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 large scale — shape, volume, density, temperature change — to what the particles are doing on the microscopic scale.

A macroscopic property is a measurable property of a whole sample, such as mass, volume, temperature or pressure. A microscopic property describes the particle scale, such as particle spacing, particle speed or intermolecular force. Much of thermal physics comes down to translating 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, giving the sample a fixed shape and a fixed volume. The particles are not stationary. At any temperature above absolute zero they are moving, though mainly by vibrating 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’s 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 expands to fill its container and is much more compressible than a solid or a liquid.

So the main physical differences are particle arrangement and particle motion. In solids, particles are close with restricted motion; in liquids, they are close but can rearrange; in gases, they are widely separated and move rapidly at random. This molecular model comes back in gas laws, electric conduction, nuclear binding ideas and many other parts of physics. The particles may change, but the modelling habit is 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’s why two objects can share the same mass but take up very different volumes, or have the same volume with very different masses.

[\rho=\frac{m}{V}]

where (\rho) is density (kg m(^{-3})), (m) is mass (kg) and (V) is volume (m(^3)).

For a regular solid, volume usually comes from measuring its dimensions. With an irregular solid, water displacement in a measuring cylinder is often the cleaner method. For liquids, measure the container plus liquid, then subtract the mass of the empty container. The common experimental slip is unit conversion: cm(^3) must be converted to m(^3) when 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 that shows the thermal state of a body. It also tells us the direction of net thermal energy transfer when bodies interact. In everyday laboratory work, we use the Celsius scale; in thermal equations, we usually use the Kelvin scale.

The Celsius scale uses the unit degree 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. Its unit is the kelvin, written K, and its zero is absolute zero. Don’t put a degree sign with kelvin: write 300 K, not 300 °K.

The conversion is

[T= heta+273]

where (T) is absolute temperature (K) and ( heta) is Celsius temperature (°C). 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

[\Delta T=\Delta heta]

where (\Delta T) is temperature change on the Kelvin scale (K) and (\Delta heta) is temperature change on the Celsius scale (°C or degree).

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; don’t add 273 for temperature differences.

A thermometer calibration is a procedure that assigns temperature values to a measurable physical property. For example, to calibrate a thermocouple, keep one junction in ice water as a reference and measure 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 directly measures the average translational kinetic energy of particles in an ideal gas. Average matters here: at the same temperature, the particles have a spread of speeds, but their mean kinetic energy is set by the temperature.

[\overline{E}_k=\frac{3}{2}k_B T]

where (\overline{E}_k) is the average translational kinetic energy of a particle (J), (k_B) is the Boltzmann constant (J K(^{-1})) and (T) is absolute temperature (K).

This equation gives one of the neatest links in the course between a macroscopic measurement and a microscopic property. Once you measure temperature, you can work out the average kinetic energy of the particles. It’s a good example of using one physical quantity to determine another.

Watch the 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 particles, 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 translational, rotational and vibrational motion of particles. The potential part depends on how neighbouring particles interact: attractions and repulsions make the stored energy change when particle separation changes.

A system is a chosen collection of interacting objects or particles that we treat as one unit. In this topic, that could be a block, a cup of water or a gas in a container. Physics uses the same idea elsewhere 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, particles usually sit far apart, so intermolecular potential energy is often small compared with random kinetic energy. In solids and liquids, the particles are close enough 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 comes back later for gravitational and nuclear binding energy.

Hot to cold, until equilibrium

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

Thermal equilibrium is the state in which bodies in thermal contact have the same temperature, with no resultant thermal energy transfer between them. Energy can still be exchanged microscopically, but the two-way transfers balance.

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

Notice the word “resultant”. 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 it, 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 instead of temperature.

During a phase change, energy is still being transferred, but the temperature stays constant. It can feel strange at first. At the particle level, though, the transferred energy changes intermolecular potential energy, not the average random kinetic energy. Since temperature is linked to average kinetic energy, 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, the phase change of water to steam is especially useful. Energy transferred to water first raises its temperature, 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.

[Q=mc\Delta T]

Here, (Q) is thermal energy transferred (J), (c) is specific heat capacity (J kg(^{-1}) K(^{-1})) and (\Delta T) is temperature change (K).

If (c) is large, the material needs a lot of energy for only a small temperature rise. Water has a high specific heat capacity, so oceans moderate coastal climates and water-rich organisms resist rapid temperature changes.

To determine this for a metal block, an electric heater supplies 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 energy time to spread through the block. In a method of mixtures, a hot object is placed into cooler water. The calculation uses energy lost by the hot object = 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.

[Q=mL]

where (L) is specific latent heat (J kg(^{-1})).

