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While deriving expression for pressure of gas

We assume that gas molecule has travelled the between two opposite walls , and time taken to travel this distance is Δt= 2l/u .

Then we say that , Force is change in momentum per unit time .

The change change is momentum is 2mu and time in which momentum is changed is 2l/u

Hence force = 2mu/(2l/u)

And we further calculate pressure.

But my question is : the time ∆t= 2l/u is not the time in which momentum is changed , it is the time in which molecule had travelled between one wall to another wall. Then why we take ∆t as the time in which momentum is changed?

i.e , the time in which momentum is changed should be equal to the time for which molecule remain in contact with wall , but in this derivation we take ∆t as the time required to move between two wall.

Why is this so?

I am beginner in statistical mechanics, pls don't use high level mathematics

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  • $\begingroup$ @RogerVadim , the question suggested by you is related to with equipartition theorem. I am struggling with some another doubt $\endgroup$
    – user324098
    Dec 28, 2022 at 15:31
  • $\begingroup$ I think the question I linked and the answer given do answer your question - but they take time to read and digest. $\endgroup$
    – Roger V.
    Dec 28, 2022 at 16:37

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We are not trying to find the force exerted by a molecule as it collides with the wall, but the mean force exerted over a 'long' time by the molecule. That long time will include the very short intervals while the collisions with the wall that change the momentum by $2mu$ are taking place, but will be dominated by the long intervals while the molecule is travelling between contacts with the walls. Hence $2l/u$ is the appropriate time per collision. You might find the argument more convincing if we consider a time during which the molecule makes $n$ collisions with one of the walls... $$\text{Mean Force} = \frac {\text{total change in momentum}}{\text{total time}}=\frac{n\times 2mu}{n\times 2l/u}=\frac{mu^2}l$$ Note that this still applies if the molecule collides elastically with the 'side' walls on its journey to and fro, as the velocity component of magnitude $u$ will be conserved.

There are more general ways to derive $pV=\frac 13 Nm\overline {c^2}$ than the one you are struggling with, but the molecule banging back and forth between walls in a cuboidal container is an undemanding first method.

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  • $\begingroup$ What does 'mean force' mean? I don't understand it intuitively $\endgroup$
    – user324098
    Dec 28, 2022 at 15:29
  • $\begingroup$ It means (sorry about the pun) average force over a long period of time Bearing this in mind, may I recommend that you read my answer again carefully? $\endgroup$ Dec 28, 2022 at 16:42
  • $\begingroup$ Suppose that I gave you 10 grams of silver in one lump every hour, but that the actual passing over of the silver took 5 seconds each time. What would be my mean rate of giving silver to you? Would it be 2.0 gram per second or 0.0028 gram per second (10 gram per hour)? $\endgroup$ Dec 28, 2022 at 17:37