The particle model

A particle model treats matter as a collection of tiny interacting units, usually atoms, molecules or ions. It links what we observe on the macroscopic scale — shape, volume, density, temperature change — to what the particles are doing microscopically.

A macroscopic property is a measurable property of a whole sample, such as mass, volume, temperature or pressure. A microscopic property describes something at particle scale, such as particle spacing, particle speed or intermolecular force. In thermal physics, you spend a lot of time moving between these two descriptions.

Solids, liquids and gases

A solid is a state of matter in which particles stay close together and vibrate about fixed equilibrium positions, so the sample has a fixed shape and a fixed volume. The particles aren’t stationary — at any temperature above absolute zero they are moving — but most of that motion is vibration about positions in a structure.

A liquid is a state of matter in which particles stay close together but can move past neighbouring particles, so the sample has a fixed volume but takes the shape of its container. That is why liquids flow, while still being hard to compress compared with gases.

A gas is a state of matter in which particles are far apart and move freely between collisions, so the sample has no fixed shape and no fixed volume. A gas spreads out to fill its container and is much more compressible than a solid or a liquid.

The main physical differences come down to particle arrangement and particle motion. In solids, particles are close together and their motion is restricted; in liquids, particles are close together but can rearrange; in gases, particles are widely separated and move rapidly at random. The same molecular model comes up again in gas laws, electric conduction, nuclear binding ideas and many other parts of physics. The particles may change, but the modelling habit stays the same.

Density as mass per unit volume

Density is a scalar material property: it tells you how much mass is packed into each unit volume of a substance. That matters because two objects may have the same mass but take up very different volumes, or have the same volume with very different masses.

For a regular solid, volume usually comes from its measured dimensions. For an irregular solid, water displacement in a measuring cylinder is often the cleaner method. For liquids, find the mass by subtracting the mass of the empty container from the mass of the container plus liquid. Watch the units here: $\mathrm{cm^3}$ must be converted to $\mathrm{m^3}$ if the density is needed in SI units.

In the particle model, density depends on particle spacing and particle mass. Gases usually have much lower densities than solids and liquids because their particles are much farther apart. Solids and liquids often have similar densities since their particles are close together, even though they flow very differently.

Temperature scales

Temperature is a macroscopic property showing the thermal state of a body. It also sets the direction of net thermal energy transfer when bodies interact. In everyday laboratory work we use the Celsius scale; in thermal equations we usually need the Kelvin scale.

The Celsius scale measures temperature in degrees Celsius, written °C, and is commonly referenced to the freezing and boiling points of water at standard pressure. The Kelvin scale is the SI thermodynamic temperature scale with unit kelvin, written K, whose zero is absolute zero. Don’t put a degree sign with kelvin: write 300 K, not 300 °K.

The conversion is

More precisely, the offset is 273.15, but IB calculations normally use 273 unless told otherwise.

A temperature change is the difference between two temperatures. A change of 1 K has the same size as a change of 1 °C, so

So, if a sample is heated from 20 °C to 35 °C, the temperature change is 15 K, not 288 K. Add 273 for actual temperatures; do not add 273 for temperature differences.

A thermometer calibration is a procedure that assigns temperature values to a measurable physical property. For example, a thermocouple can be calibrated by keeping one junction in ice water as a reference and measuring the potential difference produced when the other junction is placed at known temperatures. Plotting potential difference against temperature gives a calibration graph; the quality of the thermometer depends on the range, scatter, resolution and repeatability of these measurements.

Temperature as a measure of particle motion

Kelvin temperature measures the average translational kinetic energy of particles in an ideal gas directly. Average matters here: at a given temperature, the particles don’t all move at the same speed, but temperature fixes their mean kinetic energy.

This equation gives one of the clearest links in the course between something we can measure on a large scale and a property of the particles themselves. If you measure the temperature, you can work out the particles’ average kinetic energy. It’s a neat example of using one physical quantity to determine another.

Watch the temperature scale. Doubling a Celsius temperature does not double the average kinetic energy; convert to kelvin first. A gas at 20 °C is at about 293 K; doubling the average kinetic energy would take it to about 586 K, which is about 313 °C.

