IB DP Physics · Theme B · The particulate nature of matter

Thermal energy transfers.

B.1 · Particles and temperature, heating and changes of phase, conduction, convection and radiation
Standard level and higher level

SL + HL · 6 hours B.4 Thermodynamics (HL) ↗

The journey

Four parts.

Part 01

Particles and temperature

The molecular model, the kelvin and internal energy

Part 02

Heating and changes of phase

\(Q = mc \Delta T\) and \(Q = mL\)

Part 03

Conduction and convection

Energy passed on by particles, and carried by a moving fluid

Part 04

Thermal radiation

Black bodies, luminosity, brightness and Wien’s law

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Part 01

Particles and temperature.

What a thermometer is telling you about the molecules

B.1 THERMAL ENERGY TRANSFERS · SL + HL

The molecular model

Solids, liquids and gases.

The same particles in all three phases
What changes: their spacing, their motion and how strongly the forces between them act

Forces hold a solid in shape, hold a liquid together and are negligible in a gas.

THE SAME PARTICLES, THREE ARRANGEMENTS

B.1 THERMAL ENERGY TRANSFERS · SL + HL

What the model explains

A solid keeps its shape because each particle is held in place by its neighbors and can only vibrate. A liquid flows, but keeps its volume, because the particles are still touching and still attract each other, yet are free to change places. A gas fills any container because its particles are so far apart that the forces between them hardly act: each one flies in a straight line until it hits something.

Why liquids and solids are hard to squash

Their particles are already in contact. A gas is mostly empty space, so it can be compressed easily: in air, the molecules are about ten diameters apart.

Drawing the particles of a liquid far apart. A liquid is only slightly less dense than the solid: nearly all the particles still touch.

Mass per unit volume

Density.

\(\rho = \dfrac{m}{V}\)

Unit: \(\mathrm{kg} \,\mathrm{m}^{-3}\)
Solids and liquids: particles touching, similar densities
Gases: about a thousand times less dense

An aluminum block: \(5.0 \,\mathrm{cm} \times 4.0 \,\mathrm{cm} \times 2.0 \,\mathrm{cm}\) · mass 108 g \(V = 0.050 \times 0.040 \times 0.020 = 4.0 \times 10^{-5} \,\mathrm{m}^{3}\) \(\rho = \dfrac{m}{V} = \dfrac{0.108}{4.0 \times 10^{-5}}\) \(\rho = \class{res}{2.7 \times 10^{3} \,\mathrm{kg} \,\mathrm{m}^{-3}}\)

B.1 THERMAL ENERGY TRANSFERS · SL + HL

The factor of a thousand

Water has a density of 1000 kg m⁻³ and steam at 100 °C about 0.6 kg m⁻³. The molecules are the same, so in steam each one has about 1700 times as much space. The cube root of 1700 is about 12: on average the molecules of the gas are twelve times further apart than in the liquid.

Water is unusual

Ice (917 kg m⁻³) is less dense than liquid water, because its molecules lock into an open pattern. For nearly every other substance the solid is the denser phase and sinks in its own liquid.

Converting cm³ to m³ by dividing by 100. There are \(100^{3} = 10^{6}\) cubic centimeters in a cubic meter, so \(1 \,\mathrm{g} \,\mathrm{cm}^{-3} = 1000 \,\mathrm{kg} \,\mathrm{m}^{-3}\).

Two scales with the same step

Kelvin and Celsius.

\(T / \mathrm{K} = \theta / {}^{\circ}\mathrm{C} + 273\)

The kelvin scale starts at absolute zero
One kelvin is the same size as one degree Celsius
So a temperature change is the same number on both scales

THE TWO SCALES

B.1 THERMAL ENERGY TRANSFERS · SL + HL

The story · Munich, 1798

Background story · not assessed

Count Rumford bores a cannon.

Ink drawing of Count Rumford, a man in a long coat and riding boots leaning against a rock under a tree, holding a rolled paper — **Benjamin Thompson**, Count Rumford

Engraving of a cannon barrel held in a boring machine, and below it a section through a box of water fitted round the end of the barrel — The barrel in its **box of water** (plate of 1870, after his paper)

In 1798 heat was thought to be a fluid, caloric, that could not be made or destroyed
At the Munich arsenal, a blunt borer turned in a barrel under water
The water boiled in about two and a half hours, and went on boiling

A supply that never runs out cannot be a substance. What the borer gave the metal was motion.

Portrait: Jean François Bozio, Metropolitan Museum of Art, CC0 (cropped). Engraving: The Complete Works of Count Rumford, 1870, no known restrictions.

B.1 THERMAL ENERGY TRANSFERS · SL + HL

What temperature measures

Temperature and kinetic energy.

