Question

What is the $rms$ speed of $He$ atoms at $295K$ ?

Answer 1

Hint : To solve the given problem, we should have understanding about the different speeds with which an atom can travel.
These different speeds are Most probable speed, Average speed and Root Mean Square speed.
Most probable speed is possessed by a maximum fraction of molecules.
Average speed is the arithmetic mean of speed of different molecules.
Root mean square is the square root of the mean square of different molecular speed.
$U_{ RMS } = \sqrt { \dfrac {3RT} {M} }$
$U_ { RMS } \rightarrow$ Root Mean Square
$R \rightarrow$ Universal Gas Constant
$T \rightarrow$ Temperature of Gas
$M \rightarrow$ Molar Mass of Gas.

Complete Step By Step Answer:
Step-1 :
Here, we have given a Helium atom at temperature $295K$ . The molar mass of Helium is $4gmol_{ -1 }$ .
Step-2 :
From the given options available, we have $R$ as $8.3Jmol_{ -1 } K_{ -1 }$ . For $R$ to be in $SI$ unit, the molar mass should be in $kg$ . So, Molar mass of $He$ becomes $4 \times 10^ { -3 }kg$ .
Step-3 :
Putting all the given values in the formula for Root Mean speed, we get :
$U_{ RMS } = \sqrt { \dfrac {3RT} {M} }$
$= \sqrt { \dfrac {3 \times 8.3 \times 295 } {4 \times 10^ { -3 }} }$
$\approx 1360msec^ { -1 }$

Note :
Mathematical formula for most probable speed is :
$U_{ MP } = \sqrt { \dfrac {2RT} {M} }$
For average speed, the formula is :
$U_{ AV } = \sqrt { \dfrac {8RT} { \pi M} }$
The ratio of $U_{ AV } : U_{ MP } : U_{ RMS }$ is $1 : 1.128 : 1.224$ .
They also participated in the formation of Maxwell - Boffemann Distribution Curve.