Every day you feel thermal energy on the move: a metal spoon warming in soup, warm air rising from a radiator, sunlight warming your skin across empty space. This workbook builds the particle model of matter and temperature from the ground up, then uses it to explain the three ways thermal energy travels — conduction, convection and radiation — finishing with the physics of how hot objects glow. The next two workbooks in this set, Heating and Energy Changes with Heating, build directly on the ideas developed here.
By the end of this workbook you should be able to:
Describe the arrangement, separation and motion of particles in solids, liquids and gases, and use density ρ = m/V.
Convert between Kelvin and Celsius temperatures, and explain why temperature difference determines the direction of thermal energy transfer.
Explain internal energy in terms of particle kinetic and potential energy, and describe a phase change as a change in particle behaviour at constant temperature.
State the three mechanisms of thermal energy transfer: conduction, convection and thermal radiation.
Explain conduction in terms of the kinetic energy of particles, and use ΔQ/Δt = kAΔT/Δx to calculate the rate of thermal energy transfer.
Describe convection qualitatively in terms of changes in fluid density.
Describe thermal radiation and black bodies, and apply the Stefan–Boltzmann law L = σAT⁴.
Apply Wien's displacement law λmaxT = 2.9 × 10⁻³ m·K to relate a black body's peak wavelength to its temperature.
1. Particles in solids, liquids and gases
Almost everything you will learn in this workbook rests on one idea: all matter is made of a huge number of tiny particles (atoms, ions or molecules) which are constantly moving, with forces acting between them when they get close enough. This is the kinetic theory of matter, and it explains why solids, liquids and gases behave so differently.
Kinetic theory of matter. All matter is made up of a very large number of small particles which are in constant motion, with forces acting between them.
Fig. 1.1 A particle model of the three states of matter, drawn with particles of the same size and colour throughout. Solid: a regular, touching arrangement, settled at the bottom of its container, vibrating in place. Liquid: a disordered but still-touching arrangement that flows to fill the bottom of its container (the dashed line marks its surface). Gas: widely separated particles filling the whole container, moving randomly at high speed.
Solid
Liquid
Gas
Arrangement
regular pattern
no fixed pattern
no pattern
Forces between particles
strong enough to hold particles in place
strong enough to keep particles close, but they can move past each other
negligible, except during collisions
Separation of particles
close together
close together
much further apart
Motion of particles
vibrate about fixed positions
limited random movement is possible
move randomly in all directions, usually at high speed
Density
The more massive the individual particles are, and/or the closer together they are packed, the greater the density of a substance.
density, ρ = m / V SI unit: kg m⁻³
Gases are typically around 1000 times less dense than the same substance as a solid or liquid, because their particles are so much further apart.
Worth remembering
Because density depends on the volume between particles as well as their mass, a "1000× less dense" gas does not mean the gas particles are 1000× further apart — density scales with the cube of the typical particle separation, so a 1000× drop in density corresponds to particles being roughly 10001/3 ≈ 10× further apart on average.
Worked example 1.1
A sample of olive oil has a mass of 125 g and a volume of 137 cm³. Calculate its density in SI units.
Answer:
Convert to SI units: m = 125 g = 0.125 kg; V = 137 cm³ = 137 × 10⁻⁶ m³
ρ = m/V = 0.125 / (137 × 10⁻⁶)
Check your understanding
1An aluminium bar has dimensions 4.0 cm × 4.0 cm × 20.0 cm. The density of aluminium is 2710 kg m⁻³. Calculate the mass of the bar. (Convert all lengths to metres first.)
V = 0.040 × 0.040 × 0.200 = 3.2 × 10⁻⁴ m³. m = ρV = 2710 × (3.2 × 10⁻⁴) — complete the multiplication to find the mass in kg.
2Ice floats on liquid water. Using the idea of density and particle separation, explain what this tells us about how the particles are arranged in ice compared with liquid water.
Since ice floats, ice must be less dense than liquid water. As the mass of the particles is unchanged when water freezes, this means the particles in ice must, on average, be slightly further apart (occupy a slightly greater volume) than in liquid water — an unusual property of water compared with most other substances.