\(\bar{E}_\mathrm{k} = \dfrac{3}{2} k_\mathrm{B} T\)

\(\bar{E}_\mathrm{k}\): the average random kinetic energy of one particle
\(T\) must be in kelvin
\(k_\mathrm{B} = 1.38 \times 10^{-23} \,\mathrm{J} \,\mathrm{K}^{-1}\), the Boltzmann constant

A molecule of air in a room at 27 °C \(T = 27 + 273 = 300 \,\mathrm{K}\) \(\bar{E}_\mathrm{k} = \dfrac{3}{2} k_\mathrm{B} T = \dfrac{3}{2} \times 1.38 \times 10^{-23} \times 300\) \(\bar{E}_\mathrm{k} = \class{res}{6.2 \times 10^{-21} \,\mathrm{J}}\)

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Same temperature, same kinetic energy

The equation has no mass in it. In air at 300 K a light hydrogen molecule and a heavy carbon dioxide molecule have the same average kinetic energy. The lighter one must therefore be moving faster: hydrogen at about 1900 m s⁻¹, carbon dioxide at about 400 m s⁻¹.

Why “average”

At any instant some molecules are almost at rest and a few are moving several times faster than the typical speed. They exchange energy at every collision. Temperature describes the whole crowd, not one member of it: a single molecule does not have a temperature.

Absolute zero

Put \(T = 0\) in the equation and the average random kinetic energy is zero. That is why nothing can be colder: the particles have the least energy they can have, and none is left to remove.

The energies are tiny, so they are often given in electronvolts: \(6.2 \times 10^{-21} \,\mathrm{J}\) is about 0.04 eV.

The energy inside a body

Internal energy.

Total random

Kinetic energy

From the random motion of the molecules. Rises with temperature

+

Total intermolecular

Potential energy

From the forces between the molecules. Changes with their separation, so with the phase

=

The sum, for all the molecules

Internal energy

A property of the whole body, in joules

Temperature is about the average kinetic energy of one molecule. Internal energy is the total for all of them.

B.1 THERMAL ENERGY TRANSFERS · SL + HL

A cup and a bath

A cup of water at 80 °C is hotter than a bath at 40 °C: each of its molecules has more kinetic energy on average. But the bath has far more internal energy, because it has thousands of times as many molecules. Temperature tells you which way energy will flow. Internal energy tells you how much there is.

Why “random”

Throw a ball and all its molecules gain the same extra velocity. That ordered motion is the ball’s ordinary kinetic energy and does not make the ball hotter. Internal energy counts only the disordered motion of the molecules relative to each other.

Energy without a temperature rise

When ice melts, the energy supplied pulls the molecules out of their fixed places against the forces between them. The potential energy rises while the kinetic energy, and so the temperature, stays the same.

An ideal gas has no forces between its molecules, so its internal energy is kinetic only: \(U = \frac{3}{2} N k_\mathrm{B} T\) in B.3.

Which way the energy goes

Thermal energy flows from hot to cold.

TWO BLOCKS IN CONTACTPRESS → TO PLAY

The temperature difference sets the direction of the resultant transfer
It stops when the temperatures are equal: thermal equilibrium
The final temperature is not usually half way between

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Why hot to cold

Where the two blocks touch, fast molecules of the hot block collide with slow molecules of the cold one. In a collision between a fast and a slow particle the fast one usually loses energy and the slow one gains it. Energy also passes the other way in some collisions, which is why the slide says “resultant”: more goes from hot to cold than from cold to hot.

At equilibrium nothing has stopped

The molecules are still colliding and still exchanging energy across the boundary, but now as much goes one way as the other.

Why not half way

On the graph the cold block warms much more than the hot block cools. It has the smaller mass or the smaller specific heat capacity, so the same energy changes its temperature more. Part 2 puts numbers to this.

“Heat flows until both have the same energy.” It is the temperatures that become equal. The two bodies usually end with very different internal energies.

Quick check

Particles and temperature.

01Convert 37 °C to kelvin.

310 K

02A liquid is heated by 15 °C. What is the rise in kelvin?

15 K

03The kelvin temperature of a gas doubles. The average kinetic energy of a molecule?

It doubles

04Which two kinds of energy make up internal energy?

Random kinetic and intermolecular potential

05What decides the direction of thermal energy transfer?

The temperature difference

06A 2.0 m³ tank holds 1600 kg of oil. Density?

\(\frac{1600}{2.0} = 800 \,\mathrm{kg} \,\mathrm{m}^{-3}\)

PRESS → TO REVEAL EACH ANSWER

Part 02

Heating and changes of phase.

How much energy it takes to warm a substance, and to melt or boil it

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Energy to change the temperature

Specific heat capacity.

