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Collisions with container wall

Integrating this over all appropriate velocities within the constraint v > 0 {\displaystyle v>0}, 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}}, 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time: J collision = ∫ 0 π / 2 cos (θ) sin (θ) d θ ∫ 0 π sin (θ) d θ × n v ¯ = 1 4 n v ¯ = n 4 8 k B T π m. {\displaystyle J_{\text{collision}}={\frac {\displaystyle \int _{0}^{\pi /2}\cos(\theta)\sin(\theta)\,d\theta }{\displaystyle \int _{0}^{\pi }\sin(\theta)\,d\theta }}\times n{\bar {v}}={\frac {1}{4}}n{\bar {v}}={\frac {n}{4}}{\sqrt {\frac {8k_{\mathrm {B} }T}{\pi m}}}.}