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Expression for mean free path? full derivation

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Mean Free Path - 

On the basis of the kinetic theory of gases, it is assumed that the molecules of a gas are continuously colliding against each other. So, the distance traveled by a gas molecule between any two successive collisions is known as a free path.

There are assumptions for this theory that during two successive collisions, a molecule of gas moves in a straight line with constant velocity. Now, let us discuss the formula of the mean free path - 

                                                                          

Let  \lambda _1,\lambda _2.......\lambda_n be the distance travelled by a gas molecule during n collisions respectively, then the mean free path of a gas molecule is defined as - 

                       \lambda=\frac{\text { Total distance travelled by a gas molecule between successive collisions }}{\text { Total number of collisions }}

Here, \lambda is the mean free path.

It can also be written as -        \boldsymbol{\lambda=\frac{\lambda_{1}+\lambda_{2}+\lambda_{3}+\ldots+\lambda_{n}}{n}}

Now, let us take d = Diameter of the molecule,
                          N = Number of molecules per unit volume.

 

Also, we know that,  PV = nRT 

So, Number of moles per unit volume  =  \frac{n}{V }=\frac{P}{RT}

Also we know that number of molecules per unit mole  = N_A = 6.023 \times10^{23} 

So, the number of molecules in 'n' moles = nNA

So the number of molecules per unit volume is N =  \frac{PN_A}{RT}

                                                                 So, \dpi{100} \mathbf{\lambda = \frac{RT}{ \pi d^2PN_A}=\frac{kT}{\pi d^2P}}

If all the other molecules are not at rest then, \boldsymbol{\lambda=\frac{1}{\sqrt{2} \pi N d^{2}}}\mathbf{= \frac{RT}{\sqrt{2}\lambda d^2PN_A}=\frac{kT}{\sqrt{2}\pi d^2P}}

Posted by

avinash.dongre

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