\(Q = mc \Delta T\)

\(c\): the energy to raise the temperature of 1 kg by 1 K
Unit: \(\mathrm{J} \,\mathrm{kg}^{-1} \,\mathrm{K}^{-1}\)
A property of the substance, and of its phase

Substance (approximate values)	\(c\) / J kg⁻¹ K⁻¹
Water	4200
Ice	2100
Aluminum	900
Copper	390
Lead	130

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Water is the odd one out

It takes about ten times as much energy to warm a kilogram of water by one kelvin as a kilogram of copper. That is why the sea warms and cools so much more slowly than the land, why water is the usual coolant in engines and power stations, and why a hot-water bottle stays warm for hours.

Why metals are so low

A kilogram of lead contains far fewer atoms than a kilogram of aluminum, because each lead atom is nearly eight times heavier. Each atom takes about the same energy to speed up, so the substance with fewer atoms per kilogram needs less energy per kilogram.

\(\Delta T = T_\mathrm{f} - T_\mathrm{i}\): positive for heating, negative for cooling. \(Q\) is positive into the body and negative out. Temperature differences are the same in °C and K; do not add 273.

“Specific” means “per unit mass”. The heat capacity of a whole object, \(mc\), is in J K⁻¹.

Worked example

Boiling a kettle.

1.2 kg of water at 18 °C is heated to 100 °C by a 2.0 kW element \(c = 4200 \,\mathrm{J} \,\mathrm{kg}^{-1} \,\mathrm{K}^{-1}\) \(Q = mc \Delta T\) \(Q = 1.2 \times 4200 \times (100 - 18)\) \(Q = \class{res}{4.1 \times 10^{5} \,\mathrm{J}}\)

The time it takes \(P = \dfrac{Q}{t}\) so \(t = \dfrac{Q}{P}\) \(t = \dfrac{4.1 \times 10^{5}}{2.0 \times 10^{3}}\) \(t = \class{res}{2.1 \times 10^{2} \,\mathrm{s}}\)

B.1 THERMAL ENERGY TRANSFERS · SL + HL

An electrical method

Measuring specific heat capacity.

1.0 kg block · 48 W heater for 300 s · rise of 15 K \(Pt = mc \Delta T\) \(c = \dfrac{Pt}{m \Delta T} = \dfrac{48 \times 300}{1.0 \times 15}\) \(c = \class{res}{960 \,\mathrm{J} \,\mathrm{kg}^{-1} \,\mathrm{K}^{-1}}\)

Accepted value for aluminum: 900. Energy lost to the room makes the result too high.

A HEATED METAL BLOCK

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Why too high, not too low

The calculation credits the block with all 14 400 J. In fact some of that energy leaked away, so the block received less and its temperature rose by less than it should have. A smaller \(\Delta T\) on the bottom of the fraction gives a larger \(c\). Whenever energy escapes, this method overestimates.

Reducing the error

Insulate the block. Put a drop of oil in the thermometer hole so that the thermometer reads the temperature of the metal. Keep watching after switching off and record the highest temperature reached, because the energy takes time to spread through the block. Better still, start a few degrees below room temperature and finish the same amount above it, so that the gains and losses roughly cancel.

A graph of temperature against time for a steady heater has gradient \(\frac{P}{mc}\). Using the gradient of the straight part is better than using one pair of readings.

Worked example

Hot copper into cold water.

0.20 kg of copper at 100 °C goes into 0.50 kg of water at 20 °C \(c_\mathrm{Cu} = 390\) · \(c_\mathrm{w} = 4200 \,\mathrm{J} \,\mathrm{kg}^{-1} \,\mathrm{K}^{-1}\) · final temperature \(T\) \(Q_\mathrm{Cu} + Q_\mathrm{w} = 0\) \(m_\mathrm{Cu} c_\mathrm{Cu} (T - 100) + m_\mathrm{w} c_\mathrm{w} (T - 20) = 0\) \(78 (T - 100) + 2100 (T - 20) = 0\)

\(7800 + 42\,000 = 2178\,T\) \(T = \dfrac{49\,800}{2178}\) \(T = \class{res}{23 \,{}^{\circ}\mathrm{C}}\)

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Why so close to 20 °C

Compare the two heat capacities, \(mc\). For the copper it is 78 J K⁻¹; for the water it is 2100 J K⁻¹, twenty-seven times as much. Every joule that leaves the copper cools it twenty-seven times as much as it warms the water. The copper falls 77 K while the water rises 3 K.

Setting it up

Take the copper and water together as an isolated system: no energy passes to the cup or surroundings, so \(\Delta E_\mathrm{total} = 0\) and \(Q_\mathrm{Cu} + Q_\mathrm{w} = 0\). For each body, \(Q\) is positive for heat transferred into it and negative for heat transferred out. Every temperature change is final minus initial: \(\Delta T_\mathrm{Cu} = T - 100 < 0\), while \(\Delta T_\mathrm{w} = T - 20 > 0\). With no work or phase change, use \(Q = mc\Delta T\). Add the signed heat transfers and set their sum to zero.

Check that the answer lies between the two starting temperatures, and nearer to that of the body with the larger \(mc\).

Heating ice until it is steam

Changes of phase.

