Question

The ratio of $u_{\text{avg}}$ and $u_{\text{rms}}$ of a gas at a particular temperature is

• A. 1.1086 : 1
• B. 0.92 : 1
• C. 1 : 1.1086
• D. 1:0.92

Answer 1

The correct option is B 0.92 : 1

$u_{\text{avg}}=\sqrt{\frac{8RT}{\pi M}}$
$u_{\text{rms}}=\sqrt{\frac{3RT}{M}}$
Since its for the same gas(same molar mass) at same temperature,
$\frac{u_{\text{avg}}}{u_{\text{rms}}}=\sqrt{\frac{8}{3\pi }}=\sqrt{0.848}=0.92$

Theory:

Distribution of molecular speeds :

Plot of $\frac{1}{N}dNduvsu$

Area of rectangular strip of width du :

$\frac{1}{N}dNdu\times du=\frac{dN}{N}$

Where $\frac{dN}{N}$= fraction of molecules in the range of u and u+du

$\frac{dN}{N}$=$4\pi (\frac{m}{2\pi kT}{)}^{\frac{3}{2}}{e}^{\frac{-m{u}^2}{2kT}}{u}^{2}du$

From the plot we can conclude that the number of molecules having very low or very high speed is very less.

Molecular speeds :

Types of molecular speeds :

1.Average Speed ($u_{avg}$) 2.Most probable speed ($u_{mp}$) 3.Root mean square speed ($urms$)

Average Speed ($u_{avg}$):

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

($u_{avg}=\sqrt[2]{2\frac{8RT}{\pi M}}$)

Root mean square speed ($u_{rms}$):

$\sqrt[2]{2{\overline{u}}^2}=\sqrt[2]{2\frac{u_1^2+u_2^2+u_3^2+.........+u_n^2}{N}}$

$N=Totalnumberofgasmolecules$

($u_{rms}=\sqrt[2]{2\frac{3RT}{M}}$)

Most probable speed ($u_{mp}$):

This can be understood with a simple example ,

In the given series 10,9,8,7,6,6,63.6 we see that 6 is the most probable number found in the given series.Similarly speed which is most probable to be acquired by a gas molecule in the given sample is found more frequently .

The speed possessed by the maximum number of molecules at the given temperature.

($u_{mp}=\sqrt[2]{2\frac{2RT}{M}}$)