How To Calculate Density

Table of Contents

Molecules that compose substances interact with each other and assemble in different ways depending on internal factors, specific to the atoms involved, and on external factors, like temperature or pressure. The way these molecules are “packed” at the microscopic level affects some of the properties of the macroscopic substance. 

One of these properties is density, which tells us the amount of a certain substance that is to be found per unit volume. This means, it expresses the mass per cubic meter, liter or any other volume unit of the substance of interest. 

Knowing where density comes from and how to measure it can be very helpful to understand the behavior of several substances and apparent exceptions to the laws of physics. Let’s discover more about this basic and important concept. 

How to calculate density

To determine a substance’s density follow these simple steps: 

  1. Accurately measure the mass, m, of a sample of your substance of interest.
  2. Determine the volume of the sample, V. If in liquid state, use a graduated cylinder. If in solid state, use Archimedes’ principle.
  3. Use the following equation to calculate the substance’s density, : 


=mV (1)

Density definition

Objects are made of tiny entities called atoms. A usual way of thinking about them is as a solid positively-charged nucleus made of protons and neutrons, surrounded by even smaller negatively-charged particles, called electrons, orbiting them at super high speeds. Atoms can be categorized based on specific features of this structure. One of those features is the number of protons they have, which tells us the kind of atom we are dealing with. 

For example, every oxygen atom in the universe has 8 protons. Every single one. No matter if it is floating alone or as a part of one of your body cells, this number remains constant. If it lost one of its eight protons, it would not be oxygen anymore, but nitrogen! 

Atoms make up molecules. Molecules are bigger structures composed of at least two atoms, which can be of the same or of different kind. For example, oxygen in the air is molecular, meaning it is found as a molecule made of two oxygen atoms bonded together (O2), instead of just floating independently. 

Molecules and combinations of them make up most of the objects we know. For example, the wooden chair you are probably sitting on is made of dried wood cells, which in turn are made of thousands of carbon, hydrogen, nitrogen, and oxygen atoms, among other kinds. 

When molecules come together they assemble and interact in different ways that depend on their specific characteristics. When they do so, they leave empty spaces between them. These can either stay empty or be occupied by other molecules. Look at the following image. Each circle represents an atom. If you look closely, you will notice seven of those circles make up a molecule. These molecules assemble randomly leaving empty spaces between them.

The size of the empty spaces between the molecules that compose a substance determines how tightly packed the substance is. This means that the less empty space there is per unit volume of a certain substance, the more molecules there will be inside that volume. Let’s think of an example to clarify this:

Imagine you have a cube of 1 x 1 x 1 cm of an imaginary substance which is made of imaginary molecules called imagitrons. Let’s say, under normal conditions, 12 imagitrons fit inside our 1 cubic centimeter reference volume. If, under different conditions, only 8 imagitrons fit inside the same volume, the total empty space would be bigger, since there would have to be more space between those eight imagitrons. In the first scenario we would say our imaginary substance is more tightly packed than in the second scenario, like the following image shows:

The difference between scenario 1 and 2 can be explained, for example, through temperature. The greater the temperature of a body, the greater the kinetic energy of the particles that compose it. This means that molecules vibrate, rotate and translate faster as they gain energy through a temperature increase. This causes the average distance between them to increase, which translates into a volume increase of a certain amount of substance. 

In our previous example, this means the original 12 imagitrons would take up more volume in scenario 2 than in scenario 1. This is why, only 8 imagitrons would be found inside our reference volume. In this case, our imaginary substance has a higher temperature in scenario 2.

Now, the more molecules fit inside a unit of volume of any substance, the more mass is in it. For example, if every imagitron contains 1 g of mass, then 1 cubic centimeter would contain 12 g in the first scenario, and 8 g in the second scenario. Keep in mind we are talking about mass and not weight. The former is a measurement of the amount of matter, and the latter a force generated on objects by the gravitational attraction of the Earth. 

We can now develop a measurement of how densely packed substances are. For this, we can take the ratio between the mass contained inside a certain volume of the substance of interest, as the following equation shows. The result is known as the substance’s density, and its units are therefore kg/m3.

