Kinetic theory and internal energy: particles in motion, and temperature as their average kinetic energy
By the end of this lesson you will be able to
- State the assumptions of the kinetic theory of gases and outline why each one is needed.
- Describe the arrangement and motion of particles in solids, liquids and gases, and explain the properties of each state with them.
- Explain Brownian motion and diffusion as evidence for moving particles.
- Relate temperature to the average kinetic energy of particles, convert between °C and K, and calculate how energies and speeds scale with kelvin temperature.
- Distinguish between temperature, heat and internal energy, and explain heating at particle level.
- Apply Q = mcΔT in familiar and unfamiliar contexts, with units carried through every step.
You met this before
- All matter is made of tiny particles (atoms, molecules or ions) that are always moving; in a solid they vibrate in fixed positions, in a liquid they slide past each other, in a gas they move freely and fast (Year 1).
- Temperature and heat are different: temperature is how hot something is, in °C or K; heat is energy transferred because of a temperature difference, in joules (Year 3).
- Kelvin temperature = Celsius temperature + 273. Absolute zero, 0 K or −273 °C, is the lowest temperature possible (Year 3).
- Energy needed to warm a substance: Q = m c ΔT, where c is the specific heat capacity in J/(kg °C) — 4 180 J/(kg °C) for water (Year 3).
- Kinetic energy of a moving object: Ek = ½mv² (Year 1; Unit 2 of this course).
Convert 100 °C and −40 °C to kelvin, and explain in one sentence why a temperature of −10 K is impossible.
Reveal
T = θ + 273
100 °C + 273 = 373 K
−40 °C + 273 = 233 K
A temperature of −10 K would mean the particles had less than the lowest possible energy they can have; 0 K is the floor of the scale, so no negative kelvin temperatures exist.
Revise it in full: Year 1 · Everyday materials · Lesson 1 — Solids, liquids and gases: the particle model · Year 3 · Physics · Unit 10 · Lesson 1 — Heat is energy, temperature is not: the fall of caloric · Year 3 · Physics · Unit 10 · Lesson 2 — Measuring temperature: thermometers, three scales and absolute zero
The puzzle
The nitrogen molecules in the air around you are moving at about 500 metres per second — faster than an airliner, faster than a pistol bullet. So why, when someone opens a bottle of perfume across the room, does it take a minute or more before you smell it?
In 1858 a Dutch scientist, Christophorus Buys Ballot, used exactly this argument to attack the new kinetic theory: if gas molecules really moved that fast, smells would cross a room in a fraction of a second.
Rudolf Clausius answered him within months, and his answer is one of the key ideas of this lesson: a molecule never gets far. It travels, on average, about 70 nanometres before hitting another molecule, and it collides several billion times every second, each time flying off in a new direction.
Its path is a random zig-zag that goes almost nowhere. Diffusion alone would take days to carry the perfume across a room; the draughts and convection currents of the air do it in a minute. Fast particles, slow spreading — both are true.
A sealed syringe of air is warmed from 20 °C to 40 °C. Predict: does the average speed of the molecules double? Does their average kinetic energy double? Write your prediction with a reason before opening.
Compare your answer
Neither doubles. The average kinetic energy is proportional to the kelvin temperature, not the Celsius one: 293 K rises to 313 K, a factor of 313 ÷ 293 = 1.07, so the energy rises by about 7 %.
Kinetic energy goes as speed squared, so the speed rises by √1.07 = 1.034, about 3 %. Doubling a Celsius temperature has no physical meaning — the zero of the Celsius scale is just the freezing point of water.
The science — structured notes
1.1 One model, three states
The kinetic model of matter says that everything is made of particles in constant, random motion, and that the state of a substance depends on a contest between two things: the forces holding the particles together, and the kinetic energy they have to break away.
In a solid the forces win: the particles sit close together in fixed positions (usually a regular lattice) and vibrate about those positions. In a liquid the particles are still touching, but have enough energy to slide past one another. In a gas the particles are far apart — about ten diameters on average at room conditions — and move fast in straight lines until they collide.
| Property | Solid | Liquid | Gas | Particle explanation |
|---|---|---|---|---|
| Shape | fixed | takes the container’s | fills the container | fixed positions / particles slide / particles fly freely |
| Volume | fixed | fixed | fills the container | strong forces hold solids and liquids together; in a gas the forces are negligible except in collisions |
| Compressibility | almost none | almost none | easily compressed | particles already touching / large empty spaces between gas particles |
| Density (water) | 917 kg/m³ (ice) | 1 000 kg/m³ | 0.6 kg/m³ (steam, 100 °C) | same particles, very different spacing |
Ice is slightly less dense than liquid water — an oddity caused by the open, hexagonal lattice that hydrogen bonds build in ice. Almost every other substance is densest as a solid. (Year 3, Lesson 5 explored what this does to lakes and pipes.)