At the same temperature, different gases have the same average translational kinetic energy per particle. So lighter particles have a greater average speed than heavier ones, because the same kinetic energy is associated with a smaller mass.

What internal energy includes

Internal energy is the energy stored within a system, made up of the random kinetic energies of its particles plus the intermolecular potential energies from forces between particles.

The random kinetic part comes from particles translating, rotating and vibrating. The potential part comes from interactions between neighbouring particles: attractions and repulsions change the stored energy when the particles move closer together or farther apart.

A system is a chosen collection of interacting objects or particles that we analyse as one unit. In this topic, that might be a block, a cup of water or a gas in a container. Physics uses the same idea in other places too: a planet and its moon can be a gravitational system; a gas in a cylinder can be a thermodynamic system; a nucleus can be a bound nuclear system.

In gases, the particles are usually far apart, so intermolecular potential energy is often small compared with random kinetic energy. In solids and liquids, the particles sit close enough together for intermolecular potential energy to matter strongly, especially during phase changes.

One subtle point: potential energy in bound systems is often negative relative to a separated-particle reference level. When energy is supplied to a solid or liquid, particles can move farther apart, so the potential energy increases towards the value for separated particles. The same style of thinking appears later in gravitational and nuclear binding energy.

Hot to cold, until equilibrium

Thermal energy transfer

Thermal energy transfer is energy transfer caused by a temperature difference between a system and its surroundings or between parts of a system. The transfer that results goes from the higher-temperature body or region to the lower-temperature body or region.

Thermal equilibrium

Thermal equilibrium is a state in which bodies in thermal contact have the same temperature and there is no resultant thermal energy transfer between them. Energy can still be exchanged microscopically, but the transfers in both directions balance.

Put a hot metal block in cooler water and the block cools while the water warms. The metal doesn’t “contain heat” as a substance; energy is transferred because there is a temperature difference. When both reach the same temperature, the resultant transfer stops.

The word “resultant” matters here. All bodies above absolute zero emit and absorb radiation. A hot object in a cooler room emits radiation to the room and absorbs radiation from the room, but the larger transfer is from hot to cold.

Phase changes and particle behaviour

A phase change is a change from one state of matter to another caused by energy transfer that changes particle arrangement or freedom of motion, rather than changing temperature.

During a phase change, energy transfers while the temperature stays constant. It can feel strange at first. The particle model explains it: the transferred energy changes intermolecular potential energy, not the average random kinetic energy. Temperature is linked to average kinetic energy, so the temperature does not change during the phase change.

Melting is a phase change in which a solid becomes a liquid. Freezing is a phase change in which a liquid becomes a solid. Boiling is a phase change in which a liquid becomes a gas throughout the liquid at its boiling temperature. Condensing is a phase change in which a gas becomes a liquid. Evaporation is a phase change in which molecules leave the surface of a liquid to become gas, and it can occur below the boiling temperature.

In power generation, water’s phase change to steam is especially useful. Energy transferred to water first raises its temperature and then vaporizes it; the expanding steam can drive a turbine connected to a generator. The physics is not “steam contains magic energy” — it is controlled thermal energy transfer, phase change and mechanical work in one engineered system.

Specific heat capacity

Specific heat capacity is a material property: the energy needed to raise the temperature of 1 kg of a substance by 1 K, with no change of phase.

If $c$ is large, the material takes in a lot of energy for only a small rise in temperature. Water has a high specific heat capacity, so oceans moderate coastal climates and water-rich organisms don't change temperature quickly.

To determine this for a metal block in a practical, an electric heater transfers energy to a known mass, then the temperature rise is measured. Insulate the block, make sure the heater fits well, and read the final temperature only after giving the energy time to spread through the block. In a method of mixtures, a hot object is put into cooler water. The calculation uses energy lost by the hot object equalling energy gained by the cooler one, as long as losses to the surroundings are small.

Specific latent heat

Specific latent heat is a material property: the energy needed to change the phase of 1 kg of a substance at constant temperature.

The latent heat equation has no $\Delta T$ term because the temperature does not change. You need to recognise two named values: specific latent heat of fusion, for melting or freezing, and specific latent heat of vaporization, for boiling, evaporation or condensing.