Sloping parts: the temperature rises, \(Q = mc \Delta T\)
Flat parts: the phase changes at constant temperature
The energy goes into potential energy, not kinetic
Cooling runs the graph backwards: condensing and freezing release energy

WATER HEATED AT A STEADY RATE

B.1 THERMAL ENERGY TRANSFERS · SL + HL

What the energy does on a flat part

During melting, the energy supplied frees the molecules from their fixed places. During boiling, it pulls them right away from each other, against the forces that hold the liquid together. In both cases the separation of the molecules changes, so their potential energy rises. Their average kinetic energy does not, and that is why the thermometer does not move.

Reading the slopes

The heater supplies energy at a steady rate, so a steeper line means a smaller specific heat capacity. Ice and steam both warm about twice as fast as liquid water: their specific heat capacities are about half as large. To scale, the boiling plateau would be more than five times as long as the water slope; it is drawn shorter here to fit.

The words

Solid to liquid is melting and the reverse is freezing. Liquid to gas is boiling (or evaporation) and the reverse is condensing.

Asked to explain a flat part, say three things: the energy increases the potential energy of the molecules, the kinetic energy is unchanged, so the temperature is constant.

Energy to change the phase

Specific latent heat.

\(Q = mL\)

\(L\): the energy to change the phase of 1 kg at constant temperature
Unit: \(\mathrm{J} \,\mathrm{kg}^{-1}\)
Fusion: solid ⇌ liquid. Vaporization: liquid ⇌ gas

Water	\(L\) / J kg⁻¹
Fusion, at 0 °C	\(3.34 \times 10^{5}\)
Vaporization, at 100 °C	\(2.26 \times 10^{6}\)

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Why vaporization takes so much more

Melting only loosens the molecules: they stay in contact, and nearly all the bonds between neighbors survive. Boiling separates them completely, so every bond must be broken. For water that takes nearly seven times as much energy per kilogram.

Why steam scalds

A gram of steam at 100 °C that condenses on your skin releases 2260 J before it has cooled at all. A gram of water at 100 °C cooling to skin temperature releases about 270 J. The same property is put to use in a power station: turning water into steam stores a great deal of energy in it, which the turbine then takes out.

Putting a temperature change into \(Q = mL\). There is none: the whole point is that the temperature stays constant.

\(L\) is positive. For melting or vaporization, \(Q = +mL\); for freezing or condensation, \(Q = -mL\). “Latent” means hidden: energy is transferred without a temperature change.

The story · Glasgow, about 1761

Background story · not assessed

Joseph Black and the hidden heat.

Painted portrait of Joseph Black, an older man with white hair at the sides of his head, wearing a dark gown — **Joseph Black**, painted by David Martin in 1787

A professor at Glasgow, he noticed that melting ice takes in heat without getting warmer
Boiling water does the same: more steam, no rise in temperature
He called the heat latent. He had already found that different substances need different amounts of heat to warm up
His friend at Glasgow, James Watt, went on to transform the steam engine, with Black’s support

Both \(Q = mL\) and \(Q = mc \Delta T\) go back to Black.

Portrait: David Martin, 1787, photograph CC BY-SA 4.0.

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Worked example

Ice from the freezer.

0.050 kg of ice at −18 °C becomes water at 20 °C Warm the ice to 0 °C \(Q_{1} = mc_\mathrm{ice} \Delta T = 0.050 \times 2100 \times 18 = 1.9 \times 10^{3} \,\mathrm{J}\) Melt it \(Q_{2} = mL = 0.050 \times 3.34 \times 10^{5} = 1.67 \times 10^{4} \,\mathrm{J}\) Warm the water to 20 °C \(Q_{3} = mc_\mathrm{w} \Delta T = 0.050 \times 4200 \times 20 = 4.2 \times 10^{3} \,\mathrm{J}\)

\(Q = Q_{1} + Q_{2} + Q_{3}\) \(Q = \class{res}{2.3 \times 10^{4} \,\mathrm{J}}\)

Nearly three quarters of it is the melting.

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Two ways from liquid to gas

Boiling and evaporation.

Boiling

At one temperature, throughout the liquid

Bubbles of vapor form inside the liquid. The temperature stays at the boiling point while energy is supplied

Evaporation

At any temperature, from the surface

The fastest molecules near the surface escape. Those left behind have less energy on average, so the liquid cools

Both need the specific latent heat of vaporization for every kilogram that changes phase.

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Cooling by evaporation

The molecules in a liquid have a range of speeds. Only the fastest, and only those at the surface and heading outwards, have enough energy to break free of their neighbors. Each one that leaves takes more than its share of energy with it, so the average kinetic energy of the rest falls: the liquid gets colder. This is how sweating cools you, and why you feel cold when you step out of a swimming pool into a breeze.

