Thermal energy, temperature and kinetic theory of matter, methods of heat transfer (conduction, convection, radiation), specific heat capacity $Q = mc\Delta T$, and latent heat

What this dot point is asking

QCAA wants Year 11 students to apply kinetic theory and heat transfer concepts to thermal problems, and to use the specific heat capacity formula in calorimetry.

Kinetic theory of matter

All matter is made of particles (atoms or molecules) in constant random motion. Kinetic theory links the microscopic motion of these particles to the macroscopic quantities we measure (temperature, pressure and internal energy), and it is the conceptual spine of the whole topic.

• Temperature is a measure of the average translational kinetic energy of the particles. It is an intensive property, meaning it does not depend on how much substance is present. Two beakers of water at the same temperature have the same average particle kinetic energy regardless of volume. The absolute (kelvin) scale is built so that $0\ \text{K}$ is the point of minimum particle motion, and on the kelvin scale average kinetic energy is directly proportional to absolute temperature.
• Internal energy is the total of all the kinetic and potential energies of the particles in a sample. It is an extensive property, so it scales with the amount of substance. A large mass of warm water can hold far more internal energy than a small mass of very hot water, which is the key distinction QCAA tests.
• In a solid, strong bonds hold particles in a fixed lattice and they only vibrate about fixed positions. In a liquid, particles have enough energy to slide past one another while staying close together. In a gas, particles move freely in mostly straight-line paths between collisions, with negligible intermolecular forces. Heating a substance raises the average kinetic energy; at a phase boundary the added energy goes into potential energy (breaking bonds) instead of kinetic energy, so the temperature pauses.

Heat transfer

Heat is the transfer of thermal energy from a region of higher temperature to one of lower temperature. There are three mechanisms, and QCAA scenarios usually involve all three acting together.

Conduction

Energy flows through a material by particle vibration and collision, passing kinetic energy from hot particles to neighbouring cooler ones. Solids conduct best because their particles are close-packed; metals are exceptional conductors because free (delocalised) electrons carry energy rapidly through the lattice. Gases conduct poorly, which is why trapped air is a good insulator.

Convection

Heat is carried by the bulk movement of a fluid (liquid or gas). When a fluid is heated it expands, becomes less dense and rises, while cooler denser fluid sinks to replace it, setting up a convection current. Convection drives sea breezes, ocean currents and the circulation in a heated room, and it cannot occur in a solid.

Radiation

Energy is transferred by electromagnetic waves (predominantly infrared) and, uniquely, requires no medium, so it crosses a vacuum. Every object above $0\ \text{K}$ radiates. The Stefan-Boltzmann law gives the power radiated:

where $e$ is emissivity ($0$ to $1$), $\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$, $A$ is surface area and $T$ is the absolute temperature in kelvin. Because of the fourth-power dependence, a modest rise in absolute temperature produces a large rise in radiated power.

Specific heat capacity

The specific heat capacity $c$ of a substance is the energy needed to raise the temperature of $1\ \text{kg}$ by $1\ \text{K}$ (or $1^\circ\text{C}$, since a temperature interval is identical on both scales):

where $Q$ is energy in joules, $m$ is mass in kilograms, $c$ is specific heat capacity in $\text{J kg}^{-1}\text{K}^{-1}$ and $\Delta T$ is the temperature change. Water has an unusually high value, $c = 4186\ \text{J kg}^{-1}\text{K}^{-1}$, which is why it stores large amounts of energy, moderates coastal climates and is used as a coolant. Typical comparison values are $c_{\text{ice}} = 2100$, $c_{\text{aluminium}} = 900$ and $c_{\text{copper}} = 386\ \text{J kg}^{-1}\text{K}^{-1}$. A positive $Q$ means energy absorbed and temperature rises; a negative value means energy released and temperature falls.

Latent heat

During a phase change, energy is absorbed or released without any temperature change, because the energy goes into changing the potential energy of the particles (breaking or forming intermolecular bonds) rather than their kinetic energy.

• Latent heat of fusion $L_f$ is the energy per kilogram to melt a solid (or release on freezing). For water, $L_f = 3.34 \times 10^5\ \text{J kg}^{-1}$.
• Latent heat of vaporisation $L_v$ is the energy per kilogram to boil a liquid (or release on condensing). For water, $L_v = 2.26 \times 10^6\ \text{J kg}^{-1}$.

Vaporisation requires far more energy than fusion (about seven times for water) because all the intermolecular bonds must be fully broken to separate the particles into a gas, not merely loosened.

Calorimetry

In an insulated system no energy escapes, so energy is conserved and heat lost by the hotter bodies equals heat gained by the cooler bodies:

This is the principle behind calorimetry and the standard mixing problem. Write the conservation equation, substitute $mc\Delta T$ for each body, then solve for the unknown (usually the final temperature $T_f$ or an unknown specific heat). If a phase change occurs during mixing, an extra $mL$ term must be added on the appropriate side.

Examples in context

Example 1. A Townsville solar hot-water system raises $300 \text{ L}$ of water from $20^\circ \text{C}$ to $65^\circ \text{C}$ on a sunny day. Sensible heat is $Q = mc\Delta T = 300 \times 4186 \times 45 = 5.65 \times 10^7 \text{ J}$. Convection circulates water through the rooftop collector while conduction transfers solar-heated copper to the working fluid and radiation contributes about $30$ per cent of the collector input. The kinetic theory underpins all three: hotter water molecules vibrate and translate more vigorously, raising bulk temperature.

Example 2. ANSTO Mt Cotton's calibration lab uses a temperature-stable copper block of mass $5.0 \text{ kg}$ as a thermal anchor. Heating it through $1.0 \text{ K}$ requires $Q = 5.0 \times 390 \times 1 = 1950 \text{ J}$, providing slow thermal response that smooths satellite-tracker electronics through diurnal swings. The QCAA Unit 1 dot point binds the kinetic theory (high $c$ from many vibrational modes per mole) to the engineering choice of material.

Try this

Q1. State the kinetic theory definition of temperature. [2 marks]

• Cue. Measure of the average translational kinetic energy of the constituent particles.

Q2. A $0.40 \text{ kg}$ aluminium block ($c = 900 \text{ J kg}^{-1} \text{ K}^{-1}$) absorbs $7200 \text{ J}$. Calculate the temperature rise. [2 marks]

• Cue. $\Delta T = Q/(mc) = 7200/(0.40 \times 900) = 20 \text{ K}$.

Q3. Identify and analyse the three heat-transfer mechanisms in a Queensland slate-roofed home in summer. (a) Describe each mechanism in context. (b) Calculate the radiative loss from a $30 \text{ m}^2$ roof at $60^\circ \text{C}$ to a sky at $20^\circ \text{C}$ (emissivity $0.9$, $\sigma = 5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$). (c) Recommend two design measures. [3+3+2 marks; ISMG: Knowledge and conceptual understanding, Evaluation]

• Cue. (a) Conduction through tiles, convection in cavity, radiation to sky; (b) net $\approx 8.2 \text{ kW}$; (c) reflective foil, ridge venting.