Density — mass packed into space
One equation you must recall, a clear sense of what density actually means, and the required practical for finding it — including awkward shapes.
Part 1What density tells you
The density of a material is how much mass is packed into each unit of volume. A dense material has a lot of mass squeezed into a small space; a low-density material is "lighter for its size".
Density depends on two things: how heavy the individual particles are, and how closely packed they are. That's why the same material is usually densest as a solid and least dense as a gas — the particles spread far apart in a gas.
Equation
- ρ = m / V recall
- density (kg/m³) = mass (kg) ÷ volume (m³). The symbol ρ is the Greek letter "rho". Density can also be given in g/cm³.
Worked example — density of a block
A metal block has a mass of 540 g and a volume of 200 cm³. Calculate its density.
⚠ Watch out — keep your units consistent
Don't mix grams with cubic metres. If mass is in g and volume in cm³, density comes out in g/cm³. If mass is in kg and volume in m³, you get kg/m³. Water is 1 g/cm³ (= 1000 kg/m³) — a handy check. Anything denser than water sinks in it.
A liquid has a mass of 80 g and a volume of 100 cm³. What is its density, and will it float on water?
- A0.8 g/cm³ — it floats on water
- B1.25 g/cm³ — it sinks in water
- C0.8 g/cm³ — it sinks in water
- D8 g/cm³ — it floats on water
Show answer
Part 2Measuring density
To find a density you always need two measurements: the mass (from a balance) and the volume. Getting the volume is the tricky part — and it depends on the shape.
Determining the density of solids and liquids
Aim: measure the density of a regular solid, an irregular solid and a liquid.
- Regular solid (e.g. a cube): find the mass on a balance, then measure the sides with a ruler and calculate the volume (e.g. length × width × height).
- Irregular solid: find the mass, then use a displacement (eureka) can — lower the object in and collect the water that overflows in a measuring cylinder. That volume of water equals the object's volume.
- Liquid: place an empty measuring cylinder on the balance and zero it. Pour in a known volume of the liquid and read its mass.
- For each, calculate: ρ = m ÷ V
Control / improve: dry the irregular solid before weighing, and read the measuring cylinder at eye level (bottom of the meniscus) to avoid a parallax error. Trapped air bubbles on the object make the volume read too high.
A stone is lowered into a eureka can; 30 cm³ of water overflows. The stone's mass is 90 g. Its density is:
- A0.33 g/cm³
- B3 g/cm³
- C2700 g/cm³
- D120 g/cm³
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Write the density equation and give the SI unit of each quantity.
ρ = m ÷ V. Density in kg/m³, mass in kg, volume in m³ (g/cm³ is also accepted).How do you find the volume of an irregular solid?
By displacement: lower it into a eureka can and measure the volume of water that overflows. That volume equals the object's volume.A 250 cm³ sample of oil has a mass of 230 g. Calculate its density.
ρ = m ÷ V = 230 ÷ 250 = 0.92 g/cm³. (Less than water, so the oil floats.)Why is the same material usually denser as a solid than as a gas?
In a solid the particles are packed closely together; in a gas they're far apart, so the same mass fills a much larger volume — lower density.
States of matter & changes of state
How particles are arranged in solids, liquids and gases — and why melting or boiling is a physical change where mass is conserved.
Part 1The three states
The particle model pictures matter as tiny particles. What's different between the three states is how those particles are arranged, how close they are, and how much they move.
In a solid, particles are packed in a regular pattern, touching, and only vibrate in fixed positions — so a solid keeps its shape and volume. In a liquid, particles are still close and touching but are randomly arranged and can move past each other — so a liquid flows and takes the shape of its container. In a gas, particles are far apart and move quickly in all directions — so a gas spreads to fill any space.
Naming the changes of state
- Melting / Freezing
- Solid ⇌ liquid.
- Boiling (evaporating) / Condensing
- Liquid ⇌ gas.
- Sublimation
- Solid straight to gas (and back) without passing through liquid.
⚠ Watch out — a change of state is physical, not chemical
Melting, boiling and condensing are physical changes. The substance is the same material afterwards — only the arrangement of particles has changed, so the change can be reversed. This is different from a chemical change, which makes a new substance.