3Gold can be hammered into extremely thin sheets called gold leaf. A sample of gold leaf has a mass of 1.0 g, a thickness of 1.8 × 10⁻⁷ m, and the density of gold is 19300 kg m⁻³. Determine the area of gold leaf this sample could cover. (Volume = area × thickness.)
m = 1.0 g = 1.0 × 10⁻³ kg. V = m/ρ = (1.0 × 10⁻³)/19300. Then, since V = area × thickness, area = V / (1.8 × 10⁻⁷) — complete both divisions to find the area in m².
4Explain, in terms of the kinetic theory of matter, why gases are much easier to compress than solids or liquids.
In a gas, the particles are far apart with negligible forces between them, so there is a large amount of empty space between particles that can be reduced by squeezing the gas into a smaller volume. In solids and liquids, the particles are already close together (effectively "touching"), so there is very little space left to remove, and strong forces between particles resist being pushed closer.
2. Temperature and thermal equilibrium
In everyday life, temperature is just a number from a thermometer. But physically, temperature tells us something much more important: which way thermal energy will flow.
Key idea. The resultant flow of thermal energy between two bodies is always from the one at higher temperature to the one at lower temperature.
If two objects are in thermal contact (able to transfer thermal energy between them) and there is no net flow of thermal energy between them, they are at the same temperature — they are in thermal equilibrium.
Thermal equilibrium: the state reached when two (or more) objects in thermal contact are at the same temperature, so there is no net transfer of thermal energy between them.
Live simulation: reaching thermal equilibrium
Two objects in thermal contact
A: 90 °C B: 10 °C
Fig. 2.1 Two objects, A (red) and B (blue), placed in good thermal contact. Thermal energy flows from the hotter to the cooler object until both reach the same equilibrium temperature. Set the starting temperatures and press Start.
Kelvin and Celsius scales
Two temperature scales are used throughout this course. The Celsius scale is based on the freezing point (0 °C) and boiling point (100 °C) of pure water. The Kelvin scale is the absolute temperature scale used in physics, based on absolute zero (0 K) — the theoretical temperature at which the kinetic energy of particles is a minimum.
temperature in kelvin, T/K = θ/°C + 273
Key idea. Because the kelvin and the degree Celsius represent exactly the same size of temperature step — the two scales are simply offset from each other by 273 — a change in temperature, ΔT, has the same numerical value whether it is expressed in kelvin or in degrees Celsius.
Common mistake
0 °C is simply the freezing point of pure water — it has no other special meaning, and it is not a "true zero". It would be wrong to think that 20 °C is "twice as hot" as 10 °C, since neither value is measured from an absolute zero of thermal energy. On the Kelvin scale, however, 0 K genuinely does correspond to (almost) zero particle kinetic energy, so ratios of Kelvin temperatures do have physical meaning.
Worked example 2.1
a) The boiling point of nitrogen is −196 °C. Convert this to kelvin.
b) The melting point of copper is 1358 K. Convert this to degrees Celsius.
c) Overnight, the temperature at a weather station fell from +6 °C to −9 °C. Calculate this change in temperature in kelvin.
Answer:
a) T = −196 + 273 =
b) θ = 1358 − 273 =
c) ΔT = (−9) − (+6) = (the size of this change is the same in kelvin as in °C)
Check your understanding
5Convert the following kelvin temperatures into degrees Celsius: a) 310 K b) 195 K c) 4 K.
Use θ/°C = T/K − 273: a) 310 − 273 = 37 °C b) 195 − 273 = −78 °C c) 4 − 273 = −269 °C.
6The lowest temperature ever recorded on Earth's surface was −89.2 °C (Antarctica); the highest was 56.7 °C (California).
a) State both temperatures in kelvin.
b) Calculate the difference between the highest and lowest recorded temperatures, in kelvin.
a) −89.2 + 273 = 183.8 K; 56.7 + 273 = 329.7 K. b) Difference = 56.7 − (−89.2) = 145.9 (K or °C — the same either way).
7A block of metal at 85 °C is placed in good thermal contact with an identical block of the same metal at 25 °C, and the pair is well insulated from their surroundings. Describe, in terms of thermal energy transfer, what happens to the temperature of each block over time, and state the final temperature reached (assume the blocks are identical).
Thermal energy flows from the hotter block (85 °C) to the cooler block (25 °C), so the hotter block cools down while the cooler block warms up. The rate of transfer decreases as the temperature difference decreases. Because the blocks are identical, they reach a common equilibrium temperature exactly halfway between the two starting temperatures: 55 °C.
