Temperature and energy — Temperature describes thermal state and relates to average random particle kinetic energy in the ideal-gas model. Internal energy is the total microscopic random kinetic and interaction potential energy, so it also depends on amount and physical state. Two objects at equal temperature need not have equal internal energies; net thermal transfer follows a temperature difference.
Specific heat capacity — For a temperature change without a phase transition, ΔE = mcΔθ, with c in J kg⁻¹ K⁻¹. A temperature interval of 1 K equals 1°C, although their zero points differ. State the material and conditions and distinguish energy supplied electrically from the part actually absorbed by the sample.
Heating example — An illustrative 0.20 kg sample with c = 4200 J kg⁻¹ K⁻¹ warmed by 15 K gains 0.20 × 4200 × 15 = 12600 J. A 100 W heater would supply this in 126 s only if all its energy went into the sample and the power remained constant. Container heating and heat loss lengthen the actual time.
Latent heat — During a pure material's equilibrium phase change at fixed pressure, energy transfer can change microscopic potential energy without changing temperature. Specific latent heat L is energy per unit mass transformed, ΔE = LΔm, in J kg⁻¹. Melting and boiling generally have different values; a constant-temperature plateau does not mean that no energy is transferred.A qualitative temperature-versus-supplied-energy heating curve includes a constant-temperature phase-change interval at fixed pressure. The plateau still involves energy transfer, and a real curve needs losses and material conditions considered.
CP13 melting method — For a suitable melting sample near its melting point, measure the mass melted by a heater of measured power over a stated time, and use L ≈ VIt/Δm after relevant corrections. Collect melt in a weighed vessel and compare an unheated control or baseline melt rate. Ice can melt from room heat even before the heater is switched on.
CP13 boiling method — For a boiling liquid under the approved arrangement, measure mass decrease over a timed interval with known heater power. Subtract a background evaporation contribution or compare runs at different powers if appropriate. Account for heat transferred to the container and surroundings. Hot liquid, steam and electricity require the supervised school's controls and suitable secure apparatus.
Latent-heat processing — If 6.0 W of corrected power melts 0.0018 kg in 100 s, L = 600/0.0018 = 3.33 × 10⁵ J kg⁻¹. An uncorrected heat gain from the room makes measured melted mass too large for heater energy alone and the inferred L too low. Heat loss from a boiling sample can instead make an electrical-input estimate too high.
CP12 calibration arrangement — Place an NTC thermistor in a potential divider and compare its voltage output with a reference thermometer in a stirred water bath. Change temperature gradually, wait for thermal equilibrium and record simultaneous temperature and output readings across the relevant safe range. Keep electrical power low enough to limit sensor self-heating.
CP12 thermostat response — With a thermistor in the lower arm, warming lowers its resistance and output; upper-arm placement reverses the output trend across a fixed lower resistor. Plot an empirical calibration and select a voltage corresponding to the desired switching temperature. The response may be nonlinear, so interpolate the actual calibration rather than assume a constant voltage change per kelvin.
CP12 evaluation — Check the reference thermometer, bath uniformity, thermal lag, sensor immersion and voltage stability; repeat while heating and cooling to reveal lag or hysteresis. Keep low-voltage electrical parts appropriately isolated from the water and use the approved heater. A calibration relation is useful only for the actual sensor arrangement and conditions tested.
Internal energy ideal gas and kinetic theory
Absolute temperature — Kelvin temperature T = θ°C + 273.15. Absolute zero corresponds to the lower thermodynamic temperature limit; in the ideal classical gas model average translational kinetic energy tends to zero there. Do not claim real quantum particles have no motion or that a real gas stays ideal all the way to absolute zero.
Gas-model assumptions — Treat an ideal gas as many particles with negligible volume relative to the container, no interaction forces except during collisions, random motion in all directions and elastic collisions with walls and each other. These approximations work best at sufficiently low density and away from condensation; real gases have finite size and interactions.
Pressure from collisions — Pressure arises from momentum transfer to walls. For a particle of mass m moving between parallel walls of separation l with perpendicular component cx, each elastic collision transfers momentum 2mcx. Time between hits on the same wall is 2l/cx, giving its average force mcx²/l. Include the whole population to calculate total force.A particle component reverses elastically at a wall, changing from plus mcx to minus mcx. The wall receives the opposite momentum change of magnitude twice mcx. Incoming and outgoing paths are separated vertically for clarity, not an assertion of an angled physical collision.
Kinetic derivation — For a cube of volume V = l³, dividing the total force by wall area gives p = Nm〈cx²〉/V. Random isotropic motion gives 〈cx²〉 = 〈cy²〉 = 〈cz²〉 = 〈c²〉/3. Therefore pV = ⅓Nm〈c²〉. The one-third comes from three spatial directions, not from assuming every individual particle has equal instantaneous components.