What speeds it up

A higher temperature (more molecules fast enough to escape), a larger surface (more molecules in a position to escape), and a wind (which carries the escaped molecules away before they fall back in).

Sweating 1 g of water takes about 2400 J from your body: the latent heat is a little larger at skin temperature than at 100 °C.

Quick check

Heating and changes of phase.

01Energy to warm 2.0 kg of water by 10 K? (\(c\) = 4200 J kg⁻¹ K⁻¹)

\(2.0 \times 4200 \times 10 = 8.4 \times 10^{4} \,\mathrm{J}\)

02The same energy goes into 2.0 kg of copper. Larger or smaller rise?

Larger: copper has a smaller \(c\)

03What happens to the temperature while a pure solid melts?

It stays constant

04Which energy of the molecules rises during melting?

Their potential energy

05Energy to boil away 0.10 kg of water at 100 °C? (\(L = 2.26 \times 10^{6} \,\mathrm{J} \,\mathrm{kg}^{-1}\))

\(0.10 \times 2.26 \times 10^{6} = 2.3 \times 10^{5} \,\mathrm{J}\)

06Why does evaporation cool a liquid?

The fastest molecules leave

PRESS → TO REVEAL EACH ANSWER

Part 03

Conduction and convection.

Two of the three ways thermal energy gets from one place to another

B.1 THERMAL ENERGY TRANSFERS · SL + HL

How thermal energy is transferred

Three mechanisms.

Conduction

Passed from particle to particle

The particles stay where they are. Best in solids, above all in metals

Convection

Carried by a moving fluid

Warm fluid rises and cool fluid sinks. Liquids and gases only

Radiation

Emitted as electromagnetic waves

Needs no medium at all. The only way across empty space

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Particle to particle

Conduction.

Particles at the hot end have more kinetic energy
They pass it to their neighbors through the forces between them
In a metal, free electrons carry energy much further and faster
Gases conduct poorly: the molecules are far apart

A BAR BETWEEN HOT AND COLD

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Why metals feel cold

A metal bench and a wooden bench in the same room are at the same temperature, but the metal feels colder. Your hand is warmer than both. The metal conducts energy away from your skin quickly, so your skin temperature drops; the wood conducts so slowly that the layer you touch soon warms up. Your sense of touch reports the rate at which you lose energy, not the temperature of the object.

Why trapped air insulates

Wool, feathers, foam and double glazing all work by holding air still. Air is a very poor conductor, and if it cannot move it cannot carry energy away by convection either.

Good conductors of electricity are good conductors of thermal energy, for the same reason: free electrons. Diamond is a rare exception, an electrical insulator that conducts thermal energy better than copper.

How fast energy is conducted

The rate of conduction.

\(\dfrac{\Delta Q}{\Delta t} = -kA \dfrac{\Delta T}{\Delta x}\)

\(k\): the thermal conductivity of the material, in \(\mathrm{W} \,\mathrm{m}^{-1} \,\mathrm{K}^{-1}\)
\(A\): the cross-sectional area the energy passes through
\(\dfrac{\Delta T}{\Delta x}\): the signed temperature gradient

Material (approximate values)	\(k\) / W m⁻¹ K⁻¹
Copper	400
Aluminum	240
Glass	0.80
Brick	0.60
Still air	0.025

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Each factor makes sense

Twice the area gives twice as many paths for the energy, so twice the rate. Twice the temperature difference pushes energy through twice as fast. Twice the thickness halves the rate: the same temperature difference is spread over twice the distance, so the gradient is half as steep.

The sign

The data booklet (version 1.2, B.1, page 8) includes the minus sign. Choose a positive \(x\)-direction and use \(\Delta T = T_{2} - T_{1}\) with \(\Delta x = x_{2} - x_{1}\). The minus sign makes the energy flow opposite to the temperature gradient, from hot to cold. A positive \(\frac{\Delta Q}{\Delta t}\) means transfer in the positive \(x\)-direction.

The same shape as Ohm’s law

A temperature difference drives a flow of energy through a material, just as a potential difference drives a flow of charge through a resistor. A thick layer with a low \(k\) is a large “thermal resistance”. B.5 uses the same idea for electric current.

\(\frac{\Delta Q}{\Delta t}\) is a power, in watts. To find the energy conducted in a given time, multiply by the time in seconds.

Worked example

Energy lost through a window.

