Using the Maxwell distribution laws to prove the collision flux equation

[Bipolarity]

I have for a while been trying to understand the setup of this equation in calculating the collision flux of a gas through a given rectangular surface of area A:

Number of collisions in a time interval Δt is $(\frac{N}{V})AΔt\int^{∞}_{0}v_{x}f(v_{x})dv_{x}$ where $f(v_{x})$ is the fraction of molecules with speed in the range $(v_{x},v_{x}+dv_{x})$.

My book explains it by asking to choose a particular velocity and consider the volume inhabited by all the particles which can make the collision however I don't understand why the integral comes in after that.

My book is Physical Chemistry for the Chemical and Biological Sciences by Raymond Chang

Perhaps someone can help me by offering insight into why the integral is done or break the reasoning down into simplier steps?

Thanks!

BiP

 

[Vailanor]

. The integral is necessary in order to calculate the total number of collisions in a given time interval. The integral essentially takes into consideration all possible velocities that the molecules may have, ranging from 0 to infinity. This is necessary in order to get an accurate estimate of the number of collisions in a given time interval. The equation essentially states that the number of collisions in a time interval Δt is equal to the number of particles per unit volume (\frac{N}{V}) times the area of the surface (A) times the time interval (Δt) times the integral of the speed distribution function (f(v_x)) between 0 and infinity. The speed distribution function essentially describes the fraction of molecules with a particular speed in the range (v_x, v_x+dv_x). The integral is necessary in order to take into account all possible velocities that the molecules may have. Thus, the integral essentially calculates the total number of collisions over a range of velocities, ranging from 0 to infinity.