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


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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.

  • $\begingroup$ 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. $\endgroup$
    – Farcher
    Oct 18, 2018 at 6:10
  • $\begingroup$ 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. $\endgroup$ Apr 8 at 3:55

1 Answer 1


A simple way of thinking about this is to think about the motion of particles through an imaginary plane somewhere inside the fluid. The number of particles crossing the plane from one side to the other must be the same as the number of particles crossing in the other direction. This must be the case because first of all the gas is homogeneous and it has the same properties (e.g. average kinetic energy, particle velocity, etc.) everywhere. If the particles moved from one side to the other more than in the other direction, you'd have a build-up of particles on one side, violating the second law of thermodynamics.

For particles near a wall, they don't 'know' they are near the wall until they hit it. So the same rule must apply to particles near a wall.

Viewing it from the perspective of the 2nd law might be the easiest way of thinking about it, but you can also view it purely from the basic assumptions of an ideal gas: a bunch of non-interacting particles. If you had just a single particle in the box, the same law would apply. On average, over long enough time, the particle would find itself moving toward a wall the same number of times it would find itself moving away from it.

  • $\begingroup$ So, basically it's the second law of thermodynamics that governs the fact that the particles in average move in both the directions. Otherwise it defies the spontaneous nature of air molecules i.e. to diffuse. Did I get it right? $\endgroup$
    – suiz
    Oct 18, 2018 at 4:59
  • $\begingroup$ Look at my edit. $\endgroup$
    – Al Nejati
    Oct 19, 2018 at 4:57

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