“In a solid the particles are not moving.”
Reveal answer
False. The particles of a solid vibrate continuously about fixed positions; only their average position is fixed. The vibration grows as the solid is heated, and at absolute zero it reaches its minimum — even then, quantum physics says a small “zero-point” motion remains.
The density of liquid water is about 960 kg/m³ at 100 °C and that of steam at 100 °C is 0.60 kg/m³. Use these values to estimate how much farther apart the molecules are in steam, and explain why a gas is easy to compress while a liquid is not.
Reveal answer
Volume ratio = 960 ÷ 0.60
Volume ratio = 1 600 (the same mass takes up 1 600 times the space)
Spacing ratio = ∛1 600
Spacing ratio ≈ 12
So neighbouring molecules in steam are about 12 times farther apart than in the liquid. A gas is mostly empty space, so pushing the particles closer only reduces the empty space. In a liquid the particles already touch, and pushing them closer is resisted by strong repulsive forces — so liquids are almost incompressible (the principle behind hydraulic brakes, Unit 1).
1.2 The evidence: Brownian motion and diffusion
No microscope can show you an air molecule. The evidence that particles exist and move is indirect — and the best of it came from dust.
In 1827 the botanist Robert Brown watched tiny particles released from pollen grains, suspended in water, and saw them jiggling endlessly in random directions. He showed it was not life: particles of ground glass and rock did it too.
You can see the same thing in a smoke cell: fill a small glass cell with a little smoke (from a smouldering drinking straw, say), shine a lamp through it, and look through a microscope. The smoke particles appear as bright specks that jerk about at random, and never stop.
The explanation, made quantitative by Albert Einstein in 1905: each smoke speck is bombarded from all sides by air molecules, far too small to see, moving fast and at random. At any instant the hits on one side do not quite balance the hits on the other, so the speck is pushed a little one way, then another.
A small speck is moved more than a big one; a hotter gas makes it jiggle faster. Diffusion — the spreading of a smell, or of ink in still water — is the same random motion seen in a crowd of particles rather than one.
Fat droplets in milk, seen under a microscope at ×400, jiggle about in random directions. Explain this observation, and predict how it would change if the milk were warmed.
Reveal answer
The fat droplets are suspended in water whose molecules are moving randomly and fast, but are too small to see. The water molecules collide with each droplet from all sides; at any moment the collisions on one side outnumber or outweigh those on the other, so there is a small resultant force in a random direction and the droplet moves in a random, jerky path (Brownian motion).
If the milk is warmed, the water molecules move faster (greater average kinetic energy), the collisions are harder and more frequent, and the droplets jiggle more vigorously.
1.3 The kinetic theory of gases and its assumptions
To explain the behaviour of gases with numbers, physicists build an idealised model — an ideal gas — and state exactly what they assume. The eAssessment expects you to know these assumptions and why each is needed.
- A gas consists of a very large number of identical particles (molecules) in continuous, random motion.
- The volume of the molecules themselves is negligible compared with the volume of the container.
- There are no forces between molecules except during collisions, so between collisions each molecule moves in a straight line at constant speed.
- Collisions — with each other and with the walls — are perfectly elastic: no kinetic energy is lost in them.
- The duration of a collision is negligible compared with the time between collisions.
Each assumption earns its place. Without (4) a gas in a sealed box would slowly cool and its pressure would fall, which does not happen. Without (2) and (3) the simple gas laws of Lesson 2 would not hold exactly.
Real gases are close to ideal at ordinary temperatures and pressures. They depart from it when squeezed to very high pressures (the molecules’ own volume starts to matter) or cooled close to condensation (the attractive forces start to matter — which is exactly why they can condense).
“Gas molecules gradually slow down, because every time they hit the wall of the container they lose a little kinetic energy. That is why a gas needs to be heated to keep its pressure.” Identify the error and correct it.
Reveal answer
The kinetic theory assumes collisions are perfectly elastic: in a container at the same temperature as the gas, molecules bounce off the walls with no net loss of kinetic energy (on average the wall gives back as much as it takes).
