Question

In a dilute gas at pressure $\mathrm{P}$and temperature $\mathrm{T},$ the mean time between successive collisions of a molecule varies with $\mathrm{T}$ as:

• A. $\surd \mathrm{T}$
• B. $\frac{1}{\mathrm{T}}$
• C. $\mathrm{T}$
• D. $\frac{1}{\surd \mathrm{T}}$

Answer 1

The correct option is D

$\frac{1}{\surd \mathrm{T}}$

Step 1: Expression for free meantime :

The free mean time between the successive collisions of a molecule in a gas is given as:

${\mathrm{\tau }}_{\mathrm{mean}}=\frac{{\mathrm{\lambda }}_{\mathrm{mean}}}{\mathrm{\nu }}$ (equation 1)

Where, ${\mathrm{\lambda }}_{\mathrm{mean}}$= mean free path of the gas molecule, which is given as:

${\mathrm{\lambda }}_{\mathrm{mean}}=\frac{1}{\sqrt{2}{\mathrm{\pi nd}}^2}$ (equation 2)

n= density of gas

d= diameter of the molecule

v= probable speed of the gas molecule

$\mathrm{v}=\sqrt{\frac{2\mathrm{RT}}{\mathrm{\pi M}}}$ (equation 3)

R= gas constant,

T= temperature of the gas

M= molecular mass of the gas

Step 2: Expression for the meantime in terms of temperature:

Substituting equations (2) and (3) in equation (1), we get:

$$\begin{gathered} {\mathrm{\tau }}_{\mathrm{mean}}=\frac{{\displaystyle \frac{1}{\sqrt{2}{\mathrm{\pi nd}}^2}}}{\sqrt{{\displaystyle \frac{2\mathrm{RT}}{\mathrm{\pi M}}}}} \\ \mathrm{or},{\mathrm{\tau }}_{\mathrm{mean}}=\frac{1}{\sqrt{2}{\mathrm{\pi nd}}^2}\times \sqrt{\frac{\mathrm{\pi M}}{2\mathrm{RT}}}(\mathrm{equation}4) \end{gathered}$$

Step 3: Relation between meantime and temperature:

From equation (4) we can conclude that,

The mean free time is inversely proportional to the square root of the temperature of the gas, which can be given as:

${\mathrm{\tau }}_{\mathrm{mean}}\mathrm{\alpha }\frac{1}{\sqrt{\mathrm{T}}}$

Thus, the mean time between successive collisions of a molecule varies with $\mathrm{T}$ as $\frac{1}{\surd \mathrm{T}}$. Hence, option D is correct.