Integrating this over all appropriate velocities within the constraint v > 0 , 0 < θ θ < π π / 2 , 0 < ϕ ϕ < 2 π π yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time: J collision = ∫ ∫ 0 π π / 2 cos ⁡ ⁡ θ θ sin ⁡ ⁡ θ θ d θ θ ∫ ∫ 0 π π sin ⁡ ⁡ θ θ d θ θ × × n v ¯ ¯ = 1 4 n v ¯ ¯ = n 4 8 k B T π π m .