Thermal Physics and Ideal Gases

• Temperature measures the average kinetic energy of particles
• Thermal equilibrium: no net heat flow between objects at the same temperature
• Absolute zero (0 K = −273.15°C): particles have minimum possible energy
• $T(K) = T(°C) + 273.15$

Internal energy = total kinetic energy + total potential energy of all particles.

• For an ideal gas: internal energy = kinetic energy only (no intermolecular forces)
• Increasing temperature → increases internal energy
• During a change of state: temperature constant but internal energy changes (PE changes)

$Q = mc\Delta\theta$

Where:

• $Q$ = heat energy (J)
• $m$ = mass (kg)
• $c$ = specific heat capacity (J kg⁻¹ K⁻¹)
• $\Delta\theta$ = temperature change (K or °C)

Example: $c_{\text{water}} = 4200$ J kg⁻¹ K⁻¹

$Q = mL$

• $L_f$ = specific latent heat of fusion (solid ↔ liquid)
• $L_v$ = specific latent heat of vaporisation (liquid ↔ gas)
• During phase change: temperature remains constant

Boyle's Law: $pV = \text{constant}$ (at constant $T$) → $p \propto 1/V$

Charles's Law: $V/T = \text{constant}$ (at constant $p$) → $V \propto T$

Pressure Law: $p/T = \text{constant}$ (at constant $V$) → $p \propto T$

Combined Gas Law

$\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$

Ideal Gas Equation

$\boxed{pV = nRT = NkT}$

Where:

• $n$ = number of moles
• $R = 8.314$ J mol⁻¹ K⁻¹
• $N$ = number of molecules
• $k = 1.38 \times 10^{-23}$ J K⁻¹ (Boltzmann constant)
• $k = R/N_A$ where $N_A = 6.02 \times 10^{23}$ mol⁻¹

Pressure from Molecular Motion

$\boxed{pV = \frac{1}{3}Nm\overline{c^2}}$

Where $\overline{c^2}$ = mean square speed. $c_{\text{rms}} = \sqrt{\overline{c^2}}$.

Mean Kinetic Energy

$\boxed{\frac{1}{2}m\overline{c^2} = \frac{3}{2}kT}$

Average KE per molecule depends only on temperature.

Assumptions of Ideal Gas

1. Large number of identical molecules
2. Volume of molecules << volume of container
3. Collisions are elastic
4. No intermolecular forces (except during collisions)
5. Time of collision << time between collisions
6. Random motion

• $pV = nRT = NkT$
• Kinetic theory: $pV = \frac{1}{3}Nm\overline{c^2}$; $\frac{1}{2}m\overline{c^2} = \frac{3}{2}kT$
• $Q = mc\Delta\theta$ (heating); $Q = mL$ (phase change)
• rms speed: $c_{\text{rms}} = \sqrt{3kT/m}$
• Internal energy of ideal gas = KE only