Kinetic derivation yields P = (1/3)ρ<v²> where <v²> is mean square speed and ρ = Nm/V × m (mass density).

Kinetic theory gives average translational kinetic energy per molecule <½mv²> = (3/2)k_BT (k_B = Boltzmann constant).

v_rms = √(<v²>) and using <½mv²>=(3/2)k_BT gives v_rms=√(3k_BT/m).

Most probable speed from Maxwell distribution: v_mp=√(2k_BT/m). (Note v_rms=√(3kT/m), <v> = √(8kT/πm)).

Per mole: average translational kinetic energy = (3/2)RT (since per molecule is (3/2)k_BT and R = N_Ak_B).

Mean free path λ ≈ 1/(√2 n σ) so it is inversely proportional to number density n (molecules per volume) and collision cross-section σ.

Collision frequency z ≈ n·σ·<v> (n: number density, σ: effective cross-section, <v>: mean molecular speed).

Kinetic derivation: P = (N m <v²>)/(3 V) where m is molecular mass and <v²> is mean square speed.

N denotes the number of molecules; PV = Nk_BT is equivalent to PV = nRT using n = N/N_A and R = N_Ak_B.

Because velocity components distribute equally in three orthogonal directions, <v_x²> = <v_y²> = <v_z²> = (1/3)<v²>; hence factor 1/3.

Use N = nN_A; then PV = Nk_BT = nN_Ak_BT = nRT since R = N_Ak_B.

Ideal gas model assumes elastic collisions, negligible molecular volume, and no long-range forces; molecules do NOT stick together.

We assume molecular velocities are random and isotropic (equal distribution among directions) so averages like <v_x²> = (1/3)<v²> apply.

Pressure is due to momentum transfer when molecules collide with container walls: momentum per collision × collision frequency per unit area → pressure.

For Maxwell: v_mp < <v> < v_rms. Numerically v_mp=√(2kT/m), <v>=√(8kT/πm), v_rms=√(3kT/m).

Van der Waals introduces 'a' to correct for attractions that reduce pressure and 'b' to correct for excluded volume occupied by molecules.

Z < 1 means PV < nRT — attractions reduce the effective pressure; Z > 1 indicates repulsive/excluded-volume effects dominate.

At Boyle temperature the second virial coefficient B(T)=0 and the gas shows near-ideal behaviour to first order in density.

B(T) is the second virial coefficient representing pairwise interactions; positive or negative sign indicates dominance of repulsion or attraction.

For ideal gases there are no intermolecular potential energies; internal energy equals sum of kinetic energies which depend only on T (e.g., (3/2)RT per mole for monatomic).

For an ideal gas in free (Joule) expansion into vacuum, internal energy is constant and U depends only on T → no temperature change.

Ideal gases have enthalpy depending only on T, so at constant enthalpy throttling does not change temperature → μ_JT=0.

Since <E_trans> = (3/2)k_BT, substituting gives PV = N k_B T, equivalent to ideal gas law PV = nRT.

R = N_Ak_B converts per-particle k_BT to per-mole RT because N_A particles make a mole.

Because of isotropy, <v_x²> = <v_y²> = <v_z²> = (1/3)<v²> where <v²> = <v_x²+v_y²+v_z²>.

Collision cross-section for spherical molecules of effective diameter d is σ = π d² (area for two-body collision with center-to-center distance d).

Pressure arises from net momentum transfer to walls: momentum change × collisions per time per area summed/averaged over molecules yields macroscopic P.

In kinetic derivation we assume elastic, specular collisions with container walls so velocity component perpendicular reverses sign and momentum transfer is calculable.

Total internal energy U for ideal monatomic gas is U = N⟨E⟩ = (3/2)Nk_BT = (3/2)nRT (N = nN_A, R = N_Ak_B).

Kinetic theory shows how microscopic motion (m, v distribution) and collisions produce macroscopic pressure and temperature relationships captured by PV = nRT for ideal gases.