Density is an important property of substances because it tells us not only how tightly packed their molecules are, but also how it could behave under certain circumstances. For example, when two liquids are mixed together, the one with lower density will tend to accumulate at the top of the one with higher density, as long as there are not any additional interactions between both substances. 

A simple example of this is mixing water with oil. You will notice oil accumulates on top of water, which tells us it is less dense than water. Most oils have a density between 700 and 950 kg/m3, while water has a density of 1000 kg/m3.

As mentioned before, the density of a substance is affected by temperature. In general, it decreases as temperature increases because the average separation between the particles that compose a substance increases. This applies for most substances but for one which is very important for us: water. If you are wondering why ice, which is obviously colder than liquid water, floats when introduced in a glass of water, instead of just sinking to the bottom —which means it has a lower density—, go ahead and read the section called “Density of water”.

How to measure density

To determine the density of any given substance, we only need to accurately measure the weight and the volume of a sample of it and then apply equation 2. When your sample is ready, measure its weight with an accurate device, such as a digital scale with enough sensitivity in the respective weight range. 

Then, measure the sample’s volume. If it is in liquid state, you can simply use a graduated cylinder. If it is in solid state, one way to achieve this is through Archimedes’ principle. Let’s go through the entire process: 

If you have a volume of 100 ml of any liquid inside a graduated cylinder, and then submerge your sample in it, the total volume measured on the side of the cylinder will equal 100 ml + the volume of your sample. This is correct, assuming your sample does not change volume once immersed in the liquid (this could happen, for example, if it reacts with it). Since you can measure the final total volume, and also know the liquid’s volume, you can solve the following equation to find your sample’s volume: 

Keep in mind this method only works with solid samples with higher densities than the liquid used as reference. Otherwise, your sample would not sink entirely in it, which would lead to a wrong measurement.

Finally, use equation 2 to calculate the ratio between the mass and the volume of your sample. The result is its density, which is usually expressed in kg/m3.

If you have doubts about Archimedes’ principle, go ahead and check this simulator by Stark Science, which will let you simulate the immersion of differently shaped objects made of different materials into water, and then measure the volume difference. You will also be able to weigh the objects in order to determine their densities. 

Density of water

Water is a unique substance, given that its density does not necessarily decrease with increasing temperature. When you put ice inside a glass of water, you would intuitively expect it to sink. This is because ice has a lower temperature and its molecules should therefore be more tightly packed together as when in liquid state. This is note the case, though.

Water molecules are polar molecules. This means, although their total electric charge is zero, they have a side which tends to be negatively charged and another side which tends to be positively charged. As a whole, it remains neutral. In the following image you can see how the three atoms that compose a water molecule, namely two hydrogen atoms and one oxygen atom, arrange in a way that produces a sort of spatial charge disbalance: 

Since both hydrogen atoms are located on the same side of the molecule, and their single electron has been almost entirely taken by the oxygen atom, that side of the molecule is at a higher electric potential, which is similar to it being positively charged. On the opposite side, the oxygen atom sits at a lower electric potential since it has taken two extra electrons from both hydrogen atoms. So it acts as negatively charged.

When many water molecules come together, their positively charged ends will attract other molecules’ negatively charged ends. This is called a hydrogen bond and is usual in molecules that contain this type of atom. In consequence, water molecules will tend to align with each other. Nevertheless, in liquid state, they have so much thermal energy that they move and rotate constantly, destabilizing the hydrogen bonds. Water molecules then interact and move very close to each other, but aren’t capable of creating a stable network.

On the other hand, as temperature decreases, hydrogen bonds become more stable because molecules lose thermal energy and reduce their movement. Water molecules can finally align with each other thanks to their electrostatic interactions. They then form a stable network where molecules keep a greater distance with each other. Since molecules in ice are further apart than in liquid state, as the following image shows, there will be less molecules per unit volume. This means the density of ice is lower than that of water, which explains why ice floats on water. 


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