8A student is measuring the temperature of a beaker of water using a thermometer. Explain why the student must wait for a short time, with the thermometer bulb held in the water, before taking the reading.
The thermometer needs time to reach thermal equilibrium with the water — that is, time for thermal energy to be exchanged between the thermometer and the water until they are at the same temperature. Only once this equilibrium has been reached will the thermometer's reading be an accurate measurement of the water's temperature.
3. Internal energy and phase change
All the particles in a substance are moving (or, in a solid, vibrating), so they have kinetic energy. Where there are forces between particles, there is also potential energy associated with their separation. Together, these make up the internal energy of the substance.
Key idea. The internal energy of a substance is the sum of the total random kinetic energy and total potential energy of all the particles inside it.
Key idea. A rise in the temperature of a substance corresponds to its particles gaining, on average, kinetic energy.
Extension
More precisely, the average random translational kinetic energy of particles in a gas is directly proportional to the absolute (kelvin) temperature:
Ēk = (3/2) kBT
where kB is the Boltzmann constant, kB = 1.38 × 10⁻²³ J K⁻¹. This equation only relates directly to temperature for translational kinetic energy in a gas — the internal energy of solids and liquids is more complicated because of the significant forces between particles.
Phase change
Melting, freezing, boiling and condensing are all examples of a phase change (change of state). While a phase change is happening, energy is still being transferred to (or from) the substance — but its temperature does not change.
Key idea. A phase change represents a change in particle behaviour (their arrangement and separation) arising from a change in a substance's (potential) energy, at constant temperature.
During melting, for example, energy supplied to a solid does not increase the average kinetic energy of its particles (so the temperature does not rise); instead, it is used to weaken and break the forces holding particles in their fixed positions, increasing their potential energy and allowing them to move past one another as a liquid. The same is true of boiling, but for the much larger separation needed to become a gas.
Fig. 3.1 A heating curve for a pure substance (e.g. water) warmed steadily from solid to gas. (a) solid warms, KE↑; (b) melting — solid + liquid together, energy breaks bonds so PE↑ while T stays constant; (c) liquid warms, KE↑; (d) boiling — liquid + gas together, energy separates particles further so PE↑ while T stays constant; (e) gas warms, KE↑.
Note: The quantities of energy needed for a temperature rise (specific heat capacity) and for a phase change (specific latent heat) are treated quantitatively in the next workbook in this set, Heating.
Check your understanding
9A beaker of ice is heated steadily and its temperature is monitored. Explain why the temperature of the ice/water mixture stays at 0 °C for a period of time, even though the heater is continuously supplying energy.
While melting is taking place, the energy supplied is used to break the forces holding the particles in their fixed solid positions (increasing their potential energy), rather than increasing their average kinetic energy. Since temperature depends on average kinetic energy, the temperature does not rise until melting is complete and all the energy supplied goes back into raising the kinetic energy of the (now liquid) particles.
10Using Figure 3.1, describe what is physically happening to the substance and its particles during each of the following sections of the graph: a) the rising section between 0 °C and 100 °C; b) the flat section at 100 °C.
a) The substance is entirely liquid; energy supplied increases the average kinetic energy of the particles, so the temperature rises steadily. b) The substance is boiling — both liquid and gas are present together; energy supplied is used to separate the particles much further apart (a large increase in potential energy) rather than increasing their kinetic energy, so the temperature remains constant at 100 °C until boiling is complete.
11Two identical, sealed, insulated containers hold the same gas. Container P is at 400 K; container Q is at 200 K. Using Ēk = (3/2)kBT, determine the ratio of the average translational kinetic energy of a particle in P to that in Q.
Since Ēk ∝ T, the ratio is simply the ratio of the absolute temperatures: 400/200 = 2. A particle in P has, on average, twice the translational kinetic energy of a particle in Q.
4. Three ways to transfer thermal energy
Whenever there is a temperature difference, thermal energy will flow from the hotter region to the cooler one. There are three distinct mechanisms by which this can happen, and the rest of this workbook looks at each one in turn.
Mechanism
What is transferred?
Needs a medium?
Conduction
kinetic energy, passed particle to particle
yes (solids, liquids, gases)
Convection
thermal energy, carried by the bulk movement of a fluid
yes — only in liquids and gases
Thermal radiation
electromagnetic waves, emitted from a surface
no — can cross a vacuum
Key idea. Conduction, convection and thermal radiation are the three primary mechanisms for thermal energy transfer.