Calorimetry calculations are mostly bookkeeping. Work out which parts of the system warm or cool with $Q=mc\Delta T$, which parts change phase with $Q=mL$, then use conservation of energy. In an insulated system: energy transferred from hotter parts = energy transferred to colder parts and phase changes.

To measure the latent heat of fusion of ice, dry ice at $^\circ\text{C}$ is added to warmer water in a light insulated cup. The warm water loses energy; the ice melts at $^\circ\text{C}$ first, then the melted ice warms to the final temperature. The final mass gives the mass of ice added. The main experimental worries are energy exchange with the cup and air, water sticking to the ice, and getting a reliable reading for the final equilibrium temperature.

Three routes for energy transfer

Thermal energy can be transferred in three main ways: conduction, convection and thermal radiation.

Conduction

is thermal energy transfer through a material by particle interactions, with no bulk motion of the material. It matters especially in solids.

Convection

is thermal energy transfer in which warmer or cooler parts of a fluid move in bulk because their densities differ. It happens in liquids and gases, not solids.

Thermal radiation

is thermal energy transfer by electromagnetic waves emitted by matter. No material medium is needed, so energy from the Sun can travel across the vacuum of space.

In practice, these mechanisms usually overlap. A saucepan on a hob conducts energy through its base, the water circulates by convection, and hot surfaces emit radiation. The useful physics is working out which mechanism dominates, then deciding how to reduce or increase it.

Microscopic picture of conduction

In a solid with one end hotter than the other, particles at the hotter end have greater average kinetic energy. They vibrate more strongly, passing energy to neighbouring particles through interparticle forces and collisions. Energy spreads from the hotter region to the cooler region.

Metals conduct heat especially well because they contain free electrons. These electrons gain energy in the hotter region, move through the metal, and transfer energy to ions elsewhere in the lattice. That helps explain why good electrical conductors are often good thermal conductors too, although the relationship is not universal.

Liquids and gases can also conduct by molecular collisions, but usually less effectively than solids. Their particles are less strongly linked and, in gases, farther apart. In fluids, convection often becomes the more important mechanism.

Thermal conductivity and temperature gradient

Thermal conductivity is a material property that tells us how easily energy conducts through a material in steady state. A steady state is a condition where the temperature at each point in the material stays constant with time, even while energy flows through it.

For a flat slab or uniform rod, the rate of energy transfer by conduction is

The minus sign shows that energy flows down the temperature gradient: from higher temperature to lower temperature.

For magnitudes, many calculations use

$\left|\frac{\Delta Q}{\Delta t}\right|=kA\left|\frac{\Delta T}{\Delta x}\right|$

The rate is larger for a material with a higher $k$, a larger cross-sectional area, or a steeper temperature gradient. For the same temperature difference, a thicker conducting path gives a smaller rate. That’s why a pan base should conduct well and be broad, while a handle should conduct poorly and often be narrow or long.

In experiments, good conductors need a long, thin shape so that a measurable temperature difference can develop. Insulation around the sides matters too, since the model assumes one-dimensional energy flow. Poor conductors can be tested as thin discs because even a small thickness can produce a useful temperature difference. In both cases, take readings only after steady state is reached, and measure dimensions such as diameter or thickness carefully because they enter directly into $A$ or $\Delta x$.

Natural convection

A fluid is any substance that can flow, which includes liquids and gases. Convection happens in fluids because heating changes their density.

Heat one region of a fluid and its particles, on average, move faster and usually spread farther apart. That region expands, becomes less dense, and feels an upward buoyant force from the cooler, denser fluid around it. Cooler fluid then moves in to take its place. As this rising and sinking repeats, it forms a convection current, a circulating flow of fluid driven by density differences.

Convection is why water in a saucepan eventually warms all the way through. It also explains sea breezes reversing between day and night, warm air rising above heated land, and motion in Earth’s mantle transporting energy and material over very long times. Forced convection follows the same idea, but an external flow helps it along, such as air blowing across a hot drink.

Convection cannot happen in a solid because the particles are not free to move in large-scale currents. A solid can conduct energy, but the solid itself does not circulate.

