Pressure and density

Density is defined as the mass contained in a unit volume of a substance:

where: Mass is measured in kilograms (kg) or grams (g), Volume in cubic metres (m³) or cubic centimetres (cm³), so density comes out in $\text{kg/m}^3$ (SI unit) or the commonly used $\text{g/cm}^3$ (equivalently g/mL for liquids). Density is a property of the substance, not of a particular sample — a small iron nail and a large iron rod share the same density, because doubling the mass by taking more iron also doubles the volume, leaving the ratio unchanged. What can change that ratio for a fixed piece of matter is temperature or pressure. Under pressure, the mass of a substance never changes — no particles are added or removed — but the volume can shrink if there is room for the particles to move closer together. That "room" is the interparticle space, and it is exactly where the three states of matter differ.

• Gases have particles spread far apart with huge empty gaps between them, so an applied pressure can push those particles much closer together. Volume drops sharply while mass stays fixed, so density rises sharply. This is precisely how a gas cylinder packs a large mass of gas into a small steel container.
• Liquids have particles already touching, held by moderate attraction, with only a thin margin of space left. Pressure can close that margin only a little, so liquids are called practically incompressible and their density barely rises under pressure.
• Solids have particles locked in place by strong interparticle attraction, with essentially no space left to remove. Pressure has a negligible effect on solid density.

The figure below compares the same applied pressure acting on the three states, showing why the volume shrinkage — and therefore the density increase — gets smaller as interparticle space runs out.

Every question about pressure and density in this chapter reduces to a single arithmetic step once you know the mass and the volume of a sample, so the goal of this derivation is to show precisely how the density formula is applied to a real container of gas before and after compression, and how that same one-line calculation explains why the state of matter decides the size of the density change. Start from the definition and follow the mass and the volume through a compression event.

Step 1. Write the defining relationship. $\text{Density} = \dfrac{\text{Mass}}{\text{Volume}}$. This holds at every instant for any sample — before compression, during it, and after it — because it is a definition, not a law that can be switched off.

Step 2. Fix what pressure can and cannot change. Compressing a sample with a pump or a piston does no chemistry: no particles are created or destroyed, so the mass in the numerator is locked. Only the volume in the denominator is free to change, and it changes only by however much interparticle space the substance is willing to give up.

Step 3. Compute density before compression. Take $2\ \text{kg}$ of oxygen gas filling a volume of $1.5\ \text{m}^3$ at ordinary atmospheric pressure. $\text{Density}_{\text{before}} = \dfrac{2\ \text{kg}}{1.5\ \text{m}^3} = 1.33\ \text{kg/m}^3$.

Step 4. Compute density after compression. A pump forces the same $2\ \text{kg}$ of oxygen into a steel cylinder of volume $0.02\ \text{m}^3$ — the mass is identical, but the huge interparticle gaps of the gas have been squeezed out. $\text{Density}_{\text{after}} = \dfrac{2\ \text{kg}}{0.02\ \text{m}^3} = 100\ \text{kg/m}^3$, about $75$ times denser than before.

Step 5. Contrast with a liquid or solid subjected to the same pressure. If the identical pump pressed on $2\ \text{kg}$ of water instead, the volume might shrink from $2.0\times10^{-3}\ \text{m}^3$ to perhaps $1.999\times10^{-3}\ \text{m}^3$ — a change too small to matter for a Class 8 calculation — because water's particles have almost no interparticle space left to surrender. The same formula, the same fixed mass, but a volume that refuses to move much, is why liquids (and solids even more so) are called incompressible in this chapter.

A welder fills an empty steel cylinder of volume $0.05\ \text{m}^3$ with $3\ \text{kg}$ of nitrogen gas drawn from the open atmosphere, where the same $3\ \text{kg}$ of nitrogen originally occupied $2.4\ \text{m}^3$. Find the density of the nitrogen before and after filling, and state by what factor the density has increased.

Step 1. Identify mass and volume before filling. Mass $= 3\ \text{kg}$, Volume $= 2.4\ \text{m}^3$ (nitrogen spread out at atmospheric pressure).

Step 2. Apply the density formula before filling. $\text{Density}_{\text{before}} = \dfrac{3\ \text{kg}}{2.4\ \text{m}^3} = 1.25\ \text{kg/m}^3$.

Step 3. Identify mass and volume after filling. Mass is unchanged at $3\ \text{kg}$ (no nitrogen was added or lost, only compressed); Volume $= 0.05\ \text{m}^3$, the fixed size of the steel cylinder.

Step 4. Apply the density formula after filling. $\text{Density}_{\text{after}} = \dfrac{3\ \text{kg}}{0.05\ \text{m}^3} = 60\ \text{kg/m}^3$.

Step 5. Compare the two densities. $\dfrac{60}{1.25} = 48$, so the nitrogen's density has increased $48$-fold. This large jump is possible only because nitrogen, as a gas, had abundant interparticle space to lose; the steel cylinder itself, being a solid, would show no measurable density change from the pressure the compressed gas pushes back on it.

The nitrogen's density rises from $1.25\ \text{kg/m}^3$ to $60\ \text{kg/m}^3$, a 48-fold increase, entirely because the fixed mass now occupies a far smaller volume.

• Believing pressure changes the mass of a substance — pressure only rearranges particles into a smaller (or unchanged) volume; mass stays exactly the same throughout.
• Expecting the same percentage density increase for gases, liquids, and solids under the same pressure — the increase is large for gases, small for liquids, and negligible for solids because of how much interparticle space each has to lose.
• Confusing a pressure-driven density change with a temperature-driven one — both can change density, but this chapter's pressure argument works through interparticle space, not through particle speed or thermal expansion.
• Thinking cutting a solid into smaller pieces changes its density — density is a ratio that stays fixed for a given substance unless pressure or temperature genuinely compresses or expands it.

Ask the tutor: Want to see how the temperature effect on density compares side by side with the pressure effect, or try a few more compression calculations together?