Check your understanding
12For each situation below, state which single mechanism is the primary way that thermal energy is transferred, and justify your choice:
a) sunlight warming your skin on a cold but clear day;
b) a metal teaspoon left standing in a hot cup of tea becoming warm to the touch;
c) warm air spreading upward and outward from a radiator to heat a whole room.
a) Radiation — the energy crosses the vacuum of space from the Sun to the Earth, which is only possible by electromagnetic waves. b) Conduction — kinetic energy is passed from the hot tea, through the particles of the solid metal spoon, up to the handle. c) Convection — the bulk movement of warmed, less dense air carries thermal energy around the room.
13Explain why neither conduction nor convection can be responsible for the Sun's thermal energy reaching the Earth.
Both conduction and convection require a material medium (a substance whose particles can pass on kinetic energy, or whose bulk can circulate). Almost all of the space between the Sun and the Earth is vacuum, containing no particles to carry energy this way. Only thermal radiation (electromagnetic waves) can travel through a vacuum.
5. Conduction
Imagine one end of a solid rod being heated. The particles at that end vibrate more vigorously (they gain kinetic energy). Through the forces connecting them to their neighbours, some of this extra kinetic energy is passed along the rod, particle to particle, without any of the particles moving from their fixed positions.
Key idea. Thermal conduction occurs because of the transfer of kinetic energy between particles that are in contact with each other, without the particles themselves changing position.
Solids are generally better thermal conductors than liquids, and liquids conduct better than gases — because their particles are closer together and the forces between them are stronger. Metals are by far the best conductors of all, because they also contain many free (delocalised) electrons that can carry kinetic energy through the material very quickly.
Substance
Thermal conductivity, k / W m⁻¹ K⁻¹
air
0.025
wood
0.15
water
0.59
glass
0.86
iron
84
copper
385
Thermal conductivity, k: a constant that represents how readily a substance conducts thermal energy; a larger value of k means the substance is a better conductor. A substance with a very low k is called a thermal insulator.
A quantitative treatment
The rate at which thermal energy is conducted through a material depends on the type of material, its cross-sectional area, and how steeply the temperature changes across its thickness (its temperature gradient):
rate of thermal energy transfer by conduction: ΔQ/Δt = kA · ΔT/Δx
where A is the cross-sectional area, Δx is the thickness of the material, and ΔT is the temperature difference across that thickness. Since ΔQ/Δt is energy transferred per second, it is measured in watts (W).
Worked example 5.1
A single-glazed window pane has dimensions 1.2 m × 0.8 m and a thickness of 6.0 mm. The thermal conductivity of the glass is 0.86 W m⁻¹ K⁻¹. On a winter day, the temperature inside a room is 22 °C and the outside temperature is 9 °C. Calculate the rate of thermal energy transfer through the window.
Answer:
A = 1.2 × 0.8 = 0.96 m²; Δx = 6.0 mm = 6.0 × 10⁻³ m; ΔT = 22 − 9 = 13 K
ΔQ/Δt = kA·ΔT/Δx = 0.86 × 0.96 × 13 / (6.0 × 10⁻³)
Live simulation: exploring the conduction equation
Rate of conduction: ΔQ/Δt = kAΔT/Δx
ΔQ/Δt = … W
Fig. 5.1 A simplified visual model of conduction along a bar: warmer particles (left) vibrate with greater amplitude than cooler particles (right). Choose a material and adjust ΔT, thickness and area to see how the calculated rate of energy transfer responds.
Check your understanding
14A layer of polyurethane foam insulation (k = 0.030 W m⁻¹ K⁻¹) in a cavity wall has a thickness of 7.5 cm and an area of 1.0 m². If the rate of thermal energy flow through the foam is 6.0 W, calculate the temperature difference across the foam. (Rearrange the conduction equation first.)
Rearranging: ΔT = (ΔQ/Δt) × Δx / (kA) = 6.0 × 0.075 / (0.030 × 1.0) — complete the calculation to find ΔT in K (which is numerically the same as °C).
15Frying pans are usually made from metal, but their handles are usually made from plastic or wood. Using the thermal conductivities in the table above, explain this choice of materials.