Glass: area 1.5 m² · thickness 4.0 mm · \(k = 0.80 \,\mathrm{W} \,\mathrm{m}^{-1} \,\mathrm{K}^{-1}\) Positive \(x\): from the 12 °C face to the 7.0 °C face \(\dfrac{\Delta Q}{\Delta t} = -kA \dfrac{\Delta T}{\Delta x}\) \(\dfrac{\Delta Q}{\Delta t} = -0.80 \times 1.5 \times \dfrac{7.0 - 12}{4.0 \times 10^{-3}}\) \(\dfrac{\Delta Q}{\Delta t} = \class{res}{1.5 \times 10^{3} \,\mathrm{W}}\)

In one hour: \(1500 \times 3600 = 5.4 \times 10^{6} \,\mathrm{J}\)
Glass twice as thick: half the rate
Two panes with still air between: far less

B.1 THERMAL ENERGY TRANSFERS · SL + HL

The temperatures are those of the glass

The room might be at 20 °C and the air outside at 0 °C, but the glass faces are at 12 °C and 7 °C. A thin layer of nearly still air clings to each side of the pane and, being a poor conductor, takes most of the 20 K difference. The equation needs the temperature difference across the material itself. Along positive \(x\), \(\Delta T = 7.0 - 12 = -5.0 \,\mathrm{K}\), so the transfer rate is positive outwards. The room’s energy change in one hour is \(-5.4 \times 10^{6} \,\mathrm{J}\); the surroundings gain the same amount.

Why double glazing works

A 16 mm gap of still air has one thirtieth the thermal conductivity of glass and is four times as thick as the pane, so for the same temperature difference it transfers less than one hundredth as much energy per second. The gap must be narrow enough that the air inside cannot set up a convection current.

Leaving the thickness in millimeters. 4.0 mm is \(4.0 \times 10^{-3}\) m.

Carried by a moving fluid · described, not calculated

Convection.

The heated fluid expands: it becomes less dense
Less dense fluid rises; cooler, denser fluid sinks to replace it
The fluid itself moves, carrying its energy with it
Not possible in a solid

A LIQUID HEATED FROM BELOW

B.1 THERMAL ENERGY TRANSFERS · SL + HL

It is a buoyancy effect

From A.2, a body in a fluid feels an upward buoyancy force equal to the weight of the fluid it displaces. A parcel of warm fluid displaces cooler fluid of the same volume but greater density, so the buoyancy force on it is larger than its weight and it accelerates upwards. Convection therefore needs gravity: in an orbiting spacecraft a candle flame is a dim sphere, because the hot gas does not rise away from it.

Where you meet it

Sea breezes: by day the land warms faster than the sea, air rises over the land and cooler air moves in from the sea. A room heater is placed low, and the element of a kettle at the bottom, so that the whole fluid circulates. Inside the Sun, the outer third of the radius carries energy to the surface by convection.

“Heat rises.” Thermal energy has no direction of its own. It is the warm, less dense fluid that rises.

Quick check

Conduction and convection.

01Which mechanism cannot happen in a solid?

Convection

02Why are metals the best conductors?

Free electrons carry the energy

03A wall is made twice as thick. The rate of conduction?

It halves

04What is the unit of thermal conductivity?

W m⁻¹ K⁻¹

05Why does warm fluid rise?

It expands and becomes less dense

06Why is trapped air a good insulator?

Poor conductor, and it cannot convect

PRESS → TO REVEAL EACH ANSWER

Part 04

Thermal radiation.

Every body emits electromagnetic waves because of its temperature

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Emission from a surface

Thermal radiation and black bodies.

Every surface above absolute zero emits electromagnetic waves
At everyday temperatures, almost all of it is infrared
A black body absorbs all the radiation that falls on it, and is also the best possible emitter

Near enough black

A star

Its spectrum is close to a black body’s

Near enough black

A small hole in a hot oven

Whatever goes in is absorbed inside

Far from black

A polished metal surface

Reflects most radiation and emits little

B.1 THERMAL ENERGY TRANSFERS · SL + HL

“Black” is about absorbing, not about looking dark

A body that absorbs every wavelength reflects nothing, so at room temperature it looks black. Heat it and the same surface glows: it is now emitting visible light of its own. The Sun is a very good black body and is anything but dark.

Why a good absorber must be a good emitter

Put a black object and a shiny one in an oven and wait. Both end at the temperature of the oven and stay there. The black one absorbs more of the radiation falling on it, so to stay at a steady temperature it must also emit more. If it did not, it would keep getting hotter than its surroundings by itself.

Net transfer

You emit about 500 W of infrared, but the walls of the room radiate almost as much back to you. What matters is the difference. In a cold room, or under a clear night sky, less comes back and you feel cold even in still air.

B.2 measures how far a real surface falls short of a black body with its emissivity, a number between 0 and 1.

The power a black body radiates

The Stefan–Boltzmann law.

\(L = \sigma A T^{4}\)

\(L\): the luminosity, the total power radiated, in watts
\(A\): the surface area of the body
\(\sigma = 5.67 \times 10^{-8} \,\mathrm{W} \,\mathrm{m}^{-2} \,\mathrm{K}^{-4}\)

The Sun: radius \(6.96 \times 10^{8} \,\mathrm{m}\) · surface at 5800 K \(A = 4 \pi R^{2} = 4 \pi \times (6.96 \times 10^{8})^{2} = 6.09 \times 10^{18} \,\mathrm{m}^{2}\) \(L = \sigma A T^{4}\) \(L = 5.67 \times 10^{-8} \times 6.09 \times 10^{18} \times 5800^{4}\) \(L = \class{res}{3.9 \times 10^{26} \,\mathrm{W}}\)

B.1 THERMAL ENERGY TRANSFERS · SL + HL

The fourth power is steep

Double the kelvin temperature and the power goes up by \(2^{4} = 16\). A star only 20% hotter than the Sun, with the same radius, would be twice as luminous, since \(1.2^{4} = 2.1\). This is why a small change in the temperature of a planet makes a noticeable change in the energy it radiates to space, which matters in B.2.

Two ways to be luminous

\(L\) depends on area as well as temperature. A red giant is cool, about 3500 K, but so enormous that it outshines the Sun thousands of times over. A white dwarf is very hot but the size of the Earth, and faint. E.5 uses this equation to find the radius of a star from its luminosity and temperature.

Using °C. With a fourth power the error is huge: \(293^{4}\) is about 46 000 times \(20^{4}\).

A star’s luminosity may be given in units of the Sun’s, \(L_{\odot}\).

The story · Vienna, 1879

Background story · not assessed

Stefan takes the temperature of the Sun.

Engraved portrait of Josef Stefan, a bearded man in a dark jacket and bow tie — **Josef Stefan**, director of the Physical Institute in Vienna

In 1879 Stefan studied other people’s measurements of hot bodies cooling, and spotted the pattern: power \(\propto T^{4}\)
He used it on sunlight and found the Sun’s surface to be about 5430 °C: the first sensible value
Today’s value is about 5500 °C
In 1884 his former student Ludwig Boltzmann derived the law from theory

A law found in a laboratory in Vienna measured a star 150 million kilometers away.

Portrait: K. Schönbauer, by 1893, public domain.

B.1 THERMAL ENERGY TRANSFERS · SL + HL

What reaches the observer

Apparent brightness.

\(b = \dfrac{L}{4 \pi d^{2}}\)

\(b\): the power received per unit area, in \(\mathrm{W} \,\mathrm{m}^{-2}\)
The power \(L\) spreads over a sphere of area \(4 \pi d^{2}\)
An inverse square law: twice as far, a quarter as bright

THE SAME POWER, A LARGER SPHERE

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Luminosity and brightness are different things

Luminosity belongs to the star: it is how much power the star gives out, and it is the same wherever you stand. Apparent brightness belongs to the observer: it is how much of that power lands on each square meter where you are. A faint-looking star may be a feeble one nearby or a brilliant one far away. You cannot tell which without knowing the distance.

Where the sphere comes from

The star radiates equally in all directions and nothing is absorbed on the way. By the time the light has traveled a distance \(d\), all of it is passing through the surface of an imaginary sphere of radius \(d\) centered on the star. Share the power equally over that surface and you have the brightness.

The same inverse square pattern appears for gravitational and electric field strength in D.1 and D.2, and for the same geometric reason.

Confusing \(A\) in \(L = \sigma A T^{4}\) (the surface of the star, radius \(R\)) with \(4 \pi d^{2}\) (a sphere reaching to the observer).

Worked example

Sunlight at the Earth.

\(L = 3.9 \times 10^{26} \,\mathrm{W}\) · Earth–Sun distance \(d = 1.50 \times 10^{11} \,\mathrm{m}\) \(b = \dfrac{L}{4 \pi d^{2}}\) \(b = \dfrac{3.9 \times 10^{26}}{4 \pi \times (1.50 \times 10^{11})^{2}}\) \(b = \class{res}{1.4 \times 10^{3} \,\mathrm{W} \,\mathrm{m}^{-2}}\)

This is the solar constant, \(S\): \(1.36 \times 10^{3} \,\mathrm{W} \,\mathrm{m}^{-2}\) in the data booklet
At Mars, 1.5 times as far: \(\dfrac{1}{1.5^{2}} = 0.44\) of this

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Working backwards

Astronomers do this calculation in reverse. The brightness of sunlight can be measured directly, and the distance to the Sun is known, so \(L = 4 \pi d^{2} b\) gives the Sun’s luminosity. With the Sun’s radius, the Stefan–Boltzmann law then gives its surface temperature. That chain of reasoning is what Stefan used in 1879.

What the number means

1.4 kW falls on every square meter facing the Sun at the top of the atmosphere: about the power of a small electric heater. Less reaches the ground, because some is reflected and absorbed on the way down. B.2 starts from this number to work out the temperature of the Earth.

Forgetting to square the distance, or squaring only the 1.50 and not the \(10^{11}\).

Color and temperature

The black-body spectrum and Wien’s law.

\(\lambda_\mathrm{max} T = 2.9 \times 10^{-3} \,\mathrm{m} \,\mathrm{K}\)

A black body emits a continuous spectrum
\(\lambda_\mathrm{max}\): the wavelength of the peak
Hotter: a shorter peak wavelength

THREE TEMPERATURESFROM PLANCK’S LAW

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Reading the graph

The curves never cross: a hotter black body emits more than a cooler one at every wavelength. The area under a curve is the total power from each square meter, \(\sigma T^{4}\), so the 6000 K curve encloses five times the area of the 4000 K one, since \(1.5^{4} = 5.1\).

Color as a thermometer

A piece of iron in a fire glows dull red, then orange, then yellow-white as it heats, because the peak is moving from the infrared towards the visible. Stars work the same way: Betelgeuse, at about 3500 K, is red, the Sun is yellow-white, and Rigel, at about 12 000 K, is blue-white. Measuring the peak wavelength of a star gives its surface temperature without going there.

The shape is not symmetric

The curve falls sharply on the short-wavelength side and has a long tail on the other. A body at 4000 K emits a little blue light, but it emits far more infrared than a quick look at the peak suggests.

Reading “m K” as millikelvin. It is the meter kelvin: wavelength in meters times temperature in kelvin.

Worked example

A temperature from a peak wavelength.

Sunlight is most intense at 500 nm \(\lambda_\mathrm{max} T = 2.9 \times 10^{-3} \,\mathrm{m} \,\mathrm{K}\) \(T = \dfrac{2.9 \times 10^{-3}}{\lambda_\mathrm{max}} = \dfrac{2.9 \times 10^{-3}}{500 \times 10^{-9}}\) \(T = \class{res}{5.8 \times 10^{3} \,\mathrm{K}}\)

A lamp filament at 2800 K \(\lambda_\mathrm{max} = \dfrac{2.9 \times 10^{-3}}{T} = \dfrac{2.9 \times 10^{-3}}{2800}\) \(\lambda_\mathrm{max} = \class{res}{1.0 \times 10^{-6} \,\mathrm{m}}\) In the infrared: most of its output is not light

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Why filament lamps were replaced

The filament’s peak is at 1000 nm, beyond the red end of the visible spectrum at about 700 nm. Only the short-wavelength edge of its curve falls in the visible, so roughly 95% of the electrical energy leaves as infrared. An LED emits only in the visible and needs about a tenth of the power for the same light.

You are a black body too

Skin at 305 K has \(\lambda_\mathrm{max} = \frac{2.9 \times 10^{-3}}{305} = 9.5 \times 10^{-6}\) m, far into the infrared. Thermal cameras are built to detect wavelengths around 10 μm for this reason. The Earth’s surface, at about 288 K, peaks at almost the same wavelength, which is the starting point for the greenhouse effect in B.2.

Wavelengths are usually given in nm or μm. Convert to meters before using the constant.

Quick check

Thermal radiation.

01What is a black body?

It absorbs all the radiation falling on it

02The kelvin temperature of a black body doubles. Its power?

16 times as great

03A star is moved three times as far away. Its apparent brightness?

One ninth

04Which is hotter: a red star or a blue star?

The blue star

05A star has \(\lambda_\mathrm{max}\) = 290 nm. Its surface temperature?

\(\frac{2.9 \times 10^{-3}}{2.9 \times 10^{-7}} = 1.0 \times 10^{4} \,\mathrm{K}\)

06What is the unit of apparent brightness?

W m⁻²

PRESS → TO REVEAL EACH ANSWER

Summary

Thermal energy transfers on one slide.

Temperature

\(\bar{E}_\mathrm{k} = \frac{3}{2} k_\mathrm{B} T\)

In kelvin: \(T = \theta + 273\)
Internal energy = kinetic + potential

Heating

\(Q = mc \Delta T\)

The temperature changes
The kinetic energy changes

Changing phase

\(Q = mL\)

The temperature is constant
The potential energy changes

Conduction

\(\frac{\Delta Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x}\)

Convection: a fluid moves because its density changes

Black-body radiation

\(L = \sigma A T^{4}\)

Apparent brightness \(b = \frac{L}{4 \pi d^{2}}\)

Wien’s law

\(\lambda_\mathrm{max} T = 2.9 \times 10^{-3}\) m K

Hotter: a shorter peak wavelength

B.1 THERMAL ENERGY TRANSFERS · SL + HL

Summary

Key words.

Absolute temperatureIn kelvin; proportional to the average kinetic energy of a particle

Internal energyTotal random kinetic plus intermolecular potential energy

Thermal equilibriumSame temperature: no resultant transfer

Specific heat capacityEnergy to warm 1 kg by 1 K

Specific latent heatEnergy to change the phase of 1 kg at constant temperature

Thermal conductivityHow well a material conducts

Black bodyA perfect absorber and the best emitter

LuminosityTotal power radiated, in W

Apparent brightnessPower received per unit area, in W m⁻²

B.1 THERMAL ENERGY TRANSFERS · SL + HL