IB Diploma ProgrammeHL

Thermal energy transfers

Physics HL Chapter Notes

What this chapter covers

Thermal energy transfersGreenhouse effectGas lawsThermodynamicsCurrent and circuits
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In this chapter

  1. 1Particles, Temperature, and Internal Energy
  2. 2Specific Heat Capacity
  3. 3Phase Changes and Latent Heat
  4. 4Conduction and Convection
  5. 5Thermal Radiation
  6. 6Laws of Thermal Radiation

1. Particles, Temperature, and Internal Energy

All matter is made of tiny particles (atoms and molecules) that are in constant, random motion. In solids, particles vibrate in fixed positions within a lattice. In liquids, particles can slide past one another but remain in close contact. In gases, particles move rapidly and randomly, far apart from each other. Temperature is a measure of how hot or cold an object is. On a microscopic level, the absolute temperature (measured in Kelvin, K) of a substance is directly proportional to the average random kinetic energy of its particles. The faster the particles move or vibrate on average, the higher the temperature. Internal energy is the total energy stored within a substance. It is the sum of the random kinetic energies of all its particles and the total intermolecular potential energy. Potential energy is associated with the forces between particles; it increases when particles move further apart (e.g., during melting or boiling). It's crucial to distinguish these concepts: Temperature is an average, Internal Energy is a total, and Heat is energy transferred due to a temperature difference.

T(K) = T(°C) + 273.15

Average E_k ∝ T

Key term

Internal Energy: The sum of the random distribution of kinetic and potential energies of the particles in a system.

Examiner insight

Examiners reward clear distinctions between internal energy (a total energy of particles), temperature (a measure of average kinetic energy), and heat (energy transferred due to a temperature difference).

Common pitfall

Confusing heat with internal energy. An object contains internal energy, it does not 'contain' heat. Heat is the name for energy in transit from a hot object to a cold one.

Worked example 14 marks

A beaker of water is at a temperature of 20 °C.(a) What is its temperature in Kelvin?(b) A much larger swimming pool is also at 20 °C. Compare the temperature and internal energy of the water in the beaker and the pool.

  1. 1

    (a) To convert from Celsius to Kelvin, we use the formula T(K) = T(°C) + 273.15.

  2. 2

    T(K) = 20 + 273.15 = 293.15 K. To 3 significant figures, this is 293 K.

  3. 3

    (b) Temperature: The temperature of the water in the beaker and the pool is the same (20 °C or 293 K). This means the average kinetic energy of the water molecules is the same in both.

  4. 4

    Internal Energy: The swimming pool contains vastly more water molecules than the beaker. Since internal energy is the total energy of all molecules, the swimming pool has a much greater internal energy than the beaker, even though they are at the same temperature.

Recap

  • Matter consists of particles in continuous random motion.
  • Solids have particles vibrating in fixed positions; liquids have particles sliding past each other; gases have particles moving randomly and far apart.
  • Absolute temperature is a measure of the average random kinetic energy of the particles.
  • Internal energy is the sum of the total random kinetic and potential energies of the particles.
  • Thermal energy (heat) is energy transferred from a hotter object to a colder one.

Quick check

  1. Distinguish between the internal energy of a substance and its temperature.2 marks
  2. Convert a room temperature of 25 °C to Kelvin.1 mark

2. Specific Heat Capacity

When you add thermal energy to a substance, its temperature usually increases. The specific heat capacity, 'c', tells you how much energy is needed to do this. It is defined as the thermal energy required to raise the temperature of 1 kg of a substance by 1 Kelvin (or 1 degree Celsius) without it changing state. Substances with a high specific heat capacity, like water, require a lot of energy to heat up and can store a lot of energy. The relationship is given by the formula Q = mcΔT, where Q is the thermal energy transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. This principle is the basis of calorimetry, where we measure heat transfers by mixing substances and applying the conservation of energy: the thermal energy lost by the hotter object equals the thermal energy gained by the colder object, assuming no heat is lost to the surroundings.

Q = mcΔT

Key term

Specific Heat Capacity (c): The amount of thermal energy required to raise the temperature of a unit mass (1 kg) of a substance by one degree (1 K or 1°C).

Common pitfall

Using Celsius in the formula Q = mcΔT is acceptable for the temperature change ΔT, but for other thermal physics laws (like Stefan-Boltzmann), you must use Kelvin.

Fun fact

The high specific heat capacity of water is why coastal areas have more moderate climates than inland areas; the ocean absorbs huge amounts of heat in summer and releases it slowly in winter.

Worked example 14 marks

A 0.50 kg block of aluminium is heated by an electric heater with a power of 100 W for 1 minute. The temperature of the block rises from 20 °C to 33.2 °C. Calculate the specific heat capacity of aluminium. Assume no heat is lost to the surroundings.

  1. 1

    Step 1: Calculate the energy supplied by the heater (Q).

  2. 2

    Power = Energy / time, so Energy (Q) = Power × time.

  3. 3

    Time = 1 minute = 60 s.

  4. 4

    Q = 100 W × 60 s = 6000 J.

  5. 5

    Step 2: Calculate the temperature change (ΔT).

  6. 6

    ΔT = Final temperature - Initial temperature = 33.2 °C - 20 °C = 13.2 °C (which is also 13.2 K).

  7. 7

    Step 3: Rearrange the specific heat capacity formula and solve for c.

  8. 8

    Q = mcΔT => c = Q / (mΔT).

  9. 9

    c = 6000 J / (0.50 kg × 13.2 K) = 909.09... J kg⁻¹ K⁻¹.

  10. 10

    c ≈ 910 J kg⁻¹ K⁻¹ (to 2 significant figures, matching the power).

Worked example 24 marks

200 g of water at 80 °C is mixed with 300 g of water at 20 °C in an insulated container. What is the final temperature of the mixture?

  1. 1

    Step 1: Apply the principle of conservation of energy.

  2. 2

    Heat lost by hot water = Heat gained by cold water.

  3. 3

    Let the final temperature be T_f.

  4. 4

    Step 2: Write the expressions for heat lost and heat gained.

  5. 5

    Heat lost = m_hot * c * ΔT_hot = 0.200 kg * c * (80 - T_f).

  6. 6

    Heat gained = m_cold * c * ΔT_cold = 0.300 kg * c * (T_f - 20).

  7. 7

    Step 3: Equate the expressions and solve for T_f. The specific heat capacity 'c' is the same for both and cancels out.

  8. 8

    0.200 * (80 - T_f) = 0.300 * (T_f - 20).

  9. 9

    16 - 0.200*T_f = 0.300*T_f - 6.

  10. 10

    16 + 6 = 0.300*T_f + 0.200*T_f.

  11. 11

    22 = 0.500*T_f.

  12. 12

    T_f = 22 / 0.500 = 44 °C.

Recap

  • Specific heat capacity (c) is the energy needed to heat 1 kg of a substance by 1 K.
  • The formula for thermal energy transfer during a temperature change is Q = mcΔT.
  • A change in temperature of 1°C is equal to a change in temperature of 1 K.
  • In calorimetry, for an isolated system, the heat lost by hotter objects equals the heat gained by colder objects.
  • Water has a very high specific heat capacity, making it an excellent coolant.

Quick check

  1. How much energy is needed to heat 2.0 kg of copper from 10 °C to 110 °C? (c_copper = 385 J kg⁻¹ K⁻¹)2 marks

3. Phase Changes and Latent Heat

When a substance changes phase, for example from solid to liquid (melting) or liquid to gas (boiling), thermal energy is absorbed but the temperature does not change. This energy is called latent heat. Instead of increasing the kinetic energy of the particles (which would raise the temperature), the energy is used to do work against the intermolecular forces, breaking the bonds holding the particles together and thus increasing their potential energy. The specific latent heat, 'L', is the energy required to change the state of 1 kg of a substance at a constant temperature. The formula is Q = mL, where Q is the energy transferred and m is the mass. We distinguish between the specific latent heat of fusion (L_f) for melting/freezing and the specific latent heat of vaporisation (L_v) for boiling/condensing. L_v is always significantly larger than L_f because breaking bonds completely to form a gas requires much more energy than just loosening them to form a liquid.

Q = mL

Key term

Specific Latent Heat (L): The amount of thermal energy required to change the state of a unit mass (1 kg) of a substance at a constant temperature.

Examiner insight

Students must explicitly state that during a phase change, the supplied energy increases the potential energy of the molecules while the average kinetic energy remains constant, which is why temperature stays constant.

Common pitfall

Using the specific heat capacity formula (Q=mcΔT) for a phase change, or the latent heat formula (Q=mL) for a temperature change. Always check if the state is changing or if the temperature is changing.

Worked example 12 marks

Calculate the thermal energy required to turn 0.50 kg of ice at 0 °C into water at 0 °C. The specific latent heat of fusion of ice is 3.34 × 10⁵ J kg⁻¹.

  1. 1

    Step 1: Identify the process. This is a phase change (melting) at a constant temperature.

  2. 2

    Step 2: Select the correct formula, which is Q = mL_f.

  3. 3

    Step 3: Substitute the values and calculate Q.

  4. 4

    Q = 0.50 kg × (3.34 × 10⁵ J kg⁻¹).

  5. 5

    Q = 1.67 × 10⁵ J or 167 kJ.

Worked example 25 marks

How much energy is required to convert 100 g of ice at -10 °C to steam at 100 °C? (c_ice = 2100 J kg⁻¹ K⁻¹, c_water = 4186 J kg⁻¹ K⁻¹, L_f = 3.34 × 10⁵ J kg⁻¹, L_v = 2.26 × 10⁶ J kg⁻¹)

  1. 1

    This is a multi-step problem. We must calculate the energy for each stage separately. Mass m = 100 g = 0.100 kg.

  2. 2

    Step 1: Heat ice from -10 °C to 0 °C. (Q₁ = mc_iceΔT)

  3. 3

    Q₁ = 0.100 kg × 2100 J kg⁻¹ K⁻¹ × (0 - (-10)) K = 2100 J.

  4. 4

    Step 2: Melt ice at 0 °C. (Q₂ = mL_f)

  5. 5

    Q₂ = 0.100 kg × 3.34 × 10⁵ J kg⁻¹ = 33400 J.

  6. 6

    Step 3: Heat water from 0 °C to 100 °C. (Q₃ = mc_waterΔT)

  7. 7

    Q₃ = 0.100 kg × 4186 J kg⁻¹ K⁻¹ × (100 - 0) K = 41860 J.

  8. 8

    Step 4: Boil water at 100 °C to produce steam. (Q₄ = mL_v)

  9. 9

    Q₄ = 0.100 kg × 2.26 × 10⁶ J kg⁻¹ = 226000 J.

  10. 10

    Step 5: Sum the energies for the total energy required.

  11. 11

    Q_total = Q₁ + Q₂ + Q₃ + Q₄ = 2100 + 33400 + 41860 + 226000 = 303360 J.

  12. 12

    Q_total ≈ 303 kJ.

Recap

  • During a phase change, temperature remains constant.
  • The energy supplied during a phase change increases the potential energy of the molecules, not their kinetic energy.
  • Specific latent heat (L) is the energy per unit mass for a phase change: Q = mL.
  • Latent heat of vaporisation (boiling) is much larger than the latent heat of fusion (melting).
  • A heating curve (temperature vs. energy) has flat plateaus during phase changes.

Quick check

  1. Why does a steam burn feel much worse than a burn from boiling water at the same temperature?2 marks

4. Conduction and Convection

Thermal energy can be transferred in several ways. Conduction and convection both require a medium.

Conduction is the transfer of thermal energy through a substance by particle-to-particle collisions, without any overall movement of the substance. In non-metallic solids (insulators), particles vibrate and pass this vibrational energy to their neighbours. In metals, which are excellent conductors, the process is much more efficient. The vast number of 'free' electrons can move throughout the metal, rapidly transferring kinetic energy when they collide with the fixed positive ions.

Convection is the primary method of heat transfer in fluids (liquids and gases). When a part of a fluid is heated, it expands and becomes less dense. Due to buoyancy, this hotter, less dense fluid rises. Cooler, denser fluid from above sinks to take its place, gets heated, and also rises. This process creates a continuous circulation of fluid, called a convection current, which transfers thermal energy through the bulk movement of the fluid itself.

Key term

Convection: The transfer of thermal energy through a fluid (liquid or gas) caused by the bulk movement of hotter, less dense fluid rising and cooler, denser fluid sinking.

Examiner insight

Examiners look for a clear description of the mechanism at the particle level, for example, mentioning 'free electrons' for conduction in metals or 'changes in density' for convection.

Common pitfall

Stating that 'heat rises'. It is the hot, less dense fluid that rises, carrying thermal energy with it. Heat itself does not rise.

Worked example 14 marks

Explain, in terms of particles, why a metal spoon left in a cup of hot coffee quickly becomes hot, while a plastic spoon does not.

  1. 1

    Step 1: Identify the primary heat transfer mechanism, which is conduction.

  2. 2

    Step 2: Explain conduction in the metal spoon. The metal spoon is a good conductor. It contains a lattice of positive ions and a 'sea' of free, delocalised electrons.

  3. 3

    The hot coffee transfers energy to the end of the spoon, causing the ions and free electrons to gain kinetic energy.

  4. 4

    These high-energy free electrons move rapidly throughout the spoon, colliding with ions further up the handle and transferring energy to them. This is a very efficient process.

  5. 5

    Step 3: Explain conduction in the plastic spoon. Plastic is an insulator. It does not have free electrons.

  6. 6

    Energy is transferred only by adjacent molecules vibrating and passing the energy along to their neighbours. This is a much slower and less efficient process, so the handle of the plastic spoon stays cool.

Worked example 24 marks

A domestic radiator is used to heat a room. It is placed on a wall near the floor. Explain how it heats the air in the whole room.

  1. 1

    Step 1: Identify the main mechanism for heat transfer through the air, which is convection.

  2. 2

    Step 2: Describe the process. The radiator heats the air in direct contact with it via conduction.

  3. 3

    Step 3: This heated air expands, becomes less dense than the surrounding cooler air, and therefore rises.

  4. 4

    Step 4: Cooler, denser air from the top of the room sinks to take its place near the radiator, where it is then heated.

  5. 5

    Step 5: This sets up a continuous circulation of air, known as a convection current, which distributes the thermal energy throughout the room.

Recap

  • Conduction is heat transfer via particle collisions without the substance moving.
  • Metals are good conductors due to free electrons; insulators transfer heat via lattice vibrations.
  • Convection is heat transfer in fluids via the bulk movement of the fluid.
  • Convection occurs because heating a fluid changes its density, causing it to rise or sink.
  • A convection current is the continuous circulation of a fluid as it is heated and cooled.

Quick check

  1. Why can't conduction or convection happen in a vacuum?1 mark
  2. Why are the heating elements in electric kettles placed at the bottom?2 marks

5. Thermal Radiation

Radiation is the third method of thermal energy transfer. Unlike conduction and convection, it does not require a medium and is the only way energy can travel through a vacuum. Thermal energy is transferred as electromagnetic (EM) waves, primarily in the infrared part of the spectrum. All objects with a temperature above absolute zero (0 K) emit thermal radiation. The hotter an object is, the more radiation it emits per second. The nature of an object's surface also affects how it emits and absorbs radiation. Dark, matt surfaces are both good absorbers and good emitters of radiation. Light-coloured, shiny surfaces are poor absorbers and poor emitters (they are good reflectors). This is why, for example, a black car gets hotter in the sun than a white car, and why emergency blankets are shiny to reflect body heat back to the person.

Key term

Thermal Radiation: The transfer of energy via electromagnetic waves, which does not require a medium to travel through.

Common pitfall

Confusing thermal radiation with nuclear radiation (radioactivity). They are completely different physical processes.

Fun fact

Infrared cameras work by detecting the thermal radiation emitted by objects, allowing us to 'see' temperature differences. This is used by firefighters to find people in smoke-filled rooms and by engineers to spot overheating equipment.

Worked example 13 marks

Two identical metal cubes, one painted matt black and the other painted shiny white, are heated in an oven to 100 °C. They are then removed and placed in a cool room.(a) Which cube initially cools down faster?(b) Explain your answer.

  1. 1

    (a) The matt black cube will cool down faster.

  2. 2

    (b) Explanation: Both cubes lose heat to the room by radiation. The rate of cooling depends on the rate of emission of thermal radiation.

  3. 3

    Matt black surfaces are very good emitters of thermal radiation.

  4. 4

    Shiny white surfaces are poor emitters of thermal radiation.

  5. 5

    Since the black cube emits energy at a higher rate, its temperature will fall more quickly.

Worked example 24 marks

Explain why a vacuum flask (Thermos) is effective at keeping hot liquids hot.

  1. 1

    A vacuum flask is designed to minimise all three types of heat transfer.

  2. 2
    1. Conduction/Convection: The flask has a double-walled container with a vacuum between the walls. Since conduction and convection require a medium (particles), the vacuum prevents heat transfer by these methods from the inner container to the outer casing.
  3. 3
    1. Radiation: The inner and outer walls of the vacuum gap are silvered (shiny). A hot liquid inside radiates heat. The shiny inner surface is a poor emitter, and it reflects radiated heat back towards the liquid. Any heat radiated across the vacuum is reflected by the shiny outer surface, which is also a poor absorber.
  4. 4
    1. Conduction: The stopper is usually made of plastic or cork, which are poor conductors, to reduce heat loss through the opening.

Recap

  • Thermal radiation is energy transfer by electromagnetic waves (mainly infrared).
  • Radiation is the only form of heat transfer that can occur in a vacuum.
  • All objects above absolute zero emit thermal radiation.
  • Dark, matt surfaces are good absorbers and good emitters of radiation.
  • Light, shiny surfaces are poor absorbers and poor emitters (good reflectors).

Quick check

  1. How does the Sun's energy reach the Earth?1 mark
  2. To keep a baked potato hot for as long as possible, should you wrap it in shiny aluminium foil with the shiny side facing in or out? Explain.2 marks

6. Laws of Thermal Radiation

The radiation emitted by an object is described by two key laws. An idealised object called a black body absorbs all radiation that falls on it and is also a perfect emitter. Stars are often approximated as black bodies.

Wien's Displacement Law states that the peak wavelength (λ_max) at which a black body emits the most radiation is inversely proportional to its absolute temperature (T). The formula is λ_max = constant / T. This means hotter objects emit radiation with a shorter peak wavelength, shifting from red to orange to yellow to white and finally to blue as temperature increases.

The Stefan-Boltzmann Law describes the total power radiated. It states that the total power (P) radiated by a black body is proportional to its surface area (A) and the fourth power of its absolute temperature (T⁴). The formula is P = σAT⁴, where σ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴). For real objects, we include an emissivity factor 'e' (P = eσAT⁴), where e is between 0 and 1. For a star, this total radiated power is called its Luminosity (L). As this light travels outwards, it spreads over a larger area. The power per unit area we receive is called apparent brightness (b), which follows an inverse square law: b = L / (4πd²), where d is the distance to the star.

λ_max T = 2.90 × 10⁻³ m K

P = σAT⁴ (for a black body)

L = σAT⁴

b = L / (4πd²)

Key term

Black Body: An idealised object that absorbs all incident electromagnetic radiation and emits radiation with a spectrum determined solely by its temperature.

Examiner insight

Candidates must use absolute temperature (in Kelvin) in all calculations involving Wien's Law and the Stefan-Boltzmann Law. Forgetting to convert from Celsius is a common source of lost marks.

Common pitfall

Forgetting that the Stefan-Boltzmann law depends on T⁴. Doubling the absolute temperature increases the radiated power by a factor of 2⁴ = 16, not 2 or 4.

Worked example 13 marks

The surface of the Sun has a temperature of approximately 5800 K. Calculate the peak wavelength of the radiation it emits. In which part of the electromagnetic spectrum does this lie? (Wien's constant = 2.90 × 10⁻³ m K)

  1. 1

    Step 1: State Wien's Displacement Law.

  2. 2

    λ_max T = 2.90 × 10⁻³ m K.

  3. 3

    Step 2: Rearrange the formula to solve for λ_max.

  4. 4

    λ_max = (2.90 × 10⁻³) / T.

  5. 5

    Step 3: Substitute the temperature in Kelvin and calculate.

  6. 6

    λ_max = (2.90 × 10⁻³) / 5800 = 5.0 × 10⁻⁷ m.

  7. 7

    Step 4: Convert to nanometres and identify the part of the spectrum.

  8. 8

    λ_max = 500 nm. This lies in the visible light part of the spectrum (green light).

Worked example 24 marks

The star Rigel has a surface temperature of 12,000 K and a radius of 5.4 × 10¹⁰ m. Assuming it behaves as a perfect black body, calculate its luminosity (total power output). (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

  1. 1

    Step 1: State the Stefan-Boltzmann Law for luminosity.

  2. 2

    L = σAT⁴.

  3. 3

    Step 2: Calculate the surface area (A) of the star. A = 4πr².

  4. 4

    A = 4π * (5.4 × 10¹⁰ m)² = 3.664 × 10²² m².

  5. 5

    Step 3: Substitute the values for A, T, and σ into the luminosity formula. Remember T must be in Kelvin.

  6. 6

    L = (5.67 × 10⁻⁸ W m⁻² K⁻⁴) × (3.664 × 10²² m²) × (12000 K)⁴.

  7. 7

    L = (5.67 × 10⁻⁸) × (3.664 × 10²²) × (2.0736 × 10¹⁶).

  8. 8

    L = 4.3 × 10³¹ W.

Recap

  • Wien's Law relates peak wavelength to absolute temperature: hotter objects emit at shorter wavelengths.
  • The Stefan-Boltzmann Law relates total radiated power to surface area and the fourth power of absolute temperature (P ∝ AT⁴).
  • Luminosity (L) is the total power radiated by a star.
  • Apparent brightness (b) is the power received per unit area and decreases with distance squared (b ∝ 1/d²).
  • All calculations with these laws must use absolute temperature in Kelvin.

Quick check

  1. A star has a peak emission wavelength of 290 nm. What is its surface temperature?2 marks
  2. If two stars have the same size, but Star A is twice as hot (in Kelvin) as Star B, how many times more luminous is Star A?2 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. The boiling point of nitrogen is 77 K. What is this temperature in degrees Celsius (°C)?2 marks
  2. A 2.0 kW kettle is used to heat 1.5 kg of water from 15 °C to 95 °C. Calculate the time this will take, assuming all the electrical energy is transferred to the water. (Specific heat capacity of water = 4200 J kg⁻¹ K⁻¹).4 marks
  3. A 60 W immersion heater is used to boil water in a cup. In 2 minutes, it is found that 5.0 g of water has been turned to steam. Calculate the specific latent heat of vaporisation of water.3 marks
  4. A 500 g block of an unknown metal at 100 °C is dropped into 1.2 kg of water at 20 °C in an insulated container. The final temperature of the mixture is 22.5 °C. Calculate the specific heat capacity of the metal. (Specific heat capacity of water = 4200 J kg⁻¹ K⁻¹).5 marks
  5. A student wants to make a cold drink. They add 40 g of ice at 0 °C to 200 g of a drink at 25 °C. Determine the final temperature of the mixture after all the ice has melted. (Specific heat capacity of the drink = 4200 J kg⁻¹ K⁻¹, specific latent heat of fusion of ice = 3.3 × 10⁵ J kg⁻¹).5 marks
  6. Explain the formation of a sea breeze on a hot, sunny day using the concepts of specific heat capacity and convection.4 marks
  7. The star Betelgeuse has a surface temperature of 3500 K. Calculate the peak wavelength of its emitted radiation. (Wien's constant = 2.90 × 10⁻³ m K).2 marks
  8. A person has a skin temperature of 33 °C and a body surface area of 1.7 m². Assuming the person radiates as a black body, calculate the power they radiate. (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴).4 marks
  9. The Sun has a luminosity of 3.8 × 10²⁶ W and a surface temperature of 5800 K. Calculate its radius. (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴).4 marks
  10. Compare and contrast the three mechanisms of thermal energy transfer: conduction, convection, and radiation. For each, state the medium required (if any) and the basic transfer process.6 marks

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