Question

Among the following gases which one has the lowest root mean square velocity at \[25^\circ {\rm{C}}\] ?
(1) \[{\rm{S}}{{\rm{O}}_{\rm{2}}}\]
(2) \[{{\rm{N}}_{\rm{2}}}\]
(3) \[{{\rm{O}}_{\rm{2}}}\]
(4) \[{\rm{C}}{{\rm{l}}_{\rm{2}}}\]

Answer 1

Hint: We know the formula of Root mean square(rms) velocity, that is, \[{v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \] , where, R stands for gas constant, T stands for temperature, M stands for molar mass of the gas. Now, using the formula of RMS speed, we have to compare the rms speeds of the given gases considering the molar mass values of these gases.

Complete Step by Step Solution:
In the given question, using the root rms velocity formula we can identify the gas having lowest rms speed.

rms velocity can be find out by the following formula,
 \[{v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \]
Here, given a set of gases, we have to identify the gas which possesses maximum rms speed.
As R is gas constant and given that all gases are at same temperature, molar masses of each gas decide its rms speed.

So, \[{v_{rms}} = \sqrt {\dfrac{1}{M}} \], that means, rms velocity is indirectly proportional to molar mass of gases. So, the gas that possesses the highest molar mass has the lowest rms velocity.
So, now we will calculate the molar mass of each gas one by one.
Molar mass of \[{\rm{S}}{{\rm{O}}_{\rm{2}}} = 32 + 2 \times 16 = 32 + 32 = 64\,{\rm{u}}\]
Molar mass of \[{{\rm{N}}_{\rm{2}}} = 14 \times 2 = 28\,{\rm{u}}\]
Molar mass of \[{{\rm{O}}_{\rm{2}}} = 16 \times 2 = 32\,{\rm{u}}\]
Molar mass of \[{\rm{C}}{{\rm{l}}_{\rm{2}}} = 35.5 \times 2 = 71\,{\rm{u}}\]

As we have seen, chlorine gas possesses the highest molar mass among all the gases. Therefore, chlorine gas possesses lowest rms velocity.
Hence, option (4) is right.

Note: It is to be noted that average velocity is different from rms velocity. Average velocity defines the arithmetic mean calculation of velocities of different gaseous molecules at a particular temperature. The average velocity can be found by the formula \[{v_{av}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} \] . The rms velocity is always greater than average velocity.