Ideal Gases

Overview

Ideal Gases studies gases using a simplified model in which particles obey idealised assumptions. This gives clear relationships between pressure, volume, temperature, and amount of gas.

Definition

An ideal gas is a hypothetical gas that obeys:

$$pV=nRT$$

for all states.

Equivalent molecular form:

$$pV=NkT$$

where $p$ is pressure, $V$ is volume, $n$ is amount of substance in mol, $N$ is number of molecules, $R$ is the molar gas constant, $T$ is kelvin temperature, and $k$ is the Boltzmann constant.

$$R=N_{A}k$$

Why It Matters

The ideal gas model explains pressure changes in tyres, balloons expanding at altitude, gas compression, piston-cylinder systems, scuba tanks, and engine cycles. It also supports molecular kinetic energy and thermodynamic process analysis.

Key Representations

Ideal gas assumptions:

1. Gas consists of many tiny particles.
2. Particle volume is negligible compared with container volume.
3. Particles move in constant random motion.
4. There are no intermolecular forces except during collisions.
5. Collisions are perfectly elastic.
6. Collision time is negligible.

A gas state is described by pressure $p$, volume $V$, temperature $T$, and amount $n$.

For fixed mass of gas:

$$\frac{p_1{V}_1}{T_1}=\frac{p_2{V}_2}{T_2}$$

Boyle’s law, at constant temperature:

$$pV=\text{constant}$$

Charles’ law, at constant pressure:

$$\frac{V}{T}=\text{constant}$$

Pressure law, at constant volume:

$$\frac{p}{T}=\text{constant}$$

One mole contains:

$$N_A=6.02\times 10^{23}$$

particles, so:

$$N=n{N}_A$$

Pressure is caused by molecular collisions with container walls. Temperature measures average molecular translational kinetic energy:

$$⟨E_k⟩=\frac{3}{2}kT$$

Real gases deviate from ideal behaviour at high pressure or low temperature, where intermolecular forces and molecular volume become important.

Common Exam Traps

Use kelvin, not Celsius, in gas laws.

Simple gas laws assume fixed amount of gas. If gas escapes or is added, use $pV=nRT$.

Use SI units: Pa, ${\text{m}}^3$, K, and mol.

$20^{\circ }\text{C}$ to $40^{\circ }\text{C}$ is not doubling absolute temperature.

Same pressure does not mean same number of molecules unless $V$ and $T$ are also the same.

For a fixed amount of ideal gas, internal energy depends only on temperature, not pressure alone.

Links

• Thermal Physics B
• Kinetic Theory and Ideal Gases
• Temperature and Heat
• Kinetic Theory of Matter
• Molecular Motion
• First Law and Thermodynamic Processes
• p-V Diagrams and Cycles
• p-V Diagrams and Cycles