Ideal Gases

• Amount of substance is an SI base quantity, unit mol; one mole contains N_A = 6.02 × 10²³ particles.
• An ideal gas obeys pV ∝ T (T in kelvin): pV = nRT = NkT, with k = R/N_A.
• Kinetic theory: pV = ⅓Nm<c²>, where <c²> is the mean-square speed.
• Mean translational kinetic energy of a molecule = (3/2)kT.

Counting particles

Amount of substance is one of the SI base quantities; its unit is the mole. One mole of any substance contains N_A (the Avogadro constant, 6.02 × 10²³ mol⁻¹) particles. The molar mass M is the mass of one mole, so n = mass/M and the number of particles N = nN_A.

pV = nRT

A gas for which pV ∝ T, where T is the thermodynamic temperature, is an ideal gas. Real gases behave nearly ideally at low pressure and high temperature. The equation of state can be written per mole (with the molar gas constant R = 8.31 J K⁻¹ mol⁻¹) or per molecule (with the Boltzmann constant k = R/N_A = 1.38 × 10⁻²³ J K⁻¹).

For a fixed amount of gas, pV/T is constant, so p₁V₁/T₁ = p₂V₂/T₂. Temperatures must be in kelvin.

Molecules make pressure

Assumptions: the gas is a large number of molecules in random motion; intermolecular forces are negligible except during collisions; the volume of the molecules is negligible compared with the volume of the container; collisions (with each other and the walls) are perfectly elastic; the time of a collision is negligible compared with the time between collisions.

Deriving pressure. _{x x x x} ² _x ^{2 2} _x ² _x ^{2 2 2}

The root-mean-square speed is c_r.m.s. = √<c²>. Comparing with pV = NkT gives ⅓m<c²> = kT, so the mean translational kinetic energy ½m<c²> = (3/2)kT: temperature measures average molecular kinetic energy.