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reference:concepts of physics by hc verma I have few doubts related to this derivation

  1. why collisions with other molecules not taken into account since they could also influence the time it takes for one molecule to collide with the wall
  2. how can taking average force into account and not the real forces the equation still works very well
  3. how can we assume that the velocity of particle in x y z direction is same i mean how can this be called a good approximation? Reference : concepts of physics by HC verma
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    $\begingroup$ regarding the 3rd point, it's because of statistical randomness and symmetry. In the absence of any type of fiedl/bias in a specific direction, what distinguishes x,y & z? Nothing. They are just mutually perpendicular directions, that can be interchanged. So whyshould the velocity in x differ from y or z, on average? it's like rolling a die. Sure, you may get more 2's than the other numbers in the first 10 tries due to variation in throw-speed, rotation given, side facing up etc..., but over millions of tries, statistics wins out and every number comes up roughly equal amounts $\endgroup$
    – Vulgar Mechanick
    Jan 14, 2021 at 6:05
  • $\begingroup$ Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$
    – ACuriousMind ♦
    Jan 14, 2021 at 16:36

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The answer to all the three points is randomness and symmetry.

So what the molecule represents in fig 24.1 in your derivation is nothing but the representation of the average motion of all the particles in your system. This is based on randomness. Think of it this way - if the average molecule did not perform the motion as given in the derivation, what else could it have performed?

why collisions with other molecules not taken into account since they could also influence the time it takes for one molecule to collide with the wall

They would influence the time. However, if one molecule's time to collide is increased by $"t"$, you would expect another molecule's to be decreased by $"t"$. If you average out the time over the number of molecules, you would get no net deviation in time.

how can we assume that the velocity of particle in x y z direction is same i mean how can this be called a good approximation?

Again the molecule in the derivation represents average motion. So if you are talking about averages, what special direction can you select where speed should be different? Note that all X, Y and Z are equivalent and there is no reason for us to treat anyone preferentially.

how can taking average force into account and not the real forces the equation still works very well

Again the average force represents the mean of "real" forces.

Note that this is not a strict derivation but only an attempt to justify the known results.

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