Comprehensive theory, key formulas, diagrams, and memory aids for Thermal Physics.
Temperature is a measure of the average kinetic energy of the particles in a substance. It determines the direction of spontaneous heat flow — always from higher to lower temperature. Temperature is measured on the:
Conversion: $T$(K) $= \theta$(°C) $+ 273$
At absolute zero, particle motion does not completely cease (quantum mechanics dictates a residual zero-point energy), but classical thermodynamics treats it as minimum energy.
The internal energy of a system is the sum of all the random kinetic and potential energies of its particles:
The First Law of Thermodynamics states: $$\Delta U = Q + W$$
Where $\Delta U$ = change in internal energy, $Q$ = heat energy supplied to the system (positive when added), and $W$ = work done on the system (positive when compressed). If the system does work on the surroundings, $W$ is negative in this convention.
The specific heat capacity ($c$) of a substance is the energy required to raise the temperature of 1 kg of the substance by 1 K (or 1°C):
$$Q = mc\Delta T$$
Where $Q$ = heat energy (J), $m$ = mass (kg), $\Delta T$ = temperature change (K or °C).
| Material | Specific Heat Capacity (J kg⁻¹ K⁻¹) |
|---|---|
| Water | 4200 |
| Aluminium | 900 |
| Copper | 390 |
| Iron | 450 |
| Ice | 2100 |
Water's exceptionally high specific heat capacity (4200 J kg⁻¹ K⁻¹) makes it an excellent coolant and explains why coastal climates are milder than inland climates.
Experimental determination: An electrical heater of power $P$ heats a mass $m$ for time $t$. Energy supplied = $Pt = mc\Delta T$, so $c = Pt/(m\Delta T)$. Corrections must be made for heat losses to the surroundings.
During a change of state (melting, boiling, condensing, freezing), temperature remains constant even though heat energy is being supplied or removed. This energy goes into changing the potential energy of the intermolecular bonds, not the kinetic energy.
The specific latent heat ($L$) is the energy required to change the state of 1 kg of a substance at constant temperature: $$Q = mL$$
$L_v > L_f$ because vaporisation requires separating molecules much further apart (overcoming stronger intermolecular attractions) than melting.
graph LR
A[Solid] -->|Melting: absorb Lf| B[Liquid]
B -->|Freezing: release Lf| A
B -->|Vaporising: absorb Lv| C[Gas]
C -->|Condensing: release Lv| B
A -->|Sublimation| C
An ideal gas is a theoretical model where: - Molecules are point masses with no volume. - There are no intermolecular forces except during perfectly elastic collisions. - All collisions (between molecules and between molecules and container walls) are elastic.
The three experimental gas laws, combined, give the ideal gas law: $$pV = nRT$$
Where: - $p$ = pressure (Pa) - $V$ = volume (m³) - $n$ = number of moles (mol) - $R = 8.31$ J mol⁻¹ K⁻¹ (molar gas constant) - $T$ = absolute temperature (K)
The law can also be written as: $$pV = NkT$$
Where $N$ = number of molecules and $k = 1.38 \times 10^{-23}$ J K⁻¹ (Boltzmann constant). Note: $R = N_A k$, where $N_A = 6.02 \times 10^{23}$ mol⁻¹ is the Avogadro constant.
| Law | Variables | Equation |
|---|---|---|
| Boyle's Law | Constant T, n | $pV = $ constant |
| Charles' Law | Constant p, n | $V/T = $ constant |
| Pressure Law | Constant V, n | $p/T = $ constant |
The kinetic theory derives the macroscopic gas properties from microscopic particle behaviour.
The pressure exerted by an ideal gas: $$pV = \frac{1}{3}Nm\langle c^2 \rangle$$
Where $N$ = number of molecules, $m$ = mass of one molecule, and $\langle c^2 \rangle$ = mean square speed of molecules.
Combining with $pV = NkT$: $$\frac{1}{2}m\langle c^2 \rangle = \frac{3}{2}kT$$
This is a profound result: the average translational kinetic energy of a molecule depends only on absolute temperature. At higher temperatures, molecules move faster.
The root mean square (r.m.s.) speed of molecules: $$c_{rms} = \sqrt{\langle c^2 \rangle} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$$
Where $M$ is the molar mass. At 300 K (room temperature), nitrogen molecules move at approximately 515 m s⁻¹ r.m.s. speed.
Real gases deviate from ideal behaviour at: - High pressures: Molecules are close together, so intermolecular forces and molecular volume become significant. - Low temperatures: Molecules move slowly, allowing intermolecular attractive forces to dominate.
Real gas behaviour is modelled by the van der Waals equation, which includes corrections for intermolecular attractions and finite molecular volume.