How To Calculate Mass From Density

You calculate the density of a solid object, a liquid or a gas by measuring its volume, weighing it to determine its mass and using the formula

\(\delta = \frac{m}{V}\)

where ​∂​ is density, ​m​ is mass and ​V​ is volume. It's an easy mathematical operation to rearrange this equation so you can calculate mass from density:

\(m=\delta \times V\)

Why would you want to do this? Mass should be easy to determine – all you have to do is get out your scale and do some weighing, right? Actually, you can't always do this, especially when you're dealing with a liquid or a very heavy solid that's too big for your scale. Because the densities of most solids and liquids are tabulated, you can look up the density of the substance in question. As long as you're able to measure the volume occupied by the substance, which is easy if it's in a container, you'll know its mass.

Density is a fixed quantity, and because it never changes, it can be a proportionality factor between mass and volume for any given substance. In other words, as the volume increases, so does the mass. If you plotted increasing values of volume against the corresponding mass increases on a graph, you'd get a straight line with slope equal to the density of the substance.

You usually don't need to go to the trouble of plotting a graph, though. As long as you know the composition of a solid, you can look up the density in a table. If you have a liquid, you'll want to look up its specific gravity, which is the density compared to the density of water. For example, the specific gravity of ethyl alcohol is 0.787. Since the density of water is 1 g/ml, that means that the density of alcohol is 0.787 g/ml.

If you have a solution of unknown composition, you can't look up its specific gravity, but you can measure it. The tool for doing this is called a hygrometer. You let it float in the liquid and read the specific gravity from the graduated mark that just touches the surface.

Density is measured in a variety of units, including grams/milliliter, kilograms/cubic meter and pounds/cubic foot. When you look up the density, make sure its expressed in the units you are using to measure volume, or you'll get an inaccurate value for mass. Here are some common conversion factors

1 kg/m³ = 0.001 g/ml = 0.062 lb/ft³.

If you use corresponding units for density and volume, you can calculate the mass from density and get it in corresponding units by using the equation ​m​ = ​∂V​. Once you know the mass, you can always convert it to different units if necessary.

1\. What is the mass of a 2 ml vial of carbon tetrachloride?

The specific gravity of carbon tetrachloride is 1.589. Since the volume of the sample is measured in milliliters, divide the specific gravity by the density of water in g/ml to get the density in those units. Doing this, you find the density to be 1.598 g/ml. Now it's easy to use the density to mass conversion equation to find the mass:

\(m=\delta \times V = 1.589\times 2 = 3.178\text{ grams}\)

2\. How do you find the mass of a large gold statue without weighing it?

First, measure the volume, in liters, using the water displacement method. Next, look up the density of gold, which is 19,320 kg/m³. To convert to grams per liter, you simply need to multiply by 1, so the density is 19,320 g/l. Now you can calculate the mass from the density using the formula ​m​ = ​∂V​ and obtain the answer in grams.