I Kinetic equation of gases: time between collisions instead of time of colission?

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The discussion centers on the use of time between collisions rather than the time of collision when analyzing momentum change in molecules. It emphasizes that the average rate of momentum change for a molecule is calculated based on the time interval between its collisions with a wall, similar to how one calculates average income over a month. While introductory physics textbooks present this concept effectively, the erratic paths of molecules in a gas raise questions about the simplicity of this derivation. More advanced derivations can be found in statistical mechanics literature, suggesting a need for deeper analysis. Overall, the focus is on understanding momentum change in molecular dynamics through appropriate time intervals.
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I understand everything except why do we use time between collisions instead of time of colission?

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And can it be done differently?
 
It's the average rate of change of momentum of one molecule due to collisions of the molecule against one wall that is of interest. Suppose you have a job that pays $5000 per month and you get paid on the last day of the month. So, you deposit $5000 each payday into your bank account. You wouldn't claim that your account is increasing at an average rate of $5000 per day. The average rate of increase of the account is $5000 divided by the the time between deposits (a month). Same with the molecule. Its average "deposit" of momentum-change against one wall is the momentum change of the collision divided by the time between collisions of that molecule with that wall.

You find this derivation in many introductory physics textbooks and it does get the idea across. However, we know that for a gas of many molecules, a single molecule will travel a very erratic path as it collides with other molecules. So, to me, this makes the derivation questionable. For a better derivation see for example here. More sophisticated derivations can be found in textbooks on statistical mechanics.
 
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