Internal energy, temperature and thermal equilibrium (QCE Physics Unit 1)

What this dot point is asking

QCAA wants you to use the kinetic theory of matter to describe internal energy, temperature and the approach to thermal equilibrium, and to distinguish heat (an energy transfer) from temperature (a property of a body). Both ideas appear in EA Paper 1 multiple choice and as the qualitative spine of every Topic 1 problem.

Internal energy

The internal energy ($U$) of a substance is the total of the microscopic kinetic and potential energies of all its particles.

• Kinetic part: translation, rotation and vibration of particles (atoms or molecules).
• Potential part: stored energy in bonds and intermolecular forces.

Internal energy depends on the substance, its mass, its temperature, and its phase (solid, liquid, gas).

Temperature

Temperature is a measure of the average translational kinetic energy of the particles in a substance. The relationship for an ideal gas is

where $k_B = 1.38 \times 10^{-23}$ J K$^{-1}$ is Boltzmann's constant and $T$ is absolute temperature in kelvin.

Temperature does not depend on the number of particles. A single hot drop of water and an entire bathtub at the same temperature have the same average particle kinetic energy, but very different total internal energies.

Heat is not temperature

Heat ($Q$) is energy transferred between bodies because of a temperature difference. Heat is a process, not a property. A body does not "contain" heat; it contains internal energy. Heat flows from hot to cold spontaneously.

Heat has SI unit joule (J). Temperature has SI unit kelvin (K). Mixing the two in language (saying "the room contains a lot of heat") is the most common written-exam slip QCAA penalises.

Thermal equilibrium

Two bodies in thermal contact reach thermal equilibrium when their temperatures are equal. At that point, the average translational kinetic energy per particle is the same in both. There is no net heat flow.

The zeroth law of thermodynamics: if A is in equilibrium with B and B with C, then A is in equilibrium with C. This is what allows a thermometer to measure temperature; it equilibrates with the body and reads the common temperature.

Examples in context

Example 1. A Bundaberg sugar mill juice clarifier mixes $80^\circ \text{C}$ raw juice with $20^\circ \text{C}$ wash water at a $4:1$ flow ratio. Once the streams equalise temperature ($T_{eq}$), no net heat flows between them: they are in thermal equilibrium, so each kilogram of mixture has the same average translational kinetic energy. The bulk internal energy is the sum of the two streams' initial internal energies minus heat lost to the tank walls. The QCAA Unit 1 dot point links the macroscopic thermometer reading to the kinetic theory's microscopic definition of temperature.

Example 2. ANSTO Mt Cotton's cryogenic infrared detectors sit on a stage thermally anchored to liquid nitrogen at $77 \text{ K}$. When the detector is placed on the stage, heat flows from detector to anchor (higher to lower temperature) until the detector reaches $77 \text{ K}$ and is in thermal equilibrium with the bath. Internal energy is reduced as molecular kinetic energy drops by a factor of about $3.8$ relative to room temperature ($293/77$), suppressing thermal noise. The same principle distinguishing heat (energy in transit) from temperature is what makes the experiment work.

Try this

Q1. Define internal energy and distinguish heat from temperature. [3 marks]

• Cue. Internal energy = sum of microscopic kinetic and potential energies; heat = energy transferred due to temperature difference; temperature = measure of average translational KE.

Q2. $0.50 \text{ kg}$ of water at $80^\circ \text{C}$ is poured into a calorimeter holding $0.50 \text{ kg}$ of water at $20^\circ \text{C}$. Assuming no losses, calculate the equilibrium temperature. [3 marks]

• Cue. Equal masses, equal $c$: $T_{eq} = (80 + 20)/2 = 50^\circ \text{C}$.

Q3. A $0.20 \text{ kg}$ copper block at $150^\circ \text{C}$ ($c = 390 \text{ J kg}^{-1} \text{ K}^{-1}$) is dropped into $0.80 \text{ kg}$ of water at $20^\circ \text{C}$ ($c = 4186 \text{ J kg}^{-1} \text{ K}^{-1}$). (a) Calculate the equilibrium temperature. (b) Explain the direction of heat flow at the microscopic level. (c) Discuss one source of systematic error in such a QCAA IA1 student experiment. [4+2+2 marks; ISMG: Analysis and interpretation, Evaluation]

• Cue. (a) $T_{eq} \approx 23.0^\circ \text{C}$; (b) higher-KE copper atoms transfer KE via collisions; (c) heat loss to calorimeter walls.