A sealed gas at constant temperature keeps its pressure indefinitely with no heating. Heating is needed only to keep a gas hotter than its surroundings; then the energy flows out through the walls, which is a temperature difference at work, not a property of the collisions.
1.4 Temperature is average kinetic energy
The single most important idea of this unit: the temperature of a substance is a measure of the average kinetic energy of its particles. Hotter means faster on average, not bigger particles, not more particles.
The link is simple only on the kelvin scale. The average kinetic energy of the particles is directly proportional to the kelvin temperature. Double the kelvin temperature and you double the average kinetic energy; at 0 K it would be at its minimum.
T (in K) = θ (in °C) + 273
θ (in °C) = T (in K) − 273
(The exact offset is 273.15; 273 is accurate enough for almost every calculation.) A temperature difference is the same size in both scales: a rise of 10 °C is a rise of 10 K.
The molecules in a gas do not all have the same speed. They have a spread, as the graph shows: most are near the middle, a few are very slow and a few very fast. Heating shifts the whole spread to higher speeds. The fast “tail” of this distribution is what lets a puddle evaporate at 20 °C — Lesson 3 comes back to it.
A gas is heated from 27 °C to 327 °C. Calculate by what factor (1) the average kinetic energy of its molecules and (2) their average speed increase.
T₁ = 27 + 273 = 300 K
T₂ = 327 + 273 = 600 K
(1) Energy ratio = T₂ ÷ T₁
Energy ratio = 600 K ÷ 300 K
Energy ratio = 2.0
(2) Ek = ½mv², so v ∝ √Ek
Speed ratio = √2.0
Speed ratio = 1.4
Note the trap: the Celsius temperature rose by a factor of 327 ÷ 27, about 12, which means nothing physically.
At the same temperature, all gas molecules have the same average kinetic energy (it depends only on T). Show that hydrogen molecules (H₂, relative mass 2) move about 3.7 times faster on average than nitrogen molecules (N₂, relative mass 28) at the same temperature.
Same average Ek: ½mHvH² = ½mNvN²
vH² ÷ vN² = mN ÷ mH
vH ÷ vN = √(28 ÷ 2)
vH ÷ vN = √14
vH ÷ vN = 3.7
Light molecules are fast. In the thin, hot upper atmosphere, the fastest few hydrogen molecules in the tail of the distribution exceed the escape speed of 11.2 km/s and leave the Earth for good; over billions of years almost all free hydrogen and helium has gone, while nitrogen and oxygen stay. The Moon, with a far lower escape speed, has lost nearly its whole atmosphere.
Liquid nitrogen boils at −196 °C. Convert this to kelvin and calculate the ratio of the average kinetic energy of nitrogen molecules at room temperature (20 °C) to that at −196 °C.
Reveal answer
T(cold) = −196 + 273 = 77 K
T(room) = 20 + 273 = 293 K
Ratio = 293 K ÷ 77 K
Ratio = 3.8
The molecules at room temperature have, on average, 3.8 times the kinetic energy, and √3.8 ≈ 1.95, about twice the speed.
A student says: “If I heat a gas from 10 °C to 20 °C, I double the temperature, so the molecules have twice as much kinetic energy.” What is the real factor, and what would the Celsius temperature need to be to double the kinetic energy of a gas at 10 °C?
Reveal answer
Real factor = (20 + 273) ÷ (10 + 273)
Real factor = 293 ÷ 283
Real factor = 1.035, a rise of about 3.5 %
To double: T = 2 × 283 K = 566 K
θ = 566 − 273 = 293 °C
You must go all the way to 293 °C. Energy and speed ratios always use kelvin.
1.5 Internal energy: kinetic plus potential
The particles of a substance have two kinds of energy. They have kinetic energy because they move (vibrate, slide or fly). They have potential energy because of the forces between them — like masses joined by springs, stretching or squeezing those bonds stores energy.
The internal energy of a body is the total of the random kinetic and potential energies of all its particles. It is measured in joules.
Heating a body — transferring energy to it because it is colder than its surroundings — increases its internal energy in one of two ways:
- Its temperature rises: the energy goes mainly into the particles’ kinetic energy; they vibrate or move faster.
