2.4: Density (definition and formula)

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Learning Objective

Learn about the various temperature scales that are commonly used in chemistry.
Define density and use it as a conversion factor.

Density

Density is a physical property that is defined as a substance's mass divided by its volume:

\[density= \dfrac{mass}{volume}\Rightarrow d= \dfrac{m}{v}\nonumber \]

Density is usually a measured property of a substance, so its numerical value affects the significant figures in a calculation. Notice that density is defined in terms of two dissimilar units, mass and volume. That means that density overall has derived units, just like velocity. Common units for density include g/mL, g/cm³, g/L, kg/L, and even kg/m³. Densities for some common substances are listed in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Densities of Some Common Substances
Substance	Density (g/mL or g/cm³)
water	1.0
gold	19.3
mercury	13.6
air	0.0012
cork	0.22–0.26
aluminum	2.7
iron	7.87

Because of how it is defined, density can act as a conversion factor for switching between units of mass and volume. For example, suppose you have a sample of aluminum that has a volume of 7.88 cm³. How can you determine what mass of aluminum you have without measuring it? You can use the volume to calculate it. If you multiply the given volume by the known density (from Table \(\PageIndex{1}\)), the volume units will cancel and leave you with mass units, telling you the mass of the sample:

\[7.88\,\cancel{cm^{3}}\times \dfrac{2.7\,g}{\cancel{cm^{3}}}= 21\, g \text{ of aluminium} \nonumber \nonumber \]

where we have limited our answer to two significant figures.

Example \(\PageIndex{3}\): Mercury

What is the mass of 44.6 mL of mercury?

Solution

Use the density from Table \(\PageIndex{1}\) "Densities of Some Common Substances" as a conversion factor to go from volume to mass:

\[44.6\,\cancel{mL}\times \dfrac{13.6\,g}{\cancel{mL}}= 607\,g \nonumber \nonumber \]

The mass of the mercury is 607 g.

Exercise \(\PageIndex{3}\)

What is the mass of 25.0 cm³ of iron?

Answer: 197 g

Density can also be used as a conversion factor to convert mass to volume—but care must be taken. We have already demonstrated that the number that goes with density normally goes in the numerator when density is written as a fraction. Take the density of gold, for example:

\[d=19.3\,g/mL =\dfrac{19.3\,g}{mL} \nonumber \nonumber \]

Although this was not previously pointed out, it can be assumed that there is a 1 in the denominator:

\[d=19.3\,g/mL =\dfrac{19.3\,g}{mL} \nonumber \nonumber \]

That is, the density value tells us that we have 19.3 grams for every 1 milliliter of volume, and the 1 is an exact number. When we want to use density to convert from mass to volume, the numerator and denominator of density need to be switched—that is, we must take the reciprocal of the density. In so doing, we move not only the units, but also the numbers:

\[\dfrac{1}{d}= \dfrac{1\,mL}{19.3\,g} \nonumber \nonumber \]

This reciprocal density is still a useful conversion factor, but now the mass unit will cancel and the volume unit will be introduced. Thus, if we want to know the volume of 45.9 g of gold, we would set up the conversion as follows:

\[45.9\,\cancel{g}\times \dfrac{1\,mL}{19.3\cancel{g}}= 2.38\,mL \nonumber \nonumber \]

Note how the mass units cancel, leaving the volume unit, which is what we are looking for.

Example \(\PageIndex{4}\): Calculating Volume from Density

A cork stopper from a bottle of wine has a mass of 3.78 g. If the density of cork is 0.22 g/cm³, what is the volume of the cork?

Solution

To use density as a conversion factor, we need to take the reciprocal so that the mass unit of density is in the denominator. Taking the reciprocal, we find:

\[\dfrac{1}{d}= \dfrac{1\,cm^{3}}{0.22\,g} \nonumber \nonumber \]

We can use this expression as the conversion factor. So

\[3.78\,\cancel{g}\times \dfrac{1\,cm^{3}}{0.22\,\cancel{g}}= 17.2\,cm^{3} \nonumber \nonumber \]

Exercise \(\PageIndex{4}\)

What is the volume of 3.78 g of gold?

Answer: 0.196 cm³

Care must be used with density as a conversion factor. Make sure the mass units are the same, or the volume units are the same, before using density to convert to a different unit. Often, the unit of the given quantity must be first converted to the appropriate unit before applying density as a conversion factor.

Key Takeaways

Density relates the mass and volume of a substance.
Density can be used to calculate volume from a given mass or mass from a given volume.