# Confusion while deriving kinetic-molecular theory of gases

How many of the molecules will collide within a given area A of the wall normal to the x-direction in some fixed time interval Δt ? On average half the molecules have a positive x-component of velocity. Therefore half the molecules contained inside a cylindrical volume of cross sectional area A and length vΔt will strike the wall within a given area A during the time interval Δt.

So, the number of collision with A during $$\Delta{t}$$ is

$$\frac{1}{2}\frac{NAv_x\Delta{t}}{V}$$

So my question is how can we assume that half of the molecules are moving one way and half the other? I know that we are focusing on one co-ordinate i.e. x-coordinate, thus, we are not focusing in other directions like up and down etc.Nevertheless,I wondered, if I were the physicist coming up with this derivation, why would I divide the number of collisions by 2 saying that half of the particles go one way and half the other way. How can it be just half not one third any other probability.

Thanks for the time and help.

• Another way an equivalent relation is derived is to evaluate the time taken for one molecule to go from one wall to the other wall and dark to the original wall whilst moving parallel to the x-axis $\frac{2L}{v_{\rm x}}$ where $L$ is the separation of the walls. Oct 18, 2018 at 6:10
• These approaches to deriving the theory have always bugged me because they only work for perfectly rectangular boxes where the gas is so dilute that particle collisions can be entirely ignored. A more robust approach requires you to consider the volumetric density of material and write a distribution function for the possible angular trajectories of the particles toward the wall. By integrating these out you get the number of particles hitting the wall and from this you can calculate a force and thus a pressure, which gives the ideal gas law. Apr 8 at 3:55