Density

Pick up a small iron nail and a foam packing block of about the same size, and the difference in your hand is startling — the nail feels far heavier, even though the two objects take up roughly the same amount of space. What differs is not how much room each occupies but how much mass is packed into that room. That packing ratio has a name: density, the mass present in a unit volume of a substance. Density shows up in the CBSE Class 8 chapter on solutes, solvents and solutions as the quantitative bridge between "how much matter" and "how much space" — a single number that lets you identify a substance, predict whether it will float or sink, and compare very different-looking materials on equal footing. This article builds the formula from scratch, works through it with real numbers, and flags the mix-ups that trip up most students on their first pass.

The core insight

Density is mass divided by volume, $\rho = m/V$ — a fixed property of a substance (at given temperature and pressure) that does not depend on how large or small a sample of it is.

The core idea

Density measures how tightly matter is packed into space. Two objects can occupy the same volume yet contain very different amounts of mass — that difference is exactly what density captures.

\text{Density} = \frac{\text{Mass}}{\text{Volume}}, \qquad \rho = \frac{m}{V}

where: $\rho$ (rho) is density, $m$ is the mass of the sample, and $V$ is the volume it occupies. In SI units density comes out in $\text{kg/m}^3$ , but for everyday lab-scale samples it is far more common — and just as correct — to use $\text{g/cm}^3$ . A few properties of density are worth holding onto:

It does not depend on shape or size. A one-gram sliver of aluminium and a one-kilogram block of aluminium have the same density, because density is a ratio of mass to volume, not a total amount of either. Cutting a substance in half halves both its mass and its volume, leaving the ratio unchanged.
It does depend on temperature and pressure. Heating a substance usually makes it expand — the mass stays the same but the volume grows, so density falls. Pressure has only a small effect on solids and liquids (they barely compress) but a large effect on gases, which can be squeezed into a much smaller volume.
It is characteristic of a substance. Every pure substance has its own density at a given temperature and pressure, which is why reference tables list density alongside a material's name. Measuring a sample's mass and volume and computing its density is a practical way to check what the sample is made of, or whether two similar-looking samples are really the same material.
It connects forward in the chapter. Density is the base idea behind relative density (comparing a substance's density to that of water) and behind why some objects float and others sink in a given liquid — an object floats in a liquid when its own density is lower than the liquid's.

To measure density practically you need two independent measurements, taken with different instruments: mass, found with a balance, and volume, found with a ruler for a regular solid, or by water displacement (using a graduated cylinder) for an irregular solid or a liquid. The formula only gives a correct answer once both quantities are known and expressed in matching units — grams with cubic centimetres, or kilograms with cubic metres, but never a mix of the two.

Figure

A balance beam labelled mass on the left pan and a measuring cylinder labelled volume on the right, feeding into a divide symbol that produces a density block

Derivation AGetting density and its rearrangements from mass and volume

The goal here is to establish, and then confidently rearrange, the one relationship that ties mass, volume and density together, so that any two known quantities can be used to find the third — a skill that covers every density question this chapter can ask, whether the unknown is density itself, mass, or volume.

Step 1. Start from the definition of density as mass per unit volume: it tells you the mass contained in one unit of volume of the substance. Writing this as a formula gives $\rho = \dfrac{m}{V}$ , where $\rho$ is density, $m$ is mass and $V$ is volume. This is the only relationship needed for the whole topic — everything else is algebraic rearrangement of this single equation.

Step 2. To find mass when density and volume are known, multiply both sides of $\rho = m/V$ by $V$ . This isolates $m$ on one side: $m = \rho \times V$ . In words, mass equals density multiplied by volume — useful when a substance's density is already known (say, from a table) and you want the mass of a certain volume of it.

Step 3. To find volume when density and mass are known, start again from $\rho = m/V$ , multiply both sides by $V$ to get $\rho V = m$ , then divide both sides by $\rho$ : $V = \dfrac{m}{\rho}$ . In words, volume equals mass divided by density.

