Thermal Physics and Ideal Gases

Internal energy and temperature

• Internal energy = the sum of the randomly distributed kinetic and potential energies of all the particles in a system.
• Temperature (in kelvin) is a measure of the average kinetic energy of the particles. Absolute zero (0 K = −273 °C) is the temperature at which particles have minimum kinetic energy.
• Convert: K = °C + 273.

Specific heat capacity

Energy to raise the temperature of a substance:

Q = mcΔθ

where c = specific heat capacity (J kg⁻¹ K⁻¹) — energy to raise 1 kg by 1 K.

Specific latent heat

Energy to change state (at constant temperature):

Q = mL

L = specific latent heat (J/kg) — of fusion (melting) or vaporisation (boiling). During a change of state, energy goes into breaking bonds, so temperature stays constant.

The gas laws

For a fixed mass of ideal gas:

• Boyle's law: at constant T, pV = constant (p ∝ 1/V).
• Charles's law: at constant p, V/T = constant.
• Pressure law: at constant V, p/T = constant.

Combined into the ideal gas equation:

pV = nRT

where n = moles, R = molar gas constant (8.31 J mol⁻¹ K⁻¹), T in kelvin. Alternatively pV = NkT (N = number of molecules, k = Boltzmann constant).

Kinetic theory

The behaviour of gases is modelled by assuming many small molecules in rapid random motion, with negligible volume and no intermolecular forces, undergoing elastic collisions. This leads to:

½m⟨c²⟩ = (3/2)kT

— the average kinetic energy of a molecule is proportional to the absolute temperature.

Worked example

2.0 mol of an ideal gas at 300 K occupies what volume at a pressure of 1.0 × 10⁵ Pa? (R = 8.31)

• V = nRT/p = (2.0 × 8.31 × 300) ÷ 1.0×10⁵ = 4986 ÷ 100000 = 0.050 m³. ✓

Common mistakes

• Using temperature in °C instead of kelvin in gas equations.
• Forgetting temperature stays constant during a change of state (Q = mL, not mcΔθ).
• Confusing internal energy (total) with temperature (average KE).

Exam tips

• Always convert to kelvin for gas laws and kinetic theory.
• Use Q = mcΔθ for temperature change and Q = mL for change of state.
• Learn pV = nRT and that average molecular KE ∝ absolute temperature.

Key facts to remember

• Internal energy = random KE + PE of particles; temperature (K) ∝ average kinetic energy; 0 K = −273 °C.
• Q = mcΔθ (heating); Q = mL (change of state, constant temperature).
• Ideal gas: pV = nRT (= NkT); kinetic theory gives ½m⟨c²⟩ = (3/2)kT (KE ∝ absolute T).