HookThe thermometer a physicist packed for his honeymoon
In the summer of 1847 the Manchester brewer and physicist James Prescott Joule got married, and — as Lord Kelvin loved to retell the story years later — he took a large, precise thermometer on his honeymoon to the Alps. His plan was to measure the water at the top of a waterfall near Chamonix and again at the bottom, expecting the lower pool to be a fraction of a degree warmer. His reasoning was radical for its time: as the water falls, its gravitational potential energy does not simply vanish, it becomes the random jiggling of the water's own particles — what we now call internal energy — and that shows up as a tiny rise in temperature.
Joule was right, and the number is small but real: a 50-metre fall warms the water by only about a tenth of a degree, because water is so hard to heat. That single idea — that heating, doing work and changing state are all just energy moving into or out of the store held by countless jiggling particles — runs through the whole of P3. Whether you are working out the density of a metal, why ice sits at 0°C while it melts, why steam scalds far worse than boiling water, or why a bicycle pump gets hot, you are tracking energy into and out of the kinetic and potential stores of particles. Learn to picture the particles, and the equations stop being abstract.
ModelDensity and the three states of matter
Density is how much mass is packed into a given volume: \(\rho = \dfrac{m}{V}\), mass divided by volume, usually in kg/m³ or g/cm³. It is a property of the material, not the object — a paperclip and a girder made of the same steel have the same density, because doubling the mass also doubles the volume.
Density is really a story about how tightly particles are arranged. In a solid, particles are packed closely in a regular pattern, held by strong forces, vibrating on the spot — so solids are dense, keep a fixed shape and are almost impossible to compress. In a liquid, particles are still close together but can slide past one another, so a liquid flows and takes the shape of its container while keeping a fixed volume; most liquids are only slightly less dense than the solid. In a gas, particles are far apart and move quickly in all directions, so gases are much less dense, fill any container and are easily squashed.
That spacing explains everyday density differences: steam is far less dense than water because the particles have spread out, and a hot-air balloon rises because heating the air inside makes it expand, lowering its density below the cooler air outside. When you compare states, always reach for the particle picture — spacing and arrangement, not the particles themselves changing.
A rectangular aluminium block measures 5.0 cm by 4.0 cm by 2.0 cm and has a mass of 216 g. Its volume is \(5.0 \times 4.0 \times 2.0 = 40\ \text{cm}^3\), so its density is \[\rho = \dfrac{m}{V} = \dfrac{216}{40} = 5.4\ \text{g/cm}^3.\] Wait — aluminium's density is 2.7 g/cm³, so this 'block' is twice too dense to be pure aluminium; the question is really testing whether you trust the number. Converting to SI: \(2.7\ \text{g/cm}^3 = 2700\ \text{kg/m}^3\), because there are \(10^6\) cm³ in a cubic metre and 1000 g in a kg (\(2.7 \times 1000000 \div 1000 = 2700\)). The unit conversion between g/cm³ and kg/m³ — multiply by 1000 — is the single most common slip in density questions.
DataRequired practical 5 — measuring density
Required practical 5 finds the density of solids and liquids, and the method changes with the shape. For a regular solid — a cube or cylinder — measure the mass on a balance and calculate the volume from its dimensions with a ruler or vernier callipers, then divide.
For an irregular solid, such as a stone, you cannot measure the volume with a ruler, so you use displacement. Lower the object into a measuring cylinder of water (or a displacement can) and read the rise in water level: the volume of water pushed aside equals the volume of the object. Weigh it dry first, then \(\rho = m/V\). For a liquid, weigh an empty measuring cylinder, pour in a known volume, reweigh, and the increase in mass divided by the volume gives the density.
The accuracy traps are worth learning as facts: dry the object before weighing (surface water fakes a higher mass), read the measuring cylinder at eye level to the bottom of the meniscus, and make sure the object is fully submerged with no air bubbles clinging to it, which would exaggerate the volume. Small, careful readings matter because you are dividing two measured quantities, so both errors carry through.
An irregular pebble has a mass of 78 g on the balance. Dropped into a measuring cylinder, it raises the water level from 50 cm³ to 80 cm³, so its volume is the displaced water: \(80 - 50 = 30\ \text{cm}^3\). Its density is \[\rho = \dfrac{m}{V} = \dfrac{78}{30} = 2.6\ \text{g/cm}^3.\] That is a touch above quartz sand's typical value, consistent with a small stone. Had an air bubble clung to the pebble, the level would have risen too far, the volume would read high, and the density would come out too low — which is exactly why you check for bubbles before taking the reading.
