Question

A vessel of capactity $1d{m}^3$ contains $1.03\times {10}^{23}{H}_2$ molecules exerting a pressure of $101.325\text{kPa}$. Calculate RMS speed and average speed.

• A. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 868.35{\text{m s}}^{-1}\end{array}$
• B. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 722.12{\text{m s}}^{-1}\end{array}$
• C. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 868.36{\text{m s}}^{-1}& 942.5{\text{m s}}^{-1}\end{array}$
• D. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 622.53{\text{m s}}^{-1}\end{array}$

Answer 1

The correct option is A $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 868.35{\text{m s}}^{-1}\end{array}$

For $n$ moles $u_{\text{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3pV}{nM}}$
Moles $=\frac{1.03\times {10}^{23}}{6.02\times {10}^{23}}$

$p=101.325\times {10}^3\text{Pa},V=1d{m}^3={10}^{-3}{m}^3,M=2\times {10}^{-3}\text{kg/mol}$

$\therefore u_{\text{rms}}=\sqrt{\frac{3\times 101.325\times {10}^3\times {10}^{-3}}{\frac{1.03\times {10}^{23}}{6.02\times {10}^{23}}\times 2\times {10}^{-3}}}=942.5{\text{ms}}^{-1}$

$u_{\text{av}}=\sqrt{\frac{8RT}{\pi m}},u_{\text{rms}}=\sqrt{\frac{3RT}{M}}$
$\therefore \frac{u_{\text{av}}}{u_{\text{rms}}}=\sqrt{\frac{8}{3\pi }}=0.9213$
$\therefore u_{\text{av}}=u_{\text{rms}}\times 0.9213=942.5\times 0.9213=868.35{\text{ms}}^{-1}$
Theory :

Molecular Speeds

• Molecular speed is the term used to describe the speed of a gas molecule.

• Since the gas molecules are always in continuous motion, they colloid with each other as well as with the walls of the container.

• It is not possible to measure the speed of an individual molecule.

• The speed and energy of all the molecules at some instant are not the same.

Types of molecular speeds:

1. Average speed: The arithmetic mean of the speeds of the different gas molecules is called average speed $\left(u_{avg}\right)$.

Suppose there are $1,2,3,.......,N$ number of molecules and their speeds are $u_1,u_2,u_3,.......,u_N$ respectively, then

$u_{avg}=\frac{u_1+u_2+u_3+.......+u_N}{N}$

$u_{avg}={\left(\frac{8RT}{\pi M}\right)}^{\frac{1}{2}}$

1. Most probable speed: The speed actually possessed by the maximum number of gas molecules is called most probable speed $\left(u_{mp}\right)$.

$u_{mp}={\left(\frac{2RT}{M}\right)}^{\frac{1}{2}}$

1. Root mean square speed: The square root of the mean of the squares of the speeds of different gas molecules is called root mean square speed $\left(u_{rms}\right)$.

Suppose there are $1,2,3,.......,N$ number of molecules and their speeds are $u_1,u_2,u_3,.......,u_N$ respectively, then

$u_{rms}={\left(\frac{u_1^2+u_2^2+u_3^2+.......+u_N^2}{N}\right)}^{\frac{1}{2}}$

$u_{rms}={\left(\frac{3RT}{M}\right)}^{\frac{1}{2}}$

Relationship between different molecular speeds:

$u_{rms}:u_{avg}:u_{mp}$

${\left(\frac{3RT}{M}\right)}^{\frac{1}{2}}:{\left(\frac{8RT}{\pi M}\right)}^{\frac{1}{2}}:{\left(\frac{2RT}{M}\right)}^{\frac{1}{2}}$

For a particular gas at the same temperature,

$\sqrt{3}:{\left(\frac{8}{\pi }\right)}^{\frac{1}{2}}:\sqrt{2}$

$1.224:1.128:1$

$\therefore u_{rms}>u_{avg}>u_{mp}$

Average kinetic energy of molecules:

$\text{Average K.E. of molecules}=\frac{1}{2}m\overline{u^2}$

Where, $\overline{u^2}=\frac{u_1^2+u_2^2+u_3^2+.......+u_N^2}{N}=u_{rms}^2$

$\therefore u_{rms}=\sqrt{\overline{u^2}}$