Metal has a very high thermal conductivity, so the base of the pan conducts thermal energy efficiently from the hob to the food, cooking it quickly. Plastic or wood have a very low thermal conductivity, so the handle conducts very little thermal energy from the hot pan to the cook's hand, letting it be held safely.
16Wetsuits, worn by surfers and divers, are made from neoprene foam rubber, which traps many small pockets of air. Explain how this design keeps the wearer warm in cold water.
Air has a very low thermal conductivity, so the many trapped pockets of air within the foam act as good thermal insulators, greatly reducing the rate at which thermal energy is conducted away from the wearer's body into the surrounding cold water.
17Double-glazed windows use two parallel sheets of glass with a thin gap of air (or another gas) between them, rather than one thick sheet of glass, or a very wide gap. Explain why (a) having a gap at all reduces conductive heat loss compared with a single thick pane, and (b) the gap is kept narrow rather than made very wide.
a) Air has a much lower thermal conductivity than glass, so trapping a layer of air between two panes conducts far less thermal energy than the same thickness of glass would. b) If the gap were made very wide, the trapped air could set up convection currents, carrying thermal energy across the gap by a different mechanism and reducing the benefit of the insulating layer — so the gap is kept narrow enough to limit this convective movement.
6. Convection
Conduction can occur in solids, liquids and gases — but liquids and gases have another way of transferring thermal energy, one that is not available to solids: bulk movement of the fluid itself.
Key idea. Thermal convection is the transfer of thermal energy because of the movement of a fluid, arising from differences in its density.
When part of a fluid is heated, it expands slightly and becomes less dense than the surrounding, cooler fluid. Because of its increased buoyancy, the warmer, less dense fluid rises above the cooler, denser fluid, which sinks to take its place. This sets up a continuous circulation known as a convection current.
Note: Convection cannot occur in solids, because the particles are held in fixed positions and cannot circulate as a bulk fluid.
Live simulation: convection in a gas
This tool visualises individual gas particles above a heated surface. Watch how particles near the heat source speed up and spread out (becoming less dense), rise, then cool and sink elsewhere in the container — the microscopic picture behind the convection currents described above.
Interactive simulation — open the online version of this workbook to launch it.
Simulation by Dr Dan Jones (drjonesphysics.com). If the embedded version above doesn't load, open it directly in a new tab via that link.
Convection currents explain a huge range of everyday phenomena: room heaters are placed near the floor while air-conditioning units are placed near the ceiling; the coolest part of a fridge is near the bottom while a freezer compartment is at the top; ocean currents mix water in seas and lakes; molten rock circulates in the Earth's mantle; and clouds, storms and global wind patterns are all driven by convection in the atmosphere.
Check your understanding
18The heating element of an electric kettle is placed near the bottom of the kettle, not near the top. Explain, in terms of convection, why this is an effective design.
Water heated at the bottom expands and becomes less dense, so it rises; cooler, denser water from above sinks to take its place and is, in turn, heated. This sets up a convection current that circulates thermal energy throughout the whole volume of water, heating it much more quickly and evenly than if the heater were at the top (where the warmed water would simply stay at the top, and convection could not distribute the energy downward).
19In a refrigerator, the freezer compartment is usually placed at the top, above the main storage area. Explain why this placement helps keep the food below it cool.
Air cooled by the freezer compartment becomes denser and sinks, displacing the warmer, less dense air below it, which rises to be cooled in turn. This sets up a convection current that continuously circulates cool air down through the fridge, keeping the whole compartment cold, rather than the cold air simply staying trapped at the top.
20Coastal regions often experience an onshore breeze (from the sea toward the land) during the day, because land heats up faster than the sea. Explain, with reference to thermal energy transfer, how this temperature difference produces the onshore breeze.
During the day, the land warms up faster than the sea, so the air above the land is warmed (by conduction and radiation from the ground) more than the air above the sea. This warmed air over the land expands, becomes less dense, and rises; cooler, denser air from over the sea then flows in along the surface to replace it, producing an onshore breeze — this is a large-scale convection current.
21A simple hot-air balloon rises when the air trapped inside its envelope is heated by a burner. Explain, in terms of density, why heating the air causes the balloon to rise.
Heating the air inside the envelope makes it expand, reducing its density. Once the average density of the balloon (envelope, basket and heated air) is lower than that of the surrounding, cooler outside air, the balloon experiences a net upward buoyancy force and rises — the same underlying idea as a warm parcel of fluid rising in a convection current.
