The Maxwell Distribution Laws - Practice Questions & MCQ

• 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_{r 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}} \\
& \text { Where } \\
\mathrm{R}= & \text { Universal gas constant } \\
\mathrm{M}= & \text { molar mass } \\
\mathrm{P}= & \text { pressure due to gas } \\
\rho & =\text { density }
\end{aligned}
$

2. $v_{r m s} \alpha \quad \sqrt{T}$ I.e With the rise in temperature, rms speed of gas molecules increases.
3. $v_{r m s} \alpha \frac{1}{\sqrt{M}}$ I.e With the increase in molecular weight, $r$ ss 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)

• Most probable speed-This is defined as the speed which is possessed by maximum the fraction of the total number of molecules of the gas.

$
v_{m p s}=\sqrt{\frac{2 P}{\rho}}=\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 k T}{m}}
$

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

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

Effect of temperature on velocity distribution :

With rising of temperature, the curve starts shifting right side and become broader as shown as -