Derivation of kintetic theory equation

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The discussion focuses on the derivation of the kinetic theory equation pv=1/3nMwC^2, particularly questioning the assumption that molecules travel between walls without collisions. It highlights that the factor 1/3 accounts for the random motion of molecules in three dimensions, effectively representing the average velocity along each axis. The conversation also touches on the use of root mean square speed to average the varied speeds of molecules. Participants seek a more rigorous derivation of the equation and clarification on the role of the 1/3 factor. The discussion emphasizes the complexity of molecular interactions and the statistical nature of kinetic theory.
leibo
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Hello.

My question is related to the derivation of the equation pv=1/3nMwC^2. Most of the derivations i have seen assume that the average time between colisions, t, is 2l/Vx when l is the length of the cube. When assuming this, you actually assume that each molecule travel from one wall to the other wall without any clashes with other molecules. this is an unlikely assumption, yet those derivations do succeed. my quastion is - why? and where can i find more rigorous derivation?

I am sorry for my poor english...

Thanks in advance
 
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I would say that the answer lies in the appearance of the factor 1/3 in the equation.
If you start the analysis by considering only 1 molecule bouncing between 2 opposite walls then you get Force = rate of change of momentum or
Force = change of momentum for each collision x number of collisions per second
F = 2mc x c/2L = (mc^2)/L
In fact the container contains N molecules moving in random directions and this is equivalent to a container with N/3 molecules moving in each of the x,y and z directions.
So the 1/3 in the equation covers the random motion of N molecules.
The molecules also have a range of speeds and therefore the idea of a 'root mean square' speed is used to give a representative 'average' speed.
Very much simplified...hope it helps.
 
thanks, but I am not sure I fully understand you. is the factor 1/3 really comes to "correct" the worng assumption that no clashes are made during the travel from one wall to the other? if so, maybe you know where can I find a rigorous and accurate derivation of this factor and all the equation? as far as I know, the "1/3" is not such a correction factor but it comes from the fact that the average velocity at each one of the 3 axises is equal.
 
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I would say that the 1/3 comes from there being a large number of molecules in random motion.
The average velocity is taken car of with the introduction of a 'root mean square' (rms)
Speed
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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