Derivation of Gas Pressure Equation

Consider a cube of side `L`. Take one molecule of mass `m` whose x-component of velocity is `cₓ`. For an elastic collision with a wall perpendicular to the x-axis: initial x-momentum: mcₓ final x-momentum: −mcₓ Magnitude of momentum change: Δp = 2mcₓ Time between successive collisions with the same wall: Δt = 2L/cₓ Average force from this molecule: F = ΔpΔt Therefore: F = mcₓ²/L For `N` molecules: F = (m/L)Σcₓ² Since wall area: A=L² pressure: p = FA so: p = Nm⟨cₓ²⟩/V For random isotropic motion: ⟨cₓ²⟩ = ⟨c_y²⟩ = ⟨c_z²⟩ and: ⟨c²⟩ = ⟨cₓ²⟩+⟨c_y²⟩+⟨c_z²⟩ Therefore: ⟨cₓ²⟩ = (13)⟨c²⟩ Hence: **pV = (13)Nm⟨c²⟩** Since: **ρ = Nm/V** we also get: **p = (13)ρ⟨c²⟩ pV = (13)Nm⟨c²⟩ p = (13)ρ⟨c²⟩**