So in the book concepts in thermal physics(by Stephan and Katherine Blundell) the expression of mean scattering time is derived as shown(please refer the book for more). If we replace vdt in the derivation by dx then we will get mean free path and we will find mean free path to be equal to 1/nA. n is number density and A is collision cross sectional area(denoted by symbol 'sigma' in book). However later in the chapter an additional factor of 1/under root 2 is there in the expression of mean free path. Which one is more correct and why?
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The difference is due to the fact that in the first derivation the assumption is made that all of the other molecules, the so called target molecules, other than the one under consideration are stationary whereas in the second derivation the target molecules are also moving.
Thus, instead of the relative velocity between the molecule under consideration and the target molecules being $v$, the relative velocity is $\sqrt {2}\,v$ which increases the rate of collision and hence decreases the mean free path by the factor $\sqrt {2}$.
Compare the section "Scatter Theory$ and section "Kinetic Theory of Gases" in the Wikipedia article Mean Free Path.