There is no (\Delta T) in the latent heat equation because the temperature does not change. The two named values you must recognise are specific latent heat of fusion, used for melting or freezing, and specific latent heat of vaporization, used for boiling, evaporation or condensing.

Calorimetry calculations are mostly bookkeeping. Use (Q=mc\Delta T) for parts of the system that warm or cool, use (Q=mL) for parts that change phase, then apply 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 0 °C is added to warmer water in a light insulated cup. The warm water loses energy. The ice melts at 0 °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 reading the final equilibrium temperature reliably.

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 movement of the material. It matters especially in solids.

Convection is thermal energy transfer when warmer or cooler regions of a fluid move in bulk because of density differences. It happens in liquids and gases, not solids.

Thermal radiation is thermal energy transfer by electromagnetic waves emitted by matter. It needs no material medium, so energy from the Sun can travel across the vacuum of space.

In real situations, these mechanisms often happen together. 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 and how to reduce or increase it.

Microscopic picture of conduction

In a solid where one end is hotter than the other, particles at the hot end have greater average kinetic energy. They vibrate harder, then pass 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 and move through the metal, transferring 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 conduction is usually less effective than in solids. The 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 tells us how easily a material conducts energy when conditions have reached steady state. A steady state is reached when the temperature at every point in the material stays constant with time, even while energy is still flowing through it.

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

[\frac{\Delta Q}{\Delta t}=-kA\frac{\Delta T}{\Delta x}]

where (\Delta t) is the time interval for the transfer (s), (k) is thermal conductivity (W m(^{-1}) K(^{-1})), (A) is area (m(^2)) chosen here as the cross-sectional area perpendicular to the energy flow, and (\Delta x) is distance in the direction of energy transfer (m). 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 larger (k), for a wider cross-sectional area, or for a steeper temperature gradient. For the same temperature difference, making the conducting path thicker reduces the rate. That is 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 are usually made long and thin so that a measurable temperature difference can build up. The sides also need insulation, 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, so the term covers liquids and gases. Convection happens in fluids because heating changes density.

Heat one region of a fluid and its particles move faster on average, usually spreading farther apart. That region expands, its density decreases, and the surrounding cooler, denser fluid gives it an upward buoyant effect. 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.

This is why water in a saucepan eventually warms throughout. It also helps explain why sea breezes reverse between day and night, why warm air rises above heated land, and why motion within Earth’s mantle can transport energy and material over very long times. Forced convection follows the same idea, but an external flow helps it along, such as 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 behaviour than shiny pale surfaces; shiny surfaces reflect well and do not emit as strongly.

For a black body, the total power radiated is found using the Stefan–Boltzmann law:

[L=\sigma AT^4]

where (L) is luminosity, the total power radiated by a body (W), and (\sigma) is the Stefan–Boltzmann constant (W m(^{-2}) K(^{-4})). In this equation, (A) is the emitting surface area.

The fourth power makes a huge difference. If the absolute temperature doubles, the power radiated per square metre increases by a factor of 16. That is why temperature matters so much in astrophysics and solar-energy modelling: the Stefan–Boltzmann law helps 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 equals what it emits minus what it absorbs from the surroundings. In many school-level calculations, the body is treated as a black body and other energy-transfer mechanisms are ignored. In real rooms, though, conduction and convection usually happen as well.

A straightforward practical comparison uses two identical cans filled with hot water: one matt black, one shiny white. A cooling curve of temperature against time usually shows the black can losing energy faster. To keep the test fair, use equal starting temperatures and volumes, similar surroundings, and positions where one can does not radiate directly onto the other.

What a detector receives

Apparent brightness is the power received per unit area by an observer or detector from a distant luminous body. Astronomers often write it as (b).

A star can be extremely luminous in itself and still look dim if it is a long way away. The reverse can happen too: a less luminous star may look bright simply because it is close. So apparent brightness is not only about the star. It depends on the star’s luminosity and on its distance from the observer.

In observational astronomy, we often start with the radiation that reaches the detector and work backwards. If a star’s temperature can be estimated from its spectrum, and its luminosity can be inferred or compared with known stars, its apparent brightness can then be used to determine distance. This is a useful 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 has to spread out over bigger and bigger spherical surfaces as distance increases. A sphere’s surface area grows with the square of the distance, so the power received per unit area follows an inverse-square relationship.

[b=\frac{L}{4\pi d^2}]

where (b) is apparent brightness (W m(^{-2})) and (d) is distance from the source to the observer (m).

Double the distance, and the apparent brightness falls to one quarter of its original value. The same inverse-square pattern turns up in other parts of physics whenever something spreads uniformly from a point source across 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 can 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), the Stefan–Boltzmann law together 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.