- It changes state: the energy goes into the particles’ potential energy as they are pulled apart; the temperature stays constant until the change is complete (Lesson 3).
| Quantity | What it is | Unit | Depends on |
|---|---|---|---|
| Temperature | a measure of the average kinetic energy per particle | K or °C | how fast the particles move, not how many there are |
| Internal energy | total kinetic + potential energy of all the particles | J | temperature, mass (number of particles), material and state |
| Heat (thermal energy transferred) | energy transferred because of a temperature difference | J | a process, not something a body “contains” |
A red-hot sewing needle at 800 °C and a warm bath at 40 °C. Which is at the higher temperature, and which has more internal energy? Explain at particle level.
Reveal answer
The needle is at the higher temperature: its particles have a much higher average kinetic energy. The bath has far more internal energy: it contains around 10²⁸ water molecules against about 10²¹ atoms in the needle, so although each has less energy on average, the total is enormously larger.
That is why a spark from a grinder at over 1 000 °C barely stings your skin, while a mug of 70 °C tea poured on it scalds: the spark has very little internal energy to transfer.
1.6 Heating at particle level: Q = mcΔT
When the temperature of a body changes without a change of state, the energy transferred is
Q = m c ΔT
with Q in J, m in kg, ΔT in °C (or K — the same size) and c the specific heat capacity in J/(kg °C): the energy needed to raise the temperature of 1 kg of the material by 1 °C. Rearranged:
c = Q ÷ (m ΔT)
ΔT = Q ÷ (m c)
At particle level, c tells you how much energy each kilogram must absorb to raise the average kinetic energy of its particles by a given amount.
Water’s value, 4 180 J/(kg °C), is unusually large. One reason is that a kilogram of water contains a great many light molecules, all of which need a share. Another is that part of the energy goes into stretching and breaking the hydrogen bonds between them rather than into speed.
| Material | c / J/(kg °C) | Energy to warm 1 kg by 10 °C / kJ |
|---|---|---|
| Water | 4 180 | 41.8 |
| Ice | 2 100 | 21.0 |
| Aluminium | 900 | 9.0 |
| Steel / iron | 450 | 4.5 |
| Copper | 385 | 3.85 |
| Lead | 130 | 1.3 |
A 2.2 kW kettle holds 1.5 kg of water at 20 °C. Calculate the energy needed to bring the water to 100 °C and the minimum time the kettle needs.
Q = m c ΔT
Q = 1.5 kg × 4 180 J/(kg °C) × (100 − 20) °C
Q = 1.5 × 4 180 × 80
Q = 501 600 J ≈ 5.0 × 10⁵ J
t = Q ÷ P
t = 501 600 J ÷ 2 200 W
t = 228 s ≈ 3.8 minutes
This is a minimum: some energy also warms the kettle itself and escapes to the air, so the real time is a little longer.
A 1 200 kg car travelling at 25 m/s brakes to a stop. Its four steel brake discs have a total mass of 32 kg (c = 450 J/(kg °C)). Assuming all the kinetic energy ends up as internal energy of the discs, calculate their temperature rise.
Ek = ½mv²
Ek = ½ × 1 200 kg × (25 m/s)²
Ek = 375 000 J
ΔT = Q ÷ (m c)
ΔT = 375 000 J ÷ (32 kg × 450 J/(kg °C))
ΔT = 375 000 ÷ 14 400
ΔT = 26 °C
One stop is harmless, but a long mountain descent with the brakes on repeats this again and again; the discs can pass 500 °C, and the brake fluid and pads start to fail (“brake fade”). That is why steep passes have runaway lanes and why drivers are told to use a low gear downhill.
A 0.80 kg aluminium saucepan (c = 900 J/(kg °C)) holds 0.50 kg of water. Both start at 22 °C. Calculate the total energy needed to bring the pan and water to 100 °C, and the fraction of it that goes into the pan.
Reveal answer
ΔT = 100 − 22 = 78 °C
Q(water) = 0.50 × 4 180 × 78
Q(water) = 163 020 J
Q(pan) = 0.80 × 900 × 78
Q(pan) = 56 160 J
Total = 163 020 + 56 160
Total = 219 180 J ≈ 2.2 × 10⁵ J
Fraction for the pan = 56 160 ÷ 219 180
Fraction = 0.26, about a quarter
A heavy pan “steals” a quarter of the energy — one reason a kettle, whose element sits in the water, is more efficient than a pan on a stove.
The average kinetic energy of a gas molecule is Ek = (3/2)kT, where k = 1.38 × 10⁻²³ J/K (the Boltzmann constant). Calculate the average kinetic energy of a helium atom at 300 K and a typical speed, given that the mass of a helium atom is 6.64 × 10⁻²⁷ kg.