Part 2Mass is conserved
When a substance changes state, no particles are created or destroyed — they just move apart or together. So the mass stays the same. If 18 g of ice melts, you get 18 g of water; boil it and you get 18 g of steam.
Because the particles are the same, the change can be reversed and the material recovers its original properties. The density usually changes (the particles spread out), but the mass does not.
Worked example — mass through a change of state
A sealed flask contains 250 g of water. All of it is boiled into steam, which stays trapped in the flask. What is the mass of steam?
Why is melting described as a physical change rather than a chemical change?
- ABecause energy is transferred to the substance
- BBecause a new substance is made when it melts
- CBecause the same substance remains and the change can be reversed
- DBecause the mass increases as it melts
Show answer
Describe the arrangement and movement of particles in a solid.
Packed closely in a regular pattern, touching, and vibrating about fixed positions. The solid keeps a fixed shape and volume.Name the change of state from gas to liquid, and from solid to liquid.
Gas → liquid is condensing; solid → liquid is melting.What happens to the mass of a substance when it changes state? Explain why.
It stays the same (mass is conserved). No particles are created or destroyed — they only rearrange.Why can a change of state be reversed?
It's a physical change — the substance is unchanged, so cooling or heating it back returns it to its original state with the same properties.
Internal energy & specific heat capacity
What "internal energy" really means, and the given equation that links energy, mass and temperature change.
Part 1Energy stored by particles
The internal energy of a system is the total energy stored by its particles. It's made of two parts: the kinetic energy of the particles (their movement) and the potential energy from how they're arranged (the spacing and bonds between them).
Heat a substance and you transfer energy to its particles. That energy does one of two things. Either it raises the temperature — the particles move faster, so their kinetic energy rises. Or, at a change of state, it goes into breaking the forces between particles — that's potential energy, and the temperature stays the same (you'll meet that in Topic 4).
⚠ Watch out — heat, temperature and energy aren't the same word
Temperature is a measure of the average kinetic energy of the particles. Internal energy is the total energy of all of them — so a big bath of warm water has more internal energy than a small cup of boiling water, even though the cup is hotter. Don't write "heat energy"; write "internal energy" or "energy transferred by heating".
Part 2Specific heat capacity
The specific heat capacity of a substance is the energy needed to raise the temperature of 1 kg of it by 1 °C. Water's is high (about 4200 J/kg °C), which is why it takes so long to heat up and why it's good for carrying energy around in heating systems.
Equation
- ΔE = m c Δθ given
- change in thermal energy (J) = mass (kg) × specific heat capacity (J/kg °C) × temperature change (°C). The symbol Δ (delta) means "change in".
Worked example — heating water
How much energy is needed to raise the temperature of 3 kg of water from 18 °C to 38 °C? (c = 4200 J/kg °C)
⚠ Watch out — Δθ is the change, not the final value
Δθ means the change in temperature, so subtract the start from the finish first. The unit J/kg °C is a recipe for the equation: it's joules per (kg × °C), so you multiply mass by c by Δθ.
2 kg of a metal needs 16 000 J to warm up by 20 °C. What is its specific heat capacity?
- A160 J/kg °C
- B400 J/kg °C
- C640 000 J/kg °C
- D800 J/kg °C
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What is meant by the internal energy of a system?
The total energy stored by its particles — the sum of their kinetic energy (movement) and potential energy (spacing/arrangement).Define specific heat capacity.
The energy needed to raise the temperature of 1 kg of a substance by 1 °C (unit J/kg °C).Calculate the energy to heat 0.5 kg of aluminium (c = 900 J/kg °C) by 30 °C.
ΔE = m c Δθ = 0.5 × 900 × 30 = 13 500 J.A small cup of boiling water and a large bath of warm water — which has the greater internal energy? Explain.
The bath. Internal energy is the total for all particles; the bath has far more particles, so even at a lower temperature its total energy is greater.
Specific latent heat
Why a melting ice cube stays at 0 °C even as you heat it — energy that changes state without changing temperature.
Part 1Energy with no temperature change
When a substance is melting or boiling, you keep transferring energy to it — but the temperature does not rise. All the energy goes into breaking the forces between particles (their potential energy), not into making them move faster. This energy is called latent heat ("latent" means hidden).