Mean square and rms — 〈c²〉 is the average of squared particle speeds; crms = √〈c²〉. It differs from the square of mean speed. Net mean velocity can be zero for a gas at rest even while mean speed and mean kinetic energy are nonzero. Use particle mass rather than total gas mass in the per-particle expressions.
Ideal-gas equation — pV = NkT relates absolute pressure p, volume V, particle number N and Boltzmann constant k. Use T in K and p in Pa; a gauge reading relative to atmosphere needs conversion to absolute pressure by adding atmospheric pressure. For example, a gauge pressure of 20 kPa with atmospheric pressure 101 kPa gives 121 kPa absolute. For fixed N and T, pV is constant, while at fixed N and V, pressure is proportional to absolute temperature.
Molecular energy derivation — Equating pV = NkT and pV = ⅓Nm〈c²〉 gives ½m〈c²〉 = 3kT/2. Thus average translational kinetic energy is proportional to T and independent of particle species at the same T. A lighter particle species has larger rms speed at a fixed temperature, not greater average translational energy.
Gas worked example — For N = 2.0 × 10²³ particles at 300 K in 0.010 m³, p = NkT/V = 8.28 × 10⁴ Pa with k = 1.38 × 10⁻²³ J K⁻¹. Mean translational kinetic energy is 6.21 × 10⁻²¹ J per particle. These are illustrative ideal-gas values, not a measurement of a classroom container.
CP14 pressure-volume method — Use a trapped fixed amount of gas connected to a pressure sensor and a variable-volume chamber. Vary volume slowly, allow the gas to return to the same temperature and record absolute pressure. Ensure the total gas volume includes connecting tubing or any significant dead space, gas-filled apparatus volume that remains outside the variable chamber. Do not exceed the apparatus pressure range.
CP14 graphs and evaluation — Test p against 1/V or pV against V at constant temperature; the first should be linear through the appropriate origin for correctly included absolute variables, and the second approximately constant. Rapid compression can warm the gas and make pressure higher than the isothermal prediction. Leaks change N and dead-volume omission changes the inferred relationship.A proportional pressure-versus-reciprocal-total-volume plot tests the fixed-amount isothermal ideal-gas model. Absolute pressure and all significant trapped volume must be included before judging intercept and linearity.
Thermal versus bulk motion — Heating raises random microscopic kinetic energy in the gas model; moving a sealed container at constant speed adds bulk kinetic energy relative to the laboratory without changing its temperature by that fact alone. In a monatomic ideal gas, internal energy is the total translational random kinetic energy; more complex molecules can store energy in other modes.
Black-body radiation and thermal spectra
Black-body model — A black body absorbs all incident electromagnetic radiation and is an ideal thermal emitter. It emits a continuous spectrum determined by its absolute temperature; black does not mean it emits nothing. A cavity with a small opening can approximate the model, while real surfaces may have wavelength-dependent emissivity below one. Emissivity compares a surface's emitted radiation with that of a black body at the same temperature under the stated conditions.
Radiation curves — On a spectral-intensity-versus-wavelength graph, increasing temperature raises the emitted spectrum and moves its peak towards shorter wavelength. The integrated spectral area relates to total emitted power with the appropriate plotted units. A peak is not the only emitted wavelength, and a hotter object's spectrum need not stop overlapping a cooler object's spectrum.Two schematic wavelength spectra illustrate that a hotter black body has higher spectral output and a shorter peak wavelength. Curves are qualitative, without claimed Planck-law numerical scales; both include a broad wavelength range.
Stefan-Boltzmann relation — For a black-body surface area A at temperature T, total emitted power or luminosity L = σAT⁴. For a sphere A = 4πR², not projected area πR². At fixed area doubling T raises emitted power by sixteen. This is emitted power; net exchange with surroundings needs the absorbed incoming radiation accounted for.
Radiation calculation — Using σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, a black-body area 2.0 m² at 300 K emits 918.5 W. That is not automatically a 918.5 W net loss in a 300 K room, because it also receives radiation. For a real approximately grey surface, an emissivity factor may be needed if the problem provides one.
Wien displacement — For a black-body wavelength spectrum, λmaxT = 2.898 × 10⁻³ m K. At T = 5800 K, λmax = 5.00 × 10⁻⁷ m, about 500 nm. Use metres and kelvin. A colour-temperature estimate depends on how well the observed source fits the thermal model, not a claim that all radiation sources are black bodies.
Combining radiation evidence — Given a peak wavelength, estimate temperature with Wien's law; given total emitted luminosity, infer radiating area using Stefan-Boltzmann, then radius for a spherical source. An intensity measured far away also needs distance through inverse-square spreading. Keep surface emission, total luminosity and received intensity as distinct quantities and evaluate uncertainty and model assumptions.
Test yourself
50 questions · Random sets of 10. These quick checks support revision; practise longer explanations and justified judgements too.
Mr Kidman's Classroom
