis volume, the slope is a ratio of the change in mass to change in volume, or density.

Density is an Intensive Property

Intensive properties are those that do not change regardless of the quantity of the sample. Mass and volume, the quantities used to determine density are extensive properties because they are dependent on (and change) with the quantity of sample. Since density is a ratio of mass to volume, it remains constant for a substance. This, coupled with the ease with which density can be found, makes it a great property to use to identify a substance.

Experimental Methods - Geometric Objects

Calculating density is generally as difficult as it is to find the volume of the object. This is because density can be calculated using the formula:

Consider the following problem:
A block with a mass of 253.5 g has dimensions of 2.14 cm by 5.39 cm by 4.88 cm. Find the density of the block.
Typically, the only consideration that must be made with mass is in grams. Therefore, we can move directly to the volume. The object is a block, or a rectangular prism. Its volume can be found by multiplying the dimensions together. Doing so gives a volume of 56.3 cm3. Now that both mass and volume are known, the density formula shown above can be used. Dividing the mass by volume (253.5/56.3) gives a density of 4.50. Since a cubic centimeter and milliliter are equivalent, density units are expressed most often as either g/cm3 (more common for solids) or g/mL (more common for liquids). Therefore, the density of the object can be reported as 4.50 g/cm3. Consider the following problem:
A sphere with a diameter of 5.86 cm has a mass of 178.96 g. What is the density of the sphere?
The volume of a sphere can be found from the formula V = 4πr3, where r is the radius of the sphere. The given diameter is 5.86 cm, and since diamater is twice the radius, the radius must be 2.93 cm. Using this radius and the aforementioned formula for volume, the sphere's volume is found to be 316 cm3. The density is then found by dividing the mass (178.96 g) by its volume (316 cm3), or 0.566 g/cm3.

Experimental Methods - Water Displacement

Many, if not most, objects are not a regular geometric shape. Their density cannot be found using a ruler and a geometric formula. The method of water displacement can be used to determine the volume of the object. Consider the following problem:
A 78.19 g rock is placed into a graduated cylinder filled with 50.0 mL of water. When the rock is submerged, the water level rises to 56.8 mL. What is the density of the rock?
The key to this problem is realizing that the increase in the water level is due to the addition of the rock. In fact, the increase in volume of water is exactly equal to the volume of the rock. In other words, the rock displaces a volume of water equal to itself. Subtracting the initial volume of water from the final volume of water (56.8 - 50.0) gives the volume of the rock, 6.8 mL. Now the density formula from before can be utilized to find the density of the object. Dividing 78.19 g by 6.8 mL yields a density for the rock equal to 11 g/mL.

Experimental Methods - Liquid Pressure

Imagine a bottle filled with a liquid. As more liquid is added to the bottle, two measurements change in direct relationship to each other. They are the height and the mass of the liquid in the bottle. Since gravitataional acceleration is constant (at least anywhere on Earth's surface) the increase in mass also means there is an increase in weight, because weight is the result of gravity's influence on mass. Pressure is a measure of a force exerted on the area of contact between two objects. For a liquid contained in a bottle, the pressure exerted by the liquid is equal to its weight divided by the surface area of the bottom of the bottle. The pressure of a liquid can be used to find its density by the formula:

P = hgd

In this formula, P represents the pressure of the liquid, measured in pascals (Pa). The variable h represents the height of the liquid, in meters. Acceleration due to gravity, a value of 9.80 m/s2 on Earth, is represented with the variable g. Finally, the density of the object can be found by solving for d. Realize that the units will be different from those discussed earlier. In this formula, d will be found in kg/m3.

Graphical Methods

In a graph of mass vs. volume, the slope of the best-fit line made from the points gives the density of that object. This can be deduced from the definition of slope, δy/δx. Since the mass coordinate points are plotted on the y-axis and volume coordinate points on x, substitutions can be made and the slope formula becomes δmass/δvolume. Recall that the density formula is a ratio of mass to volume, thus the slope is equivalent to the density. Consider the data below:

This data shows mass/volume data points for four different substances called A, B, C, and D. When plotted, best-fit lines are added and the slopes are displayed:

The equations for each line are displayed on the graph, from top (A) to bottom (D). A table of density values can be used to identify possible identities for each substance. A matches mercury, B matches lead, C matches zinc, and lastly D matches titanium.