7. Radiation and black bodies
Unlike conduction and convection, thermal radiation needs no medium at all. All matter, at any temperature above absolute zero, emits electromagnetic waves because of the movement of charged particles within its atoms. This is most commonly infrared radiation, but a hot enough object also emits visible light.
Key idea. All objects continuously emit (and absorb) thermal radiation. The power radiated from a surface depends on its surface temperature, its surface area, and the nature of the surface itself.
Good absorbers and good emitters
Dark, matte (dull) surfaces are good absorbers of thermal radiation — very little is reflected. White or shiny surfaces reflect and scatter radiation well, so they are poor absorbers.
Key idea. Any surface which is a good absorber of thermal radiation will also be a good emitter. Dark, matte surfaces emit and absorb radiation well; white, shiny surfaces are poor at both absorbing and emitting.
Live simulation: infrared emission
This tool lets you compare the infrared radiation emitted by different objects and surfaces directly, using a simulated thermal camera view. Explore how emission changes with temperature and with the type of surface.
Interactive simulation — open the online version of this workbook to launch it.
Simulation by Dr Dan Jones (drjonesphysics.com). If the embedded version above doesn't load, open it directly in a new tab via that link.
Black bodies
Black body: an idealised object with a surface that absorbs all of the electromagnetic radiation falling on it. A perfect black body is also a perfect emitter of thermal radiation.
No real surface is a perfect black body, but many familiar surfaces (soil, water, human skin, most vegetation) behave approximately like one. Perhaps confusingly, a "black body" does not have to look black — the Sun behaves approximately as a black body, yet it is so hot that it emits enormous quantities of visible light.
The Stefan–Boltzmann law
The total power radiated (across all wavelengths) from a black body of surface area A and absolute temperature T is given by the Stefan–Boltzmann law:
Stefan–Boltzmann law: L = σAT⁴
where L is the power radiated (sometimes called luminosity when referring to a radiating body such as a star), and σ is the Stefan–Boltzmann constant, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.
Extension
Real surfaces are not perfect black bodies. The emissivity, e, of a surface (0 ≤ e ≤ 1) compares how well it radiates to an ideal black body at the same temperature: L = eσAT⁴, with e = 1 for a perfect black body and e < 1 for a real, imperfect emitter.
Worked example 7.1
A small sphere of red-hot metal has a radius of 3.0 cm and a uniform surface temperature of 1200 °C. Assuming it behaves as a perfect black body, calculate the total power it radiates.
Answer:
Surface area of a sphere, A = 4πr² = 4π × (0.030)²
T = 1200 + 273 = 1473 K
L = σAT⁴ = (5.67 × 10⁻⁸) × [4π × (0.030)²] × (1473)⁴
Check your understanding
22Emergency "survival blankets", used to protect long-distance runners from losing body heat, are made from thin plastic sheets with a shiny, reflective (usually silver-coloured) coating. Explain how the reflective coating helps to protect the wearer from losing thermal energy by radiation.
A shiny surface is a poor emitter of thermal radiation, so it significantly reduces the rate at which thermal energy radiated by the wearer's body escapes through the blanket. (The shiny surface is also a poor absorber, but here it is the reduced emission from the inner surface that matters most.)
23A mug of coffee has a surface area of 180 cm² and a surface temperature of 60 °C. Treating it as a perfect black body, calculate the maximum power it radiates.
A = 180 cm² = 180 × 10⁻⁴ m² = 1.80 × 10⁻² m². T = 60 + 273 = 333 K. L = σAT⁴ = (5.67 × 10⁻⁸) × (1.80 × 10⁻²) × (333)⁴ — complete the calculation. (This will be an overestimate, since a real coffee mug is not a perfect black body and also loses energy by conduction and convection.)
24A metal bar is heated from a temperature of 400 K to 800 K, with its surface area unchanged. By what factor does the power it radiates increase?
Since L ∝ T⁴ (with A, σ and e unchanged), the factor is (800/400)⁴ = 2⁴ = 16. The bar radiates 16 times as much power at 800 K as it did at 400 K.
8. Wien's law and the black-body spectrum
A black body does not emit all of its radiation at one single wavelength — it emits a continuous spread of wavelengths, called its emission spectrum. The shape of this spectrum depends only on the body's temperature.
