Kinetic Theory

$\mu$ = moles, $N$ = number of molecules, $R = 8.314\text{ J/mol·K}$.

$\rho = Nm/V$ is mass density, $\langle v^2\rangle$ the mean square speed. Derived from wall collisions alone — never required molecules to collide with each other.

Depends on $T$ and molar mass $M$ ONLY — never on $P$ or $V$ separately. Squeezing a gas into a smaller box at fixed $T$ raises pressure via collision rate, not molecular speed.

Each independent quadratic degree of freedom contributes an average $\tfrac{1}{2}k_BT$, regardless of its nature or the temperature.

1. Monatomic (3 translational dof): $U = \mu\left(\tfrac{3}{2}N_Ak_BT\right) = \tfrac{3}{2}\mu RT$
2. Diatomic, rigid, moderate $T$ (3 translational + 2 rotational = 5 dof): $U = \tfrac{5}{2}\mu RT$

Vibration adds 2 more dof (7 total) at high $T$, but is quantised and stays frozen out at ordinary temperatures.

1. Monatomic: $C_v=\tfrac{3}{2}R \approx 12.5\text{ J/mol·K},\ C_p=\tfrac{5}{2}R,\ \gamma=\dfrac{5}{3}$
2. Diatomic (rigid): $C_v=\tfrac{5}{2}R \approx 20.8\text{ J/mol·K},\ C_p=\tfrac{7}{2}R,\ \gamma=\dfrac{7}{5}$

Holds for any ideal gas — the extra $R$ is the work done pushing back surroundings at constant pressure.

$d$ = molecular diameter, $n = N/V$ number density. Shrinks as pressure rises, grows with temperature at fixed pressure.