The Maxwell Distribution Laws MCQ - Practice Questions with Answers

• Root mean square speed- It is defined as the square root of the mean of squares of the speed of different molecules.

ie. $v_{m s}=\sqrt{\frac{v_1^2+v_2^2+v_3^2+v_4^2+\ldots}{N}}=\sqrt{\bar{v}^2}$
1. As the Pressure due to an ideal gas is given as

$$
\begin{aligned}
& P=\frac{1}{3} \rho v_{r m s}^2 \\
\Rightarrow & v_{m s}=\sqrt{\frac{3 P}{\rho}}=\sqrt{\frac{3 P V}{\text { Mass of gas }}}=\sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 k T}{m}}
\end{aligned}
$$

Where
$\mathrm{R}=$ Universal gas constant
$\mathrm{M}=$ molar mass
$\mathrm{P}=$ pressure due to gas
$\rho=$ density
2. $v_{m s} \alpha \sqrt{T}$ l.e With the rise in temperature, the rms speed of gas molecules increases.
3. $v_{m s} \alpha \frac{1}{\sqrt{M}}$ l.e With the increase in molecular weight, the rms speed of the gas molecule decreases.
4. The rms speed of gas molecules does not depend on the pressure of the gas (if the temperature remains constant)   

- Average speed- It is the arithmetic mean of the speeds of molecules in a gas at a given temperature.

$$
v_{a v g}=\frac{v_1+v_2+v_3+v_4+\ldots}{N}
$$

and according to the kinetic theory of gases

$$
v_{a v g}=\sqrt{\frac{8 P}{\pi \rho}}=\sqrt{\frac{8}{\pi} \frac{R T}{M}}=\sqrt{\frac{8}{\pi} \frac{k T}{m}}
$$

- The relation between RMS speed, average speed, and most probable speed

$$
V_{r m s}>V_{a v g}>V_{m p s}
$$
 

Maxwell’s Law - 

The $v_{r m s}$ (Root mean square velocity) gives us a general idea of molecular speeds in a gas at a given temperature. So, it doesn't mean that the speed of each molecule is $v_{r m s}$.

Many of the molecules have speeds more than $v_{r m s}$ and many have speeds greater than $v_{r m s}$. So, Maxwell derived an equation that describes the distribution of molecules at different speeds as -

$$
\mathrm{dN}=4 \pi \mathrm{~N}\left(\frac{\mathrm{~m}}{2 \pi \mathrm{kT}}\right)^{3 / 2} \mathrm{v}^2 \mathrm{e}^{-\frac{\mathrm{mv}^2}{2 \mathrm{kT}}} \mathrm{dv}
$$

where, $d N=$ Number of molecules with speeds between $v$ and $v+d v$

So, from this formula you have to remember a few key points -
1. $\frac{d N}{d v} \propto N$
2. $\frac{d N}{d v} \propto v^2$

Conclusions from this graph - 

1. This graph is between a number of molecules at a particular speed and the speed of these molecules.
2. You can observe that the $\frac{d N}{d v}$ is maximum at the most probable speed.
3. This graph also represents that $v_{r m s}>v_{a v}>v_{m p}$.
4. This curve is an asymmetric curve.
5. From this curve we can calculate a number of molecules corresponding to that velocity range by calculating the area bonded by this curve with the speed axis.

Effect of temperature on velocity distribution :

With the rise of temperature, the curve starts shifting right side and becomes broader as shown as -