ModelInternal energy and changes of state
The internal energy of a system is the total of two things added over every particle: the kinetic energy of the particles (how fast they move or vibrate) and the potential energy stored in the forces between them (how far apart they are held). Heating a system transfers energy to this store, and the energy goes into one of those two pots.
When you heat a substance and its temperature rises, you are increasing the particles' kinetic energy — they move or vibrate faster. But when a substance changes state — melting, boiling, freezing, condensing — its temperature holds steady while you keep supplying energy. Here the energy is going into the potential store, doing the work of pulling particles apart (or letting them come together on cooling), not speeding them up. This is why a mixture of ice and water sits stubbornly at 0°C until the last ice has melted.
A change of state is a physical change, not a chemical one, and this matters for two reasons. First, it is reversible: freeze the melted water and you recover ice with all its original properties, because no new substance was made. Second, mass is conserved — the number of particles does not change when they merely rearrange, so the mass before a state change equals the mass after. Steam has the same mass as the water it came from; it is only spread out.
MechanismSpecific heat capacity — energy into the kinetic store
When heating raises a temperature, the amount of energy needed is set by the specific heat capacity \(c\): the energy to raise 1 kg of a material by 1°C. The equation is \(\Delta E = mc\Delta\theta\), where \(m\) is mass in kg, \(\Delta\theta\) the temperature change, and \(\Delta E\) the energy in joules.
Water's specific heat capacity is unusually large, about 4200 J/kg°C — far more than metals, whose particles are heavier and more tightly bound. A large \(c\) means a material soaks up a lot of energy for only a small temperature rise, which is precisely why Joule's waterfall warmed so little, why the sea moderates coastal climates, and why water is used as a coolant and in central-heating radiators.
The deep point is that specific heat capacity is really a statement about the kinetic store: pouring energy in raises the average kinetic energy of the particles, and temperature is our everyday measure of that average. The same equation you use to heat a kettle also predicts the faint warming at the foot of a waterfall — the physics does not care whether the energy arrived electrically or by falling.
Take Joule's idea literally. Water falls down Niagara Falls, a drop of about \(h = 50\) m. All the gravitational potential energy per kilogram, \(mgh\), becomes internal energy, \(mc\Delta\theta\). Setting them equal, the mass cancels: \[mgh = mc\Delta\theta \;\Rightarrow\; \Delta\theta = \dfrac{gh}{c} = \dfrac{9.8 \times 50}{4200} = 0.12.\] So the water at the bottom is only about 0.12°C warmer than at the top — a rise so tiny that Joule needed his best thermometer to catch it, and exactly the sort of number water's huge specific heat capacity produces. For comparison, a routine heating calculation: warming 2.0 kg of water by 30°C needs \(\Delta E = mc\Delta\theta = 2.0 \times 4200 \times 30 = 252000\) J.
MechanismSpecific latent heat — energy into the potential store
During a change of state the temperature does not budge, so \(\Delta E = mc\Delta\theta\) is useless — with \(\Delta\theta = 0\) it would predict zero energy, yet you clearly must keep heating to melt ice or boil water. The energy is measured instead by the specific latent heat \(L\): the energy needed to change the state of 1 kg of a substance with no change in temperature. The equation is \(E = mL\).
There are two versions. The specific latent heat of fusion is for melting or freezing (for water, about 334000 J/kg); the specific latent heat of vaporisation is for boiling or condensing (for water, a huge 2260000 J/kg — about 2.26 MJ/kg). Vaporisation takes far more energy than fusion because boiling has to pull the particles completely apart, whereas melting only loosens them.
This is why a heating graph — temperature against time as you warm a substance steadily — has flat plateaus. The slopes are the specific-heat-capacity stages, where temperature climbs; the horizontal plateaus are the state changes, where all the energy goes into latent heat at constant temperature. A cooling graph is the mirror image, with plateaus where energy is released as the substance freezes or condenses. Reading which parts of the graph belong to which equation is a guaranteed exam skill.