Reveal answer
Ek = (3/2)kT
Ek = 1.5 × 1.38 × 10⁻²³ J/K × 300 K
Ek = 6.21 × 10⁻²¹ J
½mv² = Ek, so v = √(2Ek ÷ m)
v = √(2 × 6.21 × 10⁻²¹ J ÷ 6.64 × 10⁻²⁷ kg)
v = √(1.87 × 10⁶ m²/s²)
v ≈ 1.37 × 10³ m/s (about 1.4 km/s)
This is the “root-mean-square” speed. Every gas at 300 K has the same average kinetic energy, 6.21 × 10⁻²¹ J per molecule; helium, being light, has to move fast to have it.
Try it — heating a block: energy in, temperature up, and a number that belongs to the material
This lab is a 1 kg block (or 1 kg of water) with a 50 W immersion heater and a thermometer. You choose the heating time (the independent variable, 0–300 s), the material, and whether the block is lagged. The physics of this lesson is inside the numbers: every joule the heater supplies raises the internal energy of the block, and in a solid that means faster vibration of its particles — a higher temperature.
gradient = 16.5 °C ÷ 300 s
gradient = 0.055 °C/s
Since ΔT ÷ t = P ÷ (mc), the specific heat capacity is:
c = P ÷ (m × gradient)
c = 50 W ÷ (1.00 kg × 0.055 °C/s)
c ≈ 9.1 × 10² J/(kg °C)
This is within 1 % of the accepted 900 J/(kg °C).
Common mistakes to avoid
- Using Celsius in a ratio. Energy and speed ratios need kelvin: 20 °C to 40 °C is 293 K to 313 K, not “double”.
- “Particles in a solid do not move.” They vibrate about fixed positions.
- “Particles expand when heated.” The particles stay the same size; the spacing between them grows.
- “The hot object contains more heat.” Objects have internal energy; heat is energy transferred because of a temperature difference.
- Temperature and internal energy confused. Temperature is the average kinetic energy per particle; internal energy is the total kinetic + potential energy.
- Brownian motion “because the smoke is hot” or “alive”. It is caused by uneven collisions from invisible, fast-moving air or water molecules.
- Mass in grams in Q = mcΔT. With c in J/(kg °C), the mass must be in kg.
- Stopping at the macroscopic level. “The gas expands because it gets hotter” scores little; say what the particles do.
Study smart: how this is examined
Study routine. Today: write the five assumptions from memory and draw the three particle boxes, then explain Brownian motion aloud in two sentences. In two days: redo the kelvin-ratio and brake-disc EXAMPLES with the book closed, and the saucepan check. In a week: run the lab as a full investigation, get c from the gradient, explain the curve of the bare block at particle level, and take the quiz.
How we found out — from caloric fluid to jiggling pollen
For most of the eighteenth century, heat was thought to be a weightless fluid, caloric, that flowed from hot bodies to cold. The first blow came from cannon-boring: in 1798 Count Rumford (Benjamin Thompson) showed that a blunt borer could boil water for as long as the horses kept turning it — an apparently endless supply of “fluid” from a finite piece of metal.
Heat looked like motion, not a substance. In 1738 Daniel Bernoulli had already suggested in Hydrodynamica that the pressure of a gas is the drumming of countless tiny particles on the walls, but almost nobody followed him up for a century.
In the 1840s James Prescott Joule, a brewer’s son from Salford, near Manchester, let falling weights turn a paddle wheel in an insulated can of water and measured the tiny temperature rise. He found the same exchange rate every time — about 4.2 joules of work for each calorie of heating — whatever the method.
Heat was a form of energy, and the unit of energy now bears his name.
The kinetic theory was built in a burst. Rudolf Clausius published “The nature of the motion which we call heat” in 1857 and introduced the mean free path in 1858 to answer Buys Ballot’s perfume objection. James Clerk Maxwell worked out the spread of molecular speeds in 1860, and Ludwig Boltzmann extended it from the 1870s.
Yet many leading scientists, among them Ernst Mach and Wilhelm Ostwald, doubted that atoms were real rather than a convenient fiction. Boltzmann, who defended atoms fiercely, died in 1906, before the argument was settled.
It was settled by Brownian motion. Albert Einstein showed in 1905 how far, on average, a jiggling particle should wander in a given time, and that the answer depended on the number of molecules in a mole. From 1908 Jean Perrin tracked tiny resin spheres under a microscope and got Avogadro’s number, about 6 × 10²³, agreeing with completely different methods.