Fig. 8.1 Black-body emission spectra at three different surface temperatures. As temperature increases, the total power emitted increases sharply (area under the curve, L ∝ T⁴) and the peak wavelength, λmax, shifts to shorter wavelengths.
There is a simple inverse relationship between a black body's absolute surface temperature and the wavelength at which it emits the most power, λmax. This is known as Wien's displacement law:
Wien's displacement law: λmaxT = 2.9 × 10⁻³ m·K
Worked example 8.1
A star's spectrum peaks at a wavelength of 4.0 × 10⁻⁷ m. Determine its surface temperature.
Change a black body's temperature and watch its emission spectrum change shape in real time. Turn on "Graph Values" to read off the peak wavelength directly, and check that your readings are consistent with Wien's law.
25Using Figure 8.1, estimate the peak wavelength emitted by a black body at 4000 K, and check that your estimate is consistent with Wien's law.
From the graph, the 4000 K curve peaks at roughly λmax ≈ 0.7–0.75 × 10⁻⁶ m. Checking with Wien's law: λmax = (2.9 × 10⁻³)/4000 ≈ 7.3 × 10⁻⁷ m = 0.73 × 10⁻⁶ m, which agrees well with the graph.
26Microwave radiation with a peak wavelength of about 1.1 mm arrives at Earth uniformly from every direction in space — the Cosmic Microwave Background, left over from the early Universe. Calculate the temperature associated with this radiation.
λmaxT = 2.9 × 10⁻³. T = (2.9 × 10⁻³)/(1.1 × 10⁻³) — complete the division. (This gives a temperature only a few kelvin above absolute zero — remarkably cold, consistent with radiation left over from a Universe that has expanded and cooled enormously since the Big Bang.)
27Star X has twice the surface temperature of Star Y, but both stars have the same radius. Determine: a) the ratio of their peak emitted wavelengths, λmax,X : λmax,Y; b) the ratio of their radiated powers (luminosities), LX : LY.
a) Since λmaxT = constant, λmax ∝ 1/T. So λmax,X : λmax,Y = 1/2 : 1 = 1 : 2 — Star X's peak wavelength is half that of Star Y (it is hotter, so bluer/shorter wavelength). b) Since L = σAT⁴ and A is the same for both, L ∝ T⁴, so LX : LY = 2⁴ : 1⁴ = 16 : 1 — Star X radiates 16 times as much power as Star Y.
Glossary
Kinetic theory of matter
All matter is made up of a very large number of small particles which are in constant motion, with forces acting between them.
Density, ρ
Mass per unit volume of a substance; ρ = m/V.
Thermal equilibrium
The state reached when objects in thermal contact are at the same temperature, so there is no net transfer of thermal energy between them.
Celsius scale
Temperature scale based on the freezing point (0 °C) and boiling point (100 °C) of pure water.
Kelvin scale
The absolute temperature scale used in physics, based on absolute zero (0 K); T/K = θ/°C + 273.
Absolute zero
0 K (−273 °C); the temperature at which the kinetic energy of particles is a minimum.
Internal energy
The sum of the total random kinetic energy and total potential energy of all the particles in a substance.
Phase change
A change in particle behaviour (arrangement and separation) arising from a change in energy at constant temperature — e.g. melting, boiling.
Thermal conduction
The transfer of thermal energy through a substance, particle to particle, by the transfer of kinetic energy.
Thermal conductivity, k
A constant representing how readily a substance conducts thermal energy; a larger value means a better conductor.
Thermal insulator
A material with a low thermal conductivity that significantly reduces the flow of thermal energy.
Thermal convection
The transfer of thermal energy by the bulk movement of a fluid, arising from differences in its density.
Thermal radiation
Electromagnetic radiation emitted by all matter because of the movement of charged particles within its atoms; most commonly infrared.
Black body
An idealised object whose surface absorbs all electromagnetic radiation falling on it, and which is also a perfect emitter of thermal radiation.
Emissivity, e
A measure (0 ≤ e ≤ 1) of how well a real surface radiates compared with an ideal black body at the same temperature.
Stefan–Boltzmann law
L = σAT⁴; gives the total power radiated from a black body of surface area A and absolute temperature T.
Wien's displacement law
λmaxT = 2.9 × 10⁻³ m·K; relates a black body's peak emission wavelength to its absolute temperature.