Solids, Liquids & Gases

Density

Density tells you how much mass is packed into a given volume. A block of lead and a block of polystyrene can be exactly the same size, yet the lead has far more mass crammed in — it is denser.

The defining equation is:

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

where $\rho$ (the Greek letter "rho") is density in $\text{kg/m}^3$, $m$ is mass in $\text{kg}$, and $V$ is volume in $\text{m}^3$. You may also meet density in $\text{g/cm}^3$ — water has a density of $1\,\text{g/cm}^3$, which is the same as $1000\,\text{kg/m}^3$.

Key terms Density — the mass per unit volume of a substance, $\rho = m/V$.

Volume — the amount of space an object occupies, in $\text{m}^3$ or $\text{cm}^3$.

Measuring density

To find a density you always need two measurements: the mass and the volume.

Regular solids (cube, cuboid, sphere, cylinder): measure the mass on a balance, then measure the dimensions with a ruler or callipers and calculate the volume from the shape (e.g. $V = l \times w \times h$ for a cuboid).

Irregular solids (a stone, a key): measure the mass on a balance, then find the volume by displacement. Lower the object into a measuring cylinder containing water (or use a displacement/eureka can) and read the rise in water level. The volume of water pushed aside equals the volume of the object.

Liquids: measure the mass of an empty measuring cylinder, pour in a known volume of liquid, and measure the new mass. The mass of liquid is the difference, and you read the volume straight off the scale.

Worked example A metal cube has sides of $2.0\,\text{cm}$ and a mass of $62\,\text{g}$. Find its density.

Volume $= 2.0 \times 2.0 \times 2.0 = 8.0\,\text{cm}^3$.

$\rho = \dfrac{m}{V} = \dfrac{62}{8.0} = 7.75\,\text{g/cm}^3$.

So $\rho \approx 7.8\,\text{g/cm}^3$ (about the density of iron).

The particle model and the three states

All matter is made of tiny particles. The kinetic theory says these particles are constantly moving, and the way they are arranged and how fast they move decides whether a substance is a solid, liquid or gas.

Solids — particles are packed closely in a regular pattern and only vibrate about fixed positions. Strong forces hold them together, so a solid has a fixed shape and a fixed volume.
Liquids — particles are still close together but have no fixed pattern. They can slide past one another, so a liquid flows and takes the shape of its container, but keeps a fixed volume.
Gases — particles are far apart and move quickly in all directions. There are almost no forces between them, so a gas has no fixed shape and no fixed volume; it spreads to fill its container.

Changes of state

Heating a substance gives its particles more energy. As the energy rises, the substance can change state: solid → liquid (melting), liquid → gas (boiling/evaporating). Cooling reverses these: gas → liquid (condensing), liquid → solid (freezing).

Changes of state are physical changes — no new substance is made, and the change can be reversed. During a change of state the temperature stays constant, because the energy is used to break the bonds between particles rather than to make them move faster.

Pressure

Pressure is the force acting per unit area:

$$P = \frac{F}{A}$$

$P$ is pressure in pascals ($\text{Pa}$), $F$ is force in newtons ($\text{N}$), and $A$ is area in $\text{m}^2$. One pascal is one newton per square metre. The same force gives a larger pressure when it acts over a smaller area — which is why a sharp knife or a drawing pin works.

Exam tip Watch your units. Areas given in $\text{cm}^2$ must be converted to $\text{m}^2$ for the answer to come out in pascals ($1\,\text{m}^2 = 10\,000\,\text{cm}^2$).

Pressure in liquids

A liquid presses on anything inside or below it. The deeper you go, the greater the weight of liquid above, so the pressure increases with depth:

$$P = \rho g h$$

Here $\rho$ is the density of the liquid ($\text{kg/m}^3$), $g$ is the gravitational field strength ($10\,\text{N/kg}$ on Earth), and $h$ is the depth below the surface ($\text{m}$). This gives the extra pressure caused by the liquid. Pressure at a point acts equally in all directions.

Real world Dam walls are built much thicker at the bottom than at the top, because the water pressure is greatest at the base. Deep-sea divers feel the same rise in pressure as they descend.

Gases: pressure, volume and temperature

The particles of a gas are in constant rapid motion. Each time a particle hits the wall of its container it pushes on it. The combined effect of billions of these collisions every second is the pressure the gas exerts. This is the kinetic theory explanation of gas pressure.

Two things raise the pressure:

Squeezing the gas into a smaller volume — the particles hit the walls more often, so pressure rises.
Heating the gas — the particles move faster, so they hit the walls harder and more often, so pressure rises.

Boyle's law

At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume. Halve the volume and you double the pressure:

$$P_1 V_1 = P_2 V_2$$

A graph of pressure against volume is a smooth curve that falls steeply then levels off (a hyperbola).

Worked example A gas has a volume of $300\,\text{cm}^3$ at a pressure of $100\,\text{kPa}$. It is compressed at constant temperature to $120\,\text{cm}^3$. Find the new pressure.

$P_1 V_1 = P_2 V_2$

$100 \times 300 = P_2 \times 120$

$P_2 = \dfrac{30\,000}{120} = 250\,\text{kPa}$.

Temperature and the kelvin scale

For gas calculations, temperature must be measured on the kelvin (absolute) scale. Absolute zero ($0\,\text{K}$, about $-273\,^\circ\text{C}$) is the lowest possible temperature — the point at which particles have the least possible energy. Convert using:

$$T(\text{K}) = \theta(^\circ\text{C}) + 273$$

For a fixed mass of gas at constant volume, the pressure is proportional to the kelvin temperature: heat it up and the pressure rises in proportion. At constant pressure, the volume is proportional to the kelvin temperature instead.

Watch out Never put degrees Celsius into a gas temperature ratio. Doubling from $20\,^\circ\text{C}$ to $40\,^\circ\text{C}$ does not double the pressure — but doubling from $300\,\text{K}$ to $600\,\text{K}$ does.

Specific heat capacity

The specific heat capacity $c$ is the energy needed to raise the temperature of $1\,\text{kg}$ of a substance by $1\,^\circ\text{C}$:

$$Q = mc\Delta\theta$$

$Q$ is the energy in joules, $m$ the mass in kg, $c$ the specific heat capacity in $\text{J/(kg}\,^\circ\text{C)}$, and $\Delta\theta$ the temperature change. Water has a high value ($4200\,\text{J/(kg}\,^\circ\text{C)}$), which is why it is used in heating systems and takes a long time to boil.

Specific latent heat

During a change of state the temperature does not change, but energy is still needed to break the bonds between particles. The specific latent heat $L$ is the energy needed to change the state of $1\,\text{kg}$ of a substance without a change in temperature:

$$Q = mL$$

There is a latent heat of fusion (melting/freezing) and a latent heat of vaporisation (boiling/condensing), each measured in $\text{J/kg}$.

Key terms Absolute zero — $0\,\text{K}$ ($\approx -273\,^\circ\text{C}$), the lowest possible temperature.

Specific heat capacity — energy to raise $1\,\text{kg}$ by $1\,^\circ\text{C}$.

Specific latent heat — energy to change the state of $1\,\text{kg}$ at constant temperature.