Density: Measurement and Calculations

Density compares mass with volume

Density is mass per unit volume: it tells us how much mass is packed into a given amount of space. Two objects can occupy the same volume but have very different masses. A bag of feathers has much less mass than an equally sized bag of metal, so the feathers have a lower density.

Density is not simply another word for mass. A large object can have a large mass while being made from a low-density material. To compare densities, we must consider both mass and volume.

The relationship is:

$$\rho = \frac{m}{V}$$

Here, $\rho$ (the Greek letter rho) is density in kg/m³, $m$ is mass in kg and $V$ is volume in m³. A density of 1000 kg/m³ means that one cubic metre of the material has a mass of 1000 kg.

Rearranging gives $m = \rho V$ to find mass and $V = m/\rho$ to find volume.

Measuring the density of a regular solid

Every density measurement needs a mass and a volume. For a regularly shaped solid, such as a rectangular block, its volume can be calculated from its dimensions.

1. Zero a digital balance, place the dry object on it and record its mass once the reading is steady.
2. Measure the dimensions needed for its shape. Use a ruler for a larger block, or suitable calipers or a micrometer for smaller dimensions.
3. Repeat each dimension measurement and calculate its mean to reduce the influence of random measurement variation.
4. Calculate the volume, then divide mass by volume.

For a rectangular block, $V = lwh$, where $l$, $w$ and $h$ are its length, width and height. For a cube of side length $a$, $$. For a cylinder, , where is its radius and its height. A sphere has volume .

Keep all dimensions in the same unit. Converting lengths to metres before multiplying produces a volume in m³. Alternatively, calculate in cm³ and then convert: because $1\text{ cm} = 0.01\text{ m}$, $1\text{ cm}^3 = 0.000001\text{ m}^3$. Convert grams to kilograms by dividing by 1000.

Measuring the density of an irregular solid

A stone does not have a simple set of dimensions from which to calculate its volume. Instead, measure how much water it displaces. When an object is completely submerged, the water pushed aside has the same volume as the object.

First measure the dry object’s mass on a balance. If it fits inside a measuring cylinder, record the initial water volume, gently lower the object into the water and record the final volume. The object’s volume is:

$$V_{\text{object}} = V_{\text{final}} - V_{\text{initial}}$$

For a larger object, use a displacement or Eureka can. Fill it until water flows from its spout, then wait until dripping stops. Put a measuring cylinder beneath the spout and gently lower the object until it is fully submerged. Wait until the overflow stops and measure the collected water: this is the object’s volume.

The object must not dissolve in or absorb the water. Make sure no air bubbles are trapped against it, because these would also displace water. Read the bottom of the water’s curved surface at eye level. Divide the object’s mass by its displaced volume to obtain its density.

Measuring the density of a liquid

A liquid needs a container, but the container’s mass must not be included in the liquid’s density calculation.

1. Measure the mass of an empty, dry measuring cylinder.
2. Add the liquid and read its volume at eye level.
3. Measure the combined mass of the cylinder and liquid, waiting for a steady balance reading.
4. Subtract the empty cylinder’s mass to find the liquid’s mass.
5. Divide the liquid’s mass by its volume, using consistent units.

Repeat measurements to check their consistency. Keep the outside of the cylinder dry so that droplets do not add extra mass. In both liquid and displacement measurements, keep water away from the balance, clear up spills promptly and handle glassware and heavy objects carefully. Lower objects gently rather than dropping them into a cylinder.

When volume changes but mass stays the same

During a change of state, mass is conserved if none of the substance escapes. For example, if 20 g of liquid evaporates, the gas produced also has a mass of 20 g. In an open container, some gas may leave, so the mass remaining in the container can decrease.

A fixed mass can occupy a different volume after a change. Gas particles are much farther apart than particles in a liquid, so the same mass usually occupies a much larger volume as a gas and has a much lower density.

The equation explains the connection: at constant mass, increasing volume decreases density. If the volume doubles, density halves; if the volume halves, density doubles. For a calculation involving a change, find the conserved mass using $m = \rho_{\text{initial}}V_{\text{initial}}$, then use that same mass in $\rho_{\text{final}} = m/V_{\text{final}}$.

Equation and units

$$\rho = \frac{m}{V}, \qquad m = \rho V, \qquad V = \frac{m}{\rho}$$