Ostwald accepted that atoms existed; Perrin received the 1926 Nobel Prize in Physics.
Why did Perrin’s measurements convince sceptics who had ignored the kinetic theory for fifty years?
Reveal answer
Einstein’s theory made a quantitative prediction: the mean-square distance a particle wanders depends on the temperature, the viscosity of the liquid, the particle’s size and the number of molecules in a mole. Perrin measured the wandering and calculated Avogadro’s number from it.
His value agreed with values from quite different experiments (on radioactivity, gas viscosity and the blue of the sky). Several independent routes converging on the same number is very hard to explain unless the molecules are real — that is what changed minds.
Where it is used now
- Cooking at altitude. In Gaborone, about 1 000 m above sea level, water boils at about 97 °C; in the Lesotho highlands, near 3 000 m, at about 90 °C. Food cooks by the kinetic energy of the water molecules, so beans and maize take longer — or go into a pressure cooker, which raises the boiling point (Lesson 3).
- Liquid nitrogen in agriculture. Cattle-breeding programmes across southern Africa store semen and embryos at 77 K (−196 °C) in liquid nitrogen flasks; at that temperature the molecular motion that drives decay is so slow that samples keep for decades.
- The vaccine cold chain. Vaccines are kept at 2–8 °C, some far colder, because the chemical reactions that spoil them depend on how many molecules collide with enough energy — a number that rises steeply with temperature.
- Thermometers of every kind. A digital probe, a car’s coolant sensor and an infrared forehead thermometer all turn the average kinetic energy of particles into a number, by expansion, electrical resistance or emitted radiation.
- Planetary atmospheres. Mars, small and cold, has kept a thin carbon dioxide atmosphere; the Moon lost almost all of its gases because its escape speed is only 2.4 km/s — the kinetic theory explains which worlds can keep air.
- Nanoparticle medicine and sensors. “Nanoparticle tracking analysis” measures the Brownian motion of particles in a liquid under a microscope and, with Einstein’s equation, gives their size — used to check vaccines, inks and drug-delivery particles.
Science now — and the scientist you need to be
What is happening in 2026
Since May 2019 the kelvin itself has been defined through kinetic theory: the Boltzmann constant, which links the energy of particles to temperature, was fixed at exactly 1.380 649 × 10⁻²³ J/K, and the kelvin now follows from it rather than from the melting of ice or a special cell of water.
Metrology laboratories measure temperature by “listening” to gas molecules — the speed of sound in a gas depends on the speed of its molecules.
At the cold extreme, physicists routinely cool clouds of atoms to billionths of a kelvin, where they form a Bose–Einstein condensate. NASA’s Cold Atom Laboratory on the International Space Station has made such condensates in orbit since 2018, where free fall lets them be studied for longer.
In industry and research, computers now simulate the motion of millions of atoms directly (“molecular dynamics”), and machine-learning models trained on quantum calculations are making these simulations much faster. They are used to design battery electrolytes, heat-resistant alloys and drugs.
Careers here run from materials science and cryogenics to battery engineering and food technology.
The mindshift
What you need to be able to do
Describe any thermal effect in three steps — what the particles are doing, how that changes, what we observe. Convert to kelvin before any ratio. Keep temperature (average), internal energy (total) and heat (transfer) apart in every sentence. And treat the kinetic theory as what it is: a model with stated assumptions that tells you when it will work and when it will not.
See an example
After 3 minutes the colour had spread through about half of the hot water but only a small cloud in the cold water. The water molecules in the hot glass move faster on average, so they collide with the dye particles harder and more often and scatter them more quickly; convection currents in the hot glass help too.
The assistant correctly said “no”, but first quoted a speed rise of 41 % (√2) before correcting itself to about 3 % using kelvin — a reminder to check every step, not just the conclusion.
Key terms
| Kinetic theory | The model in which matter is made of particles in constant random motion; its properties follow from the motion and the forces between particles. |
| Ideal gas | A model gas obeying all the kinetic-theory assumptions exactly: negligible molecular volume, no forces except in collisions, elastic collisions, negligible collision time. |
| Elastic collision | A collision in which the total kinetic energy of the colliding bodies is conserved. |
| Brownian motion | The random, jerky motion of small visible particles (smoke, pollen fragments) caused by uneven bombardment by invisible, fast-moving fluid molecules. |
| Diffusion | The net spreading of particles from where they are concentrated to where they are sparse, because of their random motion. |
| Temperature | A measure of the average kinetic energy of the particles of a substance; in kelvin it is directly proportional to that average. |
| Absolute zero | 0 K (−273 °C): the lowest possible temperature, at which the particles have their minimum possible energy. |
| Kelvin scale | T (K) = θ (°C) + 273; a change of 1 K equals a change of 1 °C. |
| Internal energy | The total random kinetic energy and potential energy of all the particles in a body, in J. |
| Heat | Energy transferred from a hotter to a colder region because of their temperature difference. |
| Specific heat capacity, c | The energy needed to raise the temperature of 1 kg of a substance by 1 °C; Q = mcΔT. |
Think like a scientist
Criterion A A delivery company keeps medicines cold with dry ice and, for a few special samples, liquid nitrogen (77 K).
A driver notices three things on a hot day in Francistown: a sealed, empty metal drinks can left on the dashboard bulges slightly at its ends; a bath of warm water back at the depot “stores more energy” than a small red-hot soldering tip; and, through a phone microscope clipped over a drop of cloudy water, tiny specks keep twitching about.
(a) State two assumptions of the kinetic theory of gases. [2]
(b) Use the kinetic theory to explain why the ends of the sealed can bulge as it heats up in the sun. [3]
(c) Calculate the ratio of the average kinetic energy of nitrogen molecules at 27 °C to that at 77 K, and state by what factor their average speed differs. [3]
(d) A bath holds 150 kg of water at 40 °C; a soldering tip of 0.50 g of steel (c = 450 J/(kg °C)) is at 380 °C. Calculate the energy each would release in cooling to 20 °C, and use your answer to explain the difference between temperature and internal energy. [4]
(e) Suggest the cause of the twitching specks, and predict one change the driver would see if the water were warmer. [3]
Model answer
(a) Any two [1 each]: gas molecules are in continuous random motion; the volume of the molecules is negligible compared with the container; there are no forces between molecules except during collisions; collisions are perfectly elastic; the time of a collision is negligible compared with the time between collisions.
(b) As the can heats up, the air molecules inside gain kinetic energy and move faster on average [1]. They hit the walls of the can more often each second and with greater force (a larger change of momentum per collision) [1]. So the pressure inside rises above the pressure outside, and the resultant outward force pushes the thin ends outwards [1].
(c)
27 °C = 27 + 273 = 300 K
Ratio = 300 K ÷ 77 K [1 for using kelvin]
Ratio = 3.9 [1]
Speed ratio = √3.9
Speed ratio ≈ 2.0 [1]
(d)
Bath: Q = 150 kg × 4 180 J/(kg °C) × 20 °C
Bath: Q = 1.25 × 10⁷ J [1]
Tip: Q = 0.000 50 kg × 450 J/(kg °C) × 360 °C
Tip: Q = 81 J [1]
The tip is at the higher temperature: its atoms have a higher average kinetic energy [1]. The bath has far more internal energy because it has vastly more particles — internal energy is the total kinetic and potential energy of all the particles, temperature only the average per particle [1].
(e) The water molecules, too small to see, are moving randomly and fast and collide with the specks unevenly from different sides, so each speck is pushed in a random direction (Brownian motion) [2]. In warmer water the molecules move faster, so the specks twitch more vigorously and wander farther in the same time [1].
Probe deeper — exam-style questions
Attempt every question before opening the answer. Marks in brackets show what a full answer is worth.
State two differences between the particles in a liquid and the particles in a gas.
Reveal answer
Any two [1 each]: in a liquid the particles are touching (close together), in a gas far apart; in a liquid they slide past one another, in a gas they move freely and fast in straight lines between collisions; the forces between particles are strong in a liquid and negligible (except in collisions) in a gas.
Explain, using the kinetic theory, how a gas exerts a pressure on the walls of its container.
Reveal answer
Gas molecules move randomly and continually collide with the walls [1]; at each collision the molecule’s momentum changes, so the wall exerts a force on it and, by Newton’s third law, it exerts an equal force on the wall [1]; the very large number of collisions per second gives a steady total force, and pressure is that force divided by the area of the wall [1].
Convert 25 °C and 4 K to the other scale, and state the lowest possible temperature in both scales.
Reveal answer
25 °C + 273 = 298 K
4 K − 273 = −269 °C [1]
Absolute zero: 0 K, which is −273 °C (−273.15 °C exactly) [1].