Step 4. Keep the three forms straight by remembering they are all the same equation viewed differently: $\rho = m/V$ , $m = \rho V$ , $V = m/\rho$ . The quantity being solved for is always alone on one side; whichever of mass, volume or density is unknown, the other two combine — either multiplied or divided — to produce it. Units must always match before substituting: grams with cubic centimetres, or kilograms with cubic metres, never one system mixed with the other.

Worked example

Question: A block of a certain metal has a mass of $270\ \text{g}$ and a volume of $100\ \text{cm}^3$ . Find its density, and then find the mass of a $250\ \text{cm}^3$ piece of the same metal.

Step 1. Write down what is known for the first part: mass $m = 270\ \text{g}$ , volume $V = 100\ \text{cm}^3$ . Both are already in matching units (grams and cubic centimetres), so no conversion is needed.

Step 2. Apply the density formula: $\rho = \dfrac{m}{V} = \dfrac{270\ \text{g}}{100\ \text{cm}^3} = 2.7\ \text{g/cm}^3$ . This is, in fact, the known density of aluminium, so the metal in question is very likely aluminium.

Step 3. Now use this density to find the mass of a different-sized piece of the same metal. Since density is a fixed property of the substance, the $2.7\ \text{g/cm}^3$ value carries over unchanged to any sample of it, however large or small.

Step 4. Use the rearranged formula $m = \rho \times V$ with the new volume: $m = 2.7\ \text{g/cm}^3 \times 250\ \text{cm}^3 = 675\ \text{g}$ .

The block's density is $2.7\ \text{g/cm}^3$ , and a $250\ \text{cm}^3$ piece of the same metal has a mass of $675\ \text{g}$ .

Competitive stretch

A liquid has a density of $0.8\ \text{g/cm}^3$ . What volume does $400\ \text{g}$ of it occupy? Using $V = m/\rho = 400\ \text{g} \div 0.8\ \text{g/cm}^3 = 500\ \text{cm}^3$ . Notice this liquid is less dense than water ( $1\ \text{g/cm}^3$ ), which is why such liquids — like cooking oil — tend to float on top of water rather than sink into it.

Common traps

Confusing mass with density — mass is the total quantity of matter in a sample, while density is mass per unit volume, a ratio that stays the same no matter how big the sample is.
Thinking a larger piece of the same substance is "denser" than a smaller piece — density depends only on the substance (and its temperature and pressure), never on the amount or shape of the sample.
Mixing units when substituting — for example, using grams for mass but litres for volume without converting first; the formula only gives a correct number when the units match.
Writing the formula upside down — density is always mass divided by volume, not volume divided by mass; the swap gives a value with completely the wrong meaning and size.
Forgetting that density changes with temperature and pressure — assuming a single density value applies at every condition ignores that heating a substance usually increases its volume and lowers its density.

Ask the tutor: Want to see how density leads directly into relative density and the rule for floating and sinking, worked through with a few more numbers?

Why doesn't density change when I cut a block in half

Density is a ratio, not a total. Cutting a block in half also halves its mass and its volume, but dividing a halved mass by a halved volume gives back the same number you started with. Density describes how tightly matter is packed at every point in the substance, and that packing does not change just because you have less of the substance — only the total mass and total volume shrink together.

Does density ever change for the same substance

Yes, but not with the size or shape of the sample — it changes with temperature and pressure. Heating most substances makes them expand, so the same mass now fills a larger volume, which lowers density. Pressure barely affects solids and liquids because they resist compression, but it strongly affects gases, which compress easily into a smaller volume, raising their density. So the density values in reference tables are always quoted at a stated temperature and pressure.

How is density different from relative density

Density is mass per unit volume, expressed with units like g per cm cubed or kg per m cubed. Relative density compares a substance's density to the density of water and has no units at all, because it is a ratio of two densities. A relative density greater than 1 means the substance is denser than water and will sink in it; less than 1 means it will float. This chapter builds relative density directly on top of the density formula covered here.

Density

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads