B.1 Thermal energy transfers

This topic develops a particle-based picture of temperature, internal energy and the different ways thermal energy can move between and within objects. The goal is to connect microscopic motion to macroscopic quantities such as specific heat capacity, latent heat, luminosity and apparent brightness.

Core ideas

Molecular model — solids, liquids and gases are made of particles in constant random motion, with spacing and intermolecular forces explaining their different macroscopic properties.
Density — mass per unit volume; variations in density help explain buoyancy, layering and convection in fluids.
Temperature scales — Celsius is convenient for everyday use, while the Kelvin scale starts at absolute zero and is proportional to average random kinetic energy.
Internal energy — total microscopic energy of a system, including random kinetic and potential energy associated with the positions of its particles.

Energy transfer and phase change

Thermal energy flows from higher to lower temperature regions until thermal equilibrium is reached; the temperature difference sets the direction of transfer.
During heating without phase change, supplied energy increases internal energy by raising particle kinetic energy, so temperature rises according to the specific heat capacity.
During melting or boiling at constant temperature, added energy changes the potential energy of particles (latent heat) and enables a phase change without altering the average kinetic energy.

Conduction, convection and radiation

Conduction — energy transfer through collisions and vibrations in a material; good conductors pass on energy quickly, while insulators slow the process.
Convection — in fluids, heating reduces density so warmer regions rise and cooler, denser regions sink, creating circulation that carries energy.
Thermal radiation — all objects above absolute zero emit electromagnetic radiation; no material medium is needed for this transfer.

Stefan–Boltzmann, luminosity and Wien’s law

The Stefan–Boltzmann law shows that an ideal black body radiates power proportional to the fourth power of its absolute temperature and directly proportional to its surface area.
For stars, luminosity is the total power output, while apparent brightness at a distance drops with the square of the distance because radiation spreads over a growing spherical surface.
Wien’s displacement law links surface temperature to the wavelength at which the emission spectrum peaks, explaining why hotter stars emit more strongly at shorter wavelengths.

Important formulas

\rho = \dfrac{m}{V}

Density: mass per unit volume, useful for comparing solids, liquids and gases.

E_{\text{int}} = E_{\text{k,random}} + E_{\text{p,intermolecular}}

Internal energy as the sum of random kinetic and potential energies of particles.

E_{\text{therm}} = mc\Delta T

Thermal energy transferred when a substance of mass m changes temperature by \Delta T.

Q = mL

Energy involved in a phase change at constant temperature (latent heat).

P = \dfrac{Q}{\Delta t}

Rate of thermal energy transfer, often called thermal power.

H = kA\dfrac{\Delta T}{L}

Conduction rate through a slab of thickness L and area A with thermal conductivity k.

P = \sigma A T^{4}

Stefan–Boltzmann law for the power radiated by an ideal black body of area A.

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

Apparent brightness b of a source with luminosity L at distance d.

\lambda_{\text{max}} T = 2.9\times10^{-3}\ \text{m K}

Wien’s displacement law for the peak wavelength of black‑body radiation.

Practice problems

Density and states of matter

A solid metal block has a mass of

1.20\ \mathrm{kg}

and a volume of

1.5\times10^{-4}\ \mathrm{m^{3}}.

Calculate its density and comment on whether it is likely to be aluminium \(2.7\times10^{3}\ \mathrm{kg\,m^{-3}}\) or copper \(8.9\times10^{3}\ \mathrm{kg\,m^{-3}}\).

Kelvin scale and kinetic energy

The temperature of a gas sample increases from

15\ ^{\circ}\mathrm{C}

45\ ^{\circ}\mathrm{C}.

State the change in temperature on the Kelvin scale and explain what this implies for the average random kinetic energy of the molecules.

Heating without phase change

0.80\ \mathrm{kg}

block of ice at

-10\ ^{\circ}\mathrm{C}

is warmed to

0\ ^{\circ}\mathrm{C}

without melting. The specific heat capacity of ice is

2100\ \mathrm{J\,kg^{-1}\,K^{-1}}.

Calculate the thermal energy transferred to the ice.

Melting at constant temperature

0\ ^{\circ}\mathrm{C}

the same

0.80\ \mathrm{kg}

block of ice then melts completely to water at

0\ ^{\circ}\mathrm{C}.

The specific latent heat of fusion of ice is

3.3\times10^{5}\ \mathrm{J\,kg^{-1}}.

Calculate the additional energy required and explain why the temperature does not rise during the melting process.

Conduction through a window

A glass window is

0.80\ \mathrm{m^{2}}

in area and

5.0\ \mathrm{mm}

thick. The inner surface is at

20\ ^{\circ}\mathrm{C}

and the outer surface is at

5\ ^{\circ}\mathrm{C}.

The thermal conductivity of glass is

0.80\ \mathrm{W\,m^{-1}\,K^{-1}}.

Calculate the rate of thermal energy transfer through the window by conduction.

Apparent brightness and distance

A star has luminosity

L = 3.0\times10^{26}\ \mathrm{W}.

An observer measures its apparent brightness to be

b = 1.2\times10^{-9}\ \mathrm{W\,m^{-2}}.

Assuming the star radiates uniformly in all directions, calculate its distance from the observer.

Wien’s law and peak wavelength

A distant star can be modelled as a black body with surface temperature

T = 5800\ \text{K}.

Estimate the wavelength at which its emission spectrum has maximum intensity using Wien’s displacement law.

Clarity tip: For every thermal question, start by identifying the system, the process (heating, cooling or phase change) and the dominant transfer mechanism, then match these to the simplest energy or power relation.

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