A mug holds 250 g of water at 20 °C. Calculate the energy needed to heat it to 80 °C and the time this takes in a microwave oven that delivers 800 W to the water.
Reveal answer
Q = m c ΔT
Q = 0.250 kg × 4 180 J/(kg °C) × 60 °C [1 for kg]
Q = 62 700 J [1]
t = Q ÷ P
t = 62 700 J ÷ 800 W
t = 78 s [1]
Pollen fragments suspended in water are seen to move in a random, jerky way. Suggest the cause of this motion.
Reveal answer
Water molecules (too small to be seen) are moving randomly and fast [1]; they collide with the pollen fragments unevenly from different sides, pushing them in random directions [1].
The temperature of a gas rises from 27 °C to 127 °C. Calculate the factor by which the average kinetic energy of its molecules increases, and the factor for their average speed.
Reveal answer
T₁ = 27 + 273 = 300 K
T₂ = 127 + 273 = 400 K [1]
Energy factor = 400 ÷ 300
Energy factor = 1.33 [1]
Speed factor = √1.33
Speed factor = 1.15 [1]
(127 ÷ 27 = 4.7 scores 0.)
7. The average kinetic energy of a molecule is (3/2)kT, with k = 1.38 × 10⁻²³ J/K. Calculate a typical speed of a nitrogen molecule (mass 4.65 × 10⁻²⁶ kg) at 300 K and of a hydrogen molecule (mass 3.35 × 10⁻²⁷ kg) at the same temperature. The escape speed from the Earth is 11.2 km/s: explain why hydrogen can nevertheless escape from the upper atmosphere over time while nitrogen does not.
Reveal answer
Ek = 1.5 × 1.38 × 10⁻²³ × 300
Ek = 6.21 × 10⁻²¹ J
v(N₂) = √(2 × 6.21 × 10⁻²¹ ÷ 4.65 × 10⁻²⁶)
v(N₂) = 517 m/s [1]
v(H₂) = √(2 × 6.21 × 10⁻²¹ ÷ 3.35 × 10⁻²⁷)
v(H₂) = 1.93 × 10³ m/s [1]
Both are well below 11.2 km/s, but these are averages: molecules have a spread of speeds, and the upper atmosphere is much hotter than 300 K (hundreds to over a thousand kelvin), which raises the whole distribution [1]. A small but steady fraction of the light, fast hydrogen molecules in the tail exceed the escape speed and leave; for nitrogen, about 3.7 times slower, the fraction is negligibly small even over billions of years [1].
8. A 5.0 g lead bullet travelling at 300 m/s embeds itself in a wooden block. Assume 60 % of its kinetic energy becomes internal energy of the bullet (c for lead = 130 J/(kg °C); melting point 327 °C). Starting at 20 °C, does the bullet melt? Explain what happens to its atoms.
Reveal answer
Ek = ½ × 0.0050 kg × (300 m/s)²
Ek = 225 J [1]
Q = 0.60 × 225 J = 135 J
ΔT = Q ÷ (m c)
ΔT = 135 J ÷ (0.0050 kg × 130 J/(kg °C))
ΔT = 208 °C [1]
Final temperature = 20 + 208 = 228 °C
228 °C is below 327 °C, so it does not melt [1]. The ordered kinetic energy of the whole bullet becomes random kinetic energy of its atoms: they vibrate much harder about their lattice positions, but not hard enough to break out of the lattice [1].
Summary
- Matter is particles in constant random motion: solids vibrate in fixed positions, liquids slide past each other while touching, gases fly far apart in straight lines between collisions.
- Brownian motion (smoke, pollen fragments) and diffusion are evidence for invisible, fast, randomly moving molecules; Einstein and Perrin made it quantitative.
- Kinetic-theory assumptions: random motion; negligible molecular volume; no forces except in collisions; elastic collisions; negligible collision time.
- Temperature is a measure of the average kinetic energy of the particles, proportional to the kelvin temperature; T(K) = θ(°C) + 273; ratios need kelvin.
- Internal energy is the total kinetic + potential energy of the particles; heating raises the kinetic part (temperature rise) or the potential part (change of state).
- Q = mcΔT, with m in kg; c is the energy per kg per °C.
Next time → The gas laws from the physics side: pressure from the change of momentum of molecules hitting a wall, Boyle and Charles laws and the pressure law explained molecule by molecule, and the road to absolute zero.