Ideal Gas Law and Kinetic Model

Boyle's Law: $pV = k$ (constant $T$) Charles's Law: $V/T = k$ (constant $p$) Gay-Lussac's Law: $p/T = k$ (constant $V$)

$\boxed{pV = nRT}$

Or equivalently: $pV = NkT$

Where:

• $p$ = pressure (Pa)
• $V$ = volume (m³)
• $n$ = moles
• $R = 8.31$ J mol⁻¹ K⁻¹
• $N$ = number of molecules
• $k = 1.38 \times 10^{-23}$ J K⁻¹ (Boltzmann constant)
• $T$ = absolute temperature (K)

Assumptions

1. Large number of identical molecules in random motion
2. Volume of molecules negligible compared to container volume
3. No intermolecular forces except during collisions
4. Collisions are perfectly elastic
5. Duration of collisions is negligible

Pressure from Molecules

Pressure arises from molecules hitting container walls: $\boxed{pV = \frac{1}{3}Nm\overline{c^2}}$

Combining with $pV = NkT$: $\frac{1}{2}m\overline{c^2} = \frac{3}{2}kT$

The average KE of a molecule depends only on temperature.

RMS Speed

$c_{\text{rms}} = \sqrt{\overline{c^2}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

Where $M$ = molar mass (kg/mol).

Ideal gas behaviour is a good approximation when:

• Temperature is high (particles have high KE, forces negligible)
• Pressure is low (particles far apart, volume negligible)

Deviation occurs at high pressure and low temperature (near liquefaction).

• $pV = nRT = NkT$
• Kinetic: $pV = \frac{1}{3}Nm\overline{c^2}$; avg KE = $\frac{3}{2}kT$
• $c_{\text{rms}} = \sqrt{3kT/m}$
• Ideal gas: high $T$, low $p$ (molecules far apart, forces negligible)