Question

Average time of collision between molecules in adiabatic expansion

Answer 1

For an adiabatic process the following relation is true :-

P∝V−γP∝V−γ

where,

PP = Pressure of gas

VV = Volume occupied by the gas

γγ = Specific heat ratio of the gas

Now the mean free path of the gas (λ)(λ) is given as :-

λ∝Vλ∝V

The average time between the collisions of the gas molecules is nothing but the mean free path divided by the root mean square speed of the gas molecules.

So,

⟹Time=t=λvRMS⟹Time=t=λvRMS

Now we also know that :-

vRMS∝PV−−−√vRMS∝PV

Using the above we get :-

⟹t∝V×1PV−−−−√⟹t∝V×1PV

⟹t∝V2PV−−−−√⟹t∝V2PV

⟹t∝VP−−√⟹t∝VP

Now, using the adiabatic relationship :-

⟹t∝VV−γ−−−−√⟹t∝VV−γ

⟹t∝Vγ+12⟹t∝Vγ+12

and hence,

⟹q=γ+12


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