Thermal Energy Transfers

The kinetic model treats all matter as an enormous number of tiny particles in constant random motion, with the phase of a substance set by how the particles are spaced, ordered and able to move.

In a solid, strong intermolecular forces lock particles into a closely packed, often regular arrangement; each particle can only vibrate about a fixed position, so solids keep both their shape and volume.

In a liquid, the particles remain almost as close together, but they carry enough energy to slide past their neighbours, so liquids flow and take the shape of their container while keeping a nearly fixed volume.

In a gas, particles are typically separated by around ten particle diameters, travel rapidly in straight lines between collisions, and feel negligible intermolecular forces except at the instant of collision, so a gas expands to fill any container and has a much lower density.

• Solid: strong forces, vibration only, fixed shape and volume.
• Liquid: forces hold particles close but not in place; fixed volume, variable shape.
• Gas: negligible forces, fast random motion, no fixed shape or volume.

A concrete comparison: $1\ \text{kg}$ of steam occupies roughly a thousand times the volume of $1\ \text{kg}$ of liquid water, which the model explains purely through particle spacing.

You should be able to use particle arrangement, spacing and motion to compare densities and explain compressibility and flow in each phase.

Temperature is the macroscopic property that decides the direction of thermal energy transfer: energy always flows spontaneously from the hotter body to the colder one until both reach the same temperature (thermal equilibrium).

Microscopically, the absolute temperature of a substance is directly proportional to the average random translational kinetic energy of its particles:

$$\bar{E}_k = \frac{3}{2} k_B T$$

Doubling the Kelvin temperature therefore doubles the average kinetic energy per particle.

Internal energy is a different quantity: it is the total of the random kinetic energies *and* the intermolecular potential energies of every particle in the system.

A swimming pool at $25^\circ\text{C}$ has a far greater internal energy than a cup of tea at $90^\circ\text{C}$, simply because it contains vastly more particles, yet thermal energy would still flow from the tea to the pool.

The Kelvin scale is the absolute scale required for these proportionalities. Convert using:

$$T(\text{K}) = \theta(^\circ\text{C}) + 273$$

Absolute zero, $0\ \text{K} = -273.15^\circ\text{C}$, is the temperature at which particle kinetic energy reaches its minimum possible value.

Note that a temperature *difference* has the same numerical value in kelvin and in degrees Celsius, but any formula containing $T$ itself (such as $\bar{E}_k = \frac{3}{2} k_B T$) demands Kelvin.

Specific heat capacity $c$ measures how much thermal energy must be supplied per kilogram of a substance to raise its temperature by one kelvin, with no change of phase. The defining relationship is:

$$Q = mc\Delta T$$

where $Q$ is the energy transferred, $m$ the mass and $\Delta T$ the temperature change.

Substances differ enormously: water's value, about $4180\ \text{J kg}^{-1}\text{K}^{-1}$, is roughly ten times that of copper, which is why coastal climates are moderated by the sea and why water is used as a coolant in car engines and central heating systems.

In the laboratory, $c$ is usually found electrically: an immersion heater of known power $P$ warms a measured mass for a time $t$, so that $Pt = mc\Delta T$ once energy losses are accounted for.

Real experiments always lose some energy to the surroundings and to the container, so the measured value of $c$ tends to come out too large unless insulation or a cooling correction is used.

You must be able to:

• rearrange $Q = mc\Delta T$ for any variable
• combine it with electrical power ($Q = Pt$)
• handle mixing problems where energy lost by the hot object equals energy gained by the cold one.

During melting, freezing, boiling or condensing, a substance changes phase at a constant temperature. The energy involved is given by:

$$Q = mL$$

where $L$ is the specific latent heat: of fusion $L_f$ for solid–liquid changes, of vaporization $L_v$ for liquid–gas changes.

For water, $L_v \approx 2.26 \times 10^6\ \text{J kg}^{-1}$ is nearly seven times $L_f \approx 3.33 \times 10^5\ \text{J kg}^{-1}$, because vaporization must separate the molecules almost completely and do work pushing back the atmosphere.

The constant temperature is explained microscopically by energy bookkeeping. Particles in a solid oscillate in deep potential energy wells created by strong electrostatic attraction; supplying thermal energy normally increases the amplitude of these vibrations, raising the random kinetic energy and hence the temperature.

During melting, however, the supplied energy is used to do work against the attractive forces, pulling the particles slightly further apart and raising their intermolecular potential energy while their average kinetic energy stays constant.

This is why a heating curve shows a horizontal plateau at each phase change: the energy is spent climbing out of the potential energy well, not speeding the particles up.

On a heating or cooling curve, apply $Q = mc\Delta T$ on the sloped sections and $Q = mL$ on the plateaus, and be ready to chain several stages together in one calculation.

Thermal energy moves from hot to cold by three distinct mechanisms.

Conduction occurs mainly in solids: energetic particles pass on kinetic energy through collisions with neighbours, and in metals free electrons diffuse rapidly through the lattice, making metals excellent conductors. The rate of conduction through a slab is given by:

$$\frac{\Delta Q}{\Delta t} = kA\frac{\Delta T}{\Delta x}$$

Each of the following increases the rate:

• a larger area $A$
• a bigger temperature difference $\Delta T$
• a thinner layer $\Delta x$
• a higher thermal conductivity $k$

Double glazing works by trapping a low-$k$ air gap.

Convection occurs only in fluids: fluid warmed at the bottom expands, becomes less dense, and rises, while cooler denser fluid sinks to replace it, setting up a circulating current. Sea breezes and the cycling of water in a heated pan are everyday examples.

Thermal radiation is the emission of electromagnetic waves (mostly infrared at everyday temperatures) and needs no medium, which is how the Sun heats the Earth. Every object above absolute zero radiates.

The emitted power follows the Stefan-Boltzmann law: a black body radiates a total power (luminosity)

$$L = \sigma A T^4$$

while a real surface radiates $P = e\sigma A T^4$, where the emissivity $e$ compares the surface to a perfect black body ($e = 1$).

The black-body spectrum peaks at a wavelength given by Wien's law, $\lambda_{\text{max}} T = 2.9 \times 10^{-3}\ \text{m K}$, and the apparent brightness of a distant source falls with distance as $b = \frac{L}{4\pi d^2}$, where $L$ is its total radiated power (luminosity).