Why does steam at 100°C scald far worse than water at 100°C? Compare the energy each dumps into skin. Cooling 0.10 kg of boiling water by 20°C to body temperature releases only \(\Delta E = mc\Delta\theta = 0.10 \times 4200 \times 20 = 8400\) J. But 0.10 kg of steam must first condense before it can even start to cool, and condensing releases the latent heat of vaporisation: \[E = mL = 0.10 \times 2260000 = 226000\ \text{J}.\] That is about 27 times more energy, delivered straight to the skin before the water has cooled at all. The same latent-heat plateau that makes boiling slow on the hob makes steam dangerous on contact — one physics fact, two everyday consequences.
ModelParticle motion and gas pressure
In a gas the particles are far apart and in constant, random, high-speed motion. Two headline ideas follow. First, the temperature of a gas is a measure of the average kinetic energy of its particles: heat the gas and the particles move faster on average; cool it and they slow. Absolute zero is the temperature at which particle motion would be minimal.
Second, gas pressure comes from collisions. Each time a particle strikes the wall of its container it bounces off, exerting a tiny force; with astronomically many collisions every second, the total force spread over the wall's area is what we feel as pressure. More collisions, or harder ones, mean higher pressure.
Put the two together and the behaviour of gases becomes predictable. If you heat a gas held at constant volume, the particles move faster (higher average kinetic energy), so they hit the walls both harder and more often — and the pressure rises. This is why an aerosol can warns against heat, and why car tyre pressures read higher after a motorway run. The particle model turns 'pressure' from a mysterious push into simple bookkeeping of collisions.
ModelThe gas laws — squeezing and working on a gas (physics only)
Hold the temperature and mass of a gas fixed and change its volume, and pressure and volume trade off exactly: \(pV = \text{constant}\), so \(p_1V_1 = p_2V_2\). Halve the volume and you double the pressure. The particle reason is clean: squeeze the same number of particles into half the space and each one hits the walls twice as often, so the pressure doubles. Pressure is inversely proportional to volume at constant temperature.
There is a subtlety the higher-tier exam tests. When you compress a gas by pushing a piston, you are doing work on the gas — transferring energy to it mechanically. That energy goes into the gas's internal energy, and if the compression is quick enough that little heat escapes, the gas's temperature rises. This is why a bicycle pump grows warm as you inflate a tyre, and why a diesel engine can ignite fuel by compression alone, with no spark. Work done on a gas raises its internal energy; the gas laws describe the pressure-volume trade, and this rider explains the heating that comes with it.
A sealed syringe holds \(V_1 = 0.50\) m³ of gas at a pressure of \(p_1 = 100\) kPa. You slowly push the plunger — slowly, so the temperature stays constant — until the volume is \(V_2 = 0.20\) m³. The new pressure is \[p_2 = \dfrac{p_1 V_1}{V_2} = \dfrac{100 \times 0.50}{0.20} = 250\ \text{kPa}.\] The gas is now at two-and-a-half times the pressure, exactly as \(pV = \text{constant}\) demands (\(100 \times 0.50 = 250 \times 0.20 = 50\) in each case). Note the condition in bold: this law only holds at constant temperature. Push the plunger fast and you do work on the gas, its internal energy and temperature rise, and the simple pressure-volume relationship no longer tells the whole story.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
The first decision in any thermal question is which equation applies, and the test is one question: is the temperature changing, or the state? If temperature changes, use ΔE = mcΔθ (specific heat capacity). If the substance is melting, boiling, freezing or condensing at constant temperature, use E = mL (specific latent heat). Choosing the wrong one is the commonest error, and a plateau on a heating or cooling graph is always a latent-heat stage — the sloped parts are the specific-heat-capacity stages.
Density marks hinge on units and volume method. Quote ρ = m/V, keep mass and volume in matching units, and remember the ×1000 between g/cm³ and kg/m³. For an irregular solid, say 'volume equals the displaced water' and mention drying the object and checking for bubbles; for the practical generally, the interpretation is that a trapped bubble makes the measured volume too high and the density too low.
Particle-model explanations must talk about particles explicitly. For gas pressure, describe particles colliding with the walls; for a pressure rise on heating at constant volume, say the particles gain kinetic energy, move faster, and hit the walls harder and more often. On the higher tier, be ready to explain that doing work on a gas by compressing it raises its internal energy and temperature — the bicycle-pump effect — and state clearly that pV = constant only holds at constant temperature.