The specific latent heat of a substance is the energy needed to change the state of 1 kg of it with no change in temperature. There are two values: the specific latent heat of fusion (for melting/freezing) and the specific latent heat of vaporisation (for boiling/condensing).
Equation
- E = m L given
- energy for a change of state (J) = mass (kg) × specific latent heat (J/kg). Use L for fusion when melting/freezing, and L for vaporisation when boiling/condensing.
⚠ Watch out — latent heat vs specific heat capacity
Use ΔE = m c Δθ only when the temperature is changing. Use E = m L only during a change of state, where the temperature is constant. Mixing them up is a classic error — check whether the temperature is moving or holding still.
Part 2Using E = m L
Worked example — melting ice
How much energy is needed to melt 0.40 kg of ice at 0 °C? (specific latent heat of fusion of water = 334 000 J/kg)
A pan of water is boiling steadily at 100 °C while a hob keeps heating it. What happens to the temperature of the water?
- AIt keeps rising above 100 °C
- BIt stays at 100 °C until all the water has boiled away
- CIt falls because energy is being lost
- DIt rises slowly as steam forms
Show answer
Define the specific latent heat of a substance.
The energy needed to change the state of 1 kg of the substance with no change in temperature (unit J/kg).Why does the temperature stay constant while a substance melts?
The energy supplied goes into breaking the forces between particles (raising their potential energy), not into speeding them up — so the temperature doesn't change.Calculate the energy to boil away 0.20 kg of water at 100 °C. (L of vaporisation = 2 260 000 J/kg)
E = m L = 0.20 × 2 260 000 = 452 000 J.Which equation do you use when the temperature is changing, and which when it is constant during a change of state?
Changing temperature: ΔE = m c Δθ. Constant temperature (change of state): E = m L.
Particle motion in gases
How fast-moving particles create gas pressure, and why heating a gas raises its temperature and its pressure.
Part 1Temperature and particle energy
The particles in a gas are spread out and move quickly in random directions. The temperature of the gas is linked to the average kinetic energy of those particles: the higher the temperature, the faster the particles move on average, so the more kinetic energy they have.
Gas pressure comes from particles colliding with the walls of the container. Each collision pushes on the wall. The pressure is the total effect of all those collisions over the wall's area.
⚠ Watch out — pressure is collisions, not "air pushing"
Gas pressure is caused by particles colliding with the container walls, not by the gas "pressing" as a lump. To explain a pressure change, talk about how often particles hit the walls and how hard (how fast) they hit. The force from each collision acts at right angles (90°) to the wall.
Part 2Heating a gas at constant volume
Heat a gas in a sealed, rigid container (so the volume can't change) and you transfer energy to the particles. They speed up — their average kinetic energy rises, which is the same as saying the temperature rises.
Faster particles hit the walls more often and harder. More frequent, harder collisions mean a greater pressure. So at constant volume, increasing the temperature increases the pressure.
Worked example — explaining a pressure rise
A gas is sealed in a fixed-volume can and warmed. Explain, in particle terms, why the pressure rises.
A fixed amount of gas in a sealed rigid container is cooled down. What happens to the pressure, and why?
- APressure rises — particles slow down and bunch up
- BPressure falls — particles move slower, so collisions are less frequent and less forceful
- CPressure stays the same — the number of particles is unchanged
- DPressure falls — particles are destroyed by the cooling
Show answer
What causes the pressure of a gas on its container?
The collisions of gas particles with the walls of the container. Each collision exerts a small force; together they create the pressure.How is the temperature of a gas related to its particles?
Temperature is linked to the average kinetic energy of the particles — higher temperature means the particles move faster on average.A sealed rigid container of gas is heated. Explain why the pressure increases.
The particles gain kinetic energy and move faster, so they hit the walls more often and harder — increasing the pressure.In which direction does the force from a particle act when it hits a wall?
At right angles (90°) to the surface of the wall.Why does the pressure fall when a fixed mass of gas at constant volume is cooled?
Cooling lowers the average kinetic energy, so particles move slower and collide with the walls less often and with less force — lower pressure.