Radiation from surfaces

Surfaces emit thermal radiation because charged particles in matter accelerate during their random thermal motion. Since the radiation is electromagnetic, it can pass through a vacuum.

A black body is an idealized object that absorbs all incident electromagnetic radiation and emits the maximum possible thermal radiation for its temperature. Real matt black surfaces come closer to this than shiny pale surfaces; shiny surfaces usually reflect well and emit poorly.

For a black body, the total power radiated is given by the Stefan–Boltzmann law:

The fourth power has a huge effect. If the absolute temperature doubles, the power radiated per square metre increases by a factor of 16. That’s why temperature carries so much weight in astrophysics and solar-energy modelling: the Stefan–Boltzmann law lets us estimate the energy output of stars and the radiation available to planets or solar collectors.

When a body exchanges radiation with its surroundings, the net radiative power is the difference between the radiation it emits and the radiation it absorbs from the surroundings. In many school-level calculations, the body is treated as a black body and other mechanisms are ignored, though real rooms usually involve conduction and convection as well.

A simple practical comparison uses identical cans, one matt black and one shiny white, filled with hot water. A cooling curve of temperature against time usually shows the black can losing energy faster. The fair-test details matter: equal starting temperatures and volumes, similar surroundings, and positioning so one can does not radiate directly onto the other.

What a detector receives

Apparent brightness is the power an observer or detector receives per unit area from a distant luminous body. Astronomers often write this as $b$.

A star can be extremely luminous and still look dim if it is far away. The reverse can happen too: a less luminous star may look bright simply because it is close by. So apparent brightness is not a property of the star alone. It depends on the star’s luminosity and on its distance from the observer.

In observational astronomy, we often work back from the radiation we receive. If a star’s temperature is estimated from its spectrum, and its luminosity is inferred or compared with known stars, then its apparent brightness can be used to determine distance. It’s a good example of using one measured quantity to find another physical property.

Radiation spreading through space

When a body radiates uniformly into space, its total luminosity is spread across larger spherical surfaces as the distance increases. Since the surface area of a sphere grows with the square of distance, the power received per unit area follows an inverse-square relationship.

Double the distance, and the apparent brightness drops to one quarter of its original value. The same inverse-square pattern turns up in other parts of physics whenever something spreads evenly from a point source over spherical surfaces: sound intensity, light intensity, gamma radiation intensity, gravitational field strength and electric field strength from point-like sources all use the same geometrical idea.

In astronomy, luminosity may be given in watts or in solar luminosities. Solar luminosity is a comparison unit equal to the luminosity of the Sun, written $L_\odot$. If a star has luminosity $20L_\odot$, it radiates 20 times as much power as the Sun.

For a spherical black-body star of radius $R$, combining the Stefan–Boltzmann law with the surface area of a sphere gives $L=4\pi R^2\sigma T^4$, where $R$ is radius (m). So a star’s luminosity depends strongly on both its size and its surface temperature.

Shape of a black-body spectrum

An emission spectrum is a graph or distribution showing how the intensity of emitted radiation changes with wavelength. A black body gives a continuous spectrum, so it emits over a wide range of wavelengths rather than just one colour.

When the temperature of a black body rises, the spectrum changes in three ways: the whole curve gets higher, the total area under the curve increases, and the peak moves to a shorter wavelength. A cool object may emit mainly infrared radiation; a hotter object may glow red, then yellow-white as more visible wavelengths become significant.

Wien’s displacement law

Wien’s displacement law states that the peak wavelength of a black-body spectrum is inversely proportional to the absolute temperature of the body:

The unit m K means metre kelvin, not millikelvin.

We can use this law to estimate a star’s surface temperature from the wavelength where its spectrum peaks. It also explains why the Sun, with a surface temperature of about 5800 K, peaks in or near the visible region, while cooler stars peak at longer wavelengths and hotter stars peak at shorter wavelengths.

The history of black-body radiation is a lovely example of international science. Measurements of black-body spectra challenged classical physics; Wien, Rayleigh, Planck and Einstein each contributed pieces of the story, and early twentieth-century conferences brought physicists together to argue, test and refine these ideas. Modern physics grew because evidence, mathematics and collaboration